PAPEES
ON
MECHANICAL AND PHYSICAL
SUBJECTS.
HonDon: C. J. CLAY AND SONS,
CAMBRIDGE UNIVERSITY PRESS WAREHOUSE,
AVE MARIA LANE.
©lassflofo: 50, WELLINGTON STREET.
H,etp?ig: F. A. BROCKHAU8.
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Bombag: E. SEYMOUR HALE.
PAPERS
ON
MECHANICAL AND PHYSICAL
SUBJECTS
BY
OSBORNE REYNOLDS, F.R.S., MEM. INST. C.E., LL.D.,
PROFESSOR OP ENGINEERING IN THE OWENS COLLEGE, AND
HONORARY FELLOW OF QUEENS' COLLEGE, CAMBRIDGE.
REPRINTED FROM VARIOUS TRANSACTIONS AND JOURNALS.
VOLUME II
1881—1900
OF
IFO|
CAMBRIDGE:
AT THE UNIVERSITY PRESS.
1901
[All Eights reserved.]
€amirtltge :
PRINTED BY J. AND C. F. CLAY,
AT THE UNIVERSITY PRESS.
PREFACE TO VOLUME II.
THIS Volume includes the Reprint of my papers on mechanical subjects,
following on to those printed in Volume I., from the year 1881 up to date.
At the expressed wishes of the authors this Volume also includes the
Reprint of the second parts of two papers, of which I contributed the first
parts only.
One of these is the paper " On the Theory of the (Steam-Engine)
Indicator and the Errors in Indicator Diagrams." Of this the second part
was contributed by Professor A. W. Brightmore, D.Sc.
The other being the paper " On the Mechanical Equivalent of Heat,"
and of this the second part was contributed by Mr W. H. Moorby, M.Sc.
OSBORNE REYNOLDS.
19, LADYBABN ROAD,
MANCHESTER.
95916
CONTENTS OF VOL. II.
41. On the Fundamental Limits to Speed .... 1 — 24
The different limits imposed by the several properties of material —
the limits determined by the ratio of strength to heaviness — stresses
due to acceleration in the coupling rods of locomotives — the de-
structive effect of periodic forces synchronising with the natural
period of the structure — the extent to which balancing of machines
is possible.
42. On an Elementary Solution of the Dynamical Problem of
Isochronous Vibration 25 — 27
43. The Comparative Resistances and Stresses in the Cases of
Oscillation and Rotation with Reference to the Steam-
Engine and Dynamo 28 — 50
The friction in the two cases — the dynamics of oscillations controlled
by a crank — effect of reservoirs of energy — loss of energy by friction
resulting from pressures caused by inertia — application to the steam-
engine and dynamo.
44. An Experimental Investigation of the circumstances which
determine whether the Motion of Water shall be direct
or sinuous, and of the Law of Resistance in Parallel
Channels 51 — 105
Section I. — Introduction.
The failure of the theory of hydrodynamics to explain why the resist-
ance is in some cases proportional to the velocity, and in others to the
square of the velocity — direct and sinuous motion — the effect of
viscosity — character of motion dependent on dimensional properties —
the evidence of these in the equations of motion — experiments by
means of coloured bands in glass tubes prove the existence of a critical
velocity at which eddying motion begins — two streams in opposite
directions — experiments showing the resistance is constant if v a p/p . c
— results shown to agree with both Darcy's and Poiseuille's experi-
ments 51-67
Section II.
Description of the experiments in glass tubes by means of colour
bands — relations between critical velocity, size of tube and viscosity
— the sudden appearance of the eddies— effect of initial disturbances —
effect of size of disturbance on instability ...... 68 — 77
Section III.
Experiments to determine critical velocity by measuring the resistance —
the apparatus — methods of measuring the discharge and the pressures
— effect of temperature— results — general law of resistance brought out
by the method of logarithmic homologues 78 — 98
Vlll CONTENTS.
PAGES
Section IV.
Application to Darcy's experiments — their reduction by means of
logarithmic homologues — effect of temperature when the critical
velocity is passed— effect of roughness of surface of pipes . . 99—105
45. The Transmission of Energy 106 — 131
Directed and undirected energy — the sources of energy — transmission
by means of coal and corn — transmission of stored energy in the
directed form — distribution by means of compressed water, com-
pressed air, ropes and shafts.
On the Equations of Motion and the Boundary Conditions
for Viscous Fluids 132 — 137
The equations of motion shown to be at variance with the boundary
conditions — modification of the equations to satisfy the boundary
conditions.
47. On the General Theory of Thermo- Dynamics . . . 138 — 1.52
Joule's law and Carnot's ideal engine — experimental illustration of the
second law of thermodynamics by mechanisms working by means of
undirected energy — the limits of the steam-engine — possibilities of
the gas-engine.
48. On the Two Manners of Motion of Water . . . 153—162
The inadequacy of the theory of fluid motion to account for the actual
behaviour of fluids — internal motions seen by introducing colour
bands — conditions for steady motion —converging channels generally
steady, and diverging channels unsteady — parallel streams steady
below and unsteady above a certain velocity — effect of viscosity.
49. On the Theory of the Steam-Engine Indicator and the
Errors in Indicator- Diagrams ..... 163 — 180
Requirements that the diagram may be exact. — The disturbances on
the pencil — (1) disfigurement of the diagram caused by the inertia of
the mechanism — (2) the friction of the pencil — the general effect to
increase size of diagram. — The disturbances on the drum — (1) inertia
of the drum — (2) varying stiffness of the spring — (3) the friction of
the drum — shortening of diagram and reduction of mean pressure.
49A. Experiments on the Steam-Engine Indicator . . . 181 — 202
Description of the apparatus — testing of the springs — effect of oscilla-
tions of spring — stretching of indicator-cord.
50. On the Dilatancy of Media composed of Rigid Particles
in contact. With Experimental Illustrations . . 203 — 216
Any change of shape causes a change of volumes in a granular medium
— equal spheres arranged as a pile of shot have a density \/2 times
as much as when arranged in a cubical formation — condition of
maximum density very stable — friction tends to increase stability —
experiments with sand and shot contained in bags — dilatancy of
media a possible explanation of the force of attraction — also of
cohesion and chemical combination.
CONTENTS.
IX
51. Experiments showing Dilatancy, a Property of Granular^
Material, possibly connected with Gravitation
52. On the Theory of Lubrication and its Application to
Mr Beauchamp Tower s Experiments, including an
Experimental Determination of the Viscosity of Olive
Oil
217—227
228—310
Section I. — Introduction.
Discordance of experimental results — Mr Tower's discovery of the sepa-
rating film of oil, &c. — the idea of a hydrodynamical theory of
lubrication — the equation of lubrication mentioned before Section A
of the British Association at Montreal, and subsequently integrated —
the comparison of the theoretical results with experimental shows
a temperature effect — determination of the variation of viscosity of
olive oil brings the theory into complete accordance with experiments,
and shows how various circumstances affect the results — the difference
in the radii of brass and journal, and the point of nearest approach
of brass to journal, and explanation of increased heating on first
reversal — the limits of complete lubrication, incomplete lubrication,
and necking — the general arrangement of the paper .
Section II. — The Properties of Lubricants.
Definition of viscosity — the character of viscosity — the two viscosities — '
experimental determination of the value of /i for olive oil, &c., Figs.
2 and 3, Table I. — the comparative values of /* for different fluids
and different units 234—242
faction 111. — General View of the Action of Lubincation.
The case of two nearly parallel surfaces separated by a viscous fluid —
the case of revolving cylindrical surface — the effect of a limited supply
of lubricating material — the relation between resistance, load, and
speed for limited lubrication — the conditions of equilibrium with
cylindrical surfaces — the wear and heating of bearings . . . 242 — 258
Section IV. — The Equations of Hydrodynamics as Applied to Lubrication.
The complete equations for interior of viscous fluid simplified — the
boundary conditions — the first integration of the resulting equations,
equations of lubrication — the conditions under which further inte-
gration has been undertaken ........ 258 — 262
Section V. — Cases in which the Equations are Completely Integrated.
Parallel plane surfaces approaching each other, the surfaces having
elliptical boundaries — plane surfaces of unlimited length . . . 262 — 265
Section VI. — The Integration of the Equations for Cylindrical Surfaces.
General adaptation of the equations — the method of approximate inte-
gration— integration of the equations 266 — 273
Section VII. — Solution of the Equations for Cylindrical Surfaces.
273—282
282—289
c and /v/ -p small compared with unity — further approximation to the
solution of the equations for particular values of c, Figs. 18, 19 and
20 — c = '5 the limit to this method of integrating ....
Section VIII. — The Effects of Heat and Elasticity.
H and a are only to be inferred from experiments — the effect of the
load and the velocity to alter a — the effect of speed on the tempera-
ture— the formulae for temperature and friction, and interpretation
of constants — the maximum load at any speed .....
CONTENTS.
Section IX. — Application of the Equations to Mr Tower's Experiments.
References to Mr Tower's reports, Tables L, IX., and XII. — the effect
of necking the journal — first approximation to the difference in the
radii of the journal and brass No. 1 — the rise in temperature of the
film, owing to friction — the actual temperature of the film— the
variation of a with the load — application of the equations to the- — v
circumstances of Mr Tower's experiments, Table IV. — the velocity
of maximum carrying power — application of the equations to deter-
mine the oil pressure with brass No. 2 — conclusions . . . 289 — 311
V/ 53. On the Flow of Oases . 311—320
Experiments show that the flow of gas from one vessel into another is
independent of the pressures when their ratio is greater than two,
whilst according to the theory the flow diminishes and finally ceases
as that ratio is increased — this anomaly due to the assumption made
in the theory — that the pressure at the orifice is the pressure in the
receiving vessel at a distance from the orifice — the assumption avoided
by the integration of the fundamental equations of fluid motion.
54. On Methods of Investigating the Qualities of Lifeboats . 321 — 325
By using models made to scale according to the laws of dynamic
similarity, experiments could be made in ordinary weather to
correspond to the severest storms with full-sized lifeboats.
55. On certain Laws relating to the Regime of Rivers and
Estuaries, and on the possibility of Experiments on a
small Scale 326—335
The action of water to raise or lower the beds of rivers and estuaries —
shown to depend on the character of the motion, and to be inde-
pendent of the magnitude or velocity of the stream — tidal rivers —
experiments on a model of the Mersey Estuary — resemblance of the
contours to the charts of the Mersey.
56. On the Triple- Expansion Engines and Engine-Trials at
the Whitworth Engineering Laboratory, Owens College,
Manchester 336—379
Description of the engines, boilers, and connections — the appliances for
measuring the condensing water, and hot-well discharge — arrangement
of indicators — the specially designed hydraulic brakes — their advan-
tages over friction brakes — objects and methods of conducting the
trials — the checks on the heat and water afforded by the surface
condenser — the radiation — method of combining the diagrams to show
the proportion of steam condensed at all points in the expansion — the
missing quantity and the effect of the steam jackets in reducing it —
the relative effect of the jackets in the three cylinders — mean results
of the trials.
57. Report of the Committee appointed to investigate the Action
of Waves and Currents on the Beds and Foreshores of
Estuaries by means of Working Models . . . 380 — 409
Experiments to determine how the distribution of the sand is affected
by the horizontal and vertical dimensions of the models, and the tide
period — description of the models and the tide generators — method of
surveying — conditions of experiments — results— conclusions — plans.
CONTENTS.
XI
58. Second Report of the Committee appointed to investigate the
Action of Waves and Currents on the Beds and Fore-
shores of Estuaries by means of Working Models . 410 — 481
Experiments to ascertain the law of the limits for dynamic similarity —
critical values of the criterion of similarity— general distribution of
sand in V-shaped estuaries — effects of land water — the automatic tide
gauges — description of the experiments —plans.
59. Third Report of the Committee appointed to investigate the
Action of Waves and Currents on the Beds and Fore-
shores of Estuaries by means of Working Models . 482 — 518
V-shaped estuary with a long tidal river — possible condition of in-
stability— tides varying from spring to neap — effect of training
walls — effect of groins — general results of the investigation — descrip-
tion of the experiments — plans.
60. On Two Harmonic Analyzers 519 — 523
An instrument for detecting and identifying periodic vibrations in
structures — an appliance for setting up vibrations in a structure so
as to find its natural period.
61. Study of Fluid Motion by means of Coloured Bands . 524-=-534
Experiments showing how the internal motions, otherwise invisible, are
shown by the use of colour bands— the small resistance to wave
motions — internal fluid motion generally a process of mixing — an
experiment showing a straight vortex illustrating how internal waves
can exist without motion on the outside boundary.
62. On the Dynamical Theory of Incompressible Viscous Fluids
and the Determination of the Criterion . . . 535 — 577
Section I. — Introduction.
Stokes' dissipation function and the author's determination of the critical
velocity of water — considerations which show that the criterion follows
from the equations of motion— basis of the method of analysis —
summary of conclusions of the investigation ..... 535 — 544
Section II.
The mean-motion and heat-motions as distinguished by periods —
mean-mean-motion and relative-mean-motion — discriminative cause
and action of transformation — two systems of equations — a dis-
criminating equation 544 — 563
Section III.
The criterion of the conditions under which relative-mean-motion cannot
be maintained in the case of incompressible fluid in uniform sym-
metrical mean-flow between parallel solid surfaces— expression for
the resistance ........... 563 — 577
63. Experiments shotving the Boiling of Water in -an Open
Tube at Ordinary Temperatures ..... 578 — 587
Explanation of the hissing noise in the kettle — reduction of pressure in
a contracting channel sufficient to cause boiling — conditions necessary
as explained by the author's determination of the critical velocity of
water.
Xll CONTENTS.
PAGES
64. On the Behaviour of the Surface of Separation of Two
Liquids of different Densities 588 — 590
65. On Methods of Determining the Dryness of Saturated
Steam and the Condition of Steam Gas . . . 591 — 600
The conditions of steam in Regnault's experiments — wet steam — wire-
drawing calorimeters — theory of the reductions — the erroneous as-
sumption that the specific heat of steam above the temperature of
saturation is constant and equal to that of the steam gas in
Regnault's experiments — the possibility of obtaining an accurate
estimate — means of assuring the final condition, that of steam gas.
66. On the Method, Appliances and Limits of Error in the
direct Determination of the Work Expended in Raising
the Temperature of Ice-Gold Water to that of Water
Boiling under a pressure of 29'899 inches of Ice-Cold
Mercury in Manchester ...... 601 — 733
PART I.
The standard of temperature in Joule's determination — description of
the experimental steam-engine and other appliances used in this
investigation —the brake and the possible errors due to fluctuations of
the speed and the turning moment — the cyclic variations of speed —
thermometer scales avoided by working between the standard tem-
peratures 32° and 212° — the additional appliances required — con-
duction and radiation of heat — the standards of length, mass, and
temperature — air in the water — the specific heat of the water —
complete table of the corrections for all circumstances affecting the
accuracy of the results 601 — 657
PART II.
Details of the several parts of the apparatus and measuring appliances —
the system of conducting the trials — comparison of the thermometers
—adjustments of the brake —leakages — details of the different series
of trials — the correction ......... 658 — 733
67. On the Slipperiness of Ice 734 — 738
An explanation afforded by the author's theory of lubrication which
shows that continuous rectilinear motion is one of the only two
cases in which complete lubrication is possible.
41.
ON THE FUNDAMENTAL LIMITS TO SPEED.
I.
[From "The Engineer," Oct. 28, 1881.]
AMONG the facts which are so familiar to us as not to command our
attention are the limits to the rates at which we can move over the surface
of this earth, or, to put it more generally, the limits to the rate at which
terrestrial objects can move. Everyone is now familiar with the fact that
railway trains do not exceed sixty or seventy miles an hour; that steamboats
do not exceed twenty-five miles an hour ; carriages on ordinary roads, ten
or twelve. The fastest running animals rarely exceed a mile in two minutes,
or the fastest bird a mile a minute. That there are circumstances on which
these limits depend must be generally recognised ; but, while speed is the
highest of our mechanical ambitions, how many of those who find themselves
confined for nine hours between London and Edinburgh have ever asked
themselves, why should there be a limit to speed at all ?
In the early days of railroads the question as to the possibility of exceed-
ing the speed of animals was very prominent ; and many of the immediate
circumstances on which possible speed depends — such as the strength and
elasticity of the machine, and the smoothness of the road — have since
received due attention. This was a matter of necessity, just as, in attempt-
ing to gain a higher standpoint on the side of a hill, account must be taken
of the difficulties of the ground immediately above one. But such notice is
a very different thing from a general survey of the limit imposed by the
height of the hill itself. While we were still in the valley, and the immediate
difficulties of ascent were great, our aspirations might well fall short of the
top of the hill, which would not then become an object of attention. But
having toiled up a great way, and having apparently reached a flat, or
.
2 ON THE FUNDAMENTAL LIMITS TO SPEED. [41
nearly flat, plane on which we are wandering without making any consider-
able ascent, it cannot but be a matter of interest and importance to make a
more general exploration, and endeavour to ascertain what is the nature of
the country behind and above the clouds which surround us.
The greatest speeds attained have not increased now for many years.
It is probable that the run from Holyhead to London is still the fastest
journey ever accomplished over so long a distance, although the number of
instances in which this speed is approximately reached are now numerous,
and continually increasing. With animals there is no great alteration — why
should there be ? And with machines, locomotives or steamboats, the
improvement is that the average speed more nearly reaches the maximum,
rather than any extension of the maximum. Noticing this, we cannot avoid
the surmise that the obstruction to further advance arises from something
more fundamental than mere economy or imperfection of mechanical con-
trivance. The question as to how far this is the case must admit of an
answer if the circumstances can be subjected to a complete theoretical
examination. The problem is very complicated, and it may well be doubted
whether our knowledge of the circumstances and possibilities of art is
sufficient to enable us to arrive at a definite conclusion. But what we may
do is to look, in the first instance, for any circumstance which imposes
a definite limit to possible speeds, and having investigated the law of this
limit, look for other limits, and having examined each separately, endeavour
to arrive at the result when they are taken in conjunction.
To begin with, it will be well to try and catch sight of the top of the hill
from a distance. Going far away from the complexity of our immediate
problem, we may ask whence there can be any limit to possible speeds ?
Any limitations in the circumstances on which speed depends would cause a
limit to speed and, although perhaps not very obvious, consideration will
show that speed depends on certain physical and mechanical properties
of material, and that these are essentially limited. Thus the strength of
material is limited. Some materials are stronger than others, but the
strength of the strongest is easily reached, and although improvement in art
brings the stronger and more appropriate materials within reach, still by no
tittle have we been able to extend the strength of the strongest beyond
what it has been, so to speak, fixed by nature. When compared by heaviness,
natural tissues are the strongest materials. A silk cord will sustain more
than a steel wire of the same weight, and such a wire is the strongest form of
any manufactured material. To the limited strength, as compared with the
weight of material, then, we may look for a limit to possible speeds; and this
is not all. There are other limits — for instance, the limited temperature at
which material retains its strength ; in fact, the properties and powers of
material are essentially limited in all directions, and, inasmuch as speed
41] ON THE FUNDAMENTAL LIMITS TO SPEED. 3
depends on these properties, it must be limited. If we take a somewhat
closer view, the immediate conclusion is that there are at least two distinct
sources of a limit to speed. The first and most obvious of these is that the
resistance to motion requires that the moving object should be continually
urged forward by a force, and the maintenance of this force requires additions
to the weight of the moving object, which additions increase the resistance;
so that at a certain speed there will be a balance between the resistance and
the force, any increase in the force causing a still greater increase in the
resistance.
This may be illustrated by reference to a railway. The resistance of the
engine is the addition necessary to maintain the motion. Taking the best
results, the resistance of an engine at high speeds is about 45 Ib. per ton of
its weight. If, then, the locomotive weighs 20 tons it would require a steady
pull of 900 Ib. to balance its resistance. To maintain this force a certain
pressure of steam must act on the pistons. To keep up this pressure the
cylinders must be filled and emptied every revolution of the driving wheels —
say, every 2(>'4 ft., or 200 times per mile. To maintain the speed then the
boiler must supply steam enough to fill the cylinders 200 times per mile,
i.e., in whatever time the mile is run. Now the power of supplying steam by
the boiler is limited. A boiler of a certain weight cannot be made to supply
more than a certain amount of steam, and if we know the shortest time
in which the boiler will produce 200 cylinders full of steam at the pressure
required to move the engine, we know the shortest time in which it could
run a mile, or the limit of speed arising from this source. To increase the
size of the boilers would be to increase the weight and consequent resistance
of the engine, so that the only chance of extending the limit is to increase
the steam -producing power of the same weight of boiler ; and the question
whether this actual limit has been reached is a question as to whether there
still remains, after all these years, room for improvement in the best boilers —
whether, in fact, the steam-producing power of boilers has reached the limit
imposed by the limit to the strength and other properties of material of
which they may be constructed.
The case of the locomotive has been introduced here merely for the sake
of illustrating the fact that, however distant, there is a limit to possible
speeds arising from this source. As a matter of fact this limit is not actually
reached, for, as will be subsequently shown, there are other and inferior
limits which come in; that is when the engine is running without a train,
but when the train is added, as it must be from an economical point of view,
then the steam -producing power of the boiler does impose an economical
limit on the speed of the train.
The case of steamboats is somewhat different. With these the resistance
1—2
4 ON THE FUNDAMENTAL LIMITS TO SPEED. [41
increases in a high ratio with the speed, as the square of the speed, so that
not only have the cylinders to be filled at a rate proportional to the speed of
the boat, but to maintain the requisite force the size of the pistons or the
pressure of the steam must increase as the square of the speed ; so that
instead of being, as with the locomotive, nearly in the simple ratio to the
speed, the quantity of steam required in a given time varies as the cube of
the speed. Thus, in the case of steamboats, the steam-producing capacity of
a certain weight of boiler is the source of the actual as well as the economical
limit to the speed. This limit has been reached with the modern steam
launch and torpedo boat, in which as much as two-thirds of the whole weight
of the ship are given up to the engines and boilers ; the highest speeds
so attained being about twenty-five miles an hour. The action of this,
which may be called the physical limit to speed, may be traced in animals,
but the requisite data for its discussion are wanting. The second funda-
mental source of limits to speed is the strength of the parts, and the forces
holding these parts, necessary to withstand the forces to which the motion
gives rise. This may be called the dynamical limit to speed.
This source of limit has received less general notice than the preceding.
That the motion of machines and animals necessarily gives rise to forces in
and between their parts is not perhaps very obvious, on account of its being
so well known that motion itself does not give rise to force between the parts
of a moving object. But this is only when the motion is rectilinear and
uniform. To stop and start a body or to change its direction requires force
proportional to the weight of the body and the rate at which the change is
made. In order to realise how all possible motions on the earth are limited,
it must be noticed that uniform rectilinear motion is impossible. Objects on
the earth have to maintain their motion against such resistances as they
encounter by the relative and limited motions of their parts ; with animals
by the motion of their legs, wings, or fins ; in machines by the motions of
their pistons, cranks, and wheels ; and, even apart from this, uniform motion
is impossible owing to the impossibility of maintaining a direct course — for
instance, a perfectly even road.
The limit to the speed of any complex body, such as an animal, an engine,
or even a revolving wheel, will depend primarily on the manner in which the
general motion depends on or involves change in the speed or direction of
motion of any or all of the parts. For example, in the case of all carriages
the limit to the strength of the tires of the wheels would limit the speed if
there were no inferior limit. That what is called centrifugal force tends to
burst the tires must be universally known ; but there is a simplicity about
the law of this limit which marks it out as the best illustration of the class of
limits which arise from acceleration.
41] ON THE FUNDAMENTAL LIMITS TO SPEED. 5
The bursting tension of the tire caused by the revolution of the wheel is
the result of the centrifugal force acting on each elementary portion of the
tire, and is the same as if the tire were subject to an outward pressure equal
to the centrifugal force all over its inner surface. The dynamical problem of
estimating the centrifugal tension from the weight, diameter, and speed of
revolution of the tire is not difficult, but it will be sufficient here to state
the result. The tension per square inch of section of the tire is '37 multiplied
by the weight of a cubic inch of the material and the square of the velocity
in feet per second. The limit of speed is that which causes a centrifugal
tension equal to the greatest stress the material will safely bear. With iron
this is about 15,000 Ib. per square inch. A cubic inch of iron weighs '24 lb.,
so that the velocity squared is equal to 11x15,000 or 165,000, or, roughly,
the velocity equals 400 ft. per second. This, which is 270 miles an hour, is
the limit arising from centrifugal force to the safe velocity ; for steel tires,
the strength of which is about double that of iron, the limit becomes
380 miles an hour. It should be noticed that neither the diameter of the
wheel nor the thickness of the tire makes any difference to this limit, which
depends solely on the ratio between the strength and heaviness of the
material. If we could get a stronger material, then we might extend the
limit, but as natural fibres are the only materials stronger than steel,
and these do not possess the hardness necessary for tires, there is absolutely
no prospect of any extension in this direction.
The velocity of the train is the same as the velocity of the tire, so
that the figures given above show the limit to the velocity of the train
arising from the centrifugal force on the tire — that is, supposing the tire
subject to no forces but those considered. Looked at in this way, the limit
appears well away from any speeds already realised. But as the tire is
subject to forces arising from its contact with the rail and from the load on
the wheel, the margin left for centrifugal force is much less than what has
been stated, so that the actual limit, which involves complex considerations,
is really much lower.
Wheels have been here considered as affording the simplest example of
how changes in the direction, or speed of motion in the parts, of a moving
object must cause a limit to the speed at which the object can move, and not
1>« -cause the wheels are the parts which would give way first were the speed
to be increased. In the locomotive, as at present constructed, there are
parts — the coupling and connecting rods, for instance— which would give
way under these accelerations before the tires ; and it will be the object in a
subsequent article to discuss somewhat fully the limit to speed imposed by
these, as well as by othor parts of the machinery.
In the case of animals there are no wheels, but the problem does not
6 ON THE FUNDAMENTAL LIMITS TO SPEED. [41
differ greatly ; for the forces required to stop and start the limbs tax the
strength of these in much the same way as the strength of the tires is taxed
by centrifugal force. So that the conclusion is the same, that the strength,
as compared with the heaviness, of the material of the bones and tissues of
animals determines a limit to the possible speed ; which conclusion is borne
out by the fact that the strength, as compared with the heaviness of these
materials, is as high, or higher, than that of any other materials — the
strength being that required to resist the particular forces which the parts
are generally called upon to sustain, i.e., bone to resist crushing, and sinews
to resist tension.
Before closing this article, which is intended as an introduction to the
more definite discussion of certain particular cases where these limits come
in, it should be pointed out that besides the two sources of limits to speed
which have been particularly noticed, viz., those which arise from the
strength of the material, and those from the limited capacity of producing
energy, there are other sources of limits. One of these, of a physical kind,
is the inability to get rid of the heat produced at the joints by friction.
The heating of bearings, which is a very common source of the actual limit
to speed, although it has not apparently received much attention except in a
practical way, admits of theoretical consideration as being subject to definite
laws.
Another source of the limit to speed, of the greatest practical importance,
although more complex than the preceding, is the effect of the moving pieces
and the forces between these to cause unsteadiness to the motion of the
whole structure. The difficulty of keeping a railway train steady has
perhaps as much to do with the actual speed attained as any other cause.
In so far as this unsteadiness arises from the unevenness of the road, and
the mere disturbing forces caused on the frame by the moving pieces,
it belongs to the class of dynamical limits, but it depends on a particular
property of matter not involved in other cases of this class of limits. The
rocking of a structure depends on the character of its elasticity, and on the
period as well as the magnitude of the disturbing forces ; and, as a matter of
fact, the tendency to vibrate would impose a limit on the speed of most
machines, so that it is entitled to a place amongst the sources of limit, and
may be called the elastic limit.
So far, then, we see that there are four distinct sources of limits to speed.
The limited capacity of producing energy, the limited strength of the
material, the limited power of discharging the heat produced by friction,
and the elastic limit. In pointing out the general nature of these limits,
attention has been directed to objects with powers of locomotion as being
more familiar ; there are, however, the same sources of limits to the speed of
stationary machinery, such as steam-engines and tools.
41] ON THE FUNDAMENTAL LIMITS TO SPEED.
II.
[From "The Engineer," Nov. 18, 1881.]
To obtain an idea of the effect of accelerations, we may take an instance
of a moving machine, and supposing its speed to increase, consider which of
its parts would give way first. The locomotive seems to afford the best
example. Imagine, then, a locomotive to be started down a long incline with
the steam fully on ; what part of the machine would give way first ? In the
case of an engine with its wheels coupled, the question may be answered
with certainty. The coupling rods would be thrown off. Although perhaps
not generally known, this has been shown both theoretically and practically.
Anyone with the smallest mechanical insight, observing from a distance a
coupled engine in motion, cannot fail to perceive that the rapid up-and-down
motion of these rods, which are held only at the ends, must call for great
strength to prevent them breaking in the middle. That the strength so
called for approaches the actual strength of the rods can, of course, only
be ascertained by definite calculation. Six years ago the case of one of these
rods was taken as an example, to illustrate to the engineering class at Owens
College the effect of accelerations, and the result of the calculation then
made was to show that the strength called for when the engine was running
at 70 miles an hour was nearer the limit imposed by the actual strength of
the material than is usually considered safe in estimating the size of such
structures. Thus, instead of 10,000 lb., the stress amounted in this example
to 15,000 lb. The fact was surprising enough to arrest attention, and raise
a question as to the considerations which had led to the proportions of these
rods. On reference to the text-books and manuals it was found that the
effects of accelerations had no place in them, so that it would appear that
engineers have had no rule to go by but that of experience; or, in other
words, that the dimensions of these rods have been arrived at by the process
of trial and failure. All these facts considered, the matter seemed one of no
small mechanical interest. For apart from the importance of these rods and
the desirability of supplying a theoretically derived formula in place of
empirical rules, the experience of the fitness of these rods has been so ample,
that as soon as we are in a position to calculate the stresses in their material,
they furnish a very important test as to the factor of safety for such parts of
machinery. Thus, it appears that while a rule has been laid down that a
certain stress is the greatest which the iron in any important part of a
machine should bear, these very important parts have been unwittingly
8 ON THE FUNDAMENTAL LIMITS TO SPEED. [41
allowed to bear, and have borne safely, half as much again as that given
by the rule. That the stress in these rods may be as great as appeared from
theoretical consideration, or, at least, that they are the parts of the engine
which first give way when an undue speed is attained, has been confirmed by
the records of railway accidents. Shortly after the first investigations were
made, a train having on it three similar coupled engines ran away down an
incline, the brakes being overpowered, and eye-witnesses described how the
first symptom of disaster was the flying off of the coupling rods from one of
the engines, those from the others following immediately after. In 1878
attention was called to these facts at a meeting of the Manchester Literary
and Philosophical Society, and they excited the interest of Dr Joule, who has
kindly sent the author published accounts of several instances of the failure
of these rods in cases of high speeds. Amongst these was the following
extract from a letter published in the Manchester Courier. The accident
occurred on the Cheshire line from Manchester to Liverpool, on which the
speeds are very high. The author of the letter has clearly used the term
connecting rod in the sense of coupling rod. " Shortly after we had passed
one of the small stations on the way, and before reaching Warrington, the
connecting rod of the engine, or some other material -portion of that part of
the mechanism, became broken, and flew off with such force as to strike the
embankment on the near side, and thence rebound with terrible power into
the window of one of the third-class carriages immediately behind, completely
smashing in the woodwork, as well as all the glass, to the great danger of one
or more passengers within, but who escaped uninjured. I was a passenger on
another occasion, on the same journey, when the connecting rod snapped in
two, and the two pieces continued to whirl round until the train could be
stopped, to the great risk of driving the engine and carriages from the metals.
And I have heard it said that accidents of a similar kind have occurred on
other occasions."
The theory of these rods has been taught in the engineering classes at
Owens College for several years, but its first appearance in print seems to
have been in a letter in The Engineer of May 27th, 1881, signed " S. R.," dated
Manchester, May llth; and more fully in an article which appeared in The
Engineer, of Sept. 9th, 1881. Leaving what we may call the swinging forces out
of consideration, the coupling rods are designed to withstand certain forces
which cannot exceed a definite amount. This amount may be estimated for
each particular case. The utmost one rod can be called upon to do is to turn
one pair of wheels against the whole friction between the wheels and the rail,
which latter may be sanded. In such a case, F, the coefficient of friction,
would be about -3. Let R be the radius of the wheels in inches, L the
length of the cranks, P the pressure between the wheel and the rail in
pounds ; then taking T for the force in pounds, tension or compression, in the
41] ON THE FUNDAMENTAL LIMITS TO SPEED. 9
rod necessary to cause the pair of wheels to slide when the other rod is in the
line of centres,
LT=FRP,
.
Li
T may be either tension or compression, but it is the latter that is the most
important for the present consideration. If now we take the swinging action
into account, we have to add the effect of the vertical force which must
act on each point of the rod in order to change its vertical motion. Relatively
to the engine each point of the rod will describe a circle exactly similar to
that described by either crank pin. In describing its circle each portion
of the rod will be subject to centrifugal force. Consider a cubic inch of
material of weight w, the centrifugal force of this by the well-known
formula is
wv*L*_ wv*L
-
Where v is the velocity of the engine in feet per second, and <jf=32'2 the
acceleration of gravity. The direction of the centrifugal force will be parallel
to the line joining the centre of the crank shaft with the centre of the
crank pin, and consequently will be vertical and directly across the rod when
the cranks are vertically up or down.
We have then a force G, acting upwards or downwards, on each cubic inch
of the rod. When the cranks are down this force must be added to the
weight of the rod, which will then act in the same direction. Then the
effect to break the rod will be the same as if the engine were standing,
and the weight of the material of the rod were increased in the ratio
C + w
w
So that as regards this force the rod may be considered as a loaded beam.
Let the rod be of uniform section of length H, area S, and depth 2y, also let
K be the radius of gyration of the section. Then the load on the rod is
(C + w) SH, and the greatest bending moment in inch-lb. is
v SH~
(C + w) 8 ^M,
and if/ be the greatest stress in the rod, for the resistance to bending we
have the well-known formula —
10 ON THE FUNDAMENTAL LIMITS TO SPEED. [41
Comparing these two values of M we may determine f the stress due to
centrifugal force in terms of the velocity of the crank pin —
8 K*'
We have thus two independent forces, to which the material of the rod is
liable. The bending moment M and the thrust T, the stress caused by T
T
distributed uniformly over the section would be ~ . Therefore the stress due
to both these causes is equal to —
/+§,
, T c + w H'y T
and /+_ = __ J + -,
which formula will give the greatest stress which one of these rods is subject
to when the dimensions, speed, and material are known. As an example let
us suppose,
#=108 in., 2/ = 2i, S=7'87, w = -28, L = 8%,
v = 100 (70 miles an hour nearly),
R = 39 in., P = 26880 (12 tons), F= "3.
Substituting these quantities, which correspond to the dimensions of an
express passenger engine on the North British Railway described in The
Engineer, Vol. L., 1878, and we find
/= 11357,
T
- = 4700,
s
so that the greatest compressive stress in the rod when the engine is
running at seventy miles an hour is 16,000 Ib. per square inch. This stress
is applied and reversed from tension to compression every revolution of the
wheel, so that the fact that these rods do safely withstand these stresses
affords sufficient proof that the material of which they are composed will
safely bear a repeated load of 16,000 Ib. on the square inch. As they are
constructed, however, these rods clearly impose a limit on the possible speed
of the engine, and a limit very close to that which is actually attained by
passenger engines. There is no necessity, however, that this limit should be
so low. The simple bar form which is usually that given to these rods is about
the worst shape they could have to resist the centrifugal forces. By making
them hollow or with flanges, it would be perfectly easy to extend the limit
considerably without adding to the weight of the rods.
41] ON THE FUNDAMENTAL LIMITS TO SPEED. 11
The coupling rods are those parts of a locomotive in which the accelera-
tions produce the greatest effect, but all the reciprocating parts oT the engine
are subjected to similar forces. The connecting rods differ from the coupling
rods, in the fact that it is only one end that swings, and hence that the effect
of the acceleration varies from nothing at the piston end to the value given
by the formula at the crank end. Thus while the coupling rods may be
regarded as a beam loaded uniformly, the connecting rods are subject to
loads varying uniformly from the piston to the crank end. But the result
will be the same, and the liability of the connecting rod to break under its
swinging action would impose a limit to the speed were it not for the inferior
limit imposed by the coupling rods. Let alone the swinging motion, the mere
reciprocation would impose a limit to speed. Thus to stop and start the
piston and its attachments requires a force which is given by the same formula
Wv'2L
— =^- , L now being the length of the crank, and W the weight of the piston
gR-
and its attachments. At moderate speeds these forces are small compared
with the forces produced by the pressures of the steam, but increasing, as
they do, as the square of the speed, they soon leave the others behind. By
diminishing the lengths of the cranks in proportion to the diameters of the
wheels, and the consequent amplitude of reciprocation, the accelerations are
proportionally diminished ; but then, in order to transmit the same power, the
size of the pistons and the dimensions of all the parts must be proportionally
increased, and then the heating of the bearings comes in to limit the speed.
Thus with high speeds of pistons the forces arising from reciprocation limit
the speed, while with low speeds the difficulty of the bearings limits the
speed. There is, therefore, a middle course between these two extremes, and
it is this medium course to which experience has led, although the deter-
mining causes have been but very imperfectly recognised.
So far the accelerations spoken of have been those which result from the
regular motion of the internal parts of the engine. But in the case of all
carriages there is another class of accelerations, which, although less regular,
act a similar part in causing a limit to the speed, and which follow the same
laws. These are the vertical accelerations which arise from the inequalities
of the road. If the road be uneven — as all roads are, more or less — the
wheels, and to some extent the carriages, move up and down according to the
inequalities. This up and down motion, although not regular, necessitates up
and down accelerating forces, which will be proportional to the square of the
velocity of the carriage, so long as no limit comes in to prevent the wheels
following the inequalities of the road. The upward acceleration is caused by
the pressure of the road on the wheel, and the limit to this is obviously one
of strength. So long as neither the road nor the wheel gives way, the motion
must ensue. But. the downward acceleration can only result from the force
12 ON THE FUNDAMENTAL LIMITS TO SPEED. [41
of gravitation acting on the wheel, and the pressure exerted by the carriage
to keep the wheel down. Where springs are used this pressure will be
maintained nearly constant, whatever the acceleration may be ; but without
springs the greatest acceleration is that of gravity — for the carriage will
have to follow the wheel in its vertical motion, and the greatest accelera-
tion is when they are both free to fall.
If W be the weight of the wheel and C the load of the carriage, then,
without springs, the greatest acceleration is 32'2, or g ; but with efficient
C + W
springs it is -™. — g. When the speed of the carriage is such as to require
an acceleration greater than this in order to keep the wheel in contact with
the road, the wheel will bound. This practically limits the speed of carriages
without springs on ordinary or paved roads to three or four miles an hour ;
but with springs there is no difficulty in attaining speeds equal to the highest
that horses can maintain.
The use of the level iron rails maintained in their proper position
diminishes the vertical motion to such an extent, that there is no difficulty
from this cause at the highest speeds attained even at the present day. But
the difficulty of maintaining the rails, and particularly the ends of the rails,
in their places, is considerable, and one misplaced rail becomes a source of
danger, so that it cannot be said that the vertical accelerations exercise
no influence on the limit of speed. This action, however, must not be
confused with the liability of the train to rock, which, although depending
on the unevenness of the road, depends rather on the frequency of the
inequalities than on their magnitude ; and further, as has already been
pointed out, depends on the elastic properties of the train. This rocking
will be considered in another article.
III.
[From "The Engineer," Dec. 9, 1881.]
ALTHOUGH vibration is one of the greatest and most common difficulties
with which engineers have to contend, it is, perhaps, of all mechanical
phenomena least understood. It does not appear to have been made the
subject of any treatise, or to have a place in works which treat of applied
mechanics. This has doubtless arisen from the great apparent diversity in
the circumstances under which it occurs. The mechanical principles involved
are sufficiently well understood by natural philosophers ; but they have not
been applied to the practical questions. Such an application is, however,
not only possible, but the general circumstances on which vibrations depend
41] ON THE FUNDAMENTAL LIMITS TO SPEED. 13
may be apprehended without the aid of mathematical symbols. In fact an
unconscious apprehension of the principles of vibration is one of—the earliest
lessons which children learn. The act of swinging in a child's swing requires
such a knowledge, and this whether swinging oneself by motion in the
swing, or swinging another by pushing the swing. And the same may be
said of shaking an apple tree. The act of shaking a tree does not consist
simply in exerting a force first in one direction and then in the opposite.
One might do this, exerting many times the force necessary, if properly
applied, to bring not only the apples but the leaves off the trees, without
bringing down a single apple. What is required besides the alternating
force is that the alternations should be timed right. This timing of the
alternations in the direction of the force we exert comes naturally when we
are trying to shake an object; for naturally we follow the object in its
motion ; indeed it is difficult to avoid doing this. But if, instead of shaking
the tree by muscular exertion, we were to arrange a steam-engine to shake
it, then we should at once perceive that there was only one particular speed
of the engine at which the tree would shake. The general phenomenon, the
apprehension of which has been wanting to the understanding of the
circumstances on which vibration depends, is that the discovery of which led
Hooke to perceive the mechanical law which bears his name — " ut extenso
sic vis" — and also led him to construct a watch after the present method.
This phenomenon is that a fixed object will, when set in motion to a greater
or less degree, continue to rock in a particular direction, with a particular,
and only with that particular, rate of oscillation. This is no less true of
ships, bridges, and parts of machines, than of apple trees, tuning-forks, and
the balance-wheels of watches. We say continue to rock ; but it is not
meant that it will continue for ever, or for any great length of time. The
motion will gradually diminish, according to the resistance encountered from
the air and the imperfect elasticity of the structure.
The rate at which a structure will rock depends on two circumstances —
the stiffness of the attachments by the bending of which the rocking takes
place, and the magnitude and distribution of the weight to be rocked. In
the case of short, stiff objects, like the prongs of a tuning-fork, the vibra-
tions may amount to hundreds per second ; whereas in the case of trees,
ships, bridges, or steam-engines, they are often as low as two or three per
second, or even one in two or three seconds.
The period in which a body will continue to rock in any manner may be
called its period of five vibration for that manner of nicking; and having
recognised the general existence of such periods of free vibration, a general
view of the circumstances under which dangerous vibrations are likely to
occur is not difficult. Were it not for the decadence of the free vibration
when ouce set up, owing to such causes as have been already mentioned,
14 ON THE FUNDAMENTAL LIMITS TO SPEED. [41
then it is obvious that if to the swing already attained a small addition were
made, the increased swing would continue, and by continually adding fresh
swings, however small, the swing must eventually increase until some limit
was reached. Thus, one child swinging another, if there were no retardation,
would, if it continued to impart a push, however slight, each time the swing
passed, eventually send the swing completely round. As it is, however,
owing to the retardation arising from the resistance of the air and the stiffness
of the ropes, the work done by the swinger only just balances the energy
lost, and so only maintains the speed ; the greater the speed the greater the
work spent in retardation, and hence the greater the exertion on the part of
the swinger necessary to maintain it. Now, the theory of all steady vibration
is the same; whatever may be its nature, there must always be something to
act the part of the swinger, and by well-timed acceleration make good the
necessary loss. In order that the extent of vibration may be constant, the
added velocity must be exactly what is lost ; if it be too great the amplitude
will increase, or if too small, diminish. There are several things which may
thus act the part of the swinger — any reciprocating or revolving weight, any
periodic force, such as may arise from the intermittent pressure of steam on
the piston, or a periodic motion, such as is caused by the wheels of a carriage
running over the setts on the street or the sleepers on a railway ; in fact, any
periodic disturbance. But as a matter of fact, such disturbances have always
a fixed period, and as the body will only oscillate in a fixed period, it is only
in the case when the period of the disturbance exactly fits the period of
vibration that this vibration can be steady, and this rarely or never happens.
What really happens is that the period of disturbance approximates more or
less to the period of vibration, and in order to understand the theory of
vibrations under consideration it is necessary to consider how a difference in
the periods influences the vibration. The swing will enable us to do this.
Suppose the period of the swing to be two seconds, and suppose that the
swinger pulls a rope every 2'05 seconds ; the first pull will set the swing in
motion a little ; in the second swing the pull will come a little late, but still
before the forward motion has ceased, which will be '5 second from the start.
The second pull will therefore accelerate the motion, and so will the eight
succeeding pulls. After this, however, the pull will come on the backward
motion and exercise a retarding effect, and by the time ten such pulls have
been given the retarding effect will have just balanced the previous accelerat-
ing effect, and all motion will have ceased. We see, then, that the result
would be waves of vibration, ten effective pulls, and as much motion as these
would impart, and then ten retarding pulls, destroying the motion. The
number of effective pulls clearly depends on the approximation of the period
of the pulls to that of the swing. If this had been only '01 second
different, then we should have had fifty effective pulls and a corresponding
motion. The magnitude of the motion attained will thus depend on two
41] ON THE FUNDAMENTAL LIMITS TO SPEED. 15
things — the magnitude of the disturbing force or pull compared with the
weight on the swing, arid the number of effective pulls of which the dis-
crepancy of the periods admits, which number will be the whole period
divided by four times the difference of the two periods. This result, which
is obvious in the case of the swing, is equally true for all classes of vibra-
tion. When the period of a disturbing or swinging force differs from the
natural period of swing, the result will be batches of oscillations increasing
from nothing till they reach a magnitude depending on the magnitude of the
disturbing force, and the ratio of the natural period to the difference in the
two periods, the number of swings before the maximum is reached being
equal to one-fourth this ratio, which will also be the number while the
motion is diminishing. It thus appears that if the periods approximate, a
comparatively small disturbing force must produce a considerable swinging,
while if the difference in the periods is large, then the amount of motion
will be confined to that produced by the action of the single disturbance.
In the latter case the vibration is called a forced vibration. Of course,
when large disturbing forces are allowed, forced vibrations may become
important, but this seldom occurs, as large disturbing forces may generally
be avoided. Small disturbing forces, however, are almost always present
where there is periodic motion, and though the forced vibrations which
would result are unimportant, when these, owing to the near coincidence of
their period with that of force vibration accumulate, motion of almost any
extent may ensue. It is this near coincidence of the period of the disturb-
ance or free vibrations with the period of free vibration which is the
condition of danger from vibrations, and the possibility of avoiding the
danger lies in the possibility of avoiding this approximate coincidence. This
may be attempted in two ways, one by adjusting the period of disturbance,
the other by constructing the structure so as to adjust the period of free
vibration wide of that of the disturbance. The first of these methods is
seldom applicable, for the period of disturbance is generally determined by
the speed of revolution of some part of the machinery, and which speed
must vary between nothing and the highest which the limit arising from
vibration or some other cause will allow. It is, therefore, to the construction
of the structure that we must look, in order to prevent the period of disturb-
ance from reaching that of free vibration. The period of free vibration in
any structure may, and generally will, be different for different directions of
rocking, even when the structure rocks as a whole on its supports ; and when
the structure consists of many parts with more or less elastic connections,
all of these parts may have different periods of rocking. Thus each branch
of a tree if shaken separately would swing in a different period from the
tree as a whole, and each apple in a different period from the branch ; so that
if we attempt to shake the stem in a wrong period for the whole tree, we
shall probably succeed in shaking some branch.
16 ON THE FUNDAMENTAL LIMITS TO SPEED. [41
Without going too deeply into the mechanics of the subject, we may
look on a structure or a part of a structure as a solid mass on elastic
supports ; and then there will be in all six independent ways in which it can
vibrate or swing, all of which may have different periods. There will be
three linear motions — for instance, up and down, north and south, east and
west, or in whatever directions these may be, they must be at right angles to
each other ; and three circular or rotary vibrations about these axes at right
angles. Owing to a want of pliancy in the supports perceptible rocking is
seldom possible in all these directions. For instance, if we fix a hammer
with a long shaft in a vice, pinching the bottom of the shaft with the head
upwards, then the head may oscillate in two ways. If the broadest way of
the handle be east and west, then the head if set swinging would swing in
one period east and west, another north and south, but owing to the rigidity
of the shaft there could be no perceptible motion up and down ; also the
head might have a rotary motion about a vertical axis by twisting the shaft,
but the shaft would not allow of the head having any perceptible rotary
motion about any horizontal axis. In saying that these are the only three
directions of oscillation, it is not meant that the body cannot be set off
oscillating in other directions, but that it will not continue to oscillate in
other directions if started. In the case of the hammer, the head might be
set swinging south-east, but it would then change its manner of swing until
it moved in a circle, such a motion being equivalent to two motions in con-
junction, one north and south, the other west and east, which would have
different periods, and so the corresponding phases would change.
These distinct periods, which are easily conceived in the case of the
hammer, will exist more or less in all structures. In the locomotive, for
instance, the boiler is capable of rocking on its springs, with a lifting up-and-
down motion, or with a rolling motion from side to side, or with what is
called a bucking motion, one end rising and the other falling; these three
motions will have different periods. And to avoid oscillations in these
directions it is necessary that these periods should be such as not nearly to
coincide with that of the machinery when this is moving fast. But it is not
only the rocking of the engine as a whole that has to be considered ; every
part of the engine will be capable of free periodic motion, and should the
period of any forced vibrations rise into coincidence with any of these periods,
the part to which it belongs will be in danger. Such a coincidence is only
to be avoided when all the free periods are smaller than the period of any
disturbing force at the fastest speed of the engine, for since as the engine
acquires motion, the period of disturbance gradually diminishes ; this must
come into coincidence with any period of free vibration which may be greater
than that of the period of disturbance when at its smallest. The periods
of vibration of any structure may be diminished by increasing its stiffness,
41] ON THE FUNDAMENTAL LIMITS TO SPEED. 17
or the stiffness of its supports or attachments, but there is a limit to the
stiffness possible, so that the structure may fulfil its functions. For instance,
the springs of the locomotive have to allow the wheels to adapt themselves
to the inequalities of the road, and if they are too stiff they will fail to do
this.
In this way it is seen that there must be a limit to the possible smallness
of the period of free vibration of the structure and its parts, and hence to
the speed of the structure, on which the smallness of the period of dis-
turbance will depend. The stiffness of structures has for the most part been
determined by experience, and any further extensions of speeds will require
increased stiffness, and this throughout the structure ; for in any complex
structure, such as a locomotive or a railway carriage, there are so many parts
of which the free periods are small — the floor, roof, the sides, seats, and
partitions — that as the period of disturbance becomes smaller, nothing but a
general stiffening of the entire structure will prevent destructive vibrations.
Doubtless the parts may be made stiffer than they are — and this without
materially increasing their weight — which would call in other limits to speed.
Much has been done of late years, imperfectly as the theory has been under-
stood. This is very apparent when we compare the smoothness of the motion
of one of the present northern express trains with what it was some few
years ago. The carriages are, however, only stiffened up to the normal
speeds, any excess of speed becoming apparent by the tremour or vibration
which ensues ; and even at the normal speeds there is room for further im-
provement, in the accomplishment of which careful attention to the foregoing
considerations should be of the greatest use.
IV.
[From "The Engineer," March 17, 1882.]
THE inertia of the moving parts of a machine besides calling for strength
in the parts themselves, as, for instance, in the tires of wheels, often calls for
restraining forces in the supports to prevent the moving parts changing
their position. Such forces exerted on the frame when they exist will, like
those in the moving parts themselves, increase in the ratio of the square
of the speed, and hence the possible speed would in such cases be limited by
the strength of the attachments or supports of the frame. As a matter of
fact, the disturbing forces on the frame constitute one of the commonest
difficulties in the way of attaining high speeds, and demand the most careful
o. R. ii. 2
18 ON THE FUNDAMENTAL LIMITS TO SPEED. [41
consideration at the hands of engineers. These forces cannot, like the forces
in the moving parts themselves, be considered as fundamental, since, except
for the complications involved, it is always possible so to arrange the moving
pieces of a machine that their inertia shall cause no resultant disturbance on
the whole frame and its supports, the forces being confined to the moving
pieces and those portions of the frame which connect them. To accomplish
this counterweights have to be employed, and in some cases it would be
necessary to add additional moving pieces, the only function of which would
be to oppose the inertia of the parts which are necessary for the primary
purpose of the machine. When this is done the machine is said to be
perfectly balanced. There are, however, many considerations relating to the
balancing of machines which have nothing to do with a complete balance,
for, as will be presently explained, such a balance is often impracticable.
The general theory of a complete balance involves two conditions : (1) in
order that there may be no force to move the frame in any particular
direction, or that there may be no tendency to move the centre of gravity of
the frame ; (2) that there may be no tendency to turn the frame round
about its centre of gravity. The condition (1) may be simply expressed.
The moving weights must be so arranged that, however the several weights
may move, the centre of gravity of the whole system of moving pieces must
not change its position during the motion. The condition (2) may also
be simply expressed in the language of theoretical mechanics. It is that
the moving weights must at no time have any aggregate moment of accelera-
tion about any axis through the centre of gravity. To those who are not
familiar with mathematical language, this second condition as thus expressed
may not be very intelligible, nor is it easy to express the complete condition
in more general language; but as the practical examples are for the most
part very simple, it will be sufficient to explain the condition as applied
to one of these examples. Suppose the moving parts consist of two equal
weights. Then the first condition involves that the accelerations on these
weights shall be equal and in opposite directions, i.e., if the acceleration
on the one is north, that on the other must be south. But this first
condition does not require that the centres of gravity of the two weights
shall be opposite one another in the direction of acceleration ; this, however,
constitutes the second condition. For instance, in the case of a crank shaft
in uniform rotation, the centre of gravity of the shaft itself, lying in the axis,
will not move ; but the centre of gravity of the crank revolving round the
shaft will be subject to continual acceleration, directed from the axis. An
equal weight fixed at an equal distance from the axis, and on the opposite
side to the crank, will suffice to satisfy the first condition, however far along
the shaft it may be from the crank ; but to satisfy the second condition, the
centre of gravity of the counterweight and of the crank must be in a line
41] ON THE FUNDAMENTAL LIMITS TO SPEED. 19
x
perpendicular to the axis of the shaft ; and since the connecting rod occupies
the space opposite the crank, it is in general impossible to balance a crank
with a single weight, two weights having to be used, placed so that the centre
of gravity of the whole mass on one side of the crank shaft shall be opposite
to the centre of gravity of the mass on the other.
The moving parts of machines consist in general of revolving pieces, such
as crank shafts, and oscillating pieces, such as pistons and connecting rods,
the motion of which is derived from, or governed by, a revolving crank.
In the case of the revolving pieces, a complete balance may always be
effected in each piece by the addition of counterweights on the piece itself.
Thus, as far as a crank shaft in a locomotive is concerned, apart from the
connecting rods, pistons and other moving parts attached to it, the addition
of suitable weights on the driving wheels will satisfy both conditions and
prevent any disturbance on the frame arising from the revolution of the
crank shaft. Oscillating pieces, however, cannot be balanced in so simple a
manner. They require a weight or weights of which the centre of gravity is
in the line of oscillation, and oscillating in exactly the reverse manner. Now,
the manner of oscillation of, say, a piston depends not only on the motion of
the crank, but also on the length of the connecting rod, the varying obliquity
of which, when the connecting rod is short, will produce an important
effect. The only way, therefore, in which a connecting rod and piston can be
completely balanced is by oscillating weights connected with cranks on the
crank shaft, by connecting rods of such length that their obliquity is always
the same as that of the connecting rod which drives the piston. In this way,
however, a complete balance may be effected. That it is rarely or never
done is owing to the complexity and increased friction attending such an
arrangement, which renders it in other ways a greater evil than the disturb-
ances on the frame which it prevents. Practically, then, it comes to this —
that revolving pieces may be completely balanced; but, as regards oscillating
pieces, the balance cannot be made complete.
In default of a complete balance, there remains the question as to the
desirability of an imperfect balance, or what may be better described as the
introduction of other forces, so as to modify the resultant force on the frame.
The practical possibility of such modifications is limited by considerations of
complexity and friction to the addition of certain weights to the crank shaft,
which introduce forces in one direction equal to those which they balance in
the direction at right angles. But for the effect of the obliquity of the
connecting rod, the force arising from the acceleration of the piston in the
direction of its motion will at all times be the same as the component in
that direction of the centrifugal force of an equal weight revolving with the
crank and having its centre of gravity in the axis of the crank pin. The
2—2
20 ON THE FUNDAMENTAL LIMITS TO SPEED. [41
1.
centrifugal force of the revolving weight, however, would not be confined to
the direction of oscillation, so that if such a weight be used to balance the
piston, it will introduce an equal force at right angles to the direction of
oscillation. Thus if weights be added to the driving wheels of a locomotive
of such magnitude as to balance not only the weights of the cranks, but also
weights equal to the connecting rods and pistons, having their centres of
gravity in the crank pins, the only horizontal forces will be those which arise
from the effect of the obliquity of the connecting rods on the motion, while
vertical forces will have been introduced nearly equal to what the horizontal
forces arising from the pistons and connecting rods would have been. The
effect of smaller balance weights is to leave more of the horizontal forces
unbalanced, and introduce less vertical force. Such is a sketch of what may
be called the practical possibilities of balancing machines, which, like a steam
engine, involve oscillating pieces.
There will, therefore, always be disturbances in the frame, unless they
are prevented by the strength of the supports, but it is possible to so
arrange counterweights as to mitigate these forces in one direction by intro-
ducing equal forces in a direction at right angles. The problem as to
how far it is desirable to do this, is that which the practical engineer has to
solve, and which, owing to a multiplicity of considerations, can in reality
only be solved by experiment. There are, however, several leading consider-
ations, a general apprehension of which should much facilitate the task.
When there is nothing to limit the firmness and stiffness that can be
given to the frame and its supports in the directions in which the forces
which arise from the inertia of the moving parts tend to move it, there
is but little inconvenience arising from these forces. Thus in a stationary
steam engine founded in the earth steadiness may be obtained by weight
and solidity of foundations, almost the only drawback being the expense
entailed and the space occupied.
But when an engine has to be carried by a floor or on any structure more
or less elastic, then the case is different, and it becomes a question of the
greatest importance in which direction disturbing forces will produce the
least and in which the most harmful effect, it being desirable as far as
possible to balance the forces in the latter direction at the expense of
forces in other directions.
It is not, however, simply a question of stiffness, as it may happen from
various reasons that equal forces caused by the revolution of the engine
might do more harm, and under certain circumstances even cause greater
disturbance in that direction in which the supports are stiffest. The
consideration of the circumstances on which vibrations depend, as described
41] ON THE FUNDAMENTAL LIMITS TO SPEED. 21
in the preceding article, at once shows that the directions in which the
greatest disturbance of the frame is likely to result from forces caused
by the revolution of the engine, are those directions in which the
period of free vibration of the frame on its supports nearly corresponds
to the period of revolution of the engine. And where there is any
direction in which such a near coincidence occurs, it is an absolute
necessity that in this direction the balance should be as nearly perfect as
possible. It is often impossible to ascertain beforehand in which direction
such a coincidence may be expected, but its existence at once declares itself
upon the engine being put to work. Indeed, wherever an engine or revolv-
ing machinery causes a visible swinging vibration, it is in consequence
of such a coincidence of periods, and the oscillations will be found to occur
in batches, the magnitude of the oscillations and the number in each batch
increasing as the speed of the engine approaches some particular value.
This may be seen in many cranes. In such structures the period of oscilla-
tion depends upon the load suspended, and hence it will often be seen that
while the engine which works the crane will run quite steadily when the
crane is unloaded, when loaded, decided oscillations are set up ; or it may be
just the other way, and oscillations occur when there is no load, while the
structure is steady when the load is on. In almost all such cases the
oscillations might be prevented by counterweights so placed as to alter the
direction in which the forces occur.
Such oscillations are, as has been said, to be feared chiefly in cases where
the frame of the engine is carried on elastic supports. All engines supported
on springs — as portable or traction engines — are liable to them, as are also
marine engines, owing to the elasticity of the ship. And in these cases it is
only in avoiding such oscillations that counterweighting has to be studied.
In some cases, however, notably that of the locomotive, it is not only in
causing such oscillations as ensue when the directions and periods of free
vibration and of the unbalanced forces coincide, that such forces are harmful.
In the locomotive, although the frame of the engine rests on elastic supports,
namely, the springs, yet the revolving piece — the crank shaft with the
driving wheels, whence alone can arise vertical unbalanced forces — rests on
the rails, which afford a very rigid support in a downward direction, and to
prevent upward motion there is the axle-box with the weight of the
locomotive upon it. Unbalanced weights on the crank shaft cannot there-
fore cause in a vertical direction such oscillations as have just been
considered ; but they give rise to other evils. If the centrifugal force is
sufficient it will lift the axle-box against the pressure of the spring, causing
the wheel to leave the rail on to which it will return with a blow, causing
what is known as hammering ; while short of this a want of vertical balance
will cause the wheel to run with varying pressure or tread upon the rail,
22 ON THE FUNDAMENTAL LIMITS TO SPEED. [41
which will cause the wheel to wear out of the round, even if the pressure be
nowhere sufficiently relieved to allow the wheels to slip. Of these evils
hammering must be avoided, i.e., the speed of the engine must not reach
the point at which this begins, and the load and speed should not be so
great as to cause slipping. But the wear arising from unequal tread is not
so serious an evil but that it may be faced as an alternative for other evils.
A certain limited want of balance in the vertical direction is thus per-
missible.
In a horizontal direction the character of the support of the engine and
crank shaft is altogether different. In this direction the crank shaft is held
to the frame of the engine by the axle-boxes, but the frame of the engine
resting on the wheels has no backward and forward support at all, except
such as is derived from the elastic drawing apparatus which connects it with
the train. While as regards twisting about a vertical axis which causes the
engine to run with a sinuous motion, the only support is that derived from
the comparatively loose fit of the flanges of the wheels between the rails.
Thus any want of balance in a horizontal direction causes the engine to move
forward with an uneven motion or with a sinuous motion on the rails. The
last of these evils is the worst, but they are both bad according to their
degree.
As we have seen, the motions of the pistons and connecting rods intro-
duce horizontal forces such as will produce one or other, or generally both,
these evils. These horizontal forces can only be diminished by counter-
weights on the driving wheels, which introduce vertical forces equal in
magnitude to those which they balance. It is a question, therefore, between
two evils — unsteady horizontal motion or unequal tread. Experience has
shown that, up to a certain point, the latter evil is the least, and that it is
better, at least in part, to balance the horizontal forces. As to the exact
degree in which this should be done practice differs. Nor is there sufficient
data on which to lay down a general rule ; but the circumstances which in
each case should determine the balance weights are to be inferred from the
foregoing considerations. The limit to the counterweights lies in the
inequalities which they cause in the pressure of the driving wheels on
the rails ; and hence the permissible magnitude of these inequalities is
what should be ascertained in order to determine the balance weights. Or,
in other words, what is wanted to be known is the greatest proportion to
the gross load on the driving wheels that the vertical component of the
centrifugal force may be practically allowed to bear, and the counterweight
might then be designed so as to produce this force when the engine is run-
ning at its normal speed; unless, indeed — as would never happen — such
weight was more than sufficient to balance the horizontal forces.
41] ON THE FUNDAMENTAL LIMITS TO SPEED. 23
This method of arriving at the best counterweight is the only logical one.
The usual custom appears to be to balance a certain proportion of the
horizontal forces. This, however, is not logical, since but for the vertical
effect of the counterweights, the more perfectly the horizontal forces are
balanced the better, and there is no fixed ratio between the horizontal forces
and the load on the driving wheels which determines the allowable magni-
tude of the vertical forces. The common rules, too, as to the distribution of
the balance weights, are apt to be faulty, for by these rules the distribution
of counterweights is to be such as would completely balance weights centered
in the crank pins bearing a certain proportion to the oscillating weights.
So that not only does the imperfection of the horizontal balance produce
irregularities in the forward motion of the engine, but it also produces
a twisting or sinuous motion. Now, whatever proportion of the horizontal
weights may be balanced, the counterweights may be so placed on the
wheels as entirely to prevent the twisting or sinuous motion. The rule,
therefore, as far as it is possible to state it, should be to use the largest
counterweights which the load on the driving wheels will allow, and to
distribute it so as to balance all tendency to turn the crank shaft about a
vertical axis.
In the case of coupled engines, the balance weights should obviously be
equally distributed between the wheels, so that the inequality of wear may
also be distributed. In these engines it is a common custom to make the
coupling rods and the crank pins which carry them act the part of counter-
weights for the pistons and driving cranks, by placing the coupling crank
pins on the opposite side of the shaft to the driving cranks. This effects a
considerable reduction in the actual counterweight, but this appears to be
the only point gained, while in order to reduce the forces and friction on the
journals, the coupling rod crank should be on the same side of the shafts as
the driving cranks, so that the forces transmitted to the coupling rods may
not be transmitted through the journals and thus cause increased friction
on the bearings.
The disturbing effect of the inertia of the oscillating parts forced itself
into notice very early in the days of the locomotive, and immediate good was
found to result from the use of counterweights, which, by increasing the
steadiness, allowed higher speeds to be attained. It does not appear,
however, that any systematic attempts to determine the best arrangements
of the balance weights have been recorded. Experiments have been made
from which certain conclusions have been drawn, but the subject has not
received the treatment which its importance deserves. This is doubtless
because the investigation is one which involves the long-continued control,
in certain respects, of locomotive engines, while those who have had this
control have not been able to devote unclouded attention to this subject.
24 ON THE FUNDAMENTAL LIMITS TO SPEED. [41
If a railway company would engage the assistance of one of the highly-
qualified young engineers to be found at the present time, giving him the
control of the balance weights and a sufficient number of locomotives, and
power to watch the results, both as regards steadiness and wear, for a con-
siderable period, not only would they be amply repaid, but they would earn
the gratitude of all locomotive engineers, and, indeed, of the travelling
public.
42.
ON AN ELEMENTARY SOLUTION OF THE DYNAMICAL
PROBLEM OF ISOCHRONOUS VIBRATION.
[From the Twenty-second Volume of the " Proceedings of the Literary
and Philosophical Society of Manchester," Session 1882-83.]
(Read November 14, 1882.)
WHEN a heavy body is free to move in one direction, subject only to a
force which is proportional to the distance the body has moved from some
neutral position, and tends to return the body to that position, the body will,
if set in motion, vibrate about the neutral position in a period which is
independent of the magnitude of the motion.
The deduction of this theorem from the laws of motion, although well
known, is generally accomplished by the solution of a differential equation.
In some text books this is avoided by comparing the law of force on the
vibrating body with that of a component of the centrifugal force on a
revolving body ; this method involves no mathematical difficulties, but it is
indirect and hides rather than removes the dynamical difficulties. My own
experience has shown me that the mathematical difficulty or obscurity of
these methods stand very much in the way of those who are commencing the
study of practical mechanics, in which vibration and oscillation play a part of
fundamental importance. It was with a view of meeting the requirements of
such students that I sought for a method involving only Elementary Mathe-
matics, in which the solution depended directly on the principle of the
conservation of energy. Having succeeded in finding such a method, which,
although it bears a superficial resemblance to the method of the text books
already mentioned, so far as I am aware, has not hitherto been published, it
seems that it may be useful to publish it.
26
ON AN ELEMENTARY SOLUTION
[42
The method is to show that the vibrating body will at all times be
opposite, in a direction perpendicular to its path, to a body revolving
uniformly in a circle, having a diameter equal to the amplitude of oscillation,
with a velocity equal to the greatest velocity of the vibrating body.
Let 0 be the neutral position of the body considered as a point, AOB the
path described during oscillation, let p be the force
on the body when at a unit distance from 0, so that
as the force varies uniformly with the distance, px
will be the force at the distance OP = x.
Take PN perpendicular to OP and make PN
on some scale equal px, then N will lie on a straight
line COD, and the area OPN will represent the
work U done against the force in moving from 0 to
to P and
u = PNxOP=paf
Let W be the weight and v the velocity of the body, then by the conser-
vation of energy
W Wv 2
o , JJ ' V0 7/r /f)\
^r— V" + U = = til \^)t
where E is the energy of the system and VQ the velocity at 0, or substituting
from equation (1)
TFv02 Wv* px*
—5— — f)~T + ^o~ W J
but when P is at A or B, v = 0 ; put OA = a, then
-(4),
and equation (3) becomes
W
— v2 = p (a? —
9
.(5).
Describe a circle about 0 as centre with a radius a, and let PN produced
meet this circle in Q, and let QT the tangent at Q meet AB in T, then the
triangle QTP will be similar to OQP and
.(6).
PT_PQ
TQ~OQ •
Now suppose a point at Q moving with a velocity u such that it keeps
opposite to P.
42] OF THE DYNAMICAL PROBLEM OF ISOCHRONOUS VIBRATION. 27
Then the component of this velocity parallel to AB is
PT _ PQ
U TQ~UOQ'
and this is v since Q remains opposite to P.
Therefore substituting in equation (5)
W
Then since PQ2 = OQ2 - OP1 = a? - a? we have
= — u? ................................. (7).
Therefore Q moves on the circle with a velocity
which is constant. Or the motion of the vibrating body will be such that it
always keeps opposite in a direction perpendicular to its path to a body
revolving in a circle of diameter equal to the amplitude and with the greatest
velocity of the vibrating body.
This completely defines the motion of the vibrating body for starting from
A the arc described by the vibrating body after time t is a /y/^.<andthe
vibrating body will be opposite.
The time of a complete oscillation will be the time taken to complete
a revolution. If t is the time of oscillation
W,_27r
W
Therefore the time of oscillation is given by
t— 2?r A/ — (8).
V pg
43.
THE COMPARATIVE RESISTANCES AND STRESSES IN THE
CASES OF OSCILLATION AND ROTATION WITH REFER-
ENCE TO THE STEAM ENGINE AND DYNAMO.
I.
[From "The Engineer," January 5, 1883.]
1. THE two principal motions which are given to the parts of machines
are uniform rotation and oscillation. These motions are both possible, and
are both capable of performing mechanical operations ; and the question as
to why one or the other should be used gives rise to some interesting points.
In some cases, as in that of the lathe, the general purpose of the machine
renders one or other of these kinds of motion essential ; but this is not so
often the case as at first sight appears, for, if we consider, there are few
operations performed by machines which cannot be performed in some way
or another by animals, and continuous rotation is unknown in the animal
mechanics. Nature has worked entirely by oscillation, so that the use of
continuous rotation in machinery must be because, for some reason, it is
preferred to oscillation. As to the reason for this preference, animal
mechanics does not help us, for the constitutions of animals require a
certain amount of continuity in the material throughout the entire animal,
and this is inconsistent with continuous rotation. In machinery, however,
this reason for the choice of reciprocation is altogether absent, and it has
to compete with rotation on its merits in other respects.
2. The respects in which the motions of reciprocation and rotation may
be compared are numerous, and sometimes complex ; amongst the principal
are adaptability to the operation, simplicity of construction, and friction.
3. The first two of these respects are those in which the relative merits
of the two classes of motion are most obvious, and accordingly we may
43] THE COMPARATIVE RESISTANCES AND STRESSES, ETC. 29
expect to find that the choice of one or other class of motion generally
turns on their relative adaptability to a particular operation, and the
simplicity of construction of the mechanism involved. It may happen that
in both these respects the same motion is to be preferred: but in many
very important cases it seems that as regards choice of motion, adaptability
to the operation is at variance with simplicity of construction : then there
is rivalry between the two classes of motion, and the choice is not easy.
Thus we find that, although one or other class of motion has firmly
established itself for certain purposes, there are a vast number of cases in
which there has been and still is a contest more or less close. Illustrations
are not far to seek. We find reciprocating and rotary pumps and blowing
machines, reciprocating pressure engines and revolving wheels or turbines
for obtaining power from water, reciprocating and rotary saws. We might
say oscillating and rotary propellers, but the rotary motion seems to have
established itself for steam boats, although the oscillating oar holds the
advantage for manual labour. Numerous other instances might be given,
but it will be sufficient to give two, and to these attention will be chiefly
directed. The first is the steam engine, and the second the dynamo —
electric machine and electric motor.
In the steam engine, although reciprocation has the best of it, the battle
hiis never been given up. This is a case in which simplicity of construction
is apparently, at all events, at variance with adaptability to the operation.
In some cases, as in pumping engines, the operation involves or admits of
reciprocation, and, as is well known, it was to such operations only that the
steam engine was confined for about a hundred years after its invention.
For these purposes it would naturally seem that the reciprocating motion
was most applicable. But so little applicable to move revolving machinery
did it appear, that when, after the lapse of a century, Watt improved the
engine and saw the importance of applying it to revolving machinery, he
kept his improvements waiting for something like ten years while he was
attempting to find a revolving substitute for the reciprocating piston. At
last he gave up the quest, and found in the crank, or his bastard form of
it, a means of applying the reciprocating engine to purposes requiring
revolution. But although abandoned by Watt, the quest has been and
is still being followed by others. The apparently obvious advantage of a
revolving engine, and the apparent simplicity of the problem, offer so
tempting a field for invention, that probably nine out of ten of those who
commence practical mechanics engage in it until they find how thoroughly
others have been over the ground before them. So the reciprocating engine
holds its own in the long practical test. This may be said to be on account
of its simplicity of construction, and the adaptability of the reciprocating
piston to the operation of taking the work out of the steam : still nothing
30 THE COMPARATIVE RESISTANCES AND STRESSES [43
approaching a satisfactory theoretical or scientific explanation of its advantage
has been given. Thus the advantage of the reciprocating over the rotary
steam engine stands almost entirely as an empirical fact, without explanation,
and somewhat in opposition to what has been thought probable from scientific
consideration.
On the other hand if we turn to the dynamo-electric machine, we see
that the case is reversed. If there is an operation for which reciprocating
motion appears to be adapted, it is the conversion of mechanical energy
into electric currents, particularly into alternating currents, such as are
best adapted for the electric light. In this operation there is something
approaching to a necessity for continuity in the material, such as that
which determines the motion in animal mechanics to be that of oscillation.
Reciprocating motion would allow of continuity of material, whereas, in the
case of continuous revolution, continuity in the conductors is only imperfectly
secured by causing the stationary position of the conductor to press against
the moving portion. Again, the modus operandi is to cause soft iron alter-
nately to approach and recede from magnets, or to cause coils of the conducting
wire to move so that the lines of magnetic force alternately pass inside and
outside the coil.
The telephone acts by a reciprocating dynamo and motor, and its
efficiency is such as to show how perfectly the motion of reciprocation is
adapted for these purposes. Experience in the construction of the dynamo
may as yet be called small: but while the records of the "Patent Journal"
show that some of the most successful electricians have started with a belief
in the adaptability of vibration, all the numerous successful machines have
been rotary. It is probable that some reason for this has occurred to those
most deeply engaged in the subject, but I am not aware that any has been
publicly expressed : so that we may say that the advantage of rotary motion
in the dynamo is an empirical fact, and is somewhat opposite to what might
have been expected. It would seem, however, that this paradox is not so
obscure as that of the advantages of the reciprocating engine ; and it is not
improbable that the explanation of the less difficult paradox may throw some
light on that which has so long remained unsolved. In the case of the
dynamo the considerations are much narrowed down, and hence the ground
for advantage must be more distinct.
4. A careful study of the kinetics of the problem shows that there is
one important respect not specifically dealt with in the treatises on the
theory of machines, in which, as it would occur in the dynamo, reciprocation
must be at a great disadvantage as compared with rotation. This respect is
the third mentioned in | (2). Careful consideration shows that in the dynamo
reciprocation must be at a great disadvantage as regards friction. This may
43] IN THE CASES OF OSCILLATION AND ROTATION. 31
not appear to be unnatural, although the data and methods for investigating
the friction of reciprocation, as compared with rotation, have not been
formulated, there is a general impression that the balance would be against
oscillation. Indeed, it is probable that this impression is one of the reasons
which has led to the persistent attempts to produce a rotary steam engine.
But such an indefinite impression entirely fails to explain why rotary motion
should have an advantage under the circumstances of the dynamo which it
has not under those of the steam engine, or why reciprocation should be at
a disadvantage in the dynamo-electric machine when it is not in the dynamo
of the telephone. Under these circumstances it appeared desirable to
attempt a more definite study of the friction of reciprocation as applied to
circumstances such as exist in the dynamo. This brings out facts which
must be of great importance in the theory of machines, and which are
altogether in the direction of explaining the foregoing riddles.
It appears that the amount of friction which has to be overcome in
maintaining the motion of reciprocation of a particular piece of a machine
controlled as by a crank, is not, as in the case of rotation, a quantity
depending merely on the weight, manner of support, and motion of the
reciprocating piece, but depends essentially on the forces which the recipro-
cating part is transmitting during its motion : and in general diminishes as
those forces increase up to a certain point, when it vanishes. To take an
illustration — in an ordinary steam engine doing full work it can be shown that
the friction resulting purely from the motion being reciprocating is zero : but
if the load be taken off the engine and the governors act so as to control the
speed, the friction due to reciprocation will rise, and will reach a maximum
when the engine is doing no work except driving itself.
The same would be to a certain extent the case in a crank-driven recipro-
cating dynamo. When moving unexcited, i.e. with the circuit open and
doing no work, the resistance from the friction entailed by the reciprocating
motion would be a maximum. When, by closing the circuit, resistance was
thrown on to the machine, the work spent in friction from reciprocation
would diminish, but it could not altogether vanish. In order that it might
vanish altogether, the resistances encountered towards the end of the stroke
must bear a certain relation to the weight and velocity, or more correctly, to
the energy of motion of the reciprocating part, and this relation cannot be
reached under the circumstances of the practical dynamo, in which the energy
of motion of the reciprocating piece bears a much greater proportion to the
work done than in the steam engine, and in which the resistances fall off at the
end of the stroke. Thus while in the steam engine the lightness of the piston
compared to the pressure which the steam exerts upon it at the commence-
ment of the stroke, allows of its being driven at convenient speeds without
entailing — when doing work — any extra friction from the reciprocation : in
32 THE COMPARATIVE RESISTANCES AND STRESSES [43
the dynamo, owing to the smallness of the resistance at the ends of the
stroke compared with the weight of the reciprocating piece and the high
speed required to develop the power, the friction entailed by reciprocation
would be large.
In this comparison both machines are supposed to be controlled by the
crank. The friction under such circumstances is not at all the same as when
the reciprocating piece is controlled in other ways, as by a spring. In the
telephone the motion is controlled by a spring, so that the same argument
does not apply here. There are, however, certain limits to such a method of
control, which it is not unimportant to consider. In order to render
intelligible the reasoning relating to these points, it will be necessary to
enter somewhat upon the kinetics of reciprocation, and this will form the
subject of my next article.
II.
[From "The Engineer," January 19, 1883.]
5. THE object of the present article is the consideration of certain
dynamical problems presented by the oscillating pieces of machines. In
former articles under the head "Limits to Speed," it has been shown that the
resistance called forth by the inertia of the revolving and oscillating parts of
machines must, as the speed increases, reach a point beyond which the
strength of material will not allow them to go. In this respect there is but
little difference between revolving and oscillating pieces. But, as regards
friction, or the work necessary to overcome friction, it will appear that
oscillating pieces stand in a very different position to rotary pieces.
In applying the principles of mechanics to machines it is customary to
treat separately the kinematical, or purely geometrical considerations, leaving
all forces out of account. In this respect, i.e., as regards the geometry
of their motion, mechanisms, such as the crank and piston, which involve
oscillating pieces, have received their due share of attention. But considera-
tions relating to the forces in such mechanism have not received very
systematic treatment. These considerations belong to two different classes,
those which do not and those which do depend on the inertia of the moving
parts. The first of these, although applied to moving bodies, are strictly
statical, relating solely to the resolution and balance of forces ; and it is this
class which has received most attention. The considerations relating to the
inertia of the parts have been much neglected. They constitute what, a few
years ago, would have been called the dynamics of machinery, but what
43] IN THE CASES OF OSCILLATION AND ROTATION. 33
is now better expressed as the kinetics of machinery. In some few instances,
as in the case of the fly-wheel of the steam-engine, inertia is necessary to the
action of the machine ; but with the majority of moving pieces the inertia
only plays an incidental part in the action of the machine, or, in other words,
the machine would get on better if these parts could be made of matter
without inertia, and hence it has been very much the custom to leave inertia
out of consideration.
This omission to consider the effect of inertia has been one of the main
causes of the much complained of discrepancy between theory and practice,
and it is to such considerations that we must look for explanations of the
practical selection of one form of mechanism from amongst several, which, so
far as kinematics show, appear to be equally applicable, as for instance, the
reciprocating piston as against all forms of rotary engines. For some
purposes the requisite motion is so slow that the inertia and energy of
motion of the moving parts, and quantities depending on these, are so small
as to be of no account, and then kinetic considerations are of no importance
in determining the fitness of the mechanism ; but whenever it is a question
of attaining the highest possible speed, such considerations assume the first
importance.
6. The kinetics of oscillating pieces. — If treated completely by integrating
the equations of motion this would be a very difficult, if at all a possible,
subject ; only one case, that in which the motion is harmonic, has received
much, if any, attention. And this case may be dealt with by the aid of
elementary mathematics. By the laws of work and energy, however, the
kinetics of oscillation are tractable, and the results so obtained are sufficient
for the present purpose. The following notation will be used unless other-
wise stated: — v is velocity in feet per second; W weight in pounds; E energy
in foot-pounds ; g = 32, acceleration of gravity. When a heavy body is
subject to reciprocating motion its velocity will vary from some maximum
value, vu, to zero, so that E, the energy of motion is given by
E will be greatest when V* is a maximum, and at its least when v= o, E = o;
so that the body must lose and gain E0 foot-pounds of energy of motion
twice in each complete oscillation. In the case of the pendulum the energy
of motion is transformed into energy of elevation, or when the velocity is
zero the mass of the pendulum is ^- feet higher than when the velocity is v.
But in other cases, as when a piston is controlled by a crank, the energy of
motion is transferred to and from the vibrating body, or, in other words, the
o. R. ii. 3
34 THE COMPARATIVE RESISTANCES AND STRESSES [43
Wv -
body must perform and receive work to the extent of - - on each reversal
*9
of its motion.
7. It will be well to express this graphically. Let AOB be the path
of the oscillating body, and suppose it to move from A to B, and to have
Fig. l.
its greatest energy of motion E0 at 0. Then, since it starts from rest
at A, before reaching 0, it must have been subject to the action of forces
which will do E0 foot-pounds of work. These forces might, if their mag-
nitude were known at each point P of the path, be represented as in the
diagram of the steam indicator by distances PM perpendicular to AB. If
so represented, the ends M of these distances would lie on some line
AMO, which, with AO, would form the diagram of inertia, or of the force
to balance inertia from A to 0. The area of this diagram would represent
the work done on the body, and would therefore represent E0, the energy
of motion at 0. In the same way, since the body comes to rest at B, the
body must have encountered resistance or opposing force against which
it does E0 foot-pounds of work in moving from 0 to B. This is represented
by a diagram ONB, the area of which represents E0 foot-pounds of work,
and is therefore equal to the area OMA. Since the resistance from 0 to
B is in the opposite direction to the force from A to 0, N will be on the
opposite side of AB to M. When the motion takes place from A to B
the area AMO represents work done on the moving body to cause energy
of motion, and the area ONB represents work done by the moving body
to get rid of its energy of motion. Therefore, ONB would be negative
work done on the body. When the motion is from B to A the area BNO
represents work done on the body, and OMA is negative. Taking v0 for the
velocity at 0 and v for the velocity at any other point P, and supposing PM
to represent the force to the scale p Ibs. to a foot. Then, areas being in
square feet,
p x area AMO =p x area ONB = ~^- ............... (2),
«/
43]
IN THE CASES OF OSCILLATION AND ROTATION.
35
and p x area 0PM represents the work from 0 to P or P to 0. Then, by the
equation of the conservation of energy we have
Wtf
= pxAMP (3),
Wv 2
Vv°
.(4).
In cases of reciprocation it is not easy to find either the force p . PM, or the
velocity v at every point of the path, but one or other of these is always
a direct circumstance of the motion. Thus, if the body move under the
action of a spring, the stiffness of the spring determines the force p . PM,
which is thus independent of every other condition. Or if the body be moved
by a crank and connecting rod, as in the steam-engine, the velocity at each
point is a kinematical consequence of the velocity of the crauk. In every
case, therefore, either p . PM or v is a direct consequence of the circumstance
of motion. Now, whichever of these may be the direct consequence, the
other is a consequence of the equation of energy, or if we know the one as
a direct consequence, we can find the other by the equation of energy.
8. Oscillation controlled by a crank. — In this case AB will be the
diameter of the crank circle. Describe a circle with AB as diameter ; then,
neglecting the effect of the obliquity of the connecting rod, the position R of
Fig. 2.
the crank on its circle, corresponding to the position P of the piston, is given
by producing PN to meet the circle in R. Let RT be the tangent to the
circle at R, then if u is the velocity of the crank pin at R, the velocity of
Pis
TP
PR
U'TR~U'OR
(5),
3—2
3G THE COMPARATIVE RESISTANCES AND STRESSES [43
whatever may be the position of P. If u is constant all round, when P is
at 0 we have from (5)
v0 = u ....................................... (6),
and for every other point
PR
Substituting the equation of energy (4)
W W
*#= ,0PM + .u»
W OR* -PR*
W
take
gp
PN x OP
then - = OPM ........................... (10).
z
Join ON and produce it to meet perpendiculars through A and B in C
and D. Then N must lie on the line CD for all positions of P, since by (9)
PN is proportional to OP. Therefore by (10) the area 0PM is equal to the
area of the triangle OPN. Therefore M coincides with N on CD, and
the force p . PN is completely expressed. Put OR — a, then by (9)
PN
or writing e for -~- ,
OP gpa?'"
PN=e.OP ................................. (12).
So that W, u, a, being known, we have e and P^ for each value of OP.
9. Oscillation controlled by a spring. — The spring gives the force p . PM;
take the usual case in which this force p . PM is proportioned to OP ; let
p.PM = peOP ................................. (13).
Then as before M is on the line CD. And it is obvious that, the diagram of
forces being the same as before, the relation between the force and velocity
will be the same; but as, in the case already considered, the force is controlled
by the motion, and in this case the motion is controlled by the force, it
is well to make the two proofs independent.
43] IN THE CASES OF OSCILLATION AND ROTATION. 37
Let R be a point moving on the circle so as always to be opposite to P,
then, as before, we have
And from the equation of energy (4)
W( . PR* iOP.PN\ OP2
2^
When P is at A,
-u *'—— =e-p- 2
W OR*
- i,0' = ep— -.
— ' / £-*
W 2PE2 (OR?-OP*\ PR2
1 heref ore, — u2
OR2~ '\ 2 / 2
7/2_ eP# ~2_». 2 (TR\
'^w
Equation (16) shows that u is constant all round the circle, so that in the
case of a spring controlled weight the motion is such that P is always
opposite a point R revolving uniformly with a velocity F0. Thus the motion
of P is completely defined.
The two cases which have been completely considered are cases of
harmonic motion which may and have been dealt with by other methods.
The method just given, as a matter of course, leads in these cases to the
sai i ui results as other methods, but it has the advantage of being applicable
to obtain certain results when neither the law of motion nor the law of force
is completely defined, and, what is its chief advantage as regards the theory
of machines, is that the kinetic forces are represented by a diagram, which
may be at once combined with the diagram such as that of steam pressure,
representing the forces acting on the oscillating pieces; and hence a complete
diagram of transmitted forces obtained. The advantages of this will appear
in the sequel ; but first there are some other general points to be dealt with.
10. Vibration and reciprocation. — The two classes of oscillating motion
typified by the two cases considered — namely, that controlled by the crank
and that controlled by the spring, are, as regards the circumstances on which
they depend, essentially different; and although custom is not uniform in the
matter, it is well to distinguish them by different names. The class repre-
sented by the crank may be well called a motion of reciprocation, as the
body is constrained to move backwards and forwards exactly along the same
path and through the same distance, whatever may be the speed. Whereas
in the case of a vibrating body, although it moves backwards and forwards
along the same path, the distance depends on the speed. In the former
38 THE COMPARATIVE RESISTANCES AND STRESSES [43
case, that of reciprocation, the only effect of increasing the speed of motion
is to increase the rate of oscillation, whereas the effect of increasing the
speed of motion in the case of vibration is primarily to increase the length
of the path, the effect on the rate of oscillation depending on the law
of stiffness of the spring, which in the case of a normal spring is such that
the rate of oscillation is constant.
If a weight of 1 Ib. be held by a spring which requires 1'2 Ib. to deflect it
1 ft., it would vibrate in a period of one second, and through a distance
depending on the initial disturbance. A weight of 1 Ib. controlled by a
uniformly revolving crank would vibrate in the period of revolution of the
crank and through a distance of twice the length of the crank. If the crank
revolve in a period of one second, and the spring be disturbed to move
through twice the length of the crank, the two motions become identical, and
the energy of motion is the same in both cases.
The next question is, what becomes of this energy of motion ? and this
will form the subject of the next article.
III.
[From "The Engineer," February 2, 1883.]
11. The transmission of energy. — In the last article the kinetics of
vibrations were treated, so far as the oscillating body was concerned. This
is only one side of the subject. To maintain oscillation there must be some
action on the body from the outside. Without refining too much, we may
say that the energy of motion must be imparted to and taken from the
oscillating body twice every revolution by the action of other bodies. What
becomes of the energy after it leaves the vibrating body, and whence comes
the fresh supply, depend on the circumstances which maintain the motion.
These may be divided into two principal classes.
12. (1) It may be that the whole or part of the work done by the body
in stopping is done against the resistance of friction. As much of the energy
as is thus spent will be transformed into heat and lost, and to maintain the
motion a fresh supply must be drawn from some external source. (2) It
may be that the work done by the body in stopping is done upon some body
susceptible of energy, which stores the energy as it receives it, and then,
when the motion of the body is reversed, returns the greater part of it to
the reciprocating body again, thus diminishing the draught to be made upon
the fresh supply. The return can never be complete, as there will always
be some frictional resistance to motion. Probably the most complete return
43] IN THE CASES OF OSCILLATION AND ROTATION. 39
is made in the case of the balance-wheel of the watch, which does its work
in stopping against the hair-spring and receives this energy again so nearly
in full that the fresh supply added by the escapement bears a very small
proportion to the whole. When the oscillation is controlled by a crank, the
work of giving the energy of motion is done by the crank, and the crank
again receives the work done by the body in stopping. If the crank is
connected with a fly-wheel this wheel will absorb and give out energy by
a variation of its velocity, and thus the energy of motion is transferred
backward and forward between the fly-wheel and the oscillating body, just as
in the former case it was transferred between the oscillating body and the
spring. In both cases there are certain losses inherent on the transmission, and
these losses constitute the disadvantage, as regards friction, of oscillation
compared with rotation. They will be different in different cases. Before,
however, proceeding to consider these, it will be well to consider shortly the
various means of storing and re-storing energy.
13. Reservoirs of energy. — The storing and re-storing of energy is
generally accomplished by the variation in the motion of some body, as of
the fly-wheel, by the elastic deformation of some body, such as a spring, or
by the pressure of a gas ; but it may be accomplished by the raising of a
weight or by magnetic or electric actions. Whichever of these means is
used, there is a material reservoir which must have sufficient capacity under
the particular circumstance to contain the energy of motion. The capacities
of such reservoirs will depend on various circumstances ; but one factor will,
in all cases, be the amount of material: (1) If the reservoir is the motion of
matter, then its capacity will depend on the circumstances of motion ; but
if u0 and u^ are the velocities of the reservoir when charged and discharged,
the capacity is
(2) If the reservoir be a spring, then the capacity will depend on the state
of stress and elasticity of the material ; but if / be the stress in pounds on
the square inch, and E is the modulus of elasticity, the capacity is
V being the volume of the material of the spring in cubic inches, / the
stress in pounds per square inch, f~ the mean value of /2 throughout the
spring, E the modulus of elasticity. Where the amount of energy to be
stored is large, the weight and size of the reservoir are often matters of the
first importance. These depend solely on the weight and velocity of the
oscillating pieces, and these being known, the weight of the reservoir, if it
is a fly-wheel or a spring, can easily be found.
40 THE COMPARATIVE RESISTANCES AND STRESSES [43
14. The case of springs is the only one that need be considered in this
respect. In this it will appear that the storage power of steel, or any
material, is so small that the size of the reservoir becomes prohibitory for
any but very small mechanisms. In a well formed spring /2 will be ^ or 1
of the square of the greatest stress caused in the spring, according as the
spring is spiral or beam. Taking the case of the beam and assuming the
greatest stress 20,000 Ib. and e = 40,000,000, then the energy must be less
than j=-^ where V is the volume of steel in cubic inches. Now, if we have
a vibrating body making n oscillations per minute, the energy of motion
by the previous article is
=]*-^a* = -00017 WaW (18),
approximately. If, then, we take such an oscillating body as the piston of a
locomotive, let W= 300, a = 1, and n= 100.
The energy is 510 foot-pounds, and the volume of steel required to store
this would be 3672 cubic inches, nearly 1000 Ib., or about ^ ton of steel
would be required for a spring sufficient to store and re-store the energy of
motion of each piston and rods of a small locomotive when at full speed.
This sufficiently shows why oscillating pieces on a large scale cannot be
controlled by springs.
15. If air or steam be used instead of steel, then the weight required
is small, and need not be considered, although the size of cylinder for its
storage is important. In most steam-engines steam is more or less used
for this purpose ; but this will be closely considered later on.
There is, however, a physical point with regard to the use of elastic
reservoirs, which is important.
1 6. Changes of temperature in reservoirs of energy. — All bodies which
expand by heat have their temperature increased by compression and
diminished by expansion.
With such rigid bodies as steel, this change of temperature is small for
possible distortions, but in the case of gases and steam it is very large. If
air be compressed to half its volume instantaneously, its temperature rises
to 172° Fah.
This change of temperature plays an important part in the loss of energy
in transmission which we now come to consider.
17. Losses of energy in transmission in the case of a steel spring. — The
loss of energy in transmission to and from the vibrating body will be
43] IN THE CASES OF OSCILLATION AND ROTATION. 41
very small, for the spring may be united with the vibrating body and the
supports, so that there is no motion or friction at the joints, and thus the
whole loss is in the spring. Even steel may not be perfectly elastic; but
the chief loss, which is also very small, is due to the change of temperature.
The spring is heated during compression, and cooled during extension, and
then, before the restitution takes place, conduction and radiation bring the
temperature to equilibrium again, so that the force of restitution is less
than that of distortion ; but this loss is small, as is shown by the time a
spring will continue to vibrate.
18. The loss in transmitting energy to steam or gas confined in a cylinder.
—In this case the loss from variation of temperature is considerable, but
not easy to estimate, arid besides this there is the friction and leakage of the
piston. When the compression is carried to several atmospheres, as in the
steam-engine, these losses cannot be less than from 15 to 25 per cent, during
each transmission. If this were not so a piston in a closed cylinder of air
would oscillate when disturbed, but as a fact it does little more than spring
back to its initial position. Such a loss as this is fatal to the use of steam
or air as a means of maintaining oscillation, except in cases, as in the steam-
engine, where the use of steam is rendered desirable for other reasons.
Before going into these, however, there remains to be considered the friction
in the important case of the crank.
19. The loss of energy in transmission in the case of the crank-controlled
oscillation. — This is the principal means by which oscillating pieces are
controlled in machinery, and it is this kind of reciprocation that competes
with revolution. Both motions, reciprocation and oscillation, entail certain
loss of energy by friction, and it is important to distinguish between those
losses that are common to both and those which are peculiar to reciprocation.
Now the losses which are common arise mainly from the action of gravity,
and the forces of the operation performed — as, in the case of revolution, the
tension of the belt or the pressure of the teeth — to cause friction. The
losses peculiar to reciprocation are those which arise from the friction caused
by the forces due to the inertia of the reciprocating piece. The simplest
case will alone be here considered, and the forces which arise from gravity
will be left out of account as common to both reciprocation and rotation.
As a simple case we may suppose a crank and fly-wheel on a shaft, the
radius of which is rl} i\ being the radius of the crank pin, and a the length
of the crank, a foot being the unit. The reciprocating piece is supposed to
be connected to the crank by a long light connecting rod, so that the whole
weight w lies in the reciprocating piece, and the pressure on the guides is
so small that it may be neglected.
The forces which arise from the inertia of the reciprocating piece will be
THE COMPARATIVE RESISTANCES AND STRESSES
[43
transmitted through the crank pin to the bearings. These forces will give
rise to friction on the crank pin and the bearings. The forces will be
different at different parts of the revolution. Let C be the mean over the
whole revolution and / the coefficient of friction at the bearings, then the
work (L) spent in overcoming this friction during one revolution is given by
the well-known formula
L = 2-rrrfG ................................ (19).
Or, taking into account that this force acts both on the crank pin and
bearings,
+ rf) ........................... (20).
To find 0 we have in Fig. 2 the value of pPM for each position of the
crank, and to find the mean we have only to divide the crank circle into any
number of equal parts, and find the corresponding positions of the recipro-
cating piece as PjP2 ... in Fig. 3; find the corresponding values of P
Fig. 3.
and take the mean. This method may be employed whatever may be the
shape of the curve AGM.2M40. In this case it is well known that
C = -.p.AC
7T
2 v^W
IT' a g
TT a
.(21).
Where, as before, E0 is the energy of motion, C is exactly -- times the
centrifugal force of a weight revolving on an arm a with velocity v0. The
loss per revolution then becomes
L = 8f(r1 + r2) — (22).
This formula gives the loss in any actual case where ra and r2 are known.
43] IN THE CASES OF OSCILLATION AND ROTATION. 43
The values of i\ and r2 will be determined to meet the forces which fall
on the crank pin and crank shaft. If we assume that the forces arising from
inertia are paramount, then, since the maximum value of these is
ag
we shall have
.(23),
where Bl and Bz are constants, which, according to the practice in steam-
engines, may be taken to be '001 ; therefore
(24),
which gives the loss on the supposition that the machine is designed to
stand the reciprocating forces only.
The importance of this value of L would be in proportion to the work
done by the machine per revolution, and it is easy to see that since L
increases as the cube of the speed, it may be very small at speeds of, say,
100 revolutions per minute, and yet become so large as to be prohibitory
at '300 or 400 revolutions per minute.
In the steam-engines as they exist, these reciprocating forces are not
large enough to affect the size of rlt r2, which are larger than they would be
as in (23), and yet the loss as given by (22) is insignificantly small. In a
reciprocating dynamo, in order to obtain anything like the same duty per
weight of material as the present revolving dynamo gives, the weights and
speeds of their oscillating pieces would have to bear nearly the same relation
to the weights and speeds of the engines which drive them as do the
armatures of the present dynamos. This means increasing the number of
revolutions, as compared with the engines, by a quantity of the order 10,
or increasing L in the ratio 1000. The further consideration of these
matters will be undertaken in the next article.
44
THE COMPARATIVE RESISTANCES AND STRESSES
[43
IV.
[From "The Engineer," February 16, 1883.]
20. Application to the dynamo and steam-engine. — In order to arrive at
a just estimate of the friction caused by the inertia of reciprocation in
practical cases, it is necessary to consider the forces which arise from inertia
in conjunction with the working force — the force required to accelerate the
piston in conjunction with the pressure of steam.
21. The resultant of inertia and the working forces on a reciprocating
piece. — When, as is generally the case, the reciprocating piece is subject to
forces besides those which act between it and the crank, the pressure on the
crank will be the resultant of this force and the force necessary to balance
the inertia.
The working force may be represented, as in the case of the steam-engine,
by a diagram. Let p.PM' be the working force at the point P. Consider
the motion from A to B, and let M' be on the upper side of AB when the
force is in the direction of AB, and on the lower side when the force is in
Fig. 4.
the direction BA. That is, PM' is drawn on the same side as PM represent-
ing the forces to overcome the inertia. The pressure on the crank will
therefore be
p(PM'-PM)=p.M'M (25);
make PM" = MM', noticing that M" will be above AB when M is above M ',
and vice versa. In this way a line AC"M"D"B may be drawn, the distance
43]
IN THE CASES OF OSCILLATION AND ROTATION.
45
of which from AB shows the pressure on the crank ; and then the mean
pressure may be found as before, substituting PM" for PM. As far as the
friction is concerned this will be independent of the direction of the pressure
on the crank, so that in finding the mean PM" must be taken always of the
same sign.
There are two cases of special interest. First let AC' be greater than
AC, then the forces acting from A to B will be altogether in the direction
AB. Take two points P and Q (Fig. 5) on the opposite sides of 0, and at
Fig. 5.
equal distances from it. Then if the two triangles OAG and BAG' are
similar, and therefore C'M'B parallel to CD,
(26).
Therefore the mean of the pressure at these two points on the crank will be
unaltered by the inertia, and as at these points the crank is making equal
angles with AB, in opposite directions, the mean pressure on the crank will
be identically the same as would arise from the working forces, or, in other
words, the inertia of the reciprocating piece will cause no extra friction ; and
this, as will be shown, is practically the case in the steam-engine. Second,
let the inertia be paramount, i.e., AG greater than AG', and let the acting
forces be symmetrical about 0, as shown by the curve AJM'N'B in Fig. 6.
Let the curve CO cut the curve AM'B in J\ draw ,TI perpendicular to
AB and take OK = 01. Then as before if P lies between / and K
(27).
46
THE COMPARATIVE RESISTANCES AND STRESSES
[43
So that between / and K the mean pressures on the crank will be the same
as if caused by the working forces PM' only.
Fig. 6.
When P lies between A and I, then
PM+QN=MM' + NN' (28),
or the mean pressure will be the same as if only the forces of inertia acted ;
and this, as will be shown, is the case of the dynamo machine.
22. Application to the dynamo machine. — Since there has been no
experience with oscillating dynamo machines, the formula obtained in the
last article can only be applied to an assumed case. The revolving dynamo
machine may be made to furnish the data for an oscillating dynamo machine ;
a, the length of the crank, may be taken equal to the mean radius of the
armature, W equal the weight of the armature, and the time of an oscillation
the same as the time of a revolution.
Taking a particular dynamo driven by 6-horse power, it appears that
W = 200 lb:
a = -3
n = 1000
/=-05
.(29).
So that v0 = 30 approximately ;
.(30).
This gives for L 1000 foot-pounds, in round numbers. This is the loss per
revolution. Per minute the loss would be about 1,000,000 foot-pounds, or
30-horse power. So that the loss due to the friction arising from the inertia
43] IN THE CASES OF OSCILLATION AND ROTATION. 47
of the reciprocating armature would be five times greater than the work
done in creating a current. Put this way, even supposing the assumed data
admit of considerable modification, it is clear that the friction arising from
reciprocation is prohibitory in the case of a dynamo machine. But before
adopting this view, it is well to see how far this loss might be modified by
the work which the dynamo machine was doing. 6-horse power, with a stroke
of '6 and 1000 revolutions per minute, would be equivalent to a uniform
Wv2
resistance on the vibrating body of 165 Ib. The force = 20,000.
\j
So that if in the diagram AC=l", and p . AC represent 20,000 Ib.,
_p = 20,000; and if ^C"=1601b., ^C" = -008". This is too small to be
drawn to scale; but if drawn, the line // in Fig. 6 would be '008 AC and
01 would be -008 AO. Therefore the amount of work represented by the
diagram ILLK would be '008 x 160, or 1'3 foot-pounds, and this may be
neglected. And for the rest of the diagram, as shown in Section 21, the
mean pressure on the crank pin will be the same as if the forces of inertia
were alone to be considered. In this case, therefore, where the forces of
inertia are paramount, the friction is determined almost entirely by the
forces of inertia, the working forces neither adding to nor subtracting from the
friction.
It thus appears that there is no chance for the reciprocating dynamo
machine, driven by a crank, and it will appear equally clear that there is no
chance for a reciprocating dynamo machine driven direct from the piston of
a steam-engine, for in this case the energy of motion, which, as in the last
example, is 3000 foot-pounds, would have to be stored by cushioning steam,
that is to say, 3000 foot-pounds would have to be transmitted to and from
the steam twice each revolution ; the entire transmission therefore would be
12,000 foot-pounds. Now, taking the smallest estimate of loss in this
transmission, namely, 15 per cent., we have a loss of 1800 foot-pounds per
revolution, nearly double as great as with the crank.
If we substitute a steel spring for the cushioning, then the weight of
steel, which, estimated as before, would be 6 tons, is prohibitory.
Thus in every case we have amply sufficient reasons for the non -applica-
bility of reciprocation to the dynamo machine.
These results are sufficiently striking in themselves, but they become still
more so when compared with the corresponding results for the steam-engine.
23. Application to the steam-engine. — For the sake of comparison the
circumstances of the engine may be taken similar to those of the dynamo
machine just considered. Thus, the weight of piston and reciprocating parts
48
THE COMPARATIVE RESISTANCES AND STRESSES
[43
is taken at 200 lb., and the length of crank, '3. This would only give a 7 in.
stroke, which is somewhat out of proportion, considering that 200 lb. would
correspond with a piston some 15 in. in diameter, that is, considering the
shortness of the stroke. This will be a convenient size to assume for the
piston, taking the initial pressure of steam 120 lb. on the square inch ; since
this over a 15 in. piston is 21,1 20 lb., which is just about the same as the
c&c'
Fig. 7.
force of inertia, and in the diagram AC = AC' , or C and G' coincide, i.e. — if,
for the sake of comparison, the number of revolutions is taken the same as
before n = 1000.
Since W, n, and a are the same as before, the crank pin will be subject
to the same pressures, on account of inertia ; and since we may assume, from
expansion, the pressure of steam to fall towards B, the greatest pressures on
the crank pin will not exceed the greatest forces of inertia; therefore r^rz
may be taken to have the same value as before. And considering only the
force of inertia, we should find as before —
L = 1000 foot-pounds,
n L = 1,000,000,
or the loss would be at the rate of 30-horse power. But even supposing this
loss to take place, it bears a very different comparison to the work done by
the steam-engine from what it did to the dynamo machine. With an initial
pressure of 120, the steam being used as in the locomotive, the mean
pressure would be, say, 70 lb.; this would give the work per stroke 15,000
foot-pounds, so that the loss would only be one-fifteenth, or between 6 and 7
per cent., instead of 500 per cent, in the case of the dynamo machine.
43] IN THE CASES OF OSCILLATION AND ROTATION. 49
As a matter of fact, however, there would be no such loss in the engine
when doing its full work. This appears on compounding the diagrams of
inertia and working pressure as in Fig. 5, Art. 21, for since AC' is not greater
than AC,
PM' + QN' = MM' + M'N (31),
throughout the diagram, or the mean pressure on the crank taken all round
is not affected by the inertia of the piston ; and hence whatever loss the
friction arising from the pressure may cause, it will be due entirely to the
acting pressure of steam, and so long as this remains unaltered, the loss per
revolution will be the same at all speeds up to 1000 revolutions. Considering
that the speed of piston here taken 1800 ft. per minute, and the number of
revolutions, 1000, are well outside all practical values, this example shows
that in whatever other ways the forces arising from the inertia of reciprocation
act to limit the speed of the steam-engine, they need not affect the friction of
the engine, either directly or indirectly by requiring larger bearings, even
should the speed of the steam-engines reach values five or six times greater
than the present values. Thus, although, as we have seen in the case of
the dynamo machine, there are circumstances in which the friction arising
from the inertia of the reciprocating force is so large compared with the
acting forces as to be prohibitory to oscillating motion, yet in the case of the
steam-engine these forces give rise to no loss whatever, and do not place the
reciprocating engine at a disadvantage as compared with the rotary engine.
It seems, then, that we have a good reason for the general impression in
favour of rotary motion as compared with reciprocating motion, and also a
good reason why the impression is erroneous as applied to the steam-engine.
Before closing these articles, it may be well to refer shortly to cushioning,
or compression, as used in reference to the steam-engine.
24. Cushioning. — The useful purposes attributed to this are these : —
(1) Cushioning is supposed to save steam by filling the passages to
the ports and other necessary clearance, so that this has not to be filled with
fresh steam which does no work in filling them.
(2) Cushioning is often supposed by relieving the crank from the duty
of stopping the piston, and so by diminishing the pressure on the crank pin
and bearings, to diminish the friction.
(3) Cushioning is found by experience to be necessary in the case of all
high-speed engines, to prevent a sudden shock attending the admission of
steam.
Now, the last of these advantages is a matter of experience, and is alone
sufficient to warrant a certain amount of cushioning. If, when running at
o. R. ii. 4
50 RESISTANCES AND STRESSES IN THE CASES OF OSCILLATION, ETC. [43
its greatest speed, an engine knocks or bumps in its bearings, it is a sign
that it is insufficiently cushioned. This admits of theoretical explanation.
If cushioned, as the piston approaches the end of its stroke A it will be
stopping itself driving the crank, the force arising from inertia being at its
greatest. Thus the force will have a tendency to close all the joints between
the piston and the bearings in the direction BA, opening them in the
direction AB. On the admission of the steam, owing to the small clearance
to be filled, the pressure suddenly rises to a greater value than the force of
inertia, and the piston is, as it were, shot back by the pressure of the steam
and the elasticity of the engine against the force of its inertia. The joints
thus close towards B with a bump. This bump could not have occurred had
not the reversal of the direction of the combined force and inertia been
sudden when the joints were open towards A. By cushioning, the pressure
of the steam which balances the inertia rises gradually, so that the joints
which are at first open towards A close gradually.
As regards the first two advantages, the first of these must be regarded
as hypothetical, or rather, as theoretical, and the second as imaginary.
The steam with which the clearance is filled is not all gain. This is well
known. The work done in compression has to be deducted from the work
done by the forward pressure of the steam, or the power of the engine will
be diminished by the power spent in compression, while the entire friction
and the losses by condensation remain the same. As these losses appear to
be something like 40 per cent, of the theoretical power of the steam as used
in the engine, there cannot be much margin for gain of steam. The
advantage may be a little one way or the other, but it is not worth
mentioning.
The second assumed advantage of cushioning, namely, the diminution of
the mean pressure on the engine, vanishes when it is perceived that it is the
working pressure of the inertia that is diminished. This assumption amounts
to nothing more or less than assuming that the moving energy of the piston
might be more efficiently stored and restored by compressing steam than it
is by the crank. It has, however, been shown that the crank performs this
work in the steam-engine with no loss, whereas in compressing steam there
will probably be a loss of from 15 to 25 per cent, of the energy stored.
This is the loss which has been shown to balance the gain in steam in (1).
In respect of (2), therefore, the cushioning is a disadvantage. That this has
not been practically perceived is because, as long as cushioning is only
carried to the extent of filling the necessary clearance, then the loss and the
gain, as in (1), are nearly balanced, as has already been shown.
The conclusion is, therefore, that cushioning should not be carried further
than is sufficient to prevent bumping.
44.
AN EXPERIMENTAL INVESTIGATION OF THE CIRCUMSTANCES
WHICH DETERMINE WHETHER THE MOTION OF WATER
SHALL BE DIRECT OR SINUOUS, AND OF THE LAW OF
RESISTANCE IN PARALLEL CHANNELS.
[From "The Philosophical Transactions of the Royal Society," 1883.]
(Received and Read March 15, 1883.)
SECTION I.
Introductory.
1. Objects and results of the investigation. — The results of this investi-
gation have both a practical and philosophical aspect.
In their practical aspect they relate to the law of resistance to the
motion of water in pipes, which appears in a new form, the law for all
velocities and all diameters being represented by an equation of two
terms.
In their philosophical aspects these results relate to the fundamental
principles of fluid motion ; inasmuch as they afford for the case of pipes
a definite verification of two principles, which are — that the general
character of the motion of fluids in contact with solid surfaces depends
on the relation between a physical constant of the fluid, and the product
of the linear dimensions of the space occupied by the fluid, and the
velocity.
The results as viewed in their philosophical aspect were the primary
object of the investigation.
4—2
52 ON THE MOTION OF WATER, AND OF [44
As regards the practical aspects of the results it is not necessary to
say anything by way of introduction ; but in order to render the philo-
sophical scope and purpose of the investigation intelligible it is necessary
to describe shortly the line of reasoning which determined the order of
investigation.
2. The leading features of the motion of actual fluids. Although in
most ways the exact manner in which water moves is difficult to per-
ceive and still more difficult to define, as are also the forces attending
such motion, certain general features both of the forces and motions
stand prominently forth, as if to invite or to defy theoretical treatment.
The relations between the resistance encountered by, and the velocity
of, a solid body moving steadily through a fluid in which it is com-
pletely immersed, or of water moving through a tube, present themselves
mostly in one or other of two simple forms. The resistance is generally
proportional to the square of the velocity, and when this is not the case
it takes a simpler form and is proportional to the velocity.
Again, the internal motion of water assumes one or other of two
broadly distinguishable forms — either the elements of the fluid follow one
another along lines of motion which lead in the most direct manner to
their destination, or they eddy about in sinuous paths the most indirect
possible.
The transparency or the uniform opacity of most fluids renders it
impossible to see the internal motion, so that, broadly distinct as are
the two classes (direct and sinuous) of motion, their existence would not
have been perceived were it not that the surface of water, where other-
wise undisturbed, indicates the nature of the motion beneath. A clear
surface of moving water has two appearances, the one like that of plate
glass, in which objects are reflected without distortion, the other like
that of sheet glass, in which the reflected objects appear crumpled up
and grimacing. These two characters of surface correspond to the two
characters of motion. This may be shown by adding a few streaks of
highly coloured water to the clear moving water. Then although the
coloured streaks may at first be irregular, they will, if there are no
eddies, soon be drawn out into even colour bands; whereas if there are
eddies they will be curled and whirled about in the manner so familiar
with smoke.
3. Connexion between the leading features of fluid motion. These
leading features of fluid motion are well known and are supposed to be
more or less connected, but it does not appear that hitherto any very
determined efforts have been made to trace a definite connexion between
44] THE LAW OF RESISTANCE IN PARALLEL CHANNELS. 53
them, or to trace the characteristics of the circumstances under which
they are generally presented. Certain circumstances have been definitely
associated with the particular laws of force. Resistance, as the square
of the velocity, is associated with motion in tubes of more than capillary
dimensions, and with the motion of bodies through the water at more
than insensibly small velocities, while resistance as the velocity is associated
with capillary tubes and small velocities.
The equations of hydrodynamics, although they are applicable to
direct motion, i.e., without eddies, and show that then the resistance is
as the velocity, have hitherto thrown no light on the circumstances on
which such motion depends. And although of late years these equations
have been applied to the theory of the eddy, they have not been in
the least applied to the motion of water which is a mass of eddies,
i.e., in sinuous motion, nor have they yielded a clue to the cause of re-
sistance varying as the square of the velocity. Thus, while as applied
to waves and the motion of water in capillary tubes the theoretical
results agree with the experimental, the theory of hydrodynamics has so
far failed to afford the slightest hint why it should explain these pheno-
mena, and signally fail to explain the law of resistance encountered by
large bodies moving at sensibly high velocities through water, or that of
water in sensibly large pipes.
This accidental fitness of the theory to explain certain phenomena
while entirely failing to explain others, affords strong presumption that
there are some fundamental principles of fluid motion of which due
account has not been taken in the theory. And several years ago it
seemed to me that a careful examination as to the connexion between
these four leading features, together with the circumstances on which
they severally depend, was the most likely means of finding the clue to
the principles overlooked.
4. Space and velocity. The definite association of resistance as the
square of the velocity with sensibly large tubes and high velocities, and
of resistance as the velocity with capillary tubes and slow velocities,
seemed to be evidence of the very general and important influence of
some properties of fluids not recognised in the theory of hydrodynamit s.
As there is no such thing as absolute space or absolute time recog-
nised in mechanical philosophy, to suppose that the character of motion
of fluids in any way depended on absolute size or absolute velocity, would
be to suppose such motion without the pale of the laws of motion. If
then fluids in their motions are subject to these laws, what appears to
be the dependence of the character of the motion on the absolute size
54 ON THE MOTION OF WATER, AND OF [44
of the tube, and on the .absolute velocity of the immersed body, must
in reality be a dependence on the size of the tube as compared with
the size of some other object, and on the velocity of the body as com-
•< pared with some other velocity. What is the standard object, and what
v the standard velocity Avhich come into comparison with the size of the
tube and the velocity of an immersed body, are questions to which the
answers were not obvious. Answers, however, were found in the discovery
of a circumstance on which sinuous motion depends.
5. The effect of viscosity on the character of fluid motion. The small
evidence which clear water shows as to the existences of internal eddies,
not less than the difficulty of estimating the viscous nature of the fluid,
appears to have hitherto obscured the very important circumstance that
the more viscous a fluid is, the less prone is it to eddying or sinuous
motion. To express this definitely — if /z is the viscosity and p the
density of the fluid — for water /j,/p diminishes rapidly as the temperature
rises, thus at 5° C. p/p is double what it is at 45° C. What I observed
was that the tendency of water to eddy becomes much greater as the
temperature rises.
Hence connecting the change in the law of resistance with the birth
and development of eddies, this discovery limited further search for the
standard distance and standard velocity to the physical properties of the
fluid. To follow the line of this search would be to enter upon a molecular
theory of liquids, and this is beyond my present purpose. It is sufficient
here to notice the well-known fact that
^ or//
P
is a quantity of the nature of the product of a distance and a velocity.
It is always difficult to trace the dependence of one idea on another.
But it may be noticed that no idea of dimensional properties, as indi-
cated by the dependence of the character of motion on the size of the
tube and the velocity of the fluid, occurred to me until after the com-
pletion of my investigation on the transpiration of gases, in which was
established the dependence of the law of. transpiration on the relation
between the size of the channel and the mean range of the gaseous
molecules.
G. Evidence of dimensional properties in the equations of motion. The
equations of motion had been subjected to such close scrutiny, particu-
larly by Professor Stokes, that there was small chance of discovering
anything new or faulty in them. It seemed to me possible, however,
44] THE LAW OF RESISTANCE IN PARALLEL CHANNELS. 55
that they might contain evidence which had been overlooked, of the
dependence of the character of motion on a relation between the dimen-
sional properties and the external circumstances of motion. Such evidence,
not only of a connexion but of a definite connexion, was found, and this
without integration.
If the motion be supposed to. depend on a single velocity para-
meter U, say the mean velocity along a tube, and on a single linear
parameter c, say the radius of the tube ; then having in the usual manner *
eliminated the pressure from the equations, the accelerations are expressed
in terms of two distinct types. In one of which
U*
c3
is a factor, and in the other
£U
pC*
is a factor. So that the relative values of these terms vary respectively
as U and
£
cp'
This is a definite relation of the exact kind for which I was in
search. Of course without integration ; the equations only gave the
relation without showing at all in what way the motion might depend
upon it.
It seemed, however, to be certain, if the eddies were due to one par-
ticular cause, that integration would show the birth of eddies to depend
on some definite value of
cpU
7. The cause of eddies. There appeared to be two possible causes for
the change of direct motion into sinuous. These are best discussed in
the language of hydrodynamics, but as the results of this investigation
relate to both these causes, which, although the distinction is subtle, are
fundunicntally distinct and lead to distinct results, it is necessary that
they should be indicated.
The general cause of the change from steady to eddying motion was
in 1843 pointed out by Professor Stokes, as being that under certain
circumstances the steady motion becomes unstable, so that an indefinitely
small disturbance may lead to a change to sinuous motion. But the causes
56
ON THE MOTION OF WATER, AND OF
[44
above referred to are of this kind, and yet they are distinct, the distinction
lying in the part taken in the instability by viscosity.
If we imagine a fluid free from viscosity .and absolutely free to glide over
solid surfaces, then comparing such a fluid with a viscous fluid in exactly
the same motion —
(1) The frictionless fluid might be instable and the viscous fluid stable.
Under these circumstances the cause of eddies is the instability as a perfect
fluid, the effect of viscosity being in the direction of stability.
(2) The frictionless fluid might be stable and the viscous fluid unstable,
under which circumstances the cause of instability would be the viscosity.
It was clear to me that the conclusions I had drawn from the equations
of motion immediately related only to the first cause ; nor could I then
perceive any possible way in which instability could result from viscosity.
All the same I felt a certain amount of uncertainty in assuming the first
cause of instability to be general. This uncertainty was the result of various
considerations, but particularly from my having observed that eddies
apparently come on in very different ways, according to a very definite
circumstance of motion, which may be illustrated.
When in a channel the water is all moving in the same direction, the
velocity being greatest in the middle and diminishing to zero at the sides, as
indicated by the curve in Fig. 1, eddies showed themselves reluctantly and
Fig. i.
Fig. 2.
irregularly ; whereas when the water on one side of the channel was moving
in the opposite direction to that on the other, as shown by the curve in
Fig. 2, eddies appeared in the middle regularly and readily.
44] THE LAW OF RESISTANCE IN PARALLEL CHANNELS. 57
8. Methods of investigation. — There appeared to be two ways of proceed-
ing — the one theoretical, the other practical.
The theoretical method involved the integration of the equations for
unsteady motion in a way that had not been accomplished and which,
considering the general intractability of the equations, was not promising.
The practical method was to test the relation between U, nfp, and c ; this,
owing to the simple and definite form of the law, seemed to offer, at all
events in the first place, a far more promising field of research.
The law of motion in a straight smooth tube offered the simplest possible
circumstances and the most crucial test.
The existing experimental knowledge of the resistance of water in tubes,
although very extensive, was in one important respect incomplete. The
previous experiments might be divided into two classes: (1) those made
under circumstances in which the law of resistance was as the square of the
velocity, and (2) those made under circumstances in which the resistance
varied as the velocity. There had not apparently been any attempt made
to determine the exact circumstances under which the change of law took
place.
Again, although it had been definitely pointed out that eddies would
explain resistance as the square of the velocity, it did not appear that any
definite experimental evidence of the existence of eddies in parallel tubes had
been obtained, and much less was there any evidence as to whether the birth
of eddies was simultaneous with the change in the law of resistance.
These open points may be best expressed in the form of queries to which
the answers anticipated were in the affirmative.
(1) What was the exact relation between the diameters of the pipes and
the velocities of the water at which the law of resistance changed ?
Was it at a certain value of
(2) Did this change depend on the temperature, i.e., the viscosity
of water ? Was it at a certain value of
(3) Were there eddies in parallel tubes ?
(4) Did steady motion hold up to a critical value and then eddies come
in?
58 ON THE MOTION OF WATER, AND OF [44
(5) Did the eddies come in at a certain value of
pcU,
P
(6) Did the eddies first make their appearance as small and then
increase gradually with the velocity, or did they come in suddenly ?
The bearing of the last query may not be obvious ; but, as will appear in
the sequel, its importance was such that, in spite of satisfactory answers to
all the other queries, a negative answer to this, in respect of one particular
class of motions, led me to the reconsideration of the supposed cause of
instability.
The queries, as they are put, suggest two methods of experimenting :—
(1) Measuring the resistances and velocities of different diameters, and
with different temperatures of water.
(2) Visual observation as to the appearance of eddies during the flow of
water along tubes or open channels.
Both these methods have been adopted, but, as the questions relating to
eddies had been the least studied, the second method was the first adopted.
9. Experiments by visual observation. — The most important of these
experiments related to water moving in one direction along glass tubes.
Besides this, however, experiments on fluids flowing in opposite directions in
the same tube were made, also a third class of experiments, which related
to motion in a flat channel of indefinite breadth.
These last-mentioned experiments resulted from an incidental observation
during some experiments made in 1876 as to the effect of oil to prevent wind
waves. As the result of this observation had no small influence in directing
the course of this investigation, it may be well to describe it first.
10. Eddies caused by the wind beneath the oiled surface of water. — A few
drops of oil on the windward side of a pond during a stiff breeze, having
spread over the pond and completely calmed the surface as regards waves,
the sheet of oil, if it may be so called, was observed to drift before the wind,
and it was then particularly noticed that while close to, and for a considerable
distance from the windward edge, the surface presented the appearance
of plate glass ; further from the edge the surface presented that irregular
wavering appearance which has already been likened to that of sheet glass,
which appearance was at the time noted as showing the existence of eddies
beneath the surface.
Subsequent observation confirmed this first view. At a sufficient distance
44] THE LAW OF RESISTANCE IN PARALLEL CHANNELS. 59
from the windward edge of an oil-calmed surface there are always eddies
beneath the surface even when the wind is light. But the distance from the
edge increases rapidly as the force of the wind diminishes, so that at a
limited distance (10 or 20 feet) the eddies will come and go with the wind.
Without oil I was unable to perceive any indication of eddies. At first I
thought that the waves might prevent their appearance even if they were
there, but by careful observation I convinced myself that they were not there.
It is not necessary to discuss these results here, although, as will appear,
they have a very important bearing on the cause of instability.
11. Experiments by means of colour bands in glass tubes. — These were
undertaken early in 1880; the final experiments were made on three tubes,
Nos. 1, 2, and 3. The diameters of these were nearly 1 inch, | inch, and
\ inch. They were all about 4 feet 6 inches long, and fitted with trumpet
mouthpieces, so that the water might enter without disturbance.
The water was drawn through the tubes out of a large glass tank, in
which the tubes were immersed, arrangements being made so that a
streak or streaks of highly coloured water entered the tubes with the clear-
water.
The general results were as follows : —
(1) When the velocities were sufficiently low, the streak of colour
extended in a beautiful straight line through the tube, Fig. 3.
Fig. 3.
(2) If the water in the tank had not quite settled to rest, at sufficiently
low velocities, the streak would shift about the tube, but there was no
appearance of sinuosity.
(3) As the velocity was increased by small stages, at some point in the
tube, always at a considerable distance from the trumpet or intake, the
Fig. 4.
colour band would all at once mix up with the surrounding water, and
fill the rest of the tube with a mass of coloured water, as in Fig. 4.
60 ON THE MOTION OF WATER, AND OF [44
Any increase in the velocity caused the point of break down to approach
the trumpet, but with no velocities that were tried did it reach this.
On viewing the tube by the light of an electric spark, the mass of colour
resolved itself into a mass of more or less distinct curls, showing eddies, as in
Fig. 5.
The experiments thus seemed to settle questions 3 and 4 in the affirma-
tive, the existence of eddies and a critical velocity.
They also settled in the negative question 6, as to the eddies coming in
gradually after the critical velocity was reached.
In order to obtain an answer to question 5, as to the law of the critical
velocity, the diameters of the tubes were carefully measured, also the
temperature of the water, and the rate of discharge.
(4) It was then found that, with water at a constant temperature, and
the tank as still as could by any means be brought about, the critical
velocities at which the eddies showed themselves were almost exactly in the
inverse ratio of the diameters of the tubes.
(5) That in all the tubes the critical velocity diminished as the tempera-
ture increased, the range being from 5° C. to 22° C. ; and the law of this
diminution, so far as could be determined, was in accordance with Poiseuille's
experiments. Taking T to express degrees centigrade, then by Poiseuille's
experiments,
^ oc P = (1 + 0-0336^+ 0-00221 172)"1-
P
Taking a metre as the unit, Us the critical velocity, and D the diameter of
the tube, the law of the critical point is completely expressed by the formula
tf-1*
8~ BSD
where Bs = 43'79
log Bs= 1-64139.
This is a complete answer to question 5.
During the experiments many things were noticed which cannot be
mentioned here, but two circumstances should be mentioned as emphasising
44] THE LAW OF RESISTANCE IN PARALLEL CHANNELS. 61
the negative answer to question 6. In the first place, the critical velocity
was much higher than had been expected in pipes of such magnitude,
resistance varying as the square of the velocity had been found at very much
smaller velocities than those at which the eddies appeared when the water in
the tank was steady; and in the second place, it was observed that the
critical velocity was very sensitive to disturbance in the water before entering
the tubes ; and it was only by the greatest care as to the uniformity of the
temperature of the tank and the stillness of the water that consistent results
were obtained. This showed that the steady motion was unstable for large
disturbances long before the critical velocity was reached, a fact which agreed
with the full-blown manner in which the eddies appeared.
12. Experiments with two streams in opposite directions in the same
tube. — A glass tube, 5 feet long and T2 inch in diameter, having its ends
slightly bent up, as shown in Fig. 6, was half filled with bisulphide of carbon,
Fig. 6.
and then filled up with water and both ends corked. The bisulphide was
chosen as being a limpid liquid but little heavier than water and completely
insoluble, the surface between the two liquids being clearly distinguishable.
When the tube was placed in a horizontal direction, the weight of the
bisulphide caused it to spread along the lower half of the tube, and the
surface of separation of the two liquids extended along the axis of the tube.
On one end of the tube being slightly raised the water would flow to the
upper end and the bisulphide fall to the lower, causing opposite currents
along the upper and lower halves of the tube, while in the middle of the
tube the level of the surface of separation remained unaltered.
The particular purpose of this investigation was to ascertain whether
there was a critical velocity at which waves or sinuosities would show them-
selves in the surface of separation.
It proved a very pretty experiment and completely answered its purpose.
When one end was raised quickly by a definite amount, the opposite
velocities of the two liquids, which were greatest in the middle of the tube,
attained a certain maximum value, depending on the inclination of the tube.
When this was small no signs of eddies or sinuosities showed themselves ;
but, at a certain definite inclination, waves (nearly stationary) showed them-
selves, presenting all the appearance of wind waves. These waves first made
62 ON THE MOTION OF WATER, AND OF [44
their appearance as very small waves of equal lengths, the length being
comparable to the diameter of the tube.
Fig. 7.
When by increasing the rise the velocities of flow were increased, the
waves kept the same length but became higher, and when the rise was
sufficient the waves would curl and break, the one fluid winding itself into
the other in regular eddies.
Whatever might be the cause, a skin formed slowly between the bisulphide
and the water, and this skin produced similar effects to that of oil and water;
the results mentioned are those which were obtained before the skin showed
itself. When the skin first came on regular waves ceased to form, and in
their place the surface was disturbed, as if by irregular eddies, above and
below, just as in the case of the oiled surface of water.
The experiment was not adapted to afford a definite measure of the
velocities at Avhich the various phenomena occurred ; but it was obvious that
the critical velocity at which the waves first appeared was many times smaller
than the critical velocity in a tube of the same size when the motion was in
one direction only. It was also clear that the critical velocity was nearly,
if not quite, independent of any existing disturbance in the liquids ; so that
this experiment shows —
(1) That there is a critical velocity in the case of opposite flow at which
direct motion becomes unstable.
(2) That the instability came on gradually and did not depend on the
magnitude of the disturbances, or in other words, that for this class of motion
question 6 must be answered in the affirmative.
It thus appeared that there was some difference in the cause of instability
in the two motions.
13. Further study of the equations of motion. — Having now definite data
to guide me, I was anxious to obtain a fuller explanation of these results
from the equations of motion. I still saw only one way open to account for
the instability, namely, by assuming the instability of a Irictionlcss fluid to
be general.
Having found a method of integrating the equations for frictionless fluid
as far as to show whether any particular form of steady motion is stable for
44] THE LAW OF RESISTANCE IN PARALLEL CHANNELS. 63
;i small disturbance, I applied this method to the case of parallel flow in
a frictionless fluid. The result, which I obtained at once, was that flow
in one direction was stable, flow in opposite directions unstable. This was
not what I was looking for, and I spent much time in trying to find a way
out of it, but whatever objections my method of integration may be open to,
I could make nothing less of it.
It was not until the end of 1882 that I abandoned further attempts with
a frictionless fluid, and attempted by the same method the integration of a
viscous fluid. The change was in consequence of a discovery that in pre-
viously considering the effect of viscosity I had omitted to take fully into
account the boundary conditions which resulted from the friction between
the fluid and the solid boundary.
On taking these boundary conditions into account, it appeared that
although the tendency of internal viscosity of the fluid is to render direct or
steady motion stable, yet owing to the boundary condition resulting from the
friction at the solid surface, the motion of the fluid, irrespective of viscosity,
would be unstable. Of course this cannot be rendered intelligible without
going into the mathematics. But what I want to point out is that this
instability, as shown by the integration of the equations of motion, depends
on exactly the same relation,
u*£,
cp
as that previously found.
This explained all the practical anomalies and particularly the absence of
eddies below a pure surface of water exposed to the wind. For in this case
the surface being free, the boundary condition was absent, whereas the film
of oil, by its tangential stiffness, introduced this condition ; this circumstance
alone seemed a sufficient verification of the theoretical conclusion.
But there was also the sudden way in which eddies came into exist-
ence in the experiments with the colour band, and the effect of disturb-
ances to lower the critical velocity. These were also explained, for as
long as the motion was steady, the instability depended upon the boundary
action alone, but once eddies were introduced, the stability would be broken
down.
Jt thus appeared that the meaning of the experimental results had been
ascertained, and the relation between the four leading features and the
circumstances on which they depend traced tor the case of water in parallel
flow.
But as it appeared that the critical velocity in the case of motion in one
direction, did not depend on the cause of instability, with a view to which it
64
ON THE MOTION OF WATER, AND OF
[44
was investigated, it followed that there must be another critical velocity,
which would be the velocity at which previously existing eddies would die
out, and the motion become steady as the water proceeded along the tube.
This conclusion has been verified.
14. Results of experiments in the law of resistance in tubes. — The
existence of the critical velocity described in the previous article, could only
be tested by allowing water in a high state of disturbance to enter a tube,
and after flowing a sufficient distance for the eddies to die out, if they were
going to die out, to test the motion.
As it seemed impossible to apply the method of colour bands, the test
applied was that of the law of resistance as indicated in questions (1) and
(2) in § 8. The result was very happy.
Two straight lead pipes No. 4 and No. 5, each 16 feet long and having
diameters of a quarter and a half inch respectively, were used. The water
was allowed to flow through rather more than 10 feet before coming to the
first gauge hole, the second gauge hole being 5 feet further along the pipe.
The results were very definite, and are partly shown in Fig. 8, and more
fully in diagram 1, page 90.
Fig. 8.
(1) At the lower velocities the pressure was proportional to the velocity,
and the velocities at which a deviation from the law first occurred were in
exact inverse ratio of the diameters of the pipes.
(2) Up to these critical velocities the discharge from the pipes agreed
exactly with those given by Poiseuille's formula for capillary tubes.
(3) For some little distance after passing the critical velocity, no very
simple relations appeared to hold between the pressures and velocities. But
by the time the velocity reached 1*2 (critical velocity) the relation became
again simple. The pressure did not vary as the square of the velocity, but
as 1722 power of the velocity ; this law held in both tubes and through
velocities ranging from 1 to 20, where it showed no signs of breaking down.
(4) The most striking result was that not only at the critical velocity,
44]
THE LAW OF RESISTANCE IN PARALLEL CHANNELS.
65
but throughout the entire motion, the laws of resistance exactly corresponded
for velocities in the ratio of
f»'
This last result was brought out in the most striking manner on reducing the
results by the graphic method of logarithmic homologues as described in my
paper on Thermal Transpiration.* Calling the resistance per unit of length
as measured in the weight of cubic units of water *, and the velocity v, log i
is taken for abscissa, and log v for ordinate, and the curve plotted.
In this way the experimental results for each tube are represented as a
curve ; these curves, which are shown as far as the small scale will admit in
Fig. 9, present exactly the same shape, and only differ in position.
Fig. 9.
Pipe. Diameter.
m.
No. 4, Lead 0'00615
„ 5, „ 0-0127
A, Glass 0-0490
B, Cast-iron 0'188
D, „ 0-5
C, Varnish 0'196.
Either of the curves may be brought into exact coincidence with the
other by a rectangular shift, and the horizontal shifts are given by the differ-
ence of the logarithms of
Phil. Tram. 1879, Part n. p. 40.
O. K. Jl.
66 ON THE MOTION OF WATER, AND OF [44
for the two tubes, the vertical shifts being the difference of the logarithms of
D
The temperatures at which the experiment had been made were nearly
the same, but not quite, so that the effect of the variations of fi showed
themselves.
15. Comparison with Darcy's experiments. — The defmiteness of these
results, their agreement with Poiseuille's law, and the new form which they
more than indicated for the law of resistance above the critical velocities, led
me to compare them with the well-known experiments of Darcy on pipes
ranging from 0*014 to 0'5 metre in diameter.
Taking no notice of the empirical laws by which Darcy had endeavoured
to represent his results, I had the logarithmic homologues drawn from his
published experiments. If my law was general then these logarithmic curves,
together with mine, should all shift into coincidence, if each were shifted
horizontally through
P*'
and vertically through
P'
In calculating these shifts there were some doubtful points. Darcy's
pipes were not uniform between the gauge points, the sections varying
as much as 20 per cent, and the temperature was only casually given.
These matters rendered a close agreement unlikely. It was rather a question
of seeing if there was any systematic disagreement. When the curves came
to be shifted the agreement was remarkable. In only one respect was there
any systematic disagreement, and this only raised another point ; it was only
in the slopes of the higher portions of the curves. In both my tubes the
slopes were as 1'722 to 1 ; in Darcy's they varied according to the nature of
the material, from the lead pipes, which were the same as mine, to T92 to 1
with the cast-iron.
This seems to show that the nature of the surface of the pipe has an
effect on the law of resistance above the critical velocity.
16. The critical velocities. — All the experiments agreed in giving
1 P
Vc ~ 278 D
as the critical velocity, to which corresponds as the critical slope of pressure
. _J P>
*c " 47700000 D3 '
44] THE LAW OF RESISTANCE IN PARALLEL CHANNELS. 67
the units being metres and degrees centigrade. It will be observed that
this value is much less than the critical velocity at which steady motion
broke down ; the ratio being 437 to 278.
17. The general law of resistance. — The logarithmic homologues all
consist of two straight branches, the lower branch inclined at 45 degrees and
the upper one at n horizontal to 1 vertical. Except for the small distance
beyond the critical velocity these branches constitute the curves. These two
branches meet in a point on the curve at a definite distance below the
critical pressure, so that, ignoring the small portion of the curve above the
point before it again coincides with the tipper branch, the logarithmic
homologue gives for the law of resistance for all pipes and all velocities
. D' . /RD \»
A — i = (B-v\ ,
P" V p )
where n has the value unity as long as either number is below unity, and
then takes the value of the slope n to 1 for the particular surface of the pipe.
If the units are metres and degrees centigrade,
4 = 67,700,000,
B = 396,
P = (1 + 0-0336 T+ 0-000221 T*)~\
This equation then, excluding the region immediately about the critical
velocity, gives the law of resistance in Poiseuille's tubes, those of the present
investigation and Darcy's, the range of diameters being
from 0-000013 (Poiseuille, 1845)
to 0-5 (Darcy, 1857),
and the range of velocities
from 0-0026)
7 [ metres per sec., 188,3.
TiO / Uv J
This algebraical formula shows that the experiments entirely accord with
the theoretical conclusions.
The empirical constants are A, B, P, and n ; the first three relate solely
to the dimensional properties of the fluid summed up in the viscosity, and it
seems probable that the last relates to the properties of the surface of the
pipe.
Much of the success of the experiments is due to the care and skill of
Mr Foster, of Owens College, who has constructed the apparatus and assisted
me in making the experiments.
5—2
68
ON THE MOTION OF WATER, AND OF
[44
SECTION II.
Experiments in glass tubes by means of colour bands.
18. In commencing these experiments it was impossible to form any very
definite idea of the velocity at which eddies might make their appearance
with a particular tube. The experiments of Poiseuille showed that the law
of resistance varying as the velocity broke down in a pipe of say 0'6 millim.
diameter ; and the experiments of Darcy showed this law did not hold in a
half-inch pipe with a velocity of 6 inches per second.
These considerations, together with the comparative ease with which
experiments on a small scale can be made, led me to commence with the
smallest tube in which I could hope to perceive what was going on with the
naked eye, expecting confidently that eddies would make their appearance at
an easily attained velocity.
19. The first apparatus. — This consisted of a tube about £ inch or
6 millims. in diameter. This was bent into the siphon form having one
straight limb about 2 feet long and the other about 5 feet (Fig. 10).
Fig. 10. Fig. KX.
The end of the shorter limb was expanded to a bell mouth, while the
44]
THE LAW OV RESISTANCE IN PARALLEL CHANNELS.
69
longer end was provided with an indiarubber extension on which was a screw
clip.
The bell-mouthed limb was held vertically in the middle of a beaker,
with the mouth several inches from the bottom as shown in Figs. 10 and 10'.
A colour tube about 6 millims. in diameter, also of siphon form, was
placed as shown in the figure, with the open end of the shorter limb just
under the bell mouth, the longer limb communicating through a controlling
clip with a reservoir of highly coloured water placed at a sufficient height.
A supply-pipe was led into the beaker for the purpose of filling it ; but not
with the idea of maintaining it full, as it seemed probable that the inflowing
water would create too much disturbance, experience having shown how
important perfect internal rest is to successful experiments with coloured
water.
20. The first experiment — The vessels and the siphons having been filled
and allowed to stand for some hours so as to allow all internal motion to
cease, the colour clip was opened so as to allow the colour to emerge slowly
below the bell (Fig. 11).
Fig. 11.
Fig. 12.
Then the clip on the running pipe was opened very gradually. The
water was drawn in at the bell mouth, and the coloured water entered, at
first taking the form of a candle flame (Fig. 12), which continually elongated
until it became a very fine streak, contracting immediately on leaving the
colour-tube, and extending all along the tube from the bell mouth to the
outlet (Fig. 10). On further opening the regulating clip so as to increase
70 ON THE MOTION OF WATER, AND OF [44
the velocity of flow, the supply of colour remaining unaltered, the only effect
was to diminish the thickness of the colour band. This was again increased
by increasing the supply of colour, and so on until the velocity was the
greatest that circumstances would allow — until the clip was fully open.
Still the colour band was perfectly clear and definite throughout the tube.
It was apparent that if there were to be eddies it must be at a higher
velocity. To obtain this about 2 feet more were added to the longer leg of
the siphon, which brought it down to the floor.
On trying the experiment with this addition, the colour band was still
clear and undisturbed.
So that for want of power to obtain greater velocity this experiment
failed to show eddies.
When the supply pipe which filled the beaker was kept running during
the experiment, it kept the water in the beaker in a certain state of disturb-
ance. The effect of this disturbance was to disturb the colour band in the
tube, but it was extremely difficult to say whether this \yas due to the
wavering of the colour band or to genuine eddies.
21. The final apparatus. — This was on a much larger scale than the
first. A straight tube, nearly 5 feet long and about an inch in diameter,
was selected from a large number as being the most nearly uniform, the
variation of the diameter being less than l-32nd of an inch.
The ends of this tube were ground off plane, and on the end which
appeared slightly the larger was fitted a trumpet mouth of varnished wood,
great care being taken to make the surface of the wood continuous with
that of the glass (Fig. 13).
The other end of the glass pipe was connected by means of an indiarubber
washer with an iron pipe nearly 2 inches in diameter.
The iron pipe passed horizontally through the end of a tank, 6 feet long,
18 inches broad and 18 inches deep, and then bent through a quadrant so
that it became vertical, and reached 7 feet below the glass tube. It then
terminated in a large cock, having, when open, a clear way of nearly a square
inch.
This cock was controlled by a long lever reaching up to the level of the
tank. The tank was raised upon trestles about 7 feet above the floor, and
on each side of it, at 4 feet from the ground, was a platform for the observers.
The glass tube thus extended in a horizontal direction along the middle of
the tank, and the trumpet mouth was something less than a foot from the
end. Through this end, just opposite the trumpet, was a straight colour
44]
THE LAW OF RESISTANCE IN PARALLEL CHANNELS.
71
tube three-quarters of an inch in diameter, and this tube was connected, by
means of an indiarubber tube with a clip upon it, with a reservoir of colour,
which for good reasons subsequently took the form of a common water bottle.
Fig. 13.
With a view to determining the velocity of flow, an instrument was fitted
for showing the changes of level of the water in the tank to the 100th of
an inch (Fig. 14). Thermometers were hung at various levels in the tank.
'22. The final experiments. — The first experiment with this apparatus
was made on 22nd February, 1880.
By means of a hose the tank was filled from the water main, and having
been allowed to stand for several hours, from 10A.M. to 2 P.M., it was then
found that the water had a temperature of 46° F. at the bottom of the tank,
and 47° F. at the top. The experiment was then commenced in the same
ON THE MOTION OF WATER, AND OF
[44
manner as in the first trials. The colour was allowed to flow very slowly,
and the cock slightly opened. The colour band established itself much as
before, and remained beautifully steady as the velocity was increased until,
Fig. 14.
all at once, on a slight further opening of the valve, at a point about two
feet from the iron pipe, the colour band appeared to expand and mix with the
water so as to fill the remainder of the pipe with a coloured cloud, of what
appeared at first sight to be of a uniform tint (Fig. 4, p. 59).
Closer inspection, however, showed the nature of this cloud. By moving
the eye so as to follow the motion of the water, the expansion of the colour
band resolved itself into a well-defined waving motion of the band, at first
without other disturbance, but after two or three waves came a succession
of well-defined and distinct eddies. These were sufficiently recognisable by
following them with the eye, but more distinctly seen by a flash from a
spark, when they appeared as in Fig. 5, p. 60.
The first time these were seen the velocity of the water was such that
the tank fell 1 inch in 1 minute, which gave a velocity of Om-627, or 2 feet
per second. On slightly closing the valve the eddies disappeared, and the
straight colour band established itself.
Having thus proved the existence of eddies, and that they came into
existence at a certain definite velocity, attention was directed to the relations
between this critical velocity, the size of the tube, and the viscosity.
Two more tubes (2 and 3) were prepared similar in length and mounting
to the first, but having diameters of about one-half and one-quarter inch
respectively.
44] THE LAW OF RESISTANCE IN PARALLEL CHANNELS. 73
In the meantime an attempt was made to ascertain the effect of viscosity
by using water at different temperatures. The temperature of the water
from the main was about 45°, the temperature of the room about 54° ; to
obtain a still higher temperature, the tank was heated to 70° by a jet of
steam. Then taking, as nearly as we could tell, similar disturbances, the
experiments which are numbered 1 and 2 in Table I., page 74, were made.
To compare these for the viscosity, Poiseuille's experiments were available,
but to prevent any accidental peculiarity of the water being overlooked,
experiments after the same manner as Poiseuille's were made with the water
in the tank. The results of these however agreed so exactly with those of
Poiseuille that the comparative effect of viscosity was taken from Poiseuille's
formula
P-I = 1 + 0-03368T + 0-000221 T\
where P x. fj, with the temperature and T is temperature centigrade.
The relative values of P at 47° and 70° Fah. are as
1-3936 to 1,
while the relative critical velocities at these temperatures were as
1-45 to 1,
which agreement is very close considering the nature of the experiments.
But whatever might have been the cause of the previous anomalies,
these were greatly augmented in the heated tank. After being heated
the tank had been allowed to stand for an hour or two, in order to become
steady. On opening the valve it was thought that the eddies presented a
different appearance from those in the colder water, and the thought at once
suggested itself that this was due to some source of initial disturbance.
Several sources of such disturbance suggested themselves — the temperature
of the tank was 11° C. above that of the room, and tne cooling arising from
the top and sides of the tank must cause circulation in the tank. A few
streaks of colour added to the water soon showed that such a circulation
existed, although it was very slow. Another source of possible disturbance
was the difference in the temperature at the top and bottom of the tank,
this had been as much as 5°.
In order to get rid of these sources of disturbance it was necessary to
have the tank at the same temperature as the room, about 54° or 55°.
Then it was found by several trials that the eddies came on at a fall of
about 1 inch in 64 seconds, which, taking the viscosity into account, was
higher than in the previous case, and this was taken to indicate that there
was less disturbance in the water.
As it was difficult to alter the temperatures of the building so as to obtain
experiments under like conditions at a higher temperature, and it appeared
74
ON THE MOTION OF WATER, AND OF
[44
that the same object would be accomplished by cooling the water to its
maximum density, 40°, this plan was adopted and answered well, ice being
used to cool the water.
Experiments were then made with three tubes 1, 2, 3, at temperatures
of about 51° and 40°. All are given in Table I.
TABLE I.
Experiments with Colour Bands — Critical Velocities at which Steady
Motion breaks down.
Pipe No, 1, glass. — Diameter 0'0268 metre; log diameter 2*42828.
„ No. 2, „ „ 0-01527 „ „ 2-18400.
„ No. 3, „ „ 0-007886 „ „ 3'89783.
Discharge, cub. metre = '021237 ; log = 2-32709.
Date, 1880
Kefer-
ence
num-
ber
Pipe
Tem-
pera-
ture,
centi-
grade
Time
of dis-
charge
Velocity,
metres
log time
-logP
log V
log Be
1 March
1
No. 1
8-3
60
0-6270
1-77815
0-11242
T-79729
1-66200
3 „
2
11
21 87
0-4325
1-93959
0-25654
1 -63593
1-67930
25 „
3
I.') 70
0-5374
1-84500
0-19198
1-73035
1-64936
21 April
4
15
12 60
0-6270
1-77815 0-15712
1 -79729
1-61730
11
.">
11
13 64
0-5878
1-80618
0-16882
1-76926
1 -64464
11
6
11
13 67
0-5614
1-82617
0-16882
1-74927
1-65363
V
7
11
13 64
0-5878
1-80618
0-16882
1-76926
1-64464
8
11
5 54
0-6967
1-73239
0-06963
1 -84305
1 -65898
11
9
*
5
52
0-7235
1-71600
0-06963
1 -85940
1-64269
22 „
10
11
10
62
0-6068
1-79239
0-13319
1-78305
1-65546
11
11
M
11
64
0-5870
1-80613
0-14525
1-76931
1-65716
25 March
12
No. 2
* 22
1.-).-,
0-7476
2-19033
0-26710
1-87367
1-67523
23 April
13
11
11
110
1-052
2-04139
0-14525
0-02261
1-64814
„
14
11
11
108
1-072
2-03342
0-14525
0-03058
1-64017
11
15
11
4
83
1-396
1-91907
0-05621
0-14493
1-61486
»
16
11
4 83
1-396
1-91907
0-05621
0-14493
1-61486
11
17
11
4 83
1-396
1-91907
0-05621
0-14493
1-61486
11
18
55
6 86
1-348
1-93449
0-08278
0-12951
1-59371
»>
19
11
6 85
1-362
1-92941
0-08278
0-13459
1-59863
24 „
20
No. 3
11
220 1-967
2-34242
0-14525
0-29392
1-66300
11
21
5>
10-5
224 1-932 j 2-35024
0-13920
0-28610
1-67687
11
22
55
11
218 1-982 2-33845
0-14525 i 0-29789
1-65903
11
23
55
11
116 2-004 2-33445
0-14525 0-30189
1-65503
2:, „
24
1)
4
164 2-637 2-21484
0-05621 0-42150
1-62446
11
25
11
4
172 2-517 2-23552
0-05621
0-40082
1 -64514
11
26
11
6
176
2-460 2-24551
0-08278
0-39083 1-62856
11
27
11
6
176
2-460 2-24551
0-08278 0-39083 : 1-62856
11
28
»
6
174
2-488 2-24054
0-08278 0-39580 ! 1-62359
11
29
»
6
177
2-446
2-24791
0-08278
0-38837
1-63102
This gives the mean value for log£, TG4139 ; and Ua =
44] THK LAW OF RESISTANCE IN PARALLEL CHANNELS. 75
In reducing the results the unit taken has been the metre and the tem-
perature is given in degrees centigrade.
The diameters of the three tubes were found by filling them with water.
The time measured was the time in which the tank fell 1 inch, which in
cubic metres is given by
Q = -021237.
In the table the logarithms of P, v, and Bs are given, as well as the natural
numbers for the sake of reference.
The velocities v have been obtained by the formula
_
Bs being obtained from the formula
The filial value of Bs is obtained from the mean value of the logarithm
of Bt.
23. . The results. — The values of log Bs show a considerable amount of
regularity, and prove, I think conclusively, not only the existence of a critical
velocity at which eddies come in, but that it is proportional to the viscosity
and inversely proportional to the diameter of the tube.
The fact, however, that this relation has only been obtained by the utmost
care to reduce the internal disturbances in the water to a minimum must not
be lost sight of.
The fact that the steady motion breaks down suddenly shows that the
rluid is in a state of instability for disturbances of the magnitude which
cause it to break down. But the fact that in some conditions it will break
down for a large disturbance, while it is stable for a smaller disturbance shows
that there is a certain residual stability so long as the disturbances do not
exceed a given amount.
The only idea that I had formed before commencing the experiments was
that at some critical velocity the motion must become unstable, so that any
disturbance from perfectly steady motion would result in eddies.
I had not been able to form any idea as to any particular form of dis-
turbance being necessary. But experience having shown the impossibility of
obtaining absolutely steady motion, I had not doubted but that appearance
of eddies would be almost simultaneous with the condition of instability.
76 ON THE MOTION OF WATER, AND OF [44
I had not, therefore, considered the disturbances except to try and diminish
them as much as possible. I had expected to see the eddies make their
appearance as the velocity increased, at first in a slow or feeble manner,
indicating that the water was but slightly unstable. And it was a matter
of surprise to me to see the sudden force with which the eddies sprang into
existence, showing a highly unstable condition to have existed at the time
the steady motion broke down.
This at once suggested the idea that the condition might be one of
instability for disturbance of a certain magnitude and stable for smaller
disturbances.
In order to test this, an open coil of wire, as in Fig. 15, was placed in the
tube so as to create a definite disturbance.
Fig. 15.
Eddies now showed themselves at a velocity of less than half the previous
critical velocity, and these eddies broke up the colour band, but it was
difficult to say whether the motion was really unstable or whether the eddies
were the result of the initial disturbance, for the colour band having once
broken up and become mixed with the water, it was impossible to say whether
the motion did not tend to become steady again later on in the tube.
Subsequent observation however tended to show that the critical value of
the velocity depended to some extent on the initial steadiness of the water.
One phenomenon in particular was very marked.
Where there was any considerable disturbance in the water of the tank
and the cock was opened very gradually, the state of disturbance would first
show itself by the wavering about of the colour band in the tube ; sometimes
it would be driven against the glass and would spread out, and all without
a symptom of eddies. Then, as the velocity increased but was still com-
paratively small, eddies, and often very regular eddies, would show themselves
along the latter part of the tube. On further opening the cock these eddies
would disappear and the colour band would become fixed and steady right
through the tube, which condition it would maintain until the velocity
reached its normal critical value, and then the eddies would appear suddenly
as before.
Another phenomenon very marked in the smaller tubes, was the inter-
44] THE LAW OF RESISTANCE IN PARALLEL CHANNELS. 77
mittent character of the disturbance. The disturbance would suddenly come
on through a certain length of the tube and pass away and then come on
again, giving the appearance of flashes, and these flashes would often
commence successively at one point in the pipe. The appearance when the
flashes succeeded each other rapidly was as shown in Fig. 16.
Fig. 16.
This condition of flashing was quite as marked when the water in the
tank was very steady, as when somewhat disturbed.
Under no circumstances would the disturbance occur nearer to the trumpet
than about 30 diameters in any of the pipes, and the flashes generally, but
not always, commenced at about this distance.
In the smaller tubes generally, and with the larger tube in the case of
the ice-cold water at 40°, the first evidence of instability was an occasional
flash beginning at the usual place and passing out as a disturbed patch two
or three inches long. As the velocity was further increased these flashes
became more frequent until the disturbance became general.
I did not see a way to any very crucial test as to whether the steady
motion became unstable for a large disturbance before it did so for a small
one ; but the general impression left on my mind was that it did in some
way — as though disturbances in the tank, or arising from irregularities in
the tube, were necessary to the existence of a state of instability.
But whatever these peculiarities may mean as to the way in which eddies
present themselves, the broad fact of there being a critical value for the
velocity at which the steady motion becomes unstable, which critical value is
proportional to
£
pc'
where c is the diameter of the pipe and fi/p the viscosity by the density, is
abundantly established. And cylindrical glass pipes for approximately steady
water have for the critical value
V =
where in metres Bs = 4370 about.
78 ON THE MOTION OF WATER, AND OF [44
SECTION III.
Experiments to determine the critical velocity by means of resistance in
the pipes.
24. Although at first sight such experiments may appear to be simple
enough, yet when one began to consider actual ways and means, so many
uncertainties and difficulties presented themselves, that the necessary courage
for undertaking them was only acquired after two years' further study of the
hydrodynamical aspect of the subject, by the light thrown upon it by the
previous experiment with the colour bands. This has been already explained
in Art. 13. Those experiments had shown definitely that there was a
critical value of the velocity at which eddies began, if the water were
approximately steady when drawn into the tube, but they had also shown
definitely, that at such critical velocity, the water in the tube was in a highly
unstable condition ; any considerable disturbance in the water causing the
break down to occur at velocities much below the highest that could be
attained when the water was at its steadiest ; suggesting that if there were
a critical velocity at which, for any disturbance whatever, the water became
stable, this velocity was much less than that at which it would become
unstable for infinitely small disturbances ; or, in other words, suggesting that
there were two critical values for the velocity in the tube, the one at which
steady motion changed into eddies, the other at which eddies changed into
steady motion.
Although the law for the critical value of the velocity had been suggested
by the equations of motion, it was, as already explained, only at the
beginning of this year that I succeeded in dealing with these equations so
as to obtain any theoretical explanation of the dual criteria ; but having at
last found this, it became clear to me that, if in a tube of sufficient length
the water were at first admitted in a high state of disturbance, then as the
water proceeded along the tube, the disturbance would settle down into a
steady condition, which condition would be one of eddies or steady motion,
according to whether the velocity was above or below what may be called the
real critical value.
The necessity of initial disturbance precluded the method of colour bands,
so that the only method left was to measure the resistance at the latter
portion of the tube in conjunction with the discharge.
The necessary condition was somewhat difficult to obtain. The change
in the law of resistance could only be ascertained by a series of experiments
44]
THK LAW OF RESISTANCE IN PARALLEL CHANNELS.
79
which had to be carried out under similar conditions as regards temperature,
kind of water, and condition of the pipe; and in order that the experiments
might be satisfactory, it seemed necessary that the range of velocities should
extend far on each side of the critical velocity. In order to best ensure
these conditions, it was resolved to draw the water direct from the
Manchester main, using the pressure in the main for forcing the water
through the pipes. The experiments were conducted in the workshop in
Owens College, which offered considerable facilities owing to arrangements
for supplying and measuring the water used in experimental turbines.
Fig. 17.
25. The apparatus is shown in Fig. 17.
80 ON THE MOTION OF WATER, AND OF [44
As the critical value under consideration would be considerably below
that found for the change for steady motion into eddies, a diameter of about
half an inch (12 millims.) was chosen for the larger pipe, and one quarter of
an inch for the smaller, such pipes being the smallest used in the previous
experiments.
The pipes (4 and 5) were ordinary lead gas, or water pipes. These,
which, owing to their construction, are very uniform in diameter, and when
new, present a bright metal surface inside, seemed well adapted for the
purpose.
Pipes 4 (which was a quarter-inch pipe) and 5 (which was a half-inch)
were 16 feet long, straightened by laying them in a trough formed by two
inch boards at right angles. This trough was then fixed so that one side of
the trough was vertical and the other horizontal, forming a horizontal ledge
on which the pipes could rest at a distance of 7 feet from the floor ; on the
outflow ends of the pipes cocks were fitted to control the discharge, and at
the inlet end the pipes were connected, by means of a T branch, with an
indiarubber hose from the main ; this connexion was purposely made in such
a manner as to necessitate considerable disturbance in the water entering
the pipes from the hose. The hose was connected, by means of a quarter-
inch cock, with a four-inch branch from the main. With this arrangement
the pressure on the inlet to the pipes was under control of the cock from the
main, and at the same time the discharge from the pipes was under control
from the cocks on their ends.
This double control was necessary owing to the varying pressure in the
main, and after a few preliminary experiments a third and more delicate
control, together with a pressure gauge, were added, so as to enable the
observer to keep the pressure in the hose, i.e., on the inlets to the pipes,
constant during the experiments.
This arrangement was accomplished by two short branches between the
hose and the control cock from the main, one of these being furnished with
an indiarubber mouthpiece with a screw clip upon it, so that part of the
water which passed the cock might be allowed to run to waste, the other
branch being connected with the lower end of a vertical glass tube, about
6 millims. in diameter and 30 inches long, having a bulb about 2 inches
diameter near its lower extremity, and being closed by a similar bulb at
its top.
This arrangement served as a delicate pressure gauge. The water
entering at the lower end forced the air from the lower bulb into the upper,
causing a pressure of about 30 inches of mercury. Any further rise increased
this pressure by forcing the air in the tubes into the upper bulb, and by the
THE LAW OF RESISTANCE IN PARALLEL CHANNELS. 81
weight, of water in the tube. During an experiment the screw clip was con-
tinually adjusted, so as to keep the level of the water in the glass tube
between the bulbs constant.
26. The resistance gauges. — Only the last 5 feet of the tube was used
for measuring the resistance, the first 10 or 11 feet being allowed for the
acquirement of a regular condition of flow.
It was a matter of guessing that 10 feet would be sufficient for this, but
since, compared with the diameter, this length was double as great for the
smaller tube, it was expected that any insufficiency would show itself in a
greater irregularity of the results obtained with the larger tube, and as no
such irregularity was noticed it appears to have been sufficient.
At distances of 5 feet near the ends of the pipe, two holes of about
1 inillim. were pierced into each of the pipes for the purpose of gauging the
pressures at these points of the pipes. As owing to the rapid motion of
the water in the pipes past these holes, any burr or roughness caused in
the inside of the pipe in piercing these holes would be apt to cause a
disturbance in the pressure, it was very important that this should be
avoided. This at first seemed difficult, as owing to the distance — 5 feet — of
one of the holes from the end of pipes of such small diameter, the removal
of a bun-, which would be certain to ensue on drilling the holes from the
outside, was difficult. This was overcome by the simple expedient suggested
by Mr Foster of drilling holes completely through the pipes and then plugging
the side on which the drill entered. Trials were made, and it was found that
the burr thus caused was very slight.
Before drilling the holes short tubes had been soldered to the pipes, so
that the holes communicated with these tubes ; these tubes were then con-
nected with the limbs of a siphon gauge by indiarubber pipes.
These gauges were about 30 inches long ; two were used, the one con-
taining mercury, the other bisulphide of carbon.
These gauges were constructed by bending a piece of glass tube into
a U form, so that the two limbs were parallel and at about one inch
apart.
Glass tubes are seldom quite uniform in diameter, and there was a
difference in the size of the limbs of both gauges, the difference being con-
siderable in the case of the bisulphide of carbon.
The tubes were fixed to stands with carefully graduated scales behind
them, so that the height of the mercury or carbon in each limb could be
read. It had been anticipated that readings taken in this way would be
o. 11. ii. 6
82 ON THE MOTION OF WATER, AND OF [44
sufficient. But it turned out to be desirable to read variations of level of
the smallness of j^ooth of an inch or J^th of a millimetre.
A species of cathetometer was used. This had been constructed for my
experiments on Thermal Transpiration, and would read the position of the
division surface of two fluids to Toio^n mcn (Pa£e 258, Vol. I.).
The water was carefully brought into direct connexion with the fluid in
the gauge, the indiarubber connexions facilitating the removal of all air.
27. Means adopted in measuring the discharge. — For many reasons it
was very desirable to measure the rate of discharge in as short a time as
possible.
For this purpose a species of orifice or weir gauge was constructed,
consisting of a vertical tin cylinder two feet deep, having a flat bottom,
being open at the top, with a diaphragm consisting of many thicknesses of
fine wire gauze about two inches from the bottom ; a tube connected the
bottom with a vertical glass tube, the height of water in which showed the
pressure of water on the bottom of the gauze ; behind this tube was a scale
divided so that the divisions were as the square roots of the height. Through
the thin tin bottom were drilled six holes, one an eighth of an inch diameter,
one a quarter of an inch, and four of half an inch.
These holes were closed by corks so that any one or any combination
could be used.
The combinations used were :
Gauge No. 1. The ^ inch hole alone.
No. 2. The £ inch hole alone.
No. 3. A £ inch hole alone.
No. 4. Two \ inch holes.
No. 5. Four \ inch holes.
According to experience, the velocity with which water flows from a still
vessel through a round hole in a thin horizontal plate is very nearly propor-
tional to the area of the hole and the square root of the pressure, so that
with any particular hole the relative quantities of water discharged would be
read off at the variable height gauge. The accuracy of the gauge, as well
as the absolute values of the readings, was checked by comparing the
readings on the gauge with the time taken to fill vessels of known capacity.
In this way coefficients for each one of the combinations 1, 2, 3, 4, 5 were
obtained as follows : —
44] THE LAW OF RESISTANCE IN PARALLEL CHANNELS.
TABLE II.
83
No. of Gauge
Readings on Time
Gauge
Quantity
Coefficient
Logarithmic
coefficient
Seconds
c.c.
Gauge No. 1
ib.
19-55
61
59
1160
1160
j -966
T-985
No. 2
5-3
54
1160
4-055
•608
ib.
15-3 full
—
A
4-055
—
No. 3
15
360
A
16-220
1-210
No. 4
15
178
A
32-440
1-511
No. 5
15
90
A
64-880
1-812
From this table it will be seen that the absolute values of the coefficients
were obtained from experiments on the gauges No. 1 and No. 2, the co-
efficients for the gauges 3, 4, and 5 being determined by comparison of the
times taken to fill a vessel of unknown capacity, which stands in the Table
as A. The relative value of these coefficients came out sensibly proportional
to the squares of the diameters of the apertures.
For the smaller velocities it was found that the gauge No. 1 was too
large, and in order not to delay the experiment in progress, two glass flasks
were used : these are distinguished as flasks (1) and (2) ; their capacities,
as subsequently determined with care, were 303 and 1160 c.c. The dis-
charge as measured by the times taken to fill these flasks are reduced to
c.c. per second by dividing the capacities of the flasks by the times.
28. The method of carrying out the experiments was generally as
follows : — My assistant, Mr Foster, had charge of the supply of water from
the main, keeping the water in the pressure gauge at a fixed level.
The tap at the end of the tube to be experimented upon being closed,
the zero reading of the gauge was carefully marked, and the micrometer
adjusted so that the spider line was on the division of water and fluid in
the left-hand limb of the gauge. The screw was then turned through one
entire revolution, which lowered the spider line one-fiftieth of an inch ; the
tap at the end of the pipe was then adjusted until the fluid in the gauge
came down to the spider line; having found that it was steady there, the
discharge was measured.
This having been done, the spider line was lowered by another complete
revolution of the screw, the tap again adjusted, and so on, for about 20
midingH, which meant about half an inch difference in the gauge. Then
the readings were taken for every five turns of the screw until the limit of
the range, about 2 inches, was reached. After this, readings were taken by
6—2
84 ON THE MOTION OF WATER, AND OF [44
simple observation of the scale attached to the gauge. In taking these
readings the best plan was to read the position of the mercury or carbon
in both limbs of the gauge, but this was not always done, some of the
readings entered in the notes referred to one or other limb of the gauge,
care having been taken to indicate which.
In the Tables III., IV., and V. of results appended, the noted readings
are given and the letters r, I, and b signify whether the reading was on the
right or left limb, or the sum of the readings on both limbs.
The readings marked I and r are reduced by the correction for the
difference in the size of the limbs as well as the coefficient for the particular
fluid in the gauge.
Thus it was found with the mercury tube that when the left limb had
moved through 39 divisions on the scale the right had moved through 41,
so that to obtain the sum of these readings, the readings on the left, or
those marked I, had to be multiplied by 2'05, and those on the right by
1-95.
With the bisulphide of carbon gauge, 11 divisions on the left caused 9
on the right, so that the correction for the reading on the left was T8 and
on the right 2*2.
29. Comparison of the pressure gauges. — The pressures as marked by
the gauges were reduced to the same standard by comparing the gauges ;
thus '25 of the left limb of the mercury corresponded with 24 inches on
both limbs of the bisulphide. Therefore to reduce the readings of the
bisulphide of carbon to the same scale as those of the mercury they were
multiplied by
•25 x 20-5
24
= 0-0213.
This brought the readings of pressure to the same standard, i.e.,
an inch of mercury, but these were further reduced by the factor 0'00032 to
bring them to metres of water.
As it was convenient for the sake of comparison to obtain the differences
of pressure per unit length of the pipe, the pressures in metres of water
have been divided by 1'524, the length in metres between the gauge holes,
and these reductions are included in the tables of results in the column
headed i.
From the discharges, as measured by the various gauges, reduced to
cubic centimetres, the mean velocity of the water was found by dividing
by the area of the section of the pipe.
44] THE LAW OF RESISTANCE IN PARALLEL CHANNELS. 85
30. Sections and diameters of the pipes. — The areas were obtained by
carefully measuring the diameters by means of fitting brass plugs into the
pipes, and then measuring the plugs. In this way the diameters were
found to be —
Diameter, No. 4 pipe, '242 inch, 6'15 millims.
No. 5 pipe, -498 inch, 127 millims.
These gave the areas of the sections —
Section, No. 4 pipe, 297 square millims.
„ No. 5 pipe, 125 square millims.
The discharge in cubic centimetres, divided by the area of section in
square millimetres, gave the mean velocity in metres per second, as given
in the Tables III., IV., and V.
The logarithms of i and v are given for the sake of comparison.
31. The temperature. — The chief reason why the water from che main
had been used, was from the necessity of having constant temperature
throughout the experiments, and my previous experience of the great
constancy of the temperature of the water in the mains, even over a period
of some weeks.
At the commencement of the experiments the temperature of the water
when flowing freely was found to be 5° C. or 41° F., and it remained the
same throughout the experiments. Nevertheless, a fact which had been
overlooked caused the temperature in the pipes to vary somewhat and in a
manner somewhat difficult to determine.
This fact, which was not discovered until after the experiments had been
reduced, was that the temperature of the workshop being above that of the
main, the water would be warmed in flowing through the pipes to an extent
depending on its flow. The possibility of this had not been altogether
overlooked, and an early observation was made to see if any such warming
occurred, but as it was found to be less than half a degree no further notice
was taken until on reducing the results it was found that the velocities
obtained with the very smallest discharges presented considerable discre-
pancies in various experiments; this suggested the cause.
The discrepancies were not serious if explained, so that all that was
necessary was to carefully repeat the experiments at the lower velocities
observing the temperatures of the effluent water. This was done, and
further experiments were made (see Art. 33).
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90
ON THE MOTION OF WATER, AND OF
[44
32. The results of the experiments. — A considerable number of preliminary
experiments were made until the results showed a high degree of consistency.
Then a complete series of experiments were made consecutively with each
tube. The results of these are given in Tables III. and V.
33. The critical velocities. — The determination of these, which had been
the main object of the experiments, was to some extent accomplished
directly during the experiments, for starting from the very lowest velocities,
it was found that the fluid in the differential gauge was at first very steady,
lowering steadily as the velocity was increased by stages, until a certain
point was reached, when there seemed to be something wrong with the
gauge. The fluid jumped about, and the smallest adjustment of the tap
controlling the velocity sent the fluid in the gauge out of the field of the
microscope. At first this unsteadiness always came upon me as a matter of
surprise, but after repeating the experiments several times, I learnt to know
exactly when to expect it. The point at which this unsteadiness is noted is
marked in the tables.
It was not, however, by the unsteadiness of the pressure gauge that the
critical velocity was supposed to be determined, but by comparing the ratio
of velocities and pressures given in the columns v and i in the tables. This
comparison is shown in diagram I. below, the values of i being abscissae and
v ordinates. It is thus seen that for each tube the points which mark the
experiments lie very nearly in a straight line up to definite points marked C,
at which divergence sets in rapidly.
The points at which this divergence occurs correspond with the experi-
ments numbered 6 and 59, which are immediately above those marked
unsteady.
Thus the change in the law of pressure agrees with the observation of
unsteadiness in fixing the critical velocities.
DIAGRAM 1.
CURVES OF PRESSURE AND VELOCITY IN PIPES.
-| ] Ko.4 nianifU-r O'"Oo6lS ,,l 5°C. ,nij
J — -I A\>5 Dianiitcr O'"OI2J.
0'" 1 00100200300400900600700800901 Oil 012 0 13 0 14 0 15 0 16 0 17
S/u/wu//V«siiri- III KVi/ir
According to my assumption, the straightness of the curves between the
origin and the critical points would depend on the constancy of temperature,
44] THE LAW OF RESISTANCE IN PARALLEL CHANNELS. 91
and it was the small divergences observed that suggested a variation of
temperature which had been overlooked. This variation was confirmed by
further experiments, amongst which are those contained in Table IV.
These showed that the probable variation of the temperature was in Table
III. from 12° C. to 9'C. at the critical point, and from 12° C. to 8° C. in
Table V., which variations would account for the small deviation from the
straight.
It only remained, then, to ascertain how far the actual values of vc, the
velocity at the critical points, corresponded with the ratio -~ or -~ .
For tube 4 from the Table III.
D = 0-006 15 metres,
vc = (V4426 metres per second at 9° C.,
at this temperature P = '757 (see p. 73).
p
Hence putting Bc = — ^ ,
VglJ
we have Bc = 279 7.
Again, for tube 5, Table V., at 8° C.
£ = •0127,
vc = -2260,
P = -7796,
whence Bc = 272'0.
The differences in the values of Bc thus obtained, would be accounted
for by a variation of a quarter of a degree in temperature, and hence the
results are well within the accuracy of the experiments.
To each critical velocity, of course, there corresponds a critical value of
the pressure. These are determined as follows.
The theoretical law of resistance for steady motion may be expressed by
Ac£Pi = BcPv.
And multiplying both sides by ^,
This law holds up to the critical velocity, and then the right-hand
number is unity, and, if Bc has the values just determined :
c~
92 ON THE MOTION OF WATER, AND OF [44
by Table III.
ie = -0516,
P2=-573,
D3 = -000,000,232,
which give Ac = 47,750,000.
By Table V.
i= -00638,
P2 = -607,
D* = -00000205,
which give Ae = 46,460,000,
which values of Ac differ by less than by what would be caused by half a
degree of temperature.
The conclusion, therefore, that the critical velocity would vary as j. is
abundantly verified.
34. Comparison with the discharges calculated by Poiseuille's formula. —
Poiseuille experimented on capillary tubes of glass between '02 and '1 millim.
in diameter, and it is a matter of no small interest to find that the formula
of discharges which he obtained from these experiments is numerically exact
for the bright metal tubes 100 times as large.
Poiseuille's formula is —
777)4
Q = 1836-724 (1 + 0-0336793 T + 0-000220992 T8) ^- ,
T = temperature in degrees centigrade.
H = pressure in millims. mercury.
D — diameter in millims.
L = length in millims.
Q = discharge in millims. cubed.
Putting i= -
P = (1 + 0-336793 T + 0-000220992 T*)~\
44] THE LAW OF RESISTANCE IN PARALLEL CHANNELS. 93
and changing the units to metres and cubic metres this formula may be
written
47 700000 ^t = 278pv,
the coefficients corresponding to Ac and Be.
The agreement of this formula with the experimental results from tubes
4 and 5 is at once evident. The actual and calculated discharges differ by
less than 2 per cent., a difference which would be more than accounted for by
an error of half a degree in the temperature.
35. Beyond the critical point. — The tables show that, beyond the critical
point, the relation between i and v differs greatly from that of a constant
ratio ; but what the exact relation is, and how far it corresponds in the two
tubes, is not to be directly seen from the tables.
In the curves (diagram I. page 90) which result from plotting i and v,
it appears that after a period of flatness the curves round off into a parabolic
form ; but whether they are exact parabolae, or how far the two curves are
similar with different parameters, is difficult to ascertain by any actual
comparison of the curves themselves, which, if plotted to a scale which will
render the small differences of pressure visible, must extend 10 feet at least.
36. The logarithmic method. — So far the comparison of the results has
been effected by the natural numbers, but a far more general and clearer com-
parison is effected by treating the logarithms of I and v.
This method of treating such experimental results was introduced in my
paper on Thermal Transpiration, page 283, Vol. I.
Instead of curves, of which i and v are the absciss* and ordinates, logt
and log v are taken for the absciss* and ordinates, and the curve so obtained
is the logarithmic homologue of the natural curve.
The advantage of the logarithmic homologues is that the shape of the
curve is made independent of any constant parameters, such parameters
affecting the position of all points on the logarithmic homologue similarly.
Any similarities in shape in the natural curves become identities in shape
in the logarithmic homologues. How admirably adapted these logarithmic
homologues are for the purpose in hand is at once seen from diagram II.,
which contains the logarithmic homologues of the curves for both pipes 4
and 5.
A glance shows the similarity of these curves, and also their general
character. But it is by tracing one of the curves, and shifting the paper
94
ON THE MOTION OF WATER, AND OF
[44
rectangularly until the traced curve is superimposed on the other, that the
exact similarity is brought out. It appears that, without turning the paper
at all, the two curves almost absolutely fit.
It also appears that the horizontal and vertical components of the shift
are —
Horizontal shift '913
Vertical shift '294
which are, within the accuracy of the work, respectively identical with the
J)3 J)
differences of the logarithms of -^ and -,j for the two tubes.
37. The general law of resistance in pipes. — The agreement of the
logarithmic homologues shows that not only at the critical velocities, but
for all velocities in these two pipes, pressure which renders (Z)3//*2) i the same
in both pipes corresponds to velocities which render (D/fi) v the same in both
pipes. This may be expressed in several ways. Thus if the tabular value
44] THE LAW OF RESISTANCE IN PARALLEL CHANNELS. 95
of i for each pipe, plotted in a scale, be multiplied by a number propor-
tional to D3/P- tor that particular pipe, and the values of v by a number
proportional to D/P, then the curves which have these reduced values of i
and v for abscissae and ordinates will be identical.
A still more general expression is that if
expresses the relation between i and v for a pipe in which D=l, T= 0, P= 1,
P
expresses the relation for every pipe and every condition of the water.
The determination of the relation between circumstances of motion and
the physical condition of the water in such a general form was not contem-
plated when the experiments were undertaken, and must be considered as
a result of the method of logarithmic homologues, which brought out the
relation in such a marked manner that it could not be overlooked. Nor
is this all.
It had formed no part of my original intention to re-investigate the law
of resistance in pipes for velocities above the critical value, as this is ground
which had been very much experimented upon, and experiments seemed to
show that the law was either indefinite or very complex — a conclusion which
did not seem inconsistent with the supposition that above this point the
resistance depended upon eddies which might be somewhat uncertain in
their action. But although it was not my intention to investigate laws, I
had made a point of continuing the experiments through a range of pressures
and velocities very much greater I think than had ever been attempted in
the same pipe.
Thus it will be noticed that in the larger tube the pressure in the last
experiment is four thousand times as large as in the first. In choosing the
great range of pressures I wished to bring out what previous experiments
had led me to expect, namely, that in the same tube for sufficiently small
pressures the pressure is proportional to the velocity, and for sufficiently
great pressures, the pressure was proportional to the square of the velocity.
Had this been the case not only would the lowest portion of the logarithmic
homologues up to the critical point have come out straight lines inclined
at 45 degrees, but the final portion of the curve would have come out a
straight line at half this inclination, or with a slope of two horizontal to
one vertical.
96 ON THE MOTION OF WATER, AND OF [44
The near approach of the lower portions of the curve to the line at 45°
led me, as I have already explained, to discover that the temperatures had
risen at the lower velocities, and to make a fresh set of experiments, some
of which are given in Table IV., in which, although the temperatures were
not constant, they were sufficiently different from the previous ones to show
that the discrepancy in the lower portions of the curves might be attributed
to variations of temperature, arid the agreement with the line of 45° con-
sidered as within the limits of accuracy of experiment.
When the logarithms of the upper portions of the curve came to be
plotted, the straightness and parallelism of the two lines was very striking.
There are a few discrepancies which could not be in any way attributed
to temperature, as with so much water moving this was very constant, but
on examination it was seen that these discrepancies marked the changes of
the discharge gauges. The law of flow through the orifices not having been
strictly as the square roots of the heights, the manner in which the gauges
had been compared forbade the possibility of there being a general error
from this cause ; the middle readings on the gauge were correct, so that the
discrepancies, which are small, are mere local errors.
This left it clear that whatever might be their inclination the lines
expressed the laws of pressures and velocities in both tubes, and since the
lines are strictly parallel, this law was independent of the diameter of the
tube. This point has been very carefully examined, for it is found that the
inclination of these lines differs decidedly from that of 2 to 1, being T723
to 1, and so giving a law of pressures through a range 1 to 50 of
i oc v1'723.
This is different from the law propounded by any of the previous experi-
menters, who have adhered to the laws
i = v2,
or i = Av + Bv2.
That neither of these laws would answer in case of the present experiments
was definitely shown, for the first of these would have a logarithmic homo-
logue inclined at 2 to 1, and the second would have a curved line. A
straight logarithmic homologue inclined at a slope T723 to 1 means no
other law than
i oc v1'733.
I have therefore been at some pains to express the law deduced from my
experiments on the uniform pipes so that it may be convenient for application.
This law as already expressed is simply
Dv\
44] THE LAW OF RESISTANCE IN PARALLEL CHANNELS. 97
where / is such that
is the equation to the curve which would result from plotting the resistance
and velocities in a pipe of diameter 1 at a temperature zero.
The exact form of / is complex, this complexity is however confined to
the region immediately after the critical point is passed.
Up to the critical point
D3 . Dv
c p2 l ~~ c p '
After the critical point is passed the law is complex until a velocity
which is l'325vc is reached. Then as shown in the homologues the curve
assumes a simple character again,
that is, the logarithmic homologue becomes a straight line inclined at 1723
to 1.
Referring to the logarithmic homologues (diagram 2, page 94), it will be
seen that although the directions of the two straight extremities of the curve
do not meet in the critical point, their intersection is at a constant distance
from this point, which in the logarithmic curves is, both for ordinates and
abscissa?,
0154.
These points o are therefore given by
log pV^og^T + 0'154
Dve Dv0
log -p- = log -p — h
Therefore putting
P2 P
A = ^, B=f-
Dhn Dvn
log A =
log B = log Bc + 0-154
and by the values of Ac and Bc previously ascertained (Art. 33, p. 92),
Iog4 = 7-8311, 4 = 67,700,000
log B = 2-598, B = 396-3
For feet log A = 6'28414, A = l ,935,000
log £ = 1-56603, 5= 36-9.
O. R. II. 7
98 ON THE MOTION OF WATER, AND OF [44
We thus have for the equation to the curves corresponding to the upper
straight branches
, D3 .
And if n have the value 1 or T722 according as either member of this
equation is < or > 1 the equation
. D3 . _ /BDv\n
is the equation to a curve which has for its logarithmic homologue the two
straight branches intersecting in o, and hence gives the law of pressures
and velocities, except those relating to velocities in the neighbourhood" of
the critical point, and these are seldom come across in practice.
Dv
By expressing n as a discontinuous function of Bc -p the equation may
be made to fit the curve throughout.
38. The effect of temperature. — It should be noticed that although the
range is comparatively small, still the displacement of the critical point in
Tables III. and IV. is distinctly marked. The temperatures were respectively
9° C., 5° C.
At 9° log P-^ 01 2093
At 5° log P-1 = 0-06963
Difference = '05130
This should be the differences in the values of log vc in Tables III.
and IV. The actual difference is '062. Also the differences in log ic
should be the differences in P2 or '10260, whereas the actual difference
is 121.
The errors correspond to a difference of about 1° C., which is a very
probable error.
It would be desirable to make experiments at higher temperature, but
there were great difficulties about this which caused me, at all events for
the time, to defer the attempt.
44] THE LAW OF RESISTANCE IN PARALLEL CHANNELS. 99
SECTION IV.
Application to DARCY'S experiments.
39. DARCY'S experiments. The law of resistance came out so definitely
from my experiments that, although beyond my original intention, I felt
constrained to examine such evidence as could be obtained of the actual
experimental results obtained by previous experimenters.
The lower velocities, up to the critical value, were found, as has already
been shown (Art. 35), to agree exactly with Poiseuille's formula.
For velocities above the critical values the most important experiments
were those of Darcy — approved by the Academy of Sciences and published
1845 — on which the formula in general use has been founded. Notwith-
standing that the formula as propounded by Darcy himself could not by
any possibility fit the results which I have obtained, it seemed possible that
the experiments on which he had based his law might fit my law. A com-
parison was therefore undertaken.
This was comparatively easy, as Darcy's experimental results have been
published in detail.
Altogether he experimented on some 22 pipes, varying in diameter from
about the size of my largest, Om-0014 up to Om'5. They were treated in
several sets, according to the material of which they were composed —
wrought- iron gas-pipes, lead pipes, varnished iron pipes, glass pipes, new
cast-iron and old rusty pipes.
The method of experimenting did not differ from mine except in scale,
the distance between Darcy's gauge points being 50"' instead of 5 feet in
my case. The great length between Darcy's gauge points entailed his
having joints in his pipes between these points, and the nature of his
pipes was such as to preclude the possibility of a very uniform diameter.
His experiments appear to have been made with extreme care and very
faithfully recorded, but the irregularity in the diameters, which appears to
have been as much as 10 per cent., and the further irregularity of the joints,
preclude the possibility of the results of his experiments following very
closely the law for uniform pipes. Another important matter to which
Darcy appears to have paid but little attention was temperature. It is
true that in many instances he has given the temperature, but he does
not appear to have taken any account of it in his discussion of his results,
although it varied as much as 20° C. in the cases where he has given it,
and as his pipes, 300 metres long, were in the open air, the effect of the
sun on the pipes would have led to still larger differences.
7—2
100
ON THE MOTION OF WATER, AND OF
[44
The effect of these various causes on his results may be seen, as he took
the precaution to use two pressure gauges on separate lengths of 50"' of
his pipes, and the records from these two gauges by no means always agree,
particularly for the lower velocities. In one case the results are as wide
apart as 15 to 7, and often 10 or 15 per cent. In arriving at tabular values
for i he has taken the mean of the two gauges.
Taking these things into account, I could not possibly expect any close
agreement with my results ; still, as experiments on pipes of such large
diameters are not likely to be repeated, at any rate with anything like the
same care and success, they offered the only chance of proving that my law
was general.
40. Reduction of the experimental results. Rejecting all the experi-
ments on rusty and rough pipes, i.e., selecting the lead, the varnished, the
glass, and new cast-iron pipes, which ranged from half-an-inch to twenty
inches diameter, I had the logarithmic homologues drawn. These are
shown on diagram 3. In the case of two of the smaller pipes the
Lines show calculated remits.
Dots iliow experimental result
•3\ -2 -1
Diameter Temp. Surface
A Omm. 014 10° C. Glass)
B 0
C 0
D 6
E 12
F 14
G 27
H 41
I 26
J 82
270
lf
}Poiseuille
L
285
650
,,
M
81
15
5 L
ad No. 4
N
137
70
5
No. 5
O
188
00
X
}
P
500
00
X
Q
243
00
X
f Darcy
R
244
00
12° C. Varnished
S
49
60
21° C.
J
Diameter Temp. Surface
K 196mm. 00 x Varnished
00 21° C.
90 15° C. Cast Iron new
00 15°
00
00
20
70
68
C. I. incrusted
ib. cleaned
Glass
Darcy
smallest velocity is well below the critical point, and in several of the other
pipes the smallest velocity is near the critical velocity. This accounts for
the lower ends of the logarithmic curves being somewhat twisted ; for the
remainder of the logarithmic homologues are nearly straight; some are
slightly bent one way and some another, but they are none of them more
bent than may be attributed to experimental inaccuracy.
44] THE LAW OF RESISTANCE IN PARALLEL CHANNELS. 101
The inclinations of the upper ends of the lead and bituminous pipes is
1746, slightly greater than mine ; but in the cases of the glass pipes and the
cast-iron pipes the slopes are T82 and T92 respectively.
So much appeared from the logarithmic homologues themselves, but the
most important question was, would the curves agree with the results
calculated from the formula
41. Comparison with the law of resistance. In applying this test I was
at first somewhat at a loss on account in some cases of the want of any
record of the temperature, and the doubt as to such temperatures as had
been recorded being the temperature of the water in the pipes between
the gauges.
The dates at which the experiments were made to a certain extent
supplied the deficiency of temperature, the temperatures given fixing the
law of temperature, so that the probable temperature could be assumed
where it was not given.
Assuming the temperature, the values of
. _P«_
l°~AD3'
P_
v°~ BD'
were calculated for each tube, using the values of A and B as already
determined, \ogi0 and v0 are the co-ordinates of 0 the intersection of the
two straight branches of the logarithmic curves, so that the application of
the formula to the results was simply tested by continuing the straight
upper branches of the logarithmic homologues to see whether they passed
through the corresponding point 0.
The agreement, which is shown in diagram 3, page 100, is remarkable.
There are some discrepancies, but nothing which may not be explained by
inaccuracies, particularly inaccuracies of temperature.
42. The effect of the temperature above the critical point. — It is a fact of
striking significance, physical as well as practical, that while the temperature
of the fluid has such an effect at the lower velocities that, ccvteris panbus,
the discharge will be double at 45° C. what it is at 5° C., so little is the
effect at the higher velocities that neither Darcy nor any other experimenter
seems to have perceived any effect at all.
In my experiments the temperature was constant, 5° C. at the higher
velocities, so that I had no cause to raise this point till I came to Darcy 's
result, and then, after perplexing myself considerably to make out what the
102 ON THE MOTION OF WATER, AND OF [44
temperatures were, I noticed the effect of the temperature is to shift the
curves 2 horizontal to 1 vertical, which corresponds with a slope of 2 to 1,
and so nearly corresponds with the direction of the curves at higher velocities
that variations of 5° or 10° C. produce no sensible effect; or, in other words,
the law of resistance at the higher velocity is sensibly independent of the
temperature, i.e., of the viscosity.
Thus not only does the critical velocity at which eddies come in, diminish
with the viscosity, but the resistance after the eddies are established is
nearly, if not quite, independent of the viscosity.
43. The inclinations of the logarithmic curves. — Although the general
agreement of the logarithmic homologues completely establishes the relations
between the diameters of the pipes, the pressures, and velocities, for each of
the four classes of pipes tried, viz., the lead, the varnished pipes, the glass
pipes, and the cast-iron, there are certain differences in the laws connecting
the pressures and velocity in the pipes of different material. In the
logarithmic curves this is very clearly shown as a slight but definite differ-
ence between the inclination of the logarithmic homologues for the higher
velocities.
The variety of the pipes tried reduces the possible causes of this difference
to a small compass. It cannot be due to any difference in diameters, as at least
three pipes of widely different diameters belong to each slope. It is not due
to temperature. This reduces the cause for the different values of n to the
irregularity in the pipes owing to joints and other causes, and the nature of
the surfaces.
The effect of the joints on the values of n seems to be proved by the fact
that Darcy's three lead pipes gave slightly different values for n, while my
two pipes without joints gave exactly the same value, which is slightly less
than that obtained from Darcy's experiments.
Darcy's pipes were all of them uneven between the gauge points, the
glass and the iron varying as much as 20 per cent, in section. The lead
were by far the most uniform, so that it is not impossible that the differences
in the values of n may be due to this unevenness.
But the number of joints and unevenness of the tarred pipes corresponded
very nearly with the new cast-iron, and between these there is a very decided
difference in the value of n. This must be attributed to the roughness of
the cast-iron surface.
44. Description of Diagram 3.
Diagram 3. — In this diagram the experiments of Poiseuille and Darcy
are brought into comparison with those of the present investigation.
44] THE LAW OF RESISTANCE IN PARALLEL CHANNELS. 103
In consequence of the number of lines, the general aspect of _the diagram
is somewhat confused, but such confusion vanishes so soon as it is clearly
perceived that each line of dots indicates the logarithmic homologue for
some particular pipe as determined by experiment, reduced and plotted
in exactly the same manner as for diagram 2, page 94 ; DD and EE being
exact repetitions of the logarithmic homologue for pipes 4 and 5, on a
somewhat smaller scale.
It is at once apparent from diagram 3 how, for the most part, the
experiments have been well below or well above the critical values. In the
small pipes of Poiseuille the velocities were below the critical values, and
hence lie in straight lines inclined at 45°.
The smallest pipe on which Poiseuille experimented had a diameter of
0'014 million. ; only one experiment, marked A, is shown in the diagram, as
the remaining three extended outside the range of the plate. They fall
exactly on the dotted line through A, and do not reach the critical value.
The same is true of all the rest of Poiseuille's experiments, except those
made on a much larger pipe, diameter 0*65 millim., hence it is thought
sufficient to plot only one, namely EE.
CO shows the experimental results obtained with the pipe 0'65 millim.
diameter, and these reach the critical value as given by the formula, and
then diverge from the line.
It is important to notice, however, that the points are not taken directly
from Poiseuille's experiments, which have been subjected to a correction
rendered necessary by the fact that Poiseuille did not measure the resistance
by ascertaining the pressure at two points in the pipe, but by ascertaining
the pressure in the vessels from which and into which the water flowed
through the pipe, so that his resistance includes, besides the resistance of
the pipe, the pressure necessary to impart the initial velocity to the water.
This fact, which appears to have been entirely overlooked, had a very
important influence on many of Poiseuille's results. Poiseuille endeavoured
to ascertain what was the limit to the application of his law, and, with the
exception of his smallest tubes, succeeded in attaining velocities at which
the results were no longer in accordance with his law.
When I first examined his experiments I expected to find these limiting
velocities above the critical velocities as given by my formula. In all
cases, however, they were very much below, and it was then I came to
see that Poiseuille had taken no account of the pressure necessary to start
the fluid.
It then became interesting to see how far the deviations were to be
explained in this way.
104 ON THE MOTION OF WATER, AND OF [44
In pipes of sensible size the pressure necessary to start the fluid lies
between
v2 v2
- and 1-505 1- ,
2# 2#
according to whether the mouthpiece is trumpet-shaped or cylindrical.
Poiseuille states that he was careful to keep both ends of his pipe cylindrical,
hence according to the law mouthpieces of sensible size, the pressures which
v2
he gives should be corrected by T505 -~- .
if
This correction was made, and it was then found that with all the smaller
tubes Poiseuille's law held throughout his experiments, and with the larger
pipe it held up to the critical value and then diverged in exact accordance
with my formula, as shown by the line CG.
Darcy's experiments in the case of three tubes F, G, I, fall below the
critical value, and in all these cases agree very well with the theoretical
curve as regards both branches.
This, however, must be looked upon as accidental, as at the lower
velocities Darcy had clearly reached the limit of sensitiveness of his pressure
gauges ; thus, for instance, the experiment close by the letter F is the mean
of two readings which are respectively 7 and 15; there is a tendency through-
out the entire experiments to irregularity in the lower readings, which may
be attributed to the same cause, and this seems to explain the somewhat
common deviation of the one or two lower experiments from the line given
by the middle dots.
A somewhat similar cause will explain cases of deviation in the one
or two upper experiments, for the discrepancy in the two gauges here again
becomes considerable.
For these reasons the intermediate experiments were chiefly considered
in determining the slopes of the theoretical lines.
These slopes were obtained as the mean of each class of tubes : —
Lead jointed T79
Varnished 1 -82
Glass 1-79
New cast-iron T88
Incrusted pipe 2'
Cleaned pipe T91
44] THE LAW OF RESISTANCE IN PARALLEL CHANNELS. 105
and then in the cases in which the temperature was given, /, /, L, M, N, the
points 0 having been determined by the formulae :
Log i'0 = 2 log P - 3 log D - 7-851
Log v0 = log P - log D - 2-598
the lines having the respective slopes were drawn through these points, and
in all cases agreed closely with the experiments.
In the cases where the temperature was not given, the values of log i0 and
log v0 were calculated for 5° C., these are shown along the line marked " line
of intersections at 5°," through these points lines are shown drawn at an
inclination of 2 to 1, which are the lines on which 0 would lie whatever
might be the temperature. These with the respective slope lines were drawn
so as most nearly to agree with the experiments, these intersect the lines at
2 to 1 in the points 0 which indicate the temperatures, and considering the
extremely small effect of the temperature these are all very probable
temperatures with the exception of G, H, and S, in which cases 0 is above
the line for 5° C. This indicates strongly that in these cases there must
have been a small error, 2 or 3 per cent., in determining the effective
diameter of the pipes.
It seemed very probable that roughness in the pipes, such as might arise
from incrustation or badly formed joints, would affect the logarithmic
homologues, and for this reason only the smoother classes of pipes were
treated ; but with a view to test this idea, the homologues Q and R, which
related to the same incrusted pipe before and after being cleaned were
drawn, and their agreement is such as to show that for such pipe the effect of
incrustation is confined to the effect on the diameter of the pipe, and on the
value of n which it raises to 2. This, however, was a large pipe, and the
velocities a long way above the critical velocity, so that it is quite possible
that the same incrustation in a smaller pipe would have produced a some-
what different effect.
The general result of this diagram is to show that throughout the entire
range — from pipes of 0-'"000014 to 0-m5 in diameter, and from slopes of
pressure ranging from 1 to 700,000 — there is not a difference of more than
10 per cent, in the experimental and calculated velocities, and, with very few
exceptions, the agreement is within 2 or 3 per cent., and it does not appear
that there is any systematic deviation whatever.
45.
THE TRANSMISSION OF ENERGY.
"Cantor Lectures delivered before the Society of Arts in 1883,"
I.
(Delivered April 23, 1883.)
SOME few days ago, during a conversation with a friend, I remarked that
I was going to give some lectures at the Society of Arts upon the trans-
mission of energy, whereupon my friend inquired, "Is that the transmission
of energy by electricity?" To this I replied, "No." The fact is that we
have heard so much about electricity that I began to think it was time to
recall attention to the fact that there are other means of performing
mechanical operations.
I am not sure whether, during the various lectures which have been given
in this room on electricity, the actual term, transmission of energy, has been
used. But whether it has or not, some of the leading ideas connected with
it have been before you.
I think it may be said that the great interest which the public has mani-
fested in the recent advance in the arts relating to electricity has arisen, in
a large measure, from the cry of joy with which Faure's battery was received.
A cry which said, in so many words, " Here we have at last a means of
utilising our waterfalls and natural sources of power in a way that may
relieve us of all the anxiety about our coal-fields." To those who had studied
the subject it was evident at the time that this cry was premature. And
to some of us, at all events, it seems to be a mistake to encourage false
hopes, or, rather, knowingly to base hopes on a false foundation, to hold out
as a means of replacing our coal what was, in all probability, only another
45] THE TRANSMISSION OF ENERGY. 107
means of increasing its rate of consumption, for every step in art which
facilitates the application of power must increase the demand on the acting
sources.
But this is not all ; the exaggerated claim set up for electricity, diverted,
for a time at all events, attention from the true claim, which would have
been sufficient in itself had it not thus been put out of sight. It is not our
object at present to save our coal, but to turn it to the best advantage, to get
the greatest result we can, and if Faure's battery or any subsequent advance
in this direction conduces to this, it is no small matter. Now, during the
last ten or fifteen years an entirely new aspect has been given to mechanics
by the general recognition of the physical entity which we call energy, in
different forms.
We recognise the one thing under different forms in the raised hammer,
the bent spring, the compressed air, the moving shot, the charged jar, the
hot water in the boiler, and the separate existence of coal, corn, or metals,
and oxygen. We see in the revolution of the shafts and the travel of belts
in our mills, the passage of water, steam, and air along pipes, the conveyance
of coal, corn, and metals, and the electric currents, the transmission of this
same thing — energy — from one place to another; and in all mechanical
actions we perceive but the change of form of the same thing.
Taking this general or energy point, of view we may get rid of all the
complication arising from special purpose, and recognise nothing but the
form of energy in its source, the distance it has to be transmitted, and the
special form that must be given to it for its application. And this view,
although not the best in which to study the special purpose of mechanics or
contrivances, is of great importance, inasmuch as it has revealed many
general laws, and many fundamental limits to the possibilities of extension
in certain directions.
My object in these lectures is to direct your attention to some of the
leading mechanical facts and limits revealed by this view.
There is one general remark I would wish to make, by way of caution.
I hope nothing I may say will be interpreted by any of my hearers into a
prediction as to what may happen in the future. I have to deal with facts,
and I shall try to deal with nothing but facts. Many of these facts, or the
conclusions to be immediately drawn from them, may appear to bear on the
possibilities — or, rather, the impossibilities — of art. But in the Society of
Arts I need not point out that art knows no limit ; where one way is found
to be closed, it is the function of art to find another. Science teaches us
the results that will follow from a known condition of things; but there is
always the unknown condition, the future effect of which no science can
108 THE TRANSMISSION OF ENERGY. [45
predict. You must have heard of the statement in 1837, that a steam
voyage across the Atlantic was a physical impossibility, which was said to
have been made by Dr Lardner. What Dr Lardner really stated, according
to his own showing, was that such a voyage exceeded the then present
limits of steam-power. In this he was within the mark, as anyone would
be if he were to say now that conversation between England and America
exceeded the limit of the power of the telephone. But to use such an
argument against a proposed enterprise, is to ignore the development of art
to which such an enterprise may lead.
I wish to do nothing of this kind, and if, in following my subject, I have
to point out circumstances which limit the possibilities of present art, and
even seek to define the limits thus imposed, it is in the hope of concentrating
the efforts of art into what may be possible directions, by pointing out the
whereabouts of such barriers as science shows to be impassable.
Although the terms energy and power are in continual, we might almost
say familiar, use, such use is seldom in strict accordance with their scientific
meaning. In many ways the conception of energy has been rendered
popular, but a clear idea of the relation of energy to power is difficult.
This arises from the extreme generality of the terms ; in any particular case
the distinction is easy. I was going to say that it is easiest to express this
distinction by an analogy, but, as a matter of fact, everything that seems
analogous is really an instance of energy. Power may be considered to be
directed energy ; and we may liken many forms of energy to an excited
mob, while the directed forms are likened to a disciplined army. Energy in
the form of heat is in the mob form ; while energy in the form of a bent
spring, or a raised weight, matter moving in one direction, or of electricity,
is in the army form. In the one case we can bring the whole effect to bear
in any direction, while in the other case we can only bring a certain portion
to bear, depending on its concentration. Out of energy in the mob form we
may extract a certain portion, depending on its intensity and surrounding
circumstances, and it is only this portion which is available for mechanical
operations.
Now energy in what we may call its natural sources has both these
forms. All heat is in the mob form, hence all the energy of chemical
separation, which can only be developed by combustion, is in the mob form ;
and this includes the energy stored in the medium of coal. The combustion
of 1 Ib. of coal yields from ten to twelve million foot-pounds of energy in
the mob form of heat ; under no circumstances existing at present can all
this be directed, nor have we a right, as is often done, to call this the power
of coal. What the exact possible power is we do not know, but probably
about four-fifths of this, that is to say, from eight to ten million foot-pounds
45] THE TRANSMISSION OF ENERGY. 109
of energy per pound of coal is the extreme limit it can yield under the
present conditions of temperature at the earth's surface. But before this
energy becomes power, it must be directed. This direction is at present
performed by the steam-engine, which is the best instrument art has yet
devised, but the efficiency of which is limited by the fact that before the
very intense mob energy of the fire is at all directed, it has to be allowed to
pass into the less intense mob energy of hot water or steam. The relative
intensities of these energies are something like twenty-five to nine. The very
first operation of the steam-engine is to diminish the directable portion of
the energy of the pound of coal from nine million to three millions. In
addition to this there are necessary wastes of directable energy, and a con-
siderable expenditure of already directed energy in the necessary mechanical
operations. The result is that, as the limit, in the very highest class engines
the pound of coal yields about one and a-half millions of foot-pounds; in
what are called "first-class engines," such as the compound engines on
steamboats, the pound of coal yields one million, and in the majority of
engines, about five or six hundred thousand foot-pounds. These quantities
have been largely increased during the last few years; as far as science
can predict, they are open to a further increase. In the steam-engine art is
limited to its three million foot-pounds per pound of coal ; but gas-engines
have already made a new departure, and there seems no reason why art
should stop short of a large portion of the nine millions.
Other important natural sources of mechanical powers are energy in an
already directed or army form, wind and water power. Here the power needs
no development, but merely transmission and adaptation, and hence it has
one important advantage over the energy of chemical separation. These
have both been, and are, good servants to man. But there appears to be
what are greater drawbacks — in the irregularity of these forces as regards
time, and the distribution as regards space.
The application of the power of the wind to the propulsion of ships has,
doubtless, influenced the economy of the world more than any other
mechanical feat ; and, not very long ago, water-power played no relatively
unimportant part of the work of the world. But it would seem that both
these have had their day, and are now relegated to work of a secondary
kind. Some further development of art might however bring them to a
foremost place again, by developing their use to a hitherto unprecedented
extent. Hitherto both wind and water have only had a local application —
that is to say, they were used where and when they were wanted. Wind
was only used in the sailing of ships on voyages, and for mills, distributed so
as to be within range of such corn as was too far from water ; while water-
power, though very valuable to a certain limited extent, when near habitable
country, was otherwise allowed to run to waste ; and these wastes included
110 THE TRANSMISSION OF ENERGY. [45
by far the larger sources of this power — the larger rivers and waterfalls, the
tidal estuaries, and last, but not least, the waves of the sea, a source which
has never been utilised for good. A modern idea is, that it needs nothing
but a possible development of art to render these larger sources not only
available for power in their immediate neighbourhood, but available to
supply power wherever it is wanted, and so displace the coal, or replace the
power as coal becomes exhausted. The desirability of such a result fully
explains the entertainment of the pleasant idea ; but, unfortunately, when we
come to look closer into the question, the probability of its accomplishment
diminishes rapidly. Many of the considerations of which I shall have to
speak bear directly on this question ; so that I shall now defer its further
consideration, merely pointing out that, to accomplish this result, the power
must not only be extracted from the water on the spot and at the same
time, but it must be transmitted over hundreds or thousands of miles, and
must be stored till it is wanted.
It may well be thought that energy in a directed form, or in the army
form, may be better transmitted than in the undirected or mob form. As a
matter of fact, however, energy has never been and never can be transmitted
as mechanical power in large quantities, over more than trifling distances,
say, as a limit, twenty or thirty miles. I say never can, because such trans-
mission depends on the strength of material ; and unless there is some other
material on the earth of Avhich we know nothing, we know the limit of this.
This is a part of rny subject into which I shall enter more closely in my
second and third lectures.
In deprecating the idea that wind and water will ever largely supply the
place of coal, I do not for a moment wish it to be thought that I take a
gloomy view of the mechanical future of the earth. This, I believe, admits
of immense development, and will not for long depend, as it does at present,
on the adjacency of coal-fields. This will be explained as I proceed.
It must not be forgotten that, after all, the most important source of
energy is not coal, but corn and vegetable matter. The power developed in
the labour of animals exceeds the power derived from all other sources,
including coal, in the ratio of, probably, 20 or 30 to 1 ; so that, after all, if
we could find the means of employing such power for the purposes for
which coal is specially employed — such as driving our ships, and working our
locomotives — an increase of 10 per cent, in the agricultural yield of the
earth would supply the place of all the coal burnt in engines. The energy
which may be derived from the oxidisation of corn has as yet only been
artificially developed in the form of heat, and this may be the only possible
way ; but physiology has not yet advanced to the point of explaining the
physical process of the development of energy consequent on the oxidisation
45] THE TRANSMISSION OF ENERGY. Ill
of the blood ; and it is at all events an open question whether the energy of
corn may not be really a form of directed energy, in which case^corn would
yield six or eight times as much energy as coal does at present, consumed in
our engines. As consumed in animals, it yields a larger proportion of
energy — two or three times as much, and may be more — whereas by burning
it in steam-engines, we cannot get half as much. Should we find an
artificial means of developing anything like the full directable power of
corn — a problem which has not yet been attempted — coal would no longer
be necessary for power. I do not mention this as a prediction, but as
showing that there are, besides wind and water, other, and as yet untried,
directions from which mechanical energy may be derived in the future.
Electricity is not a natural source of energy, for the simple reason
that the metals have mostly been burnt or oxidised during the past history
of the earth. But still it is important, at this stage of my lecture, to point
out that the energy consequent on the separate existence of metals and
oxygen can be developed without combustion, in a totally directed form, i.e.,
totally available for power.
There are many peculiarities which distinguish the group of elementary
substances we call metals, but there is no more distinctive feature than this.
This is not a primary source of power, but. as it at present appears, it
promises to become the most important secondary source. We cannot find
metals existing in a separate form but by the use of power ; where and when
it exists, we can separate them from the salts, and so store the energy in a
form completely available for power. The economical questions relating to
such storage of energy will be considered in their place later in the course.
It is not, however, only as effecting storage of power that electricity
demands our attention, it also affords a means of transmitting power, which
has long held an important place in art, and to which all eyes have been
recently turned in expectation of something new and startling.
Before considering the developments of art, and the circumstances on
which their further development depends, I shall turn, for a moment, to the
processes of nature. The mechanics of the universe, no less than those
relating to human art, depend on the transmission of energy. In nature
energy is transmitted in all its forms and under all circumstances, both those
which we can imitate in art, and those we can not.
The most important point with regard to the artificial transmission of
energy is the proportion of power spent in effecting the transmission, and
the circumstances on which this proportionate loss depends. Is such loss
universal ? So far as we know, it is attendant in a greater or less degree on
all artificial means of transmission, and on all transmissions effected by
112 THE TRANSMISSION OF ENERGY. [45
nature on the surface of the earth. If it were not, this earth would be no
place to live upon. No motion would ever cease. As it is, the winds and
waters are rapidly brought to rest by the friction which they encounter.
Currents of wind and currents of water form the principal means by which
energy is transmitted over the surface of the earth. But there are other
means which experience less resistance. Oscillatory waves, those of sound,
are a very efficient means of transmitting energy. Sounds are not trans-
mitted to an unlimited distance, chiefly because by the spreading of the wave
the sound becomes weaker and weaker as it proceeds. It is also destroyed
by the friction of the solid surface of the earth. Hence the sounds which
reach us from bodies high up, as the explosion of a meteor, are heard much
further than such sounds made at the surface of the earth, although there
are two records of artillery having been heard two hundred miles. Owing to
such incidental destruction of sound we cannot say from experience that
sound waves in air are destroyed, but from the physical properties of gases
we know they are.
Waves on the sea are another very efficient means of transmitting power,
a means which may be called nature's mill. The waves which take up the
energy or power from the wind in mid ocean travel onwards, carrying this
energy, and experience such slight resistance that they will, after travelling
hundreds or thousands of miles, destroy the shores on which they expend
the last of their energy. If we could find a means of utilising the energy
of waves, we should not only save our coal, but also save our country from
the waves ; still, water waves experience resistance which we can better
estimate theoretically than practically.
These are the principal ways in which energy is transmitted from one
part of the earth to another. There are others, such as earthquakes, but
they all show the same thing, that power is spent in the transmission of
energy.
If we look away into interstellar space, the case is altered. Here we see
two ways in which energy is transmitted — heat, or light, and the motion of
the heavenly bodies. In neither of these can we see any direct evidence of
resistance or loss of power ; and, as judged by any terrestrial measure, there
certainly is none. The distance at which we see stars is a sufficient proof of
the freedom with which a wave of light travels ; while the regularity of the
motion of the planetary bodies shoAvs that they encounter no sensible
resistance. Yet, although not directly perceivable, there are circumstances
that strongly suggest that in both these forms, transmission of energy is
resisted. If space is unlimited, and there are stars throughout it, why do
not we see them at greater distances than we do ? Under these circum-
stances there could be no spot in the heavens at which at a sufficiently
45] THE TRANSMISSION OF ENERGY. 113
great distance there was not a star, so that, if the light were not stopped,
the whole heavens would be one fiery envelope as bright "as "the sun.
This is a question which philosophers have not decided. But one, and the
favourite, way out of the difficulty, is to suppose that the light does en-
counter resistance, even in interstellar space. This is a subject on which
your Chairman of Council has boldly launched ; and whether his hypothesis
be right or wrong, it has brought to the front a very interesting subject.
With regard to the resistance encountered by the planetary bodies,
our evidence is even slighter. A few domesticated comets seem to
diminish their speed ; and it is not so long since we were all on the
qui vive, by the promise of the spectacle of an old friend, who seemed
to have come earlier than he was expected, on purpose to verify a pre-
diction of plunging into the sun, but instead of doing so he passed away
and was pronounced a stranger, to the joy of the nervous, but some-
what to the discomfiture of astronomers.
The energy which we derive from the sun comes to us in the form
of sunshine, in a highly directed but extremely scattered form, being
uniformly distributed all over the illuminated disc of the earth. It reaches
the outer atmosphere nearly in the same condition as it left the sun,
having traversed ninety odd millions of miles without any sensible ex-
penditure of power. In the twenty or thirty miles of the lower atmo-
sphere, however, it encounters very great, but variable, resistance. Sometimes
half of it, or three-quarters of it, may reach the earth's surface. This is
rare in our country, and on the average not more than a very small fraction
ever reaches the surface.
When the sun does shine, the sunshine is a form of energy which
may be, and is, very largely directed so as to yield power. Any such
direction which may be accomplished by human art is undertaken at an
enormous disadvantage, on account of the scattered manner in which the
energy reaches us. The sunshine must be collected before we can make
any mechanical use of it.
In the abstract, there are two methods. The one would be to accu-
mulate the energy of sunshine on a given place, over a long time. This
is nature's method. The energy on each portion of the earth's surface,
during days, weeks, the whole year, or many years, is accumulated by
the growth of vegetables. Corresponding to this, however, art has as yet
developed no means whatever. If we don't use the sunshine as it falls,
energy is lost for all mechanical purposes. I say if we don't, not that
we do use it, but because we can, and have done so in a small way.
By means of a lens, or reflectors, the sunshine which falls on a certain
o. R. ii. 8
114 THE TRANSMISSION OF ENERGY. [45
place may be concentrated on to a smaller space, and so be sufficient to
perform some mechanical operation. In this way small vapour engines
have been worked by sunshine. But the cost of the apparatus necessary
for such concentration is out of all proportion to the result accomplished,
and shows the art difficulties must be got over by a new departure.
There is the further consideration that sunshine on land is too valuable
for the maintenance of vital energy to allow of its being devoted to
mechanical purposes.
As regards the perfectness of nature's method, so far as I know, no
attempts have even been made to test this. It is probably very wasteful,
as are all nature's methods, but it is effective. In the first instance, the
energy of sunshine is stored on the spot where it falls, in the tissues,
but chiefly in the sap of the grass and vegetation. If this is not re-
moved, a large portion of the energy of the year's growth, that which
is in the sap, is stored in the seed, and the rest, although apparently
again scattered on the decay of the tissues, is to some extent preserved
in the ground, and either forwards the next year's crop, or takes the
permanent form of peat; and our coal-fields are but evidence of the way
in which the directable energy of sunshine has been stored under cir-
cumstances where there was no immediate purpose for which to apply it.
Under present circumstances, however, this energy is almost everywhere
too valuable to admit of secular storage.
It is either removed directly by nature's method, the teeth of animals,
or allowed to accumulate for a longer period, and then removed by human
industry. The further aggregation of this energy involves the transmission
of energy in a mechanical sense, and hence involves the expenditure of
power. Nature works by means of directly converting this energy into
power. The plant accumulates the energy of sunshine, the animal collects
and appropriates this energy. This collection is accomplished by the ex-
penditure of power, which means a redistribution of that portion of the
energy which is capable of direction. The scheme of nature, therefore, is
a cycle. The vegetation accumulates the energy, as far as time is con-
cerned, leaving it in a scattered form, requiring power to collect it ; this
power is in the grass, and only wants direction; this it receives in the
animal, which again expends some of the energy in the operation of
collecting. If vegetable energy be supplied to the animal in a collected
form, then a large portion of the directed energy is available for mechanical
purposes. And in this way we may form a rough estimate of the directed
energy to be obtained from sunshine in this country. The common agri-
cultural rule is one horse or bullock to two acres, such a horse pulling
120 Ibs. at a rate of 3'6 feet per second for eight hours a day. That is
a nominal horse.
45] THE TRANSMISSION OF ENEHGY. 115
We thus get something like 3,000,000,000 over and above the energy
necessary for the energy spent in eating the corn and moving itself,
which we must put down as at least equal in amount. Taking only the
available portion, we have the equivalent per acre of nearly three tons
of coal burnt in such steam-engines as exist at present. Now the
average weight of the vegetable produce from one acre, taking the form
of straw and corn, would be about two tons. So that, as far as mechanical
power is concerned, coal burnt in our present steam-engines, and corn
and straw eaten by horses, yield about the same energy, weight for
weight.
The energy which we derive from sunshine is scattered all over the
earth, and if it is to be utilised at any spot other than that at which
the sunshine falls, it must be transmitted by the expenditure of power.
The energy required for immediate operations of agriculture absorbs
a large proportion of the actual energy grown. The surplus is available
for purposes of art, and we may say that the primary object of man has
been to render this surplus as large as possible. This is accomplished,
in the first instance, by applying the residue of energy to so ameliorate
the conditions of agriculture as to increase the yield and diminish the
labour. In this way the land is levelled, enclosed, and drained ; buildings
are erected, and finally, but most important of all, roads are made. The
effect of roads in increasing the surplus energy is probably greater than
any other human accomplishment. The only means of transmitting for
purposes of collection or other purpose aggregate energy in the shape of
corn, without roads, is on the backs of animals. In this way two or three
hundred miles was the absolute limit to the distance an animal could
proceed, carrying its own food. On a good road a horse will draw a ton
of food at twenty miles a day, which would mean that it would proceed
800 miles before it had exhausted its supply, or whatever surplus energy
there might be available on one spot, half this would be available at 400 miles
distance. The labour of maintaining the roads should, of course, be de-
ducted, but this is very small.
The labour of constructing canals is very great, but the result is
equal ; a horse can move 800 tons twenty miles a day ; or a horse could
draw his own food for 80,000 miles on a canal. That is to say, with a
canal properly formed, a horse could go five times round the world without
consuming more energy than was in the boat behind it. Or corn could be
sent round the world with a consumption of one-fifth. On railways, at
low speeds, the force required is about ten times greater than on a canal,
so that the expenditure in going round the world would be about equal
to the total energy drawn. If for a moment we replace the horse by
8—2
116 THE TRANSMISSION OF ENERGY. [45
the steam-engine, and the corn by coal, we have to add the weight of the
engine to the coal, and diminish the efficiency by one-third ; we so get
that the consumption of coal for the same load of coal as of corn, would
be about double, or an engine would go about one-fourth round the world,
consuming in coal the net weight in the train, that is exclusive of carriages
and engine. Or for every thousand miles corn is carried by rail, some-
thing like 10 per cent, of the energy of the corn is expended in draft.
This is exclusive of the expenditure in wear and repairs, which will be
certainly equal, if not greater. Taking, then, the mean distance by rail
between London and the West of America, as 2,000 miles, the present
expenditure in the energy of corn in transit is somewhere about 20 per cent.
The expenditure of energy on the ocean varies, but if transported by steam
it would be probably 10 per cent, more, so that at the present time we
are actually receiving available mechanical energy, transported in the form
of corn, over 2,000 miles of land and 3,000 miles of sea, entirely by
artificially directed power, with an expenditure of less than 20 per cent. ;
a proportion which 200 years ago would have had to have been spent in
transmitting it, fifty miles over land ; a result which has been accomplished
by the employment in the meantime of the residual energy over and
above that necessary for agriculture, together with a further supply drawn
from our coal beds.
Turning now our consideration to coal, we find that per weight as used
at present, this yields rather less power than corn, but not less than two-
thirds, and it then appears that coal may be transmitted at the present time,
between any two places on the earth which are connected by rail and water,
with an expenditure of less than 50 per cent.
In instituting this comparison, the standard has been the actual available
power, as developed in our present engines and in horses, with which, weight
for weight, there is not much difference. But the adaptability of this energy,
so developed for particular purposes, renders the one medium much more
valuable than the other. Thus for agricultural purposes, weight for weight,
horse food is worth in money ten times as much as coal. This shows the
extreme difference in the value of energy according to its adaptability ; and
the extension, for which there is unlimited scope, of the adaptability of steam
power, may render it ten times as valuable as at present ; nor would this be
any small proportion compared with the total energy employed in the work
of the world. In this country there are said to be between two and three
million horses, and we may put the labouring men down at five millions, or
the total power derived from corn as over three million horses. From
the best information going, the work done by steam in this country does not
exceed the labour of two million horses, so that more than half the energy
is still derived from corn. A greater proportion of the actual corn used for
45] THE TRANSMISSION OF ENERGY. 117
horse food comes across the Atlantic ; and for many years maize was sold
in this country at an average price of £6 or £7 a ton, the cost of transit
being a very small matter. Of course the same cost, say £1 per ton, applied
to coal would be a serious matter, considering the low price of the latter.
But if, in the present state of our art, energy can be transmitted by corn
from any part of the world to this country with an insensible rise, there is
no reason to suppose but that, with the advance which science shows us,
there is every reason to expect coal may be transmitted with a corresponding
small increase in its cost, wherever the demand for it is sufficient to recom-
pense the outlay necessary for opening the roads or canals.
II.
(Delivered April 30, 1883.)
In my last lecture I dealt with the transmission of energy through the
means of coal and corn, showing that by either of these means power may
be transmitted by rail; with an expenditure of 1/1 2,000th per mile, or by
water of l/120,OOUth per mile, this either through the agency of horse or
steam.
This ease of transmission, however, depends entirely on the railroad or
water, and is only possible between places so connected. Hence such means
are only applicable to what may be called the mains of power.
We come to-day to consider other means of transmitting energy in
smaller quantities applicable to its distribution for immediate application.
Such transmission is not a matter of secondary importance, although the
distances over which it is transmitted may be comparatively insignificant.
To emphasise this, I may recall what was previously mentioned, namely,
that the relative price of corn and coal shows that the power given out by
horses is at least ten times as valuable as that of steam, for more than half
the purposes for which energy is used ; or that it answers better to burn
our coal in bringing corn from America to plough in England, than to use
the coal here for ploughing.
In fact, for most of the detailed purposes for which power is used, to
draw it from a large source (such as a steam-engine), distribute it and
adapt it to its purpose, is ten or twenty times more costly than its trans-
portation in large quantities over thousands of miles.
Now the means of artificially transmitting power may be considered as
three. The power may be stored in matter in various ways, and the matter
118 THE TRANSMISSION OF ENERGY. [45
with the energy transported — as, for instance, in our watch-springs. The
second means is the transmission of power by moving matter, without
actually storing the power in the matter — as in shafts and belts, hydraulic
connection, &c. And the third method, which is distinct from the others,
is the transmission of energy, in the form of heat or electricity, by the now
of currents through conductors; in this way all the power in the steam
passes through the boiler-plates from the furnace into the boiler. Of course,
each one of these means includes an infinite variety of detailed contrivances,
more or less dissimilar. But there is good reason for classing them under
these three heads, for all the contrivances under each of these heads are
subject to the same general limits, whether those of efficiency or distance.
There is one thing in common to all these means of transmission, and
that is that they all involve a material medium. The quantity of matter
required constitutes a primary consideration in all of them. This quantity
of matter is fixed by what we may call the properties of matter, one of the
most important of which, as regards the first two means, is the possible
strength of material. Looking round, we see the effect of the limited
strength of material in all nature's works. Of course it may be that we
shall be able to work with stronger materials than we have at present.
Organic materials, such as the feathers and tissues of animals, are stronger
than steel, weight for weight, so that there is a possibility of improvement,
but that man will go beyond nature in constructing organic fibre seems
improbable, and such possibility of improvement as exists may be discounted.
At present we may set down our strongest working material as steel, the
art of working in which is so perfect, that we may calculate on nearly the
greatest strength for all purposes. I have taken fifteen tons on the square
inch as the limit of safe working tension, in making the estimates which I
shall now bring before you. First of all, I will ask your attention to the
possibilities of transporting power in a stored form.
The question of economy in the conveyance of energy in a stored form is
simply one of the intensity with which it can be stored. If we want to
carry energy about, we must have it stored in some material form — and
this material has to be carried by ordinary means — so that the question of
economy is simply the amount of available energy that we can store in a
given amount of material.
If energy, stored in a particular manner, is more readily available for
some special purpose than that stored in another, then it may, on the whole,
be more economical to carry it in that form. This is abundantly illustrated
in our watch-springs.
The greatest amount of energy that can be stored in a given weight of
steel is very small, compared with other means. To take a familiar unit, to
45] THE TRANSMISSION OF ENERGY. 119
store the energy necessary to maintain one horse-power for one hour would
require no less than fifty tons of steel — that is to say, fifty tons of steel in
the form of watch-springs, all fresh wound-up, would not supply one horse-
power for one hour ; and yet this is the commonest form in which energy is
carried about.
It is the adaptability of the spring, and the readiness with which energy
can be put in and taken out, which recommend the steel spring.
India-rubber will store much more energy than the same weight of any
other material, say, eight or ten times as much as steel ; but of this, several
tons would be required to store the horse-power for one hour. A much
more capacious reservoir, according to its weight, is compressed air. There
are certain difficulties in getting the energy in and out without loss ; but
with air, compressed to four times the pressure of the atmosphere, we should
only require about 20 Ibs. of air to yield the amount of one horse-power for
one hour. Of course, if we were going to carry this air about, to the weight
of the air would have to be added the weight of a case to contain it, and
such a case, in the form of steel tubes, would weigh something like 230 Ibs.;
so that, in any form in which we can carry compressed air about, we shall
have about 300 Ibs. to carry for each horse-power per hour.
Another means of storing energy, very largely used, is hot water. This
is largely used in a way not always recognised. The boiler serves another
purpose besides that of converting the energy of the furnace into the power
of the steam. It stores the power, and equalises the stream between the
fire and the engine, a function the importance of which has been brought
to the front in the recent efforts to apply electricity for communication of
power, where the want of a similar reservoir between the generator and the
motor has, in many cases, proved fatal to the enterprise, a want which
secondary batteries are now being used to meet. Hot water has also been
employed as an independent reservoir, and as such it is better in some
respects than compressed air. The fundamental limits are of much the
same kind. In this case, however, the absolute limit is temperature. The
vessel in which the water is carried must be strong enough to withstand
the pressure, and all materials lose their strength as they get hot. The
considerations are here much the same as in the steam-engine, and 400° Fah.
appears to be about the limit. At this temperature, for every 4 Ibs. of water
the cases would weigh 1 lb., and there would be no advantage of large over
small cases ; except as a matter of construction, the proportionate weight
would be the same. The gross power of a pound of water, the steam being
used without condensation, is about 20,000 foot-pounds, or we should require
50 Ibs. to store 1,000,000; this is the extreme limit again. The present
accomplishment would be about 150 Ibs. per 1,000,000 foot-pounds stored—
120 THE TRANSMISSION OF ENERGY. [45
rather less than compressed air. The only other means of packing power,
that is at present looked to, is that of the much talked about secondary
battery. Here there is a great deal of doubt as to what is actually ac-
complished ; take the most reliable statements, from which it seems that
in order to get 1,000,000 foot-pounds, something like 100 Ibs. of battery
is required, which will make this means of storing energy very much the
same as compressed air or hot water.
It is important to notice that the initial cost of the energy stored by
these means differs considerably. This cost is rather difficult to estimate ;
but a practical estimate may be formed in this way : —
Taking the power, as delivered by the steam-engine, as 1 , how much of
this power will be given out after secondary storage ? Here the hot water
has an advantage, for it is heated directly by the coal, and is all on its way
to the steam-engine.
With compressed air, there are three operations, each as costly as the
steam-engine, and at least half the initial power is spent during the com-
pression, storage, and expansion ; so that the energy is at least double as
costly in coal, and six times as costly in machinery. I have put it down as
three times as costly as the energy in hot water, but this is considerably
below the mark. The electricity has also to go through three operations,
and cannot be less costly than compressed air.
Now, if we revert for one moment to the consideration of the main
transmission of power, we see at what an immense disadvantage any form of
packed energy is, compared with coal or corn ; as at present packed it
weighs at least 100 times as much.
While the limits imposed by the strength of material render it certain,
as far as compressed air and hot water are concerned, that the weight can
never be reduced by more than half, these limits are sufficient to show that
packed energy cannot be transported over long distances, even if it can be
obtained directly from such falls as Niagara. But this is no argument
against the importance of these means for short distances and special
purposes. As I have already pointed out, our watches show that circum-
stances may render the very heaviest means the best for particular purposes.
And if in any of its forms packed energy were directly available for house-
hold purposes, though it cost ten or twenty times as much as power direct
from the steam-engine, its use would still be assured.
One fact should be noticed, that in all these forms the power is packed,
and needs nothing but drawing off, whereas corn or coal do not contain the
power. The oxygen is an equally essential ingredient. In this fact lies the
45] THE TRANSMISSION OF ENERGY. 121
great advantage of corn and coal for transportation. They are_really, so to
speak, but cheques for power, which can be cashed at any spot where a bank,
in the form of a steam-engine or a horse, exists. But, of course, not being
energy, they are not generally current — in fact they are worthless, except
where the bank exists, and even there when they represent such small
amounts that the banks refuse them. Now these forms of packed power are,
so to speak, generally current; that is to say, they are available under almost
all circumstances, and in greater or less degrees of smallness; from the very
smallest, which is the watch-spring in our pockets, which supplies a con-
tinuous stream of power in less than one ten thousand millionth of a horse-
power ; or the Whitehead torpedo, which carries some million foot-pounds of
energy under the sea. Perhaps the most pressing purpose for which these
forms of packed energy are wanting is that of locomotion.
The distance which a locomotive body, be it animal or machine, can
travel, loaded or free, is limited by the ratio of the power which it carries to
its gross weight. The speed which it can attain is limited by the rate at
which it can use its energy compared with its weight. Hence there are two
particulars in which we can compare the different forms of stored energy for
locomotive purposes.
Let us take the horse and the locomotive. A full-sized horse weighs,
say, 1,500 Ibs., and, at a rate of 2£ miles an hour, will go five hours without
food, doing about 10,000,000 foot-pounds of work, including the work neces-
sary to move itself; this represents the largest result, or about 150 Ibs.
per 1,000,000 foot-pounds. If the horse is put to ten miles an hour, it will
not do more than 1'5 million foot-pounds in a single journey, besides moving
itself. Probably the greatest rate at which a horse can use its energy is
about 4,000,000 foot-pounds per hour, or 750 Ibs. per horse-power.
A locomotive with its tender, say, weighing sixty tons, exerts 500 horse-
power gross — 270 Ibs. per horse-power; so that a first-class locomotive
with tender is about one-fifth as heavy for its power as the horse ; but then
the horse cannot go more than ten miles an hour.
Now, in a general way, passenger locomotives carry coal and water for
eighty or one hundred miles, i.e. two hours; or the locomotive already
mentioned expends at one run about 2,000,000,000 foot-pounds; which
means that the gross weight of the locomotive is about 60 Ibs. or 70 Ibs.
per 1,000,000 foot-pounds of power with which the locomotive starts.
In thus taking the gross weight of the horse or locomotive, we must
remember that this includes the weight of carriage and machinery, and that
in whatever form the energy is carried, this weight must be added. In the
locomotive the weight of water and coal in the tender for two hours' journey
122 THE TRANSMISSION OF ENERGY. [45
weighs about one-quarter the gross load ; and if we add the weight of the
boiler, we may consider the carriage and machinery at one-half to one-third
the gross load. Taking the latter, and substituting for the boiler, coal, and
water, energy in either of the above forms, the coal, water, and boiler would
be about 40 Ibs. per 1,000,000 : so that, if we took compressed air instead,
we should have one-fourth the power ; or the engine would run for thirty
minutes instead of two hours, a distance of twenty-five miles instead of
a hundred. A fireless locomotive might do more than this, say, thirty-five
minutes, or thirty miles, at the same speed as the locomotive. Faure's
battery, if it could be made to work at all, would carry the locomotive forty-
eight minutes, or thirty-five to forty miles.
These figures seem to show that the locomotive has little to fear from
any of these rivals, that is, under circumstances where the smoke and steam
are no harm, and where a full-sized locomotive is required. But there are
already some cases where the locomotive is required and where the burning
of coal is impossible. Should the Channel Tunnel be made, there will be a
great field for some form of packed energy. As regards horses, however,
there is nothing to show why the horse should not be rivalled by some one of
the forms of packed energy. There have been inventors who have constructed
carriages to go by clockwork. This has now become possible, substituting
hot water, compressed air, or a battery for the spring, and such means have
already rivalled the horse on tramways. The fact that horses are at all used
for tramcars is a matter of as much surprise as that steam should be used on
underground railways. For locomotives driven by compressed air might
certainly be made cheaper and better in every way.
At the present time it would probably answer well, from a pecuniary point
of view, to supply in compressed air energy at the rate of 2d. or 3d. per
million foot-pounds, provided a sufficient quantity could be required ; so that
if simple and efficient means of applying such energy to perform the heavier
part of manual labour could be found, we might get as much power for 6d.
as a man will do in a day at 2s. But it is the means of applying it that is
wanting.
Even for horse work — except where there is a railway or tramway — the
mechanical means are wanting. We have no mechanical substitute for the
horse's foot. So that there are more than a million horses in this
country continually engaged in the operations of husbandry, where they
work in groups so as to get three or four horse-power at one operation,
an amount of power not too small for the direct application of steam power ;
and although for twenty-five years steam-engine makers have been doing
their very best to adapt the power of the steam-engine to this labour, which
exceeds any other actual application of power, the possibility of steam
45] THE TRANSMISSION OF ENERGY. 123
ploughing with economy is still a question. The use of steam on paved or on
macadam roads is much the same, so that, until steam has been applied to
such purposes, there is little hope for other forms of stored energy.
Coming back for a moment to Faure's battery, I would carefully point out
that the result which I have put down — 100 Ibs. per 1,000,000 foot-pounds
of energy — refers to what has been already accomplished, and not to any
possible limit. The principles involved in the chemical action of these
batteries, in fact in all batteries, are well understood ; and so far as these
principles are involved, it is easy to define limits ; but there are a number of
secondary actions which are not so well understood, and which have hitherto
prevented any approach to the theoretical limits. In the Faure's battery,
the theoretical limits are about 3 Ibs. per 1,000,000 foot-pounds. That is to
say, the oxidisation of 1 Ib. of lead to litharge, and the deoxidisation of 1 Ib.
of peroxide, together, yield 360,000 foot-pounds. How far, at present,
Faure's battery is within this limit, at once appears something like twenty-
four times. Should this be accomplished, power could be packed at the rate
of 1,000,000 foot-pounds for 3 Ibs., or say 6 Ibs. weight, to allow for wastes,
a result which would most certainly displace steam in the locomotive, but
which would still leave coal and corn six times the lightest vehicle of power.
It should be noticed, however, that although the means of doing so are
still entirely wanting, could other metals, such as iron or zinc, be used
instead of lead, the results would be much greater. This is shown by the
relative amount of power necessary to oxidise or deoxidise these materials,
which we see for iron and zinc are five or six times greater than for lead ;
here is an apparent opportunity for art.
Should this be realised, then, indeed, coal might be displaced as the
cheapest medium for the transmission of power, but that would be a small
matter compared with the change that would occur in our ways of applying
power. For the dream of Jules Verne, of 20,000 miles under the sea, would
become a reality, and, instead of steamboats, we should travel in submarine
monsters as yet unnamed, which we may call steam-fish.
But if science as yet imposes no limits beyond those I have mentioned, at
the same time it holds out no prospect. The chemistry of these batteries
has been very deeply considered, and those who have studied the subject
most deeply apparently see no direction in which to direct their efforts ;
so that any great advance in this art must entail a great discovery in science.
There now only remains for me to consider the transmission of power as
power, or by electricity, a most important branch of my subject, which I must
take in my next lecture.
124 THE TRANSMISSION OF ENERGY. [45
III.
(Delivered May 7, 1883.)
So far I have spoken only of the conveyance of power by means of
coal, corn, or in one or other of the several forms of packed energy.
To-night I come to consider the transmission of power by what are more
distinctly mechanical methods, and by currents along pipes and conductors.
These are the means by which power is almost always distributed, i.e., trans-
mitted from the acting agent, be it horse, water-wheel, or steam-engine,
to its operation, whatever it may be. In most cases the distance of such
transmission is so short as to be the subject of small consideration in de-
termining the means to be employed. That is to say, the means are chosen
rather by their adaptability to receive and render up the power than by
the efficiency with which they transmit it. Thus, if we take an ordinary
mill, the shaft which receives the power from the engine is generally
driven at that speed which is best adapted to receive the power from the
engine, and deliver it to the machinery in the mill, without considering
whether a much smaller shaft might be used if it were caused to run at
a much higher speed. Thus, in a mill driven by an engine of two or three
hundred horse-power, the shaft which receives the power will generally be
five or six inches in diameter, whereas it would be possible to use a shaft
of two inches diameter if the efficiency of the shaft were the only con-
sideration. Or, again, take a screw steamboat. The distance from the
engines to the screw may be 250 feet, the power 10,000 horse. This could
be transmitted by a shaft twelve inches in diameter, if allowed sufficient
speed, but the screw has to make sixty revolutions per minute, and this
determines the speed at which the shaft is made to run, and hence the
shaft is made thirty inches instead of twelve inches. This is because,
owing to the smallness of the distance, the efficiency of the means of
transmitting the power is a small consideration. There are, however,
many circumstances under which it is impossible to bring the source of
power close to its work, and then either mechanical power is not used, or
the efficiency of the means becomes a consideration.
In other cases it is a question whether it is better to distribute the
sources of power, such as steam-engines, so that they may be near their
work, or to use one large source, and distribute the power by some me-
chanical means. This rivalry exists in almost all engineering work which
covers a large area, and, generally, a compromise is come to, engines being
distributed about the works, and the power of these distributed to the
45] THE TRANSMISSION OF ENERGY. 125
machines by means of shafting. In many cases separate engines are used
for each machine, but not often separate boilers, the power being distributed
by steam-pipes.
Dockyards have long afforded a field for the competition of the various
means of distributing power. Here, generally, the distances between the
operating machines, such as cranes and capstans, is considerable, and the
work required from each machine very casual. And every means of dis-
tribution is or has been in use, from a separate engine and boiler to each
machine as at Glasgow, separate engines drawing their steam from central
boilers, to a complete system of hydraulic transmission from a central
pumping station, as at Grimsby or Birkenhead.
But the question between centralisation or distribution of steam-engines
is not by any means the only one, or most important one, which depends
on mechanical means of distributing power. Every improvement in the
means of distributing power from a central engine opens a fresh field for
its use.
The considerations relating to this subject are numerous. Hitherto, as
regards the main transmission of power, the principal consideration has
been the percentage of loss according to the distance ; but, as regards the
final distribution of power, the form in which it is distributed must be
such as admits of its being at once available for its purpose. Thus hydraulic
distribution is favoured in dockyards, because it is required for heavy forces
and slow motions, but where rapid motion is required, hydraulic distribution
gives place to some other.
Again, where the quantity of power that has to be distributed is a most
important consideration, the distribution by means of water or compressed
air will generally be the most efficient, whereas these would be by far the
most costly means for small quantities. It thus has to be remembered
that, besides the general question of efficiency, each means has particular
recommendations for particular purposes.
It is not, however, with these particular recommendations that I am
concerned. My object is to show the limits within which the use of each
means is confined, however fit it may be for its purpose. Taking first the
mechanical means, which are shafts and ropes, we find that the possible
limits to both these means are absolutely defined by the strength of material.
The amount of power any piece of material will transmit by motion against
resistance, is simply the mean product of the stress or force acting in the
direction of motion on the section multiplied by the velocity, so that, if
the stress is uniform over the section, the work is the product of the area
and intensity of stress and the velocity.
126 THE TRANSMISSION OF ENERGY. [45
In a revolving shaft, neither the stress nor the velocity is uniform
over the section, both varying uniformly from nothing in the middle to
their greatest value on the outside ; so that their mean product is exactly
half the product of the greatest values. The greatest power per square
unit of section a shaft can transmit is half the product of the greatest stress
into the velocity at the outside of the shaft.
Taking, then, the greatest safe working stress for steel at 15,000 Ibs.
on the square inch ; taking what is the greatest practical velocity at the
surface, 10 feet per second (the speed of railway journals); the work trans-
mitted is 75,000 foot-pounds per second per square inch of section — 135
horse-power ; so that we should have to have a shaft of upwards of 7 square
inches in section to transmit 1,000 horse-power, that is, a shaft of over
3 inch diameter. The friction between such a shaft and lubricated bearings
is well known, '04 ; so that, calculating the weight of the shaft 24 Ibs.
per foot, we have power spent in friction about 52,000 foot-pounds per
mile, that is one-tenth the total power the shaft will transmit. That is,
if we put 1,000 horse-power into a 3-inch shaft, making 500 revolutions
per minute, we ought, at the end of a mile, to be able to take 900 horse-
power out of it. If we had to go farther, the size of the shaft might be
diminished, so that in the next mile we should again lose a tenth, and if
we repeat this process seven times, we shall, at the end of seven miles, have
left about half the original power put in.
It will be thought, perhaps, that a 3-inch shaft is very small to transmit
so large a force ; this is because the speed of 500 revolutions per minute
is inconveniently high for purposes of employing the power ; but if it
were merely a question of transmission, it would be about the best speed.
This, then, shows the limit of the capacity of shafts as transmitters of
work.
Turning now to steel ropes, these have a great advantage over shafts,
for the stress on the section will be uniform, the velocity will be uniform,
and may be at least ten to fifteen times as great as with shafts — say
100 feet per second ; the rope is carried on friction pulleys, which may
be at distances of five or six hundred feet, so that the coefficient of friction
will not be more than '015, instead of '04. Taking all this into account,
and turning to actual results, the work transmitted per inch would be
1,500,000 foot-pounds per second ; or that a |-inch rope is all that is
necessary to transmit 1,000 horse-power in one direction, this would make
the loss per mile only l-60th. But in practice, rope has to be worked
backwards and forwards, and the tension in the backward portion of the
rope must be half the tension in the forward portion. This reduces the
performance from l-60th to l-20th, which would cause half the work to
45] THE TRANSMISSION OF ENERGY. 127
be lost in ten miles. If we use a bigger rope, and run at lower speed,
then the coefficient of friction would be reduced to '01, and the distance
extended to fifteen miles.
Experience with ropes is large, and they have been found, without
question, to have been the most efficient mechanical means of transmitting
power to long distances, but their use is subject to drawbacks. The ropes
wear somewhat rapidly, as do also the pulleys on which they run, and this
circumstance is very much against their use in any permanent work.
Nevertheless, they are used for working mines, steep inclines, and steam-
ploughs ; while at Schaff hausen they have been used for transmitting power
to considerable distances.
Turning to the transmission of power along pipes, we find the conditions
somewhat modified. The formula for the amount of power transmitted by
water is the same, namely, the product of the pressure and area of section into
the velocity. But the resistance obeys different laws. In the case of shafts
and ropes, we have seen that the distance is subject to an absolute limit.
In the case of fluid in pipes this is not so. No matter how long a
pipe may be, if there is no leakage, water would flow along the pipe until
the level of its surface were the same at both ends. But the rate of flow
would diminish with the length and diameter of the pipe. Thus we can
transmit power through a perfectly tight pipe, however small, and however
long; but when we come to consider the gross power that can be trans-
mitted through a given pipe, with a given percentage of loss, the question
is different. Given the size and strength of the pipe, the gross amount
of power, and the percentage of loss, and the limits are fixed. Thus, taking
a 12-inch pipe capable of standing 1,400 Ibs. on the square inch, the loss
in transmitting 1,000 horse-power would be about 5 per cent, per mile,
at first increasing — as the pressure fell to 700 Ibs. — to 10 per cent. We
should thus have lost half the power in about seven miles. We cannot
say that seven miles is the absolute limit, for with a 24 inch pipe, which
would cost four times as much per mile, we could transmit the same power
thirty times as far with the same loss. The cost of laving a 12-inch pipe
for seven miles, however, would probably be as much as even 1,000 horse-
power would stand ; while a 24-inch pipe for 200 miles would be out of
all proportion. Then there is the consideration of leakage, which, although
very small for short lengths, is larger for greater lengths.
Seven miles is at present an outside economical limit of hydraulic trans-
mission, even for such a large amount of power : but with air the case is
different. This flows so much easier than water, that the cost of trans-
mitting the same power through the same distance, with the same loss,
would be about 12 per cent., or, at the same cost per mile, the air may
128 THE TRANSMISSION OF ENERGY. [45
be transmitted 100 times as far with the same loss. The total cost,
however, would thus be 100 times as great, which would exceed the eco-
nomical limit ; but not only theory but practice has shown that power
may be economically transmitted five times as far by air as by water —
something like thirty miles. But on comparing these two means, one
circumstance must not be lost sight of, and that is, that getting the power
into the pipe in the form of compressed air, will cost twice as much as
getting it in in the form of water. This is a great advantage for water
where the distance is short, but where the distance is long, the greater
efficiency of air more than compensates for this initial loss.
Like water, air can only be transmitted economically where the quantity
is large, the friction being proportionately greater in small pipes than in
large, varying as the four-fifths power of the diameter.
This is a great drawback, both as regards hydraulic and compressed
air transmission. It does not affect ropes and shafts in the same way, but
even in these cases considerations of durability prevent these means being
used efficiently for the transmission of small quantities of power to con-
siderable distances, so that, with the possibility already mentioned, there
remains an opening for any means that will enable power to be transmitted
efficiently in small quantities, and such a means we have in the flow of
electricity along wires or conductors. In considering electricity, we may
well start with the questions, (1) Will electricity enable us to transmit power
in large quantities more efficiently than the foregoing means ? (2) Will it
enable us to transmit small quantities ? These questions may be more
definitely answered than they could a few weeks ago. Thanks to the ex-
periments of M. Deprez, who appears to have been the only one, out of all
those who are advocating the use of electricity, who has had the courage
to try and see what can be done, we can now say with certainty that a
current of electricity, equivalent to 5 horse-power, may be sent along a
telegraph wire l-6th of an inch in diameter, some ten miles long (there
and back) with an expenditure of 29 per cent, of the power, because this
has already been done. In order to do this, it would seem that M. Deprez
has perfected his apparatus so as to have nearly reached the possible limit.
Compared with wire rope, this means falls short in actual efficiency, as
M. Hirn sends 500 horse-power along a f-inch rope. To carry this amount,
as in the experiment of Deprez, one hundred telegraph wires would be
required ; these wound into a rope would make it more than T4 inches
in diameter, four times the weight of M. Hirn's rope. With the moving
rope the loss per mile is only 1*4 per cent., while with the electricity it
was nearly 6 ; so that, as regards weight of conductor and efficiency, the
electric transmission is far inferior to the flying rope. Nor is this all. With
45] THE TRANSMISSION OF ENERGY. 129
the flying belt, M. Hirn found the loss at the ends, in getting_the power
into and out of the rope, 2£ per cent. ; whereas, in M. Deprez's experiment,
30 per cent, was lost in the electric machinery alone, which is very small
as such machinery goes. But this is not all. No account is here taken
of the loss of power in the transmission to and from the electric machinery,
a matter which is, I believe, very much under-estimated.
The machines made revolutions at 1,000 and 700, much too high for
direct connection with either a steam-engine or any mechanical operator :
the power, then, had at each end to be transmitted through gearing, or
a system of belts. And supposing this alteration of speed to have been
five or six at each end, experience tells us that a loss of at least 15 per cent,
must ensue. This loss was indeed apparent, for the dynamometer was
connected with the machine with a belt, which showed a loss from this
one belt alone of 20 per cent. Taking the whole result, it does not
appear that more than 15 or 20 per cent, of the work done by the
steam-engine could have been applied to any mechanical operation at the
other end of the line, as against 90 per cent, which might have been
realised with wire rope transmission. To set off against this, electricity
has the enormous advantage in the conductor being fixed, and in the fact
that it is likely to be, if anything, less costly and more efficient for small
quantities of power than for large. These advantages will certainly insure
a very large use for electricity in the distribution of power, particularly for
high speed machinery.
There is yet another means of communicating and distributing energy
now coming rapidly into vogue. This is by the transmission of coal-gas
along pipes. The distances, often many miles, through which the gas is often
transmitted before reaching the engine, are such that, with any other means
of distributing power, would considerably enhance the cost of the power.
But in the case of gas, it does not appear that these distances are at all
a matter of consideration. This may be at once explained. It takes about
ten cubic feet of gas to develop 1,000,000 foot-pounds in a gas-engine,
whereas of air compressed in the ordinary way it would require something
like 140 cubic feet to yield the same power. Hence the comparative cost
of transmission is the cost of transmitting ten cubic feet of gas against
that of 140 cubic feet of compressed air, and these would be about as one
to twenty-five ; so, as a means of distributing energy, gas is twenty-five
times more efficient than compressed air.
I have now placed before you, as far as circumstances will allow, the
various means by which energy, in a form available for power, may be trans-
mitted over long distances, together with the circumstances which limit such
transmission. By means of the railway and steamboat, corn and coal can
o. K. ii. 9
130 THE TRANSMISSION OF ENERGY. [45
be, nay is, transmitted half-way round the earth with an expenditure of
power less than half the power represented by the coal carried, but this
can only be done where the quantity to be transmitted is very large.
At present this efficiency is unrivalled, no means of packed energy or
of current energy approaching even 1 per cent. And further, there is
apparent room for a large diminution in the present expenditure, small as
it is, in the improvement of the steam-engine as a means of directing the
energy of coal. For the distribution of power, this means ceases to be
efficient, nor can it be employed to transmit energy which has already
taken the form of power. For these purposes other means have to be
employed. These various means, although they differ greatly in efficiency,
all fall so far below the efficiency of coal and corn, that a hundred miles
appears to be the outside limit any economical transmission of power in
quantity for mechanical purposes, could be at present effected ; and hence
any power, be it derived from wind or water, must be used within this
radius of its source ; and, except in places far out of the reach of rail or
water, this limit may be divided by ten.
So far as efficiency of transmission in considerable quantities, neither
secondary batteries nor electrical transmission are more efficient than com-
pressed air or belts, but when it comes to transmitting small quantities,
then electric transmission has a decided advantage. The cost of the electric
conductor diminishes with the quantity to be transmitted, and by making
the conductor sufficiently large, its efficiency may be increased to any
extent.
At the present time, electric conductors are continuous half-way round
the world, and whenever a message is sent from England to Australia direct
energy is transmitted 10,000 miles, but in what quantity ? The energy of
the current, as it arrives, is not much more than sufficient to keep a watch
going, at any rate not more than I'lOOO millionths of horse-power. The
value of such energy, estimated at £17 per minute, would be equivalent to
a billion pounds per horse-power per hour, whereas the highest price paid
for animal labour in Australia or England is not more than 6d. per horse-
power per hour. This shows the difference between the transmission of
electricity for telegraphic purposes and its transmission for mechanical
purposes. Energy differs in value greatly, but for operations that can be
performed by men or horses, the price of energy must be regulated by the
highest price of corn.
The prosperity of any spot in the past depended on the fertility of
the adjacent soil. But the use of coal has altered this, and now the present
prosperity of this country is owing to the adjacency of our coal-fields, these
45] THE TRANSMISSION OF ENERGY. 131
having rendered it possible to bring our food across the earth.- The im-
proved means of transmitting coal and corn, it would seem, have, or may
again, change this, and if, instead of looking on the life of this country as
limited by the life of our coal-fields, we look boldly forward, and foster every
means, political, social, and mechanical, which may render this a favourite
spot to live upon, we need not fear that the necessity of bringing our coal
from a distance will make a difference which will counterbalance the ad-
vantage we shall derive from the mechanical facilities we shall have here.
9—2
46.
[Read before Section A at the "British Association," 1883.]
ON THE EQUATIONS OF MOTION AND THE BOUNDARY
CONDITIONS FOR VISCOUS FLUIDS.
TAKING the ordinary equations of motion for viscous fluid, and supposing
a tube indefinitely broad in the direction z, bounded by solid surfaces
y=±<> .................................... (i).
which tube may be supposed continued in a circle so as to make a circular
trough. Suppose it full of water, at rest, and subject to an acceleration X ,
the equation of motion gives
du d?u ,r
di=*w+x ................................ (2)>
or by altering the arrangement, instead of X we may have --- f-
Now initially u = 0, .-. $* — = 0 •
dy dy*
du
or
right up to the surface.
But by the boundary condition at the solid surface u = 0 always ;
which shows that the boundary conditions are at variance with the equation
of motion.
46] ON THE EQUATIONS OF MOTION, ETC. 133
The equations simplify to
du d*u 1 dp „
Si=l*dy>--pdx + X ........................... <3>'
with the boundary condition that u is always zero at the boundaries.
For initial conditions we will take p constant, and u uniformly zero.
The equation (2) then becomes
* .................................... w X
If now we suppose X to have a uniform value the equation of motion
gives -j- = X throughout the fluid, i.e. at the boundaries. This is contrary
to the boundary conditions, for if u is always zero at the boundaries -^- must
(TC
also be zero.
The functions wanting were rendered evident in the following manner.
By differentiating equation (3) with respect to t, remembering that X
, du du
and p are constants, and that TTT = -?r ,
ot at
d du _ d* /du\
dt~dt~t*dy*\di)"
an equation of which the integrals are well known,
ft = Z(A^»+^) .............................. (6),
and which may be determined to suit the initial and boundary conditions.
du
But it does not follow that the value of -j- in equation (6) is the same
CLt
as in (4) because the integral of (5) includes an arbitrary function of y,
du d-u
If we determine f(y) to suit equation (4) then equation (7) will not fit
the initial boundary conditions.
If however we determine f(y) so that equation (4) shall be satisfied at
some small distance r from the boundary, and the boundary condition
satisfied we have
_
X(l-e '} ........................... (8),
where p is a numerical large quantity.
134 ON THE EQUATIONS OF MOTION [46
Such a function satisfies all the boundary conditions, but the general
value of the function would be
where 2X0 = X ................................. (9).
The addition of such a function to the equation of motion would meet the
initial conditions of the case in question, in which X is independent of t,
but this is all, for the boundary conditions include that we must have all the
differential coefficients of u with respect to t zero at the boundary, to meet
which case it would be necessary to have a function
Po (y±c) Piy^c
r t) + &c ............. (10).
Instead of adding such functions, however, it seems better to consider in
what way the equation of motion can be modified so that these functions
result from integration. This would be the case if instead of equation (3)
we had the equation
du Vr0 . d* (du\ dp v d*u
-+*+'l .............. (1
where %A0 = 1.
That is, if we add the term 2 — A~, - to the equation, it becomes com-
p0 dif at
patible with the boundary conditions, and the term itself is of the same
order as others which have been neglected in constructing the equations of
motion, and the strong presumption is that such terms have been neglected.
The case pursued here is the simplest possible, but by a similar method
it may be shown that the general case for a fluid at constant density will be
met if the equations of motion be modified as follows :
du r2 _0 du _ dp
dt p* dt dx
dv r2 dv dp v _., i /19\
777 iv "35 = ~j + Y + p\-v >• (>*;.
dt p* dt dy
dw r2 dw dp „ _
dt p* dt ~ dz ^
The equations of motion were not originally the outcome of any com-
plete hypothesis of the molecular constitution of fluids. They involved
certain assumptions which would enter into such an hypothesis, but by no
means completely define it, any more than would the phenomena of
approximately steady motion suffice to define the complete phenomena of
motion.
46] AND THE BOUNDARY CONDITIONS FOR VISCOUS FLUIDS. 135
The original basis of the equations of motion for viscous fluids were
certain experimental phenomena, and it is important to notice that all these
phenomena belong to what may be called approximately steady motions.
So that neither the experimental verification of these equations, nor the
molecular hypothesis on which they were originally based, was in any sense
complete or general. And if the original framers of these equations had
attempted to carry them to the second order of small quantities, it would
only have been done by further molecular assumptions — and anything like a
complete experimental verification was entirely wanting.
This aspect of the case was changed by the foundation of a complete
molecular hypothesis of gases, for founding as it did the dynamical
theory of gases on complete fundamental assumptions, the equations of
motion followed as a consequence of these assumptions — and although not
attempted, could have been obtained to any degree of small quantities.
Maxwell contented himself with showing that the equations of motion
resulting from his assumptions agreed with the equations of motion obtained
by Stokes to the first order of small quantities*, but it was perfectly possible to
have pursued his reasoning to the second order of small quantities. Having
then found that certain terms of the second order were wanting in the
equations of motion to meet the boundary conditions as shown by experi-
i IK nt, the most probable method of defining these terms seemed to be to
• •any the dynamical theory of gases to the second order of small quantities.
For this investigation I adopted the same method as that which I have
explained in my paper on the dimensional properties of matter in the
gaseous state f, merely extending the method to meet the case of varying
motion. The result was that I found terms of the form required, but they
entered into the equations with the opposite sign to those required to meet
the boundary conditions, and would thus only introduce arbitrary constants
of a periodic character. Besides which, these terms clearly vanished at the
boundaries, i.e. if the boundary were regarded as a plane of total reflection ;
while according to the theory, as regards the first order of small quantities,
the boundary produced no tangential effects whatever.
Having considerable confidence in the method I was using in deducing
the equations of motion from the fundamental assumption; it naturally
occurred to me to re-examine the fundamental assumptions, to see if these
had been introduced into the theory in their fulness. It was then I
observed that the theory, both as applied by Maxwell, and myself, neglected
any possible dimensions of a molecule, and it became clear that by neglecting
* Phil. Trniin., 1867, p. 81.
t See Paper 33, Vol. I., pp. 257 ff.
136 ON THE EQUATIONS OF MOTION [46
this we had neglected that which made it possible for the boundary to
produce an acceleration on the fluid.
By neglecting the dimensions of a molecule, the cause of transference of
momentum across a surface reduces itself to the carriage of momentum by
the moving molecules, whereas if we take the size of molecules into account,
a certain portion of the area of any ideal surface drawn through the gas
must be occupied by the solid matter of the molecules, and the stresses in
these molecules will be the cause of the transference of momentum across
the surface. This cause of the transference of momentum across a plane
had been ignored with the dimensions of the molecule in the theory of
gases*.
It became necessary therefore to take this into account to see what effect
it had on the equations of motion.
It is clear that this effect would involve the elasticity of the molecules
themselves, as the rate at which momentum would traverse them would be
that of the propagation of sound in a solid, but considering the relative
elasticities of solids and gases, it seemed legitimate to take the elasticity of
the molecules as infinite compared with that of the gas, i.e. to assume the
molecules as absolutely rigid — and the same for groups of molecules in
contact, either directly or through other molecules with a solid surface.
Now if we imagine a surface plane for the instant to be moving with the
mean velocity of the matter which it traverses, and suppose that in molecules
are cut by an unit of this plane and if the m molecules, cut by a plane
parallel to the first and at a distance r, (the diameter of a molecule) — are
in contact with those on the first, then if we have a third plane also at a
?71
distance r from the second, — of the molecule cut by this will be directly in
772-
contact with the second, and — indirectly in contact with those on the first,
so that of the m molecules on two planes at a distance y from each other
-V JW
mq r = me r ,
where p = loge q,
will be indirectly in contact with each other.
Now since according to the assumptions, we regard this connection as
* See Paper 33, Vol. I., pp. 257 ff., "On Certain Dimensional Properties of Matter in the
Gaseous State."
46] AND THE BOUNDARY CONDITIONS FOR VISCOUS FLUIDS. 137
rigid, we see that if p is the density of the matter on any plane^this matter
is rigidly connected with matter
= pe r,
at a distance y on either side of the plane, which therefore is an expression
for the rigidity of a gas. And it may be noticed that, although the distance
to which this rigidity extends is limited by the value of - at a surface such
as y = 0, it is absolute, so that at a solid surface all the matter (not
molecules) in contact with the surface, has the mean motion of this surface
7j
whatever may be the value of -.
_py
The expression pe r has been obtained on the hypothesis of the distri-
bution of molecules in a gas, and even so, without any very great degree of
refining.
It is impossible without a more definite hypothesis than has been pro-
pounded at present as to the constitution of a liquid, to say what form the
expression for rigidity might there take, but it is reasonable to suppose that
as regards the law of molecular contact, it would be the same as that of a
gas, only p instead of being large would be very small, but as regards the
rigidity, the same assumption could not be made any more than a whole
fishing-rod can be considered rigid in the same degree as a single joint of
such a rod.
47.
ON THE GENERAL THEORY OF THERMO-DYNAMICS.
A LECTURE DELIVERED TO THE INSTITUTION OF CIVIL
ENGINEERS. 15 NOVEMBER, 1883.
From " The Proceedings of the Institution of Civil Engineers, 1883."
IN lecturing on any subject, it seems to be a natural course to begin with
a clear explanation of the nature, purpose, and scope of the subject. But
in answer to the question — What is thermo-dynamics ? I feel tempted to
reply — It is a very difficult subject, nearly, if not quite, unfit for a lecture.
The reasoning involved is such as can only be expressed in mathematical
language. But this alone should not preclude the discussion of the leading
features in popular language. The physical theories of astronomy, light,
and sound involve even more complex reasoning, and yet these have been
rendered popular, to the very great improvement of the theories. Had it
appeared to me that it was the necessity for mathematical expression which
alone stood in the way of a general comprehension of this subject, I should
have felt compelled to decline to deliver this lecture, honourable as I acknow-
ledge the task to be.
What I conceive to be the real difficulty in the apprehension of the leading
features of thermo-dynamics is, that it deals with a thing or entity (if I may
so call heat) which, although we can recognise and measure its effects, is
yet of such a nature that we cannot with any of our senses perceive its mode
of operation.
Imagine, for a moment, that clocks had been the work of Nature, and
that the mechanism had been on such a small scale as to be imperceptible
even with the highest microscope. The task of Galileo would then have
been reversed ; instead of inventing machinery to perform a certain object,
his task would have been from the observed motion of the hands to have
47] ON THE GENERAL THEORY OF THERMO-DYNAMICS. 139
discovered the mechanical principles and actions of which these motions
were the result. Such an effort of reason would be strictly parallel to
that which was required for the discovery of the mechanical principles
and actions of which the phenomena of heat were the result.
In the imaginary case of the clock, the discovery might have been made
in either of two ways. The scientific method would have been to have
observed that the motion of the hands of the clock depended on uniform
intermittent motion ; this would have led to the principle of the uniformity
of the period of vibrating bodies, and on this principle the whole theory of
dynamics might have been founded. Such a theory would have been as
obscure, but not more obscure, than the theory of thermo-dynamics. But
there was another method in the case of timekeepers, the one by which the
theory of dynamics was actually brought to light — namely, the invention of
an artificial clock, the action of which could be seen, and, so to speak, under-
stood. It was from the pendulum that the constancy of the periods of
vibrating bodies was discovered, and from this followed the dynamical
theories of astronomy, light, and sound. There is no great difficulty in the
apprehension of these theories, because they do not call for the creation of a
mental picture, but merely for the exaggeration or diminution of what we
can actually see in the clock.
As regards the mechanical theory of heat, however, no visible mechanical
contrivance was discovered or recognised which afforded an example of this
action ; apparently, therefore, the only possible method was the scientific
method — namely, the discovery of the laws of its action from the observation
of the phenomena of heat, and accepting these laws, without forming any
mental image of the dynamical origin, was the only method open. This is
what the present theory of thermo-dynamics purports to be.
But although the theory of thermo-dynamics may be said to have been
discovered in the form in which it is now put forward, this is not quite true.
For one of the discoverers of the second law, and the one who had priority
over the others, worked by the aid of a definite mechanical hypothesis as to
the actual molecular motions and forces on which the phenomena of heat
depend, and many of the most important steps in the theory are solely to be
attributed to his labours. But to return to the theory. This may be defined
as including all the reasoning based on two perfectly general experimental
laws, without any hypothesis as to the mechanical origin of heat. In this
form thermo-dynamics is a purely mathematical subject and unfit for a
lecture. But as no one who has studied the subject doubts for a moment
the mechanical origin of these laws, I shall be following the spirit, if not the
letter of my subject, if I introduce a conception of the mechanical actions
from which these laws spring. And this I shall do, although I should hardly
140 ON THE GENERAL THEORY OF THERMO-DYNAMICS. [47
have ventured, had it not been that, while considering this lecture, I hit on
certain mechanical contrivances which afford sensible examples of the action
of heat, in the same way as the pendulum is an example of the same
principles as those involved in the phenomena of sound and light. These
examples, thanks to the ready aid of Mr Forster in constructing the
apparatus, I am in a position to show you, and I am not without hope that
these kinetic engines may in a great measure remove the source of obscurity
on which I have dwelt.
The general action of heat to cause matter to expand, or to tend to
expand, is sufficiently obvious and popular. That the expanding matter will
do work is also sufficiently obvious, but the exact part which the heat plays
in doing this work is very obscure.
It is now known that heat performs two, and it may well be said three,
distinct parts in doing the work. These are —
(1) To suppty the energy equivalent to the work done.
(2) To give the matter the elasticity which enables it to expand, i.e., to
convert the inert matter into an acting machine.
(3) To convey itself (i.e., heat) in and out of the matter.
This third function is generally taken for granted in the theory of thermo-
dynamics.
In order to make any use of therrno-dynamics, a knowledge of the
experimental phenomena of heat is necessary ; but as time will not permit
of my entering largely into these, I have had some of the leading facts
suspended as diagrams. One or two it will be well to mention.
Heat as a quantity is independent of temperature, the thermal unit
taken being the amount of heat necessary to raise 1 Ib. of matter
1° Fahrenheit.
Temperature represents the intensity of heat in matter. Matter in
most of its forms expands more or less uniformly as we add heat to it ;
hence the expansion of matter measures temperature. Gases such as
air expand in absolute proportion to the heat added under a constant
pressure.
Absolute temperature is an idea derived from the observed rate of
contraction of gases ; they would vanish to nothing with the temperature
461° below zero Fahrenheit. For the other phenomena I must refer to the
diagrams as I proceed.
Our knowledge of these facts has been accumulating during the last two
hundred years, and it was in a very complete condition forty years ago,
47] ON THE GENERAL THEORY OF THERMO-DYNAMICS. 141
before ther mo-dynamics was born. The birth of this science may be con-
sidered as the result of the recognition of work — motion against resistance
as a true measure of mechanical action, and of accumulated work or energy
as the potency of all sources of power. These ideas have now become
extremely popular, and all are able to recognise in the raised weight, the
bent spring, the moving hammer, the same thing, energy, which is
measured by the amount of work which can be derived from any of these
sources.
Before the recognition of this means of measuring mechanical potency,
any definite idea of the true mechanical action of heat was impossible, for
we had not recognised the only mechanical action by which it can be
measured.
In 1843 Joule conclusively proved that, by the expenditure of 772 ft.-lbs.
a thermal unit of heat must be produced, provided all the work was spent in
producing heat. The simplicity of the ideas here involved, and the com-
pleteness of Joule's proof, acted at once to render the first law popular. No
language can be too strong in which to express the importance of this
discovery ; yet, as was long ago pointed out by Rankine, the very popularity
of Joule's law went a long way to obscure the fact that it did not constitute
the sole foundation of the theory of thermo-dynamics. Before Joule's dis-
covery it was recognised that heat acted a part in causing work to be
performed. It was clearly seen that it was heat which caused the water to
expand into steam, against the resistance of the engine, and the necessity of
heat to cause matter to expand was recognised.
To make matter do work it was only necessary to heat it. It would
expand, raising a weight ; and since after doing its work the matter was still
hot, it was supposed that the only necessity for the heat was to add increased
elasticity to matter. It was seen that the heat that had once been used was
so degraded in temperature that it could not be all used again. So that,
although there was no idea that heat was actually consumed in doing the
work, it was seen that for continuous work a continuous supply of heat at a
high temperature was necessary. As regards the exact proportion of heat
required for the supply of elasticity, to perform a certain quantity of work,
fairly clear ideas prevailed. It was seen that this depended on various
circumstances. These were formulated by Carnot, who in 1828 gave a
formula, which is equivalent to our second law of thermo-dynamics, of which
it was the parent.
Now this idea that heat merely caused work to be done was not absunl,
as is sometimes supposed. Indeed we may say that the present popular idea
that the whole heat is convertible into work is more erroneous than the old
idea in the ratio of 10 to 1 ; because the old idea that the function of heat
142 ON THE GENERAL THEORY OF THERMO-DYNAMICS. [47
is to supply elasticity was right, as far as it went. Although the present
idea that the function of heat is to supply energy from which the work is
drawn is also right, yet in any known possible heat-engine ten times more
heat is necessary for the purpose of giving elasticity to matter than is
converted into work by elasticity. This error, which seems to be very
general amongst those who have not made a special study of the subject, may,
1 think, be attributed — first, to the popularity of the first law of thermo-
dynamics, and secondly to the fact that although the second law of thermo-
dynamics is nothing more nor less than a statement of the proportion which
the quantity of heat necessary to produce elasticity bears to the quantity
which this elasticity will convert into work, yet that it is the invariable
custom in stating this law to omit all attempt to explain the purpose which
this excess of heat serves ; the reason for this omission being that experiment
only shows that this heat is necessary, and hence this is all that we have a
right to say.
If such an error prevails it is only a popular error, for it certainly did
not affect the progress of the science. No sooner did Joule's law become
known than it was taken up by Rankine, who, in 1849, published a complete
theory of thermo-dynamics, based, as I have said, on a hypothetical
constitution of matter. This was almost simultaneously followed by
theories based on an improved form of Carnot's reasoning by Thomson and
Clausius.
Rankine's theory was based on a hypothetical constitution of matter. He
invented a system of molecular motions and constraints, which he called
molecular vortices, and he then calculated the effects of these motions by
the theory of mechanics. The fact that his reasoning was based on a
hypothesis was considered by many as a fault in his reasoning. But on the
other hand the clear idea thus obtained, as to the reason of everything he
was doing, gave him such an advantage over those who were working by
experimental laws, of the meaning of which they would venture no opinion,
that he was led to make discovery after discovery in advance of his
competitors, while some of his discoveries are still beyond the reach of
experiment.
There was, however, a difficulty Rankine had to face ; some properties of
matter were pointed out which his hypothetical matter did not possess.
This was not much to be wondered at, for although Rankine had invented
machinery which would account for the mechanical action of heat, there was
no reason to suppose this to be the only machinery. Rankine, with a view
to the difficult calculations he had to make, had chosen machinery as simple
as possible. Instead, however, of trying to complicate it, he, yielding to
the opinion of his cotemporaries, adopted the general conclusions to which
47] ON THE GENERAL THEORY OF THERMO-DYNAMICS. 143
it had led him as axiomatic laws, and so cut himself adrift from his
hypothesis.
It comes to be, then, that the student of thermo-dynamics finds as a
reason why we must pass a large amount of heat through his engine,
besides that which is converted into work, he is to accept an axiomatic law
as to the greatest possible amount that can be converted under the
circumstances.
To tell a child who asks why he cannot have more food, that he can only
have 6 oz. a day, would be considered cruel. So to tell a student who wants
to know why, out of the ten million foot-lbs. in 1 Ib. of coal, a steam-engine
T — T
can only give one million as work, that he is only allowed -=^— ~j~ , is cruel,
J 1 ~T~
yet this is all he can have from the theory of thermo-dynamics based on
its experimental laws.
Rankine, when compelled to abandon his hypothesis as the foundation
of his theory by the objections justly urged against it, pointed out the
great disadvantage of a mechanical theory conveying no conception of the
mechanical basis of its laws ; and called on all those who taught the
subject, to try and find some popular means of illustrating the second law.
This call was made twenty years ago ; but, I believe, up to the present
time no such illustration has been forthcoming. When undertaking this
lecture I had no idea of such an illustration, and I did not intend to say
much as to the reason of the second law. But, as I have said, three weeks
ago an idea occurred to me. It arose in this way : Heat acts in matter to
transform heat into work by molecular mechanism. Having much studied
the subject, I have in my mind a picture, right or wrong, of the mechanism,
and the part which heat acts. The question occurred — Is there no way of
making a machine such that, although the parts are in visible motion, and
the energy transformed to work is visible energy, yet the energy supplied
shall have the characteristics of heat-energy, and the machine shall act simply
in virtue of the elasticity caused by the motion of its parts ?
The question had no sooner arisen than several ways of carrying out the
idea presented themselves.
The general idea of the mechanical condition which we call heat is, that
the particles of matter are in active motion ; but it is the motion of the
individuals in a mob, with no common direction or aim. Rankine assumed
the motion to be rotatory, but it now appears more probable that the motion
in the particles is oscillatory, undulatory, rotatory, and all kinds of motion,
whatsoever; so that the communication of heat to matter means the com-
munication of internal agitation — mob agitation. If, then, we are to make a
machine to act the part of hot matter, we must make a machine to perform
144 ON THE GENERAL THEORY OF THERMO-DYNAMICS. [47
its work in virtue of the communication of internal promiscuous motion
amongst its parts. The action of heat-mechanism to do work is simply that
of expansion of volume, or the increased effort to expand owing to increased
agitation. I first tried to think of some working arrangements of small
bodies which should forcibly expand when shaken ; but it appeared that it
would be much easier to effect a contraction. This was as good. As long as
any definite alteration in shape could be produced against resistances by a
definite amount of agitation in its parts, we should have a machine illustrat-
ing the action of the heat-engine.
Suppose we want to raise a bucket from a well. Our best way is to pull
or wind up the rope, but that is because the energy we employ is in a
completely directable form. Suppose we had no such directable energy, but
could only shake the rope, it having been first made fast at the top (Fig. 1,
next page). Then, it being a heavy rope, a chain is better ; suppose we
shake the chain laterally, waves will run down the chain, and, if we go on
shaking, the chain will assume a continuously changing sinuous form (Figs.
2 and 3); and, as the chain does not stretch, the bucket must be raised to
allow for the sinuosities. The chain will have changed its mechanical
character, and from being a tight line or tie in a vertical direction, will
possess kinetic elasticity, that is, elasticity in virtue of its motion, causing it
to contract its vertical length.
The bucket will be raised, although not to the top of the well, and work
will have been done in raising it, but the work spent in shaking the chain
will be not only the equivalent of the work spent in raising the bucket, but
also of all the kinetic agitation in the chain necessary to raise the bucket.
Having raised the bucket as far as possible with a certain power of agitation,
if the supply of agitation be cut off, then that already in the chain will
sustain the bucket until it is destroyed by friction, when the bucket will
gradually descend.
But if we want to do more work, to raise another bucket, we may take
that which is raised off at the level at which it is raised ; then, to get the
chain down again, we must allow it to cool, i.e., allow the agitation to die
out ; then, attaching another bucket, to raise this, we shall again have to
supply the same heat, perform the same work, i.e., the work to raise the
bucket, and the agitation-energy of the chain. Thus we see that the energy
necessary to the working of the machine serves two purposes, it supplies
the energy necessary to raise the bucket, and the energy necessary to
convert the chain from an inextensible tie into an elastic contracting system,
capable of raising the weight, neither of which portions of energy is again
serviceable after the bucket has been raised. The one portion is already
converted into work, and the other, although still in existence in the chain
47]
ON THE GENERAL THEORY OF THERMO-DYNAMICS.
145
as energy, can only sustain the position of the chain. Before it could be
used to do more work it must be got out of the chain and back again, which
Fig. i.
Fig. 2.
Fig. 3.
is just the thing you cannot do ; we can get some of it out and some of it
back, but not all.
It must not be supposed that this method of raising a bucket by shaking
the rope is recommended as the best means. No one would dream of using
it if we could get a direct pull, but that is nothing to the point. We are
considering the action of heat, and we have limited ourselves to using energy
of the same kind that heat supplies ; that is, energy in the form of promis-
cuous agitation, absolutely without direction, so that the question is, how
can we raise the bucket by shaking ?
I feel that there is a childish simplicity about this illustration, that may
at first raise the feeling of "Abana and Pharpar, rivers of Damascus," in the
minds of some of rny hearers, but, should this be the case, I have every
confidence that calm reflection will have the same effect as on Naaman.
The case of the shaken rope, as I have put it, is no mere illustration of
the action of heat, but an instance of the same application of the same
principles. The sensible energy in the shaking rope only differs from the
energy of heat, i.e., a bar of metal is the scale of the motion ; we see that in
the chain but not in the bar, not because the molecules of the bar are
moving slower, but because the scale of motion is infinitely smaller. The
temperature of the bar from absolute zero measures the mean square of the
velocity of all its parts, multiplied by some constant depending on the mass
of the parts which are moving together ; so the mean square of the velocity
of the chain multiplied by the weight per foot of the chain really represents
the absolute temperature of the sensible energy in the chain.
The apparatus which I have on the table is an obvious adaptation of the
rope and the bucket. There are three different illustrations apparently very
different in form, but all working by the same principle.
O. R. II. 10
146
ON THE GENERAL THEORY OF THERMO-DYNAMICS.
[47
Here is the chain (Figs. 1, 2, 3), by the shaking of which (addition of
promiscuous energy) a weight of 2 Ibs. is raised 3 feet, or 6 foot-lbs. of work
done ; here is another sort of chain, a series of parallel horizontal bars of
wood, connected and suspended by two strings (Figs. 4, 5, and 6). By giving
a circular oscillation to the upper bar, the whole apparatus is set into a
Fig. 4.
Fig. 5.
Fig. 6.
twisting motion (agitation); the strings are continually bent, and the
vertical length of the whole system is shortened, and a weight of 10 Ibs. or
the bucket of the pump is caused to rise, raising water just as if we boiled
water under the piston of a steam-engine. To get the bucket down again
for another stroke, we must quiet or cool the chain, just as we must condense
the steam, and the energy taken out of the chain in cooling corresponds
exactly with the heat that must be taken out of the steam in order to
condense it.
The waves of the sea constitute a source of energy in the form of sensible
agitation ; but this energy cannot be used to work continuously one of these
kinetic-machines, for exactly the same reason as the heat in the bodies at
the mean temperature of the earth's surface cannot be used to work heat-
engines. A chain attached to a ship's mast in a rough sea would become
elastic with agitation, but this elasticity could not be used to raise cargo
out of the hold, because it would be a constant quantity as long as the
roughness of the sea lasted.
In practical mechanics we have no source of energy consisting of sensible
agitation, besides the waves of the sea ; so that there has been no demand
for these kinetic engines to transform sensible mob-energy into work ; had
there been, I might have patented my idea, though probably it would have
long ago been discovered. But there has been a demand for what we may
call sensible kinetic elasticity, to perform for sensible motion the part which
47] ON THE GENERAL THEORY OF THERMO-DYNAMICS. 147
the heat elasticity performs in the thermometer, and for this purpose the
principle of the kinetic machine was long ago applied by Watt. The
common governor of a steam-engine acts by kinetic elasticity, which elasticity,
depending on the speed at which the governor is driven, enables the
governor to contract as the speed increases. The motion of the governor is
not of the form of promiscuous agitation, but, though systematic, all the
motion is at right angles to the direction of operation, so that the principle
of its action is the same.
The kinetic elasticity of the governor performs the same part as the
heat elasticity in the matter of the thermometer ; the first measures by
contraction the velocity of the engine, and the other measures by expansion
the velocity of the molecules of the matter by which it is surrounded, so
that we now see that while measuring the speed of sensible revolution, we
are performing on a different scale the same operation as measuring the
temperature of bodies which depends on the molecular velocities, and that
quite unconsciously we have constructed instruments to perform the two
similar operations which act by means of the same mechanical action, namely,
kinetic elasticity.
These kinetic examples of the action of heat must not be expected to
simplify the theory, except in so far as they give the mind something definite
to grasp ; what they do is to substitute something we can see for what we
can barely conceive.
The theory of thermo-dynamics can be deduced from any one of these
kinetic examples by the application of the principles of mechanics ; such
application involves complex dynamical reasoning, such as can only be
executed by the aid of mathematics, and would be altogether unfit to intro-
duce into a lecture. I shall therefore pass on to some considerations resulting
from the theory of thermo-dynamics.
The discovery of the two laws has enabled us to perfect and complete
our experimental knowledge of the phenomena of heat. But probably the
greatest practical use is that these two laws enable us to calculate with
certainty, from the experimental properties of any matter, the extreme
potency of any source of power.
Thus we find by experiment that a pound of coal burnt in a furnace
yields fourteen to sixteen thousand thermal units of heat. The first law,
Joule's law, tells us at once that this is equivalent to from 11,000,000 to
13,000,000 foot-lbs. of energy. But this is not, as seems to be generally
supposed, the power of coal. The second law of thermo-dynamics tells us
that in order that this energy might be realised, it must be capable of being
developed at an infinite temperature, whereas we know that this cannot be
10—2
148 ON THE GENERAL THEORY OF THERMO-DYNAMICS. [47
the case ; and there is a growing idea that the temperature at which coal
will burn is not so extremely high, about 3,000° Fahrenheit. Taking this
temperature, and assuming the temperature of the atmosphere to be 60°, we
have for the proportion of the heat of coal, that we could with a perfect
engine call power, j£, about 80 per cent., or from 9,000,000 to 11,000,000
O^rO J.
foot-lbs.
Again, we know the heat properties of all known liquids and gases, so
that we can, by the second law, tell the greatest possible proportion of the
heat received, which can be converted into power by any of these agents.
In the steam-engine, for instance, we see that the present limits of art
restrict the temperatures absolutely to 400°, and practically the limits are
much less; while the lowest temperature that can be worked to in a
condenser is 100°. Then, as the limit to the possibility, we have one-third
as the greatest proportion, or three out of the nine million foot-lbs.
The greatest actual achievement by Mr Perkins has been about two
millions, while the best engines in use only give us a little over one million,
or about one-ninth of the possible realizable portion between 3,000° and the
mean temperature of the earth's surface.
I cannot here enter upon these, but the reasons why higher temperatures
cannot be used in the steam-engine are obvious enough.
The same reasons do not apply to hot air as an agent. This may be
worked at much greater temperatures ; and about thirty years ago, as soon
as it appeared from the science of thermo-dynainics that the limit of
efficiency depended on the range of temperature, attention was much directed
to air as a substitute for steam. The attempts then made failed through
what were then called practical, or art difficulties.
Just at the present time the possibility of other heat-engines than
steam-engines has again come to the front ; and as this is so, it seems
desirable to call attention to a circumstance connected with heat-engines
which has as yet occupied quite a subordinate place in the theory of heat-
engines. This is the law as to the rate at which heat can be made to do
work by an agent, such as steam or air. The greatest possible efficiency of
the agent, i.e., the proportion which the work done bears to the mechanical
equivalent of the heat spent, is a matter of fundamental importance ; but
the rapidity with which the heat can be so transformed with a given amount
of apparatus, as an engine of a given weight, is a matter of at least as great
importance.
Which would be the best engine for a steamboat ; one that would develop
47] ON THE GENERAL THEORY OF THERMO-DYNAMICS. 149
20 H.P. for every ton gross weight, consuming 2 Ibs. of coal j>er H.P. per
hour, or one that only gave 2 H.P. per ton weight, and only consumed 1 Ib.
of coal ? Unquestionably the former ; yet hitherto the question of heat
economy has been considered theoretically, to the exclusion of time economy.
Yet the latter forms a legitimate part of the subject of thermo-dynamics, and
has played a greater part in the selection of stearn as the fittest agent than
the consideration of the heat-economy.
In the theory of thermo-dynamics it is assumed that the working agent,
be it water or any other, can be heated up and cooled down at pleasure,
without any consideration as to the time taken for these operations, which
are considered to be mere mechanical details.
Yet in the science of heat a great amount of labour has been spent ; a
great amount of knowledge gained as to the rate at which heat will traverse
matter. And more than this ; it is well known that heat cannot be made to
enter and leave matter without a certain loss of power, i.e., a certain lowering
of the working range of temperature. It is by heat that heat is carried
into the substance ; and hence, as I have indicated, there is a third law of
thermo-dynamics relative to this transmission. Heat only flows down the
gradient of temperature, and in any particular substance the rate at which
heat flows is proportional to the gradient of temperature. Hence to get the
heat from the source or furnace into the working substance a certain time
must be consumed, and this time diminishes as the difference of temperature
of the furnace and the working substance increases.
The examples of the kinetic engines which I have shown you well
illustrate this. If we shake the end of a chain, the wriggle passes along the
chain at a given speed. It appears that an interval must elapse between the
first shaking of the chain and the establishment of sufficient agitation to
move the bucket ; a further interval before the bucket is completely raised ;
and further still, another interval must elapse before the chain can be cooled
again for another stroke ; so that this kinetic engine will only work at a given
rate. I can increase this rate by shaking harder, but then I expend more
energy in proportion to the work done.
This exactly corresponds with what goes on in the steam-engine, only,
owing to the agent water being heated, expanded, and cooled severally in the
boiler, cylinder and condenser, the connection is somewhat confused.
But it is clear that for every H.P. something like 15 million foot-pounds
of power have to pass from the furnace into the boiler. As out of this 15
we cannot use more than 2 million, the remaining 13 are available for
forcing the heat from the products of combustion into the water, and out of
the steam into the condensing water, and they are usefully employed for
this purpose.
150 ON THE GENERAL THEORY OF THERMO-DYNAMICS. [47
The boilers are made small enough to produce sufficient steam, and this
size is determined by the difference of the internal temperature of the gases
in the furnace and the water in the boiler, and whatever diminishes this
difference would necessarily increase the size of the heating surface, i.e., the
weight of the engine. The power which this difference of temperature
represents cannot be realised in the steam-engine, so that it is most usefully
employed in diminishing the necessary size of the boiler. Still it is an
important fact to recognise that our present steam-engines require the
expenditure of more than five times as much of the power of the heat (not
of the heat) in getting the heat into the working substance as in performing
the actual operation. This loss of power does not so much occur in the
resistance of the metal which separates the furnace from the water as in the
resistance of the gases. Gas is a very bad conductor; and though a thin
layer adjacent to the plates is always considerably cooled, little further cooling
goes on until, by the internal currents, this layer is removed, and a fresh hot
layer substituted in its place.
Similar resistance would occur inside the boiler between the water and
the hot plate, nay does occur, until the water begins to boil, but then the
evaporation of the water takes place at the hot surface, and every particle of
water boiled absorbs a great deal of heat, which leaves the surface in the form
of bubbles, allowing fresh water to come up.
If we had air inside the boiler instead of water, we should require from
five to ten times the surface to carry off the same heat, which is a sufficient
reason why what are called hot-air engines cannot answer, even did not the
same argument hold with enormously greater force in the condenser.
Steam is as bad a conductor of heat as air as long as it does not condense,
but, in condensing, steam will conduct heat to a cold surface at an almost
infinite rate, for as the steam comes up to the surface it is virtually anni-
hilated, leaving room for fresh steam to follow, which it will do if necessary
with the velocity of sound. If, however, there is the least incondensable
air in the steam this will be left as a layer against the fresh steam.
Some years ago I made some experiments on this subject, which showed
that 5 or 10 per cent, of air in the steam would virtually prevent
condensation.
If a flask be boiled till all the air is out, and nothing but pure steam is
left, and if the flask be then closed and a few drops of cold water
introduced, the pressure instantly falls to zero, though it immediately
recovers from the boiling of the water in the flask. If now a little air be
admitted, and allowed to mix with the steam, the few drops of water produce
scarcely any effect.
The facility with which steam carries heat to a cold surface is both an
47] ON THE GENERAL THEORY OF THERMO-DYNAMICS. 151
enormous advantage and some drawback ; as compared with air it is an
enormous advantage in enabling the steam to be cooled in the condenser.
But during the working of the steam in the cylinder, when the steam is
wanted to keep its heat, the facility with which it condenses is a great draw-
back, and necessitates the keeping of the cylinder hotter than the steam by
a steam-jacket. For this part of its work the non-conductivity of incon-
densable air is a great advantage.
In dwelling thus on the conducting powers of air and steam, my purpose
has been to prepare the way for a few remarks I wish to make on another
form of heat-engine — the engine in which the heat is generated in the working
substance itself.
The combustion-engine, in the form of the cannon, is the oldest form of
heat-engine. Here the chemically separate elements in the form of gun-
powder are the working substances put into the cylinder ; they take in with
them the potential energy of chemical separation, which by means of a spark
take the kinetic form of heat. Here there is no conduction, the kinetic
elasticity propels the shot, and all the heat over and above that used
in imparting energy to the shot is lost. The advantages of this form of
engine are two. There is no time necessary for conduction, and as the gas
generated is not condensable, there is little loss of heat by conduction to the
cold metal.
These two advantages are very great, but I should not have mentioned
them in reference to guns were it not that there appears to be the dawning
of an idea of taming this form of engine so as to substitute it for the steam-
engine. To do this it is necessary to introduce coal or coal-gas ; — and oxygen
in the form of air in place of gunpowder. The thermo-dynamic theory
applied to such engines shows that they should possess great advantages over
the steam-engine in point of economy. And the considerations I have
brought forward as to the loss of the power of heat in the transference of
heat from the furnace to the boiler seem to promise such engines an
enormous advantage in rate of work, while the substitution of a non-con-
densable gas for steam in the cylinder seems to get over the art-difficulty of
making cylinders to work under high temperatures. We cannot expect any
piston to work in a cylinder of over 800° or 400° temperature, but with
non-condensing gases the cylinder may be kept cool with little cooling effect
on the gases contained in it, even if the temperature of these is 3,000°.
This will be the case if the gas in the cylinder is not in a violent state of
internal agitation, but it should be remembered that all internal currents
much facilitate the conveyance of heat to the walls.
There is one drawback shown by the theory of these engines. The
simple expansion of the gases resulting from combustion is not sufficient to
152 ON THE GENERAL THEORY OF THERMO-DYNAMICS. [47
cool them to anything like the temperature of 60°, and to get the greatest
economy some of the remaining heat should be used to heat the fresh
charge. To do this, however, would necessitate the extraction of the heat
from one mass of gas to communicate it to another, which would introduce
all the difficulties of the boiler, increased by having gas instead of water.
But even wasting this heat, the theory still shows a large margin of
economy for such engines over the present performance of steam-engines,
a margin which is said to have been already realised in the gas-engine, which
is a form of combustion-engine in a high state of efficiency. Now, by means
of Dowson gas, Messrs Crossley seem to have obtained 2,000,000 out of the
10,000,000 ft.-lbs. in 1 Ib. of coal. Further accomplishment in this direction
is a question of art ; but while on all other hands science shows impassable
barriers not far in advance of the present achievements of art, in this
direction thermo-dynamics extended to include the rate of operation shows
no known barriers ; while the fact that, as gas-engines, this system of com-
bustion heat-engines has already established a footing assures them continual
improvement.
In conclusion I would say, by way of caution, that the theory of thermo-
dynamics does not lead to the conclusion, which seems to be generally held
by those who have only realised the first law of the science, that the steam-
engine is a semi-barbarous machine, wasting more than it uses, very well
for those who know no science, but only waiting until those better educated
have time to turn their attention to practical matters, and then to give place
to something much better. Thermo-dynamics shows us not the faults but
the perfections of the steam-engine, in which there is no waste of power,
since all is used either in doing work or in promoting the rate at which the
work can be done. Next to the watch, the steam-engine is the highest
development of mechanical art, and the science of thermo-dynamics may be
said to be the result of the study of the steam-engine.
48.
[From the "Proceedings of the Royal Institution of Great Britain," 1884.]
(Head March 28, 1884.)
IT has long been a matter of very general regret with those who are
interested in natural philosophy, that in spite of the most strenuous efforts
of the ablest mathematicians, the theory of fluid motion fits very ill with the
actual behaviour of fluids ; and this for unexplained reasons. The theory
itself appears to be very tolerably complete, and affords the means of
calculating the results to be expected in almost every case of fluid motion,
but while in many cases the theoretical results agree with those actually
obtained, in other cases they are altogether different.
If we take a small body such as a raindrop moving through the air, the
theory gives us the true law of resistance ; but if we take a large body such
as a ship moving through the water, the theoretical law of resistance is
altogether out. And what is the most unsatisfactory part of the matter is
that the theory affords no clue to the reason why it should apply to the one
class more than the other.
When, seven years ago, I had the honour of lecturing in this room on the
then novel subject of vortex motion, I ventured to insist that the reason why
such ill success had attended our theoretical efforts was because, owing to
the uniform clearness or opacity of water and air, we can see nothing of the
internal motion ; and while exhibiting the phenomena of vortex rings in
water, rendered strikingly apparent by partially colouring the water, but
otherwise as strikingly invisible, I ventured to predict that the more general
application of this method, which I may call the method of colour-bands,
154 ON THE TWO MANNERS OF MOTION OF WATER. [48
would reveal clues to those mysteries of fluid motion which had baffled
philosophy.
To-night I venture to claim what is at all events a partial verification
of that prediction. The fact that we can see as far into fluids as into solids
naturally raises the question why the same success should not have been
obtained in the case of the theory of fluids as in that of solids ? The answer
is plain enough. As a rule, there is no internal motion in solid bodies ; and
hence our theory based on the assumption of relative internal rest applies to
all cases. It is not, however, impossible that an, at all events seemingly,
solid body should have internal motion, and a simple experiment will show
that if a class of such bodies existed they would apparently have disobeyed
the laws of motion.
These two wooden cubes are apparently just alike, each has a string tied
to it. Now, if a ball is suspended by a string you all know that it hangs
vertically below the point of suspension or swings like a pendulum. You see
this one does so. The other you see behaves quite differently, turning
up sideways. The effect is very striking so long as you do not know the
cause. There is a heavy revolving wheel inside which makes it behave like
a top.
Now what I wish you to see is, that had such bodies been a work of
nature so that we could not see what was going on — if, for instance, apples
were of this nature while pears were what they are — the laws of motion
would not have been discovered ; if discovered for pears they would not
have applied to apples, and so would hardly have been thought satis-
factory.
Such is the case with fluids : here are two vessels of water which
appear exactly similar — even more so than the solids, because you can see
right through them — and there is nothing unreasonable in supposing that
the same laws of motion would apply to both vessels. The application of
the method of colour-bands, however, reveals a secret : the water of the one
is at rest, while that in the other is in a high state of agitation.
I am speaking of the two manners of motion of water — not because there
are only two motions possible ; looked at by their general appearance the
motions of water are infinite in number; but what it is my object to make
clear to-night is that all the various phenomena of moving water may be
divided into two broadly distinct classes, not according to what with uniform
fluids are their apparent motions, but according to the internal motions
of the fluids, which are invisible with clear fluids, but which become visible
with colour-bands.
The phenomena to be shown will, I hope, have some interest in them-
48] ON THE TWO MANNERS OF MOTION OF WATER. 155
selves, but their intrinsic interest is as nothing compared to their philosophical
interest. On this, however, I can but slightly touch.
I have already pointed out that the problems of fluid-motion may be
divided into two classes: those in which the theoretical results agree with
the experimental, and those in which they are altogether different. Now
what makes the recognition of the two manners of internal motion of fluids
so important, is that all those problems to which the theory fits belong to the
one class of internal motions.
The point before us to-night is simple enough, and may be well expressed
by analogy. Most of us have more or less familiarity with the motion of
troops, and we can well understand that there exists a science of military
tactics which treats of the best manoeuvres and evolutions to meet particular
circumstances.
Suppose this science proceeds on the assumption that the discipline of
the troops is perfect, and hence takes no account of such moral effects as may
be produced by the presence of an enemy.
Such a theory would stand in the same relation to the movements of
troops, as that of hydrodynamics does to the movements of water. For
although only the disciplined motion is recognised in military tactics, troops
have another manner of motion when anything disturbs their order. And
this is precisely how it is with water: it will move in a perfectly direct
disciplined manner under some circumstances, while under others it becomes
a mass of eddies and cross streams, which may be well likened to the motion
of a whirling, struggling mob where each individual particle is obstructing
the others.
Nor does the analogy end here : the circumstances which determine
whether the motion of troops shall be a march or a scramble, are closely
analogous to those which determine whether the motion of water shall be
direct or sinuous.
In both cases there is a certain influence necessary for order : with troops
it is discipline ; with water it is viscosity or treacliness.
The better the discipline of the troops, or the more treacly the fluid, the
less likely is steady motion to be disturbed under any circumstances. On the
other hand, speed arid size are in both cases influences conducive to un-
steadiness. The larger the army, and the more rapid the evolutions, the
greater the chance of disorder ; so with fluid, the larger the channel, and the
greater the velocity, the more chance of eddies.
With troops some evolutions are much more difficult to effect with
steadiness than others, and some evolutions which would be perfectly safe
156 ON THE TWO MANNERS OF MOTION OF WATER. [48
on parade, would be sheer madness in the presence of an enemy. So it is
with water.
One of my chief objects in introducing this analogy of the troops is to
emphasise the fact, that even while executing manoeuvres in a steady manner,
there may be a fundamental difference in the condition of the fluid. This is
easily realised in the case of troops. Difficult and easy manoeuvres may be
executed in equally steady manners if all goes well, but the conditions of the
moving troops are essentially different. For while in the one case any slight
disarrangement would be easily rectified, in the other it would inevitably lead
to a scramble. The source of such a change in the manner of motion under
such circumstances, may be ascribed either to the delicacy of the manoeuvre,
or to the upsetting disturbance, but as a matter of fact, both of these
causes are necessary. In the case of extreme delicacy an indefinitely
small disturbance, such as is always to be counted on, will effect the
change.
Under these circumstances we may well describe the condition of the
troops in the simple manoeuvre as stable, while that in the delicate man-
oeuvre is unstable, i.e. will break down on the smallest disarrangement.
The small disarrangement is the immediate source of the break-down in the
same sense as the sound of a voice is sometimes the cause of an avalanche ;
but if we regard such disarrangement as certain to occur, then the source of
the disturbance is a condition of instability.
All this is exactly true for the motion of water. Supposing no disarrange-
ment, the water would move in the manner indicated in theory just as, if
there is no disturbance, an egg will stand on its end ; but as there is always
slight disturbance, it is only when the condition of steady motion is more or
less stable that it can exist. In addition then to the theories either of military
tactics or of hydrodynamics, it is necessary to know under what circum-
stances the manoeuvres of which they treat are stable or unstable. And it
is in definitely separating these conditions that the method of colour-bands
has done good service which will remove the discredit in which the theory of
hydrodynamics has been held.
In the first place, it has shown that the property of viscosity or treacliness,
possessed more or less by all fluids, is the general influence conclusive to
steadiness, while, on the other hand, space and velocity are the counter
influence ; and the effect of these influences is subject to one perfectly
definite law, which is that a particular evolution becomes unstable for a
definite value of the viscosity divided by the product of the velocity and
space. This law explains a vast number of phenomena which have hitherto
appeared paradoxical. One general conclusion is, that with sufficiently slow
motion all manners of motion are stable.
48] ON THE TWO MANNERS OF MOTION OF WATER. 157
The effect of viscosity is well shown by introducing a band of coloured
water across a beaker filled with clear water at rest. Now the water is quite
still, I turn the beaker round about its axis. The glass turns but not the
water, except that which is close to the glass. The coloured water which is
close to the glass is drawn out into what looks like a long smear, but it is
not a smear, it is simply a colour-band extending from the point in which the
colour touched the glass in a spiral manner inwards, showing that the
viscosity was slowly communicating the motion of the glass to the water
within. To prove this I have only to turn the beaker back, and the colour-
band assumes its radial position. Throughout this evolution the motion has
been quite steady — quite according to the theory.
When water flows steadily it flows in streams. Water flowing along a pipe
is such a stream bounded by the solid surface of the pipe, but if the water
be flowing steadily we can imagine the water to be divided by ideal tubes
into a fagot of indefinitely small streams, aoy of which may be coloured
without altering its motion, just as one column of infantry may be distin-
guished from another by colour.
If there is internal motion, it is clear that we cannot consider the whole
stream bounded by the pipe as a fagot of elementary streams, as the water is
continually crossing the pipe from one side to the other, any more than we
can distinguish the streaks of colour in a human stream in the corridor of
a theatre.
Solid walls are not necessary to form a stream : the jet from a fire hose,
the falls of Niagara, are streams bounded by a free surface.
A river is a stream half bounded by a solid surface.
Streams may be parallel, as in a pipe ; converging, as in a conical mouth-
piece ; or when the motion is reversed, diverging. Moreover, the streams
may be straight or curved.
All these circumstances have their influence on stability in a manner
which is indicated in the accompanying table : —
Circumstances conducive to
Direct or Steady Motion.
Viscosity or fluid friction which
continually destroys disturb-
ances.
(Treacle is steadier than water.)
2. A free surface.
3. Converging solid boundaries.
4. Curvature with the velocity
greatest on the outside.
Sinuous or Unsteady Motion.
5. Particular variation of velocity
across the stream, as when a
stream flows through still
water.
6. Solid bounding walls.
7. Diverging solid boundaries.
8. Curvature with the velocity
greatest on the inside.
158 ON THE TWO MANNERS OF MOTION OF WATER. [48
It has for a long time been noticed that a stream of fluid through fluid
otherwise at rest is in an unstable condition. It is this instability which
gives rise to the talking-flame and sensitive-jet with which you have been
long familiar in this room. I have here a glass vessel of clear water in
front of the lantern, so that any colour-bands will be projected on the
screen.
You see the ends of two vertical tubes one above the other. Nothing
is flowing through these tubes, and the water in the vessel is at rest. I now
open two taps, so as to allow a steady stream of coloured water to enter at
the lower pipe, water flowing out at the upper. The water enters quite
steadily, forms a sort of vortex ring at the end which proceeds across the
vessel, and passes out at the lower tube. Now the coloured stream extends
straight across the vessel, and fills both pipes. You see no motion ; it looks
like a glass rod. The water is, however, flowing slowly along it. The
motion is so slow, that the viscosity is paramount, and hence the stream
is steady.
I increase the speed ; you see a certain wriggling sinuous action in the
column ; faster, the column breaks up into beautiful and well-defined eddies,
and spreads out into the surrounding water, which, becoming opaque with
colour, gradually draws a veil over the experiment.
The same is true of all streams bounded by standing water. If the
motion is sufficiently slow, according to the size of the stream and the
viscosity of the fluid, it is steady and stable. At a certain critical velocity,
which is determined by the ratio of the viscosity to the diameter of the
stream, the stream becomes unstable. Under any conditions, then, which
involve a stream flowing through surrounding water, the motion will be
unstable if the velocity is sufficient.
Now, one of the most marked facts relating to experimental hydro-
dynamics is the difference in the way in which water flows along contract-
ing and expanding channels ; these include an enormously large class of the
motions of water, but a typical phenomenon is shown by the simple conical
tubes. Such a tube is now projected on the screen ; it is surrounded with
clear still water. The mouth of the tube at which the water enters is the
largest part, and it contracts uniformly for some way down the channel, then
the tube expands again gradually until it is nearly as large as at the mouth,
and then again contracts to the tube necessary to discharge the water.
I draw water through the tube, but you see nothing as to what is going on.
I now colour one of the elementary streams outside the mouth ; this colour-
band is drawn in with the surrounding water, and will show us what is going
on. It enters quite steadily, preserving its clear streak-like character until it
has reached the neck where convergence ceases ; now the moment it enters
48] ON THE TWO MANNERS OF MOTION OF WATER. 159
the expanding tube it is altogether broken up into eddies. Thus^the motion
is direct in the contracting tube, sinuous in the expanding.
The hydrodynamical theory affords no clue to the cause why ; and even
by the method of colour-bands the reason for the sinuosity is not at once
obvious. If we start the current suddenly, the motion is at first the same in
both tubes, its change in the expanding pipe seemed to imply that here the
motion was unstable. If so, this ought to appear from the equations ot
motion. With this view this case was studied, I am ashamed to say how
long, without any light. I then had recourse to the colour-bands again, to
try and see how the phenomena came on. It all then became clear : there is
an intermediate stage. When the tap is opened, the immediately ensuing
motion is nearly the same in both parts ; but while that in the contracting
portion maintains its character, that in the expanding portion changes its
character. A vortex ring is formed which, moving forward, leaves the motion
behind that of a parallel stream through the surrounding water.
If the motion be sufficiently slow, as it is now, this stream is stable, as
already explained. We thus have steady or direct motion in both the con-
tracting and expanding parts of the tube, but the two motions are not
similar : the first being one of a fagot of similar elementary contracting
streams, the latter being that of one parallel stream through the surround-
ing fluid. The first of these is a stable form ; the second an unstable form,
and, on increasing the velocity, the first remains, while the second breaks
down ; and we have, as before, the expanding part filled with eddies.
This experiment is typical of a large class of motions. Wherever fluid
flows through a narrow, as it approaches the neck it is steady, after passing,
it is sinuous. The same effect is produced by an obstacle in the middle of a
stream ; and very nearly the same thing by the motion of a solid object
through the water.
You see projected on the screen an object not unlike a ship. Here
the ship is fixed, and the water flowing past it; but the effect would be
the same if we had the ship moving through the water. In the front of
the ship the stream is steady, and so till it has passed the middle, then you
see the eddies formed behind the ship. It is these eddies which account for
the discrepancy between the actual and theoretical resistance of ships. We
see, then, that the motion in the expanding channel is sinuous because the
only steady motion is that of a stream through water. Numerous cases
in which the motion is sinuous may be explained in the same way, but
not all.
If we have a perfectly parallel channel, neither contracting nor expand-
ing, the steady moving stream will be a fagot of perfectly steady parallel
160 ON THE TWO MANNERS OF MOTION OF WATER. [48
elementary streams all in motion, but moving fastest at the centre. Here we
have no stream through steady water. Now when this investigation began it
was not known, or imperfectly known, whether such a stream was stable or not,
but there was a well-known anomaly in the resistance to motion in parallel
channels. In rivers, and all pipes of sensible size, experience had shown
that the resistance increased as the square of the velocity, whereas in very
small pipes, such as represent the smaller veins in animals, Poiseuille had
proved the resistance increased as the velocity.
Now since the resistance would be as the square of the velocity with
sinuous motion, and as the velocity, if direct, it seemed that the discrepancy
could be accounted for if the motion could be shown to become unstable for a
sufficiently large velocity. This suggested the experiment I am now about
to produce before you.
You see on the screen a pipe with its end open. It is surrounded by
clear water, and by opening a tap I can draw water through it. This makes
no difference to the appearance, until I colour one of the elementary streams,
when you see a beautiful streak of colour extend all along the pipe. The
stream has so far been running steadily, and appears quite stable. I now
merely increase the speed ; it is still steady, but the colour-band is drawn
down fine. I increase the colour and then again increase the speed. Now
you see the colour-band at first vibrates and then mixes so as to fill the tube.
This is at a definite velocity ; if the velocity be diminished ever so little the
band becomes straight and clear ; increase it again, it breaks up. This
critical speed depends on the size of the tube in the exact inverse ratio ; the
smaller the tube, the greater the velocity ; also, the more viscous the water
the greater the velocity.
We have then not only a complete explanation of the difference in the
laws of resistance generally experienced, and that found by Poiseuille, but
also we have complete evidence of the instability of parallel streams flowing
between or over solid surfaces. The cause of the instability is as yet not
explained, but this much can be shown, that whereas lateral stiffness in the
walls is unimportant, inextensibility or tangential rigidity is essential to the
creation of eddies. I cannot show you this, because the only way in which
we can produce the necessary conditions without a solid channel, is by a wind
blowing over water. When the wind blows over water, it imparts motion
to the surface of the water just as a moving solid surface ; moving in this
way, however, the water is not susceptible of eddies. It is unstable, but the
result of disturbance is waves. This is proved by an experiment long known,
but which has recently attracted considerable notice. If oil be put on the
surface it spreads out into an indefinitely thin sheet which possesses only
one of the characteristics of a solid surface, it offers resistance, very slight,
48] ON THE TWO MANNERS OF MOTION OF WATER. 161
but still resistance to extension and contraction. This, however, is sufficient
to entirely alter the character of the motion. It renders the water~unstable
internally, and instead of waves, what the wind does is to produce eddies
beneath the surface. This has been proved, although I cannot show you the
experiments.
To those who have observed the phenomena of oil preventing waves, there
is probably nothing more striking throughout the region of mechanics. A
film of oil so thin that we have no means of illustrating its thickness, and
which cannot be perceived except by its effect — which possesses no
mechanical properties that can be made apparent to our senses — is yet able
to entirely prevent an action which involves forces the strongest we can con-
ceive, which upset our ships and destroy our coasts. This, however, becomes
intelligible when we perceive that the action of the oil is not to calm the sea
by sheer force, but merely, as by its moral force, to alter the manner of motion
produced by the action of the wind, from that of the terrible waves upon the
surface, into the harmless eddies below. The wind throws the water into a
highly unstable condition, into what morally we should call a condition of
great excitement. The oil by an influence we cannot perceive directs this
excitement.
This influence, though insensibly small, is however now proved of a
mechanical kind, and to me it seems that the phenomenon of one of the
most powerful mechanical actions of which the forces of nature are capable,
being entirely controlled by a mechanical force so slight as to be other-
wise quite imperceptible, does away with every argument against the strictly
mechanical sources of what we may call mental and moral forces.
But to return to the instability in parallel channels. This has been the
most complete, as well as the most definite result of the colour-bauds.
The circumstances are such as to render definite experiments possible.
These have been made, and reveal a definite law of the instability, which law
has been tested by reference to all the numerous and important experiments
on the resistance in channels by previous observers ; whereupon it is found
that waters behave in exactly the same manner whether the channel, as
in Poiseuille's experiment, is of the dimensions of a hair, or whether it be the
size of a water main or of the Mississippi ; the only difference being that in
order that the motions may be compared, the velocity must be inversely as
the diameter of the pipe. But this is not the only point explained if we
consider other fluids than water. Some fluids, like oil or treacle, apparently
flow more slowly and steadily than water. This, however, is only in smaller
channels ; the critical velocity increases with the viscosity of the fluid.
Thus, while water in comparatively large streams is always above its critical
velocity, and the motion always sinuous, the motion of treacle in streams
of such size as we see is below its critical velocity, and the motion direct,
o. R. ii. 11
162 ON THE TWO MANNERS OF MOTION OF WATER. [48
But if nature had produced rivers of treacle the size of the Thames, for
instance, the treacle would have flowed just like water. Thus, in the lava
streams from a volcano, although looked at close the lava has the consistence
of a pudding, in the large and rapid streams down the mountain sides the
lava flows as freely as water.
I have now only one circumstance left to which to ask your attention.
This is the effect of the curvature of the stream on the stability of the
fluid.
Here again we see the whole effect altered by very slight causes.
If water be flowing in a bent channel in steady streams, the question as
to whether it will be stable or not turns on the variation in the velocity from
the inside to the outside of the stream.
In front of the lantern is a cylinder with glass ends, so that the light
passes through in the direction of the axis. The disk of light on the screen
being the light which passes through this water, and is bounded by the
circular walls of the cylinder.
By means of two tubes temporarily attached, a stream of coloured water
is introduced right across the cylinder extending from wall to wall; the
motion is very slow, and the taps being closed, and the tubes removed, the
colour-band is practically stationary. The vessel is now caused to revolve
about its axis. At first, only the walls of the cylinder move, but the colour-
band shows that the water gradually takes up the motion, the streak being
wound off at the ends into a spiral thread, but otherwise remaining still and
vertical. When the spirals meet in the middle, the whole water is in motion,
but the motion is greatest at the outside, and is therefore stable. The
vessel stops, and gradually stops the water, beginning at the outside. If the
motion remained steady, the spirals would unwind, and the streak be restored.
But the motion being slowest at the outside against the surface, you see
eddies form, breaking up the spirals for a certain distance towards the middle,
but leaving the middle revolving steadily.
Besides indicating the effect of curvature, this experiment really illustrates
the action of the surface of the earth on the air.moving over it; the varying
temperature having much the same influence as the curvature of the vessel
on stability. The air is unstable for a few thousand feet above the surface,
and the motion is sinuous, resulting in the mixing of the strata, and pro-
ducing the heavy cumulus clouds ; but above this the influence of temperature
predominates, and clouds, if there are any, are of the stratus-form, like the
inner spirals of colour. But it was not the intention of this lecture to trace
the two manners of motion of fluids in the phenomena of Nature and Art, so
I thank you for your attention.
49.
ON THE THEORY OF THE STEAM-ENGINE INDICATOR*.
[From the " Proceedings of the Institution of Civil Engineers, 1885."]
"ON THE THEORY OF THE STEAM-ENGINE INDICATOR
AND THE ERRORS IN INDICATOR-DIAGRAMS."
By OSBORNE REYNOLDS, M.A., LL.D., F.R.S., M. Inst. C.E.
SECTION I. — INTRODUCTION.
IN 1856 Hirn published an experimental comparison of the indicated
work, with the work done on the brake, and came to the conclusion that,
whatever might be the cause, the indicated work was too small, being only
just equal to the brake-work, leaving no margin for the air-pump and the
friction of the engine.
This conclusion of Hirn's seems to have excited little notice. Rankine
mentions it in " The Steam-engine," but expresses doubt whether it accords
with subsequent experience, particularly that of marine engines.
Since that time many engine-experiments have been made. It does not
appear, however, that these have been made with a view to verify the
indicator, but rather that the indicator-diagrams have been taken as data
from which to determine the efficiency of the engines; nor has, so far as
the Author is aware, any definite theory of the disturbances to which the
diagram is subjected as yet been published.
The importance of studying the disturbances, or, in other words, the
errors in the diagrams, becomes evident, when it is considered to what an
* Joint paper with A. W. Brightmore, D.Sc.
11—2
164 ON THE THEORY OF THE INDICATOR. [49
extreme extent the indicator is now trusted to give a true measure of the
work on the piston. In ninety-nine cases out of every hundred, there is
absolutely no check within 20 or 30 per cent. In some classes of engines
(winding and pumping) the work they are performing is of a measurable
kind, but rarely or never is the work measurable to within 5 or 10 per cent.
The only work which is definitely measurable is that done on the friction-
brake as used by the Royal Agricultural Society ; and even then, although
the brake may give a measure of the actual work to within 1 per cent, or
less, it does not furnish a check on the indicator to within from 5 to 20 per
cent., for between the work measured by the indicator and that measured
by the brake, is the unknown work done in overcoming the resistance of
the engine. This, which varies from 5 to 20 per cent., is an absolutely
unknown quantity, except in so far as it is found by subtracting the brake-
power from the indicated-power, and hence furnishes no check within its
own magnitude on these quantities.
There is thus absolutely no check on the indicator, which is now made
the sole standard, not only of the performance, but of the value of engines.
Considering what this means in mere money, where, as in the case of marine
engines, large sums often depend on a margin of power which is a very
small percentage of the whole, it becomes evident how important it is that
the exact extent to which these instruments can be trusted should be well
known. Yet, in spite of Hirn's warning, the results of the indicator appear
to be accepted without question, solely on the ground of their general
consistency, of the simplicity of its apparent action, and the excellence
of its construction.
On close examination, it appears in this case, as in others, that the
apparent simplicity of action is due to the obscurity of certain facts ; for
example, the possible stretching of a piece of string; and that, taking all
the circumstances which may affect the diagram into account, its action is
by no means a simple matter. It may be that, in some cases, these dis-
regarded circumstances only produce an inappreciable effect, but even this
cannot be known as long as they are disregarded.
The theory of the indicator has now been taught for many years in the
engineering classes in Owens College, Manchester, and the calculations to
a certain extent have been verified by experiments on the College engine.
This engine, though by no means of a high class, has been rendered well
adapted for this purpose by the addition of a brake-dynamometer and
a speed-indicator. It has long been the Author's intention to publish this
theory, but this has been deferred for want of time to make a sufficiently
extensive series of experiments. Last year Mr Brightmore, Berkeley
Fellow in Owens College, Manchester, undertook the experiments and
49] ON THE THEORY OF THE INDICATOR. 165
carried them out very successfully. The results of his investigation appear
to be of considerable importance, and as their interpretation depends on
the theory, an account of this is submitted, to be read in conjunction with
a Paper by Mr Brightmore.
For the diagram to be exact, it is necessary —
1. That the pencil of the indicator shall, under every change of
pressure, instantly move through a distance in exact proportion to the
change of pressure in the cylinder of the engine.
2. That the paper on which the diagram is taken shall change its
position in exact accordance with the change of position of the piston of
the engine.
The first of these is accomplished, so far as it is accomplished, by
holding the piston of the indicator by a spring, carefully adjusted, so that
the deflection is proportional to the load ; and as there is no great difficulty
in making a spring such that this proportion shall be maintained so long
as the temperature is constant, and in making the instrument so that the
temperature of the spring shall be 212° Fahrenheit, there is no reason to
suppose that the indications of the indicator are not within 1 per cent,
of the forces at each instant deflecting the spring.
But in order that these indications may correspond with the pressures
of steam, it is necessary that there should be no other forces acting on
the spring. Such forces, however, arise from the inertia of the weights
to be moved and the friction, notably that entailed by the necessity of
pressing the pencil on the paper.
In assuming the indicator as accurate, it is supposed that the forces
resulting from inertia and friction are too small to be perceived; whether
this is so or not, can only be ascertained by considering these forces.
The second of these conditions of exactness is accomplished by connect-
ing a revolving drum, by means of mechanism, with the piston of the
engine, so that, if there is no yielding in the mechanism, the drum will
revolve through distances exactly proportional to the distance moved by
the piston of the engine. There is no difficulty in arranging mechanism
which will secure the corresponding motion of two bodies, if the forces can
be kept constant on the mechanism. This is attempted in the indicator
by pulling the drum in one direction by a spring, and connecting it with
the piston by means of a cord wound round the drum, so that the spring
always keeps the string in tension. Since all strings — in fact, all matter —
is elastic, in order that the position of the drum may always correspond
with the position of the engine-piston, it is necessary that the spring shall
166 ON THE THEORY OF THE INDICATOR. [49
exert a constant force in all positions of the drum, and that there shall be
no other forces.
As a matter of fact, however, the springs used do not exert a constant
force, the force increasing as the drum is moved against the spring ; and
further, there are forces, namely, the forces arising from the inertia of the
drum and the friction of the mechanism, principally of the drum on its
supports. The diagram will, therefore, only be accurate in so far as these
unequal forces are small ; and the effect of these forces can only be
ascertained after careful consideration.
It thus appears that there are five principal causes of disturbance ; two
of these (1) and (2) affect the motion of the pencil, and three (3) (4) and
(5) the motion of the drum.
(1) The inertia of the piston of the indicator and its attached weights.
(2) The friction of the pencil on the paper and its attached mechanism.
(3) Varying action of the spring.
(4) Inertia of the drum.
(5) Friction of the drum.
These will be separately considered.
SECTION II. — DISTURBANCES ON THE PENCIL.
(1) The effect of the inertia of the Pencil and its attached Mechanism. —
This, although obvious enough in a general way, presents the same problem
as the planetary disturbances, which can only be definitely expressed by
means of some form of mathematics. As the general solution of the
problem is well known to mathematicians, and is unintelligible to those
who are not, it will be best here to omit all the steps, and to proceed at
once to the results, about which there can be no question.
These results may be best expressed in symbols, of which the meaning
is as follows ; taking Ibs., feet, and seconds as general units, then put —
i for the indicated pressure at any instant ;
p for the actual pressure corresponding to i ;
w for the weight of any particular piece of mechanism attached to
the pencil ;
r for the ratio which the motion of this weight bears to the motion
of the piston of the indicator ;
49]
ON THE THEORY OF THE INDICATOR.
167
W for 2 (r*w) where 5 expresses the sum of all the quantities in the
brackets ;
g for 32'2, the acceleration of gravitation ;
e for the number of Ibs. to the inch on the diagram ;
a for the area of the piston of the indicator in square inches ;
s for the ratio of the motion of the pencil to that of the piston of
the indicator.
In Richards' indicator —
a = 0'5 ;
F = 0'33;
s = 4.
For other indicators these may be found by measurement.
The relation between i and p, in so far as it is affected by inertia, is
expressed by the equation —
W fJ2i
- -U n ' — nn (~\ }
The general solution to this equation is well known, and without going
into detail, it will be sufficient to give the solution for the case, which is,
N being the number of revolutions of the engine per minute —
. „„ . . 27rJV,
900xl2a^!^Sm3o' + ^Sm^O-'
&c.
(7 sin
in\/
I2aesg
.t
..(2).
t expresses time in seconds ;
P! greatest pressure ;
ps least back pressure ;
A1} Az are coefficients depending on the shape of the true diagram ;
C is a constant depending on the disturbed state of the pencil.
From equation (2) it appears that the effect of inertia is to cause two
disturbances, corresponding to the two terms on the right-hand side. These
may be considered separately.
The first term has the factor
, .
A1 sm -- t
sm
.
t + &c.,
which will go through a complete cycle when t changes by
60
JV '
168 ON THE THEORY OF THE INDICATOR. [49
that is, by the time of revolution of the engine in seconds. This disturbance
will be the same during each revolution of the engine, and will be called
the cyclic disturbance.
Given the shape of the true diagram, it would be possible to determine
Alt A2 so as to find from equation (2) the value of i— p. But this would
be a very complicated piece of work for such an irregular curve as the
diagram, and as the object is not so much to find the magnitude as to find
when this is small, it is sufficient to consider a circular or elliptic diagram ;
for such a diagram it is found that the mean difference of i and p, written
i— p, is given by
— 7T22
the positive sign to be taken for the forward stroke and the negative for
the backward.
If this effect were large compared with the mean acting pressure
•^ o , then in all probability the area as well as the form of a true
diagram would be seriously disturbed ; but if this effect is small, say 1 per
cent, in the case of the oval, it will be small for the true diagram. Hence
the increase of area is less than 1 per cent, so long as
2WWTT
12 x 900cMW.gr <
and from this it is found that the cyclic disturbance may be 1 per cent,
for Richards' indicator when N and e have the values in Table I., and as
N*
this disturbance increases as --, its possible values for all other cases
may be found.
TABLE I. — ENGINE SPEEDS AT WHICH THE ENLARGEMENT OF THE DIAGRAM
BY INERTIA BECOMES 1 PER CENT. WITH THE RICHARDS' INDICATOR
USED IN THIS INVESTIGATION.
Scale of Diagram Number of
in Ibs. to an inch. Revolutions.
20 166
30 203
40 237
50 262
60 288
70 312
80 332
90 352
100 371
49]
ON THE THEORY OF THE INDICATOR.
169
In the case of the oval or circular diagram the effect of_this cyclic
disturbance would be to increase the vertical diameter, as shown by the
dotted line in Fig. 1. What it would be on the true diagram is very
difficult to express, except to say that it would be to round-off all corners
and increase its size much in the same way as in the oval.
The second term in equation (2) represents a disturbance which goes
through its cycle in an interval of T seconds, where
T =
W
ISaesg
.(4).
This may be called the vibratory disturbance. The period represented by
r is that in which the pencil vibrates when disturbed. Such disturbances
are introduced by the departure of the diagram from the true ellipse.
Fig. 1.
The result of such disturbance is shown by the waving line in Fig. 1.
The time occupied in completing each one of these waves as from pl to pt
is constant, viz. r equation (4).
Hence the number of waves in a complete revolution is given by
n =
60
N
27T
W
I2aesg
.(5).
170 ON THE THEORY OF THE INDICATOR. [49
For Richards' indicator —
In the diagram, owing to the unequal motion of the engine-piston, the
lengths of these oscillations increase from the ends to the middle. If,
however, a circle be drawn on the atmospheric line AB, having the extreme
length of the diagram as diameter, this may be taken to represent the
crank-circle on the same scale as AB represents the stroke. Then if the
points pl9 p.2 &c., in which the waving line cuts the mean line, are first
projected perpendicularly on to AB in P1} P2 &c., and then Pl} P2 projected
by means of a radius to represent the connecting-rod on to the crank-circle
in the points cl} c2 &c., it will be found that the arcs CjCa, C2c3, are all equal,
since the crank turns through equal arcs in equal times.
But for the effects of friction these oscillations, once set up, would go on
for ever ; so that even at low speeds a fair diagram would be impossible.
By friction the oscillations are gradually destroyed, so that they are
more or less localized to the neighbourhood of the points at which they
are produced, i.e., the points where the curvature in the true diagram is
sharp, particularly at the point of admission where the rise of pressure
being instantaneous acts the part of a live-load, and forces the pencil twice
as far as it ought to go. This sets up a series of oscillations.
It is seldom that the time of oscillation is exactly commensurable with
that of revolution, so that if all the oscillations set up in one revolution
are not destroyed by friction before the revolution is complete, the pencil will
not describe the same path in two successive revolutions, a fact frequently
observed in diagrams taken from locomotives at high speed.
The error which these oscillations cause in the area of the diagram
depends on their magnitude, but also, and to a greater extent, on the small-
ness of n, the number in a revolution. But the evil of these oscillations
is not so much an effect on the area, which, even did they exist to the
extent shown in Fig. 1, in which n is between six and seven, would still
be small. It is the disfigurement and the confusion they produce in the
diagram which limits the usefulness of the instrument to cases in which
they can be avoided.
So long as there are thirty of these oscillations in a cycle the necessary
fluid friction of the indicator-piston will so far reduce them as to render
a fair diagram possible, but when the number approaches fifteen it becomes
necessary to call in the aid of considerable pencil-pressure to prevent their '
destroying the form of the diagram ; and when n is as low as ten it is all
the pencil will do to prevent them upsetting the diagram. The Author
49] ON THE THEORY OF THE INDICATOR. 171
has never been able to produce a respectable diagram when th<3 number is
as low as ten, but accounts are continually published in which from the
speed of the engine and strength of the springs the value of n must be
below this. In such cases the pressure of the pencil must have been very
great, and it becomes a question how far this cure is a less evil than the
disease.
(2) The Friction arising from the Pressure of the Pencil. — This always
acts to oppose the motion of the pencil, and therefore renders it too large
during expansion and exhaust, and too small during compression and
admission, and thus the general effect is to increase the size of the diagram.
In order to understand this effect, it is necessary to notice that this
friction consists of two parts : (1) That of the pencil on the paper. (2) That
of the mechanism, caused by sustaining the pressure of the pencil.
The effect of the actual friction of the pencil is greatly reduced by the
motion of the paper. Thus, if while the drum is at rest, the pencil be
lifted quietly it will be possible for friction to hold it above or below the
atmospheric-line, by a distance depending on the pressure. If, when placed
as high or low as it will stand, the drum be moved by the cord, the pencil
at once approaches the atmospheric-line, describing a line as shown in Fig. 2
Fig. 2.
at first sloping toward the atmospheric-line at 45°, but finally becoming
parallel. Fig. 2 represents the results with a 20-lb. spring ; the distance at
starting was equal to about .4 Ibs., but eventually became about £ lb., at
which it remained constant.
The distance at starting represents the extreme friction of pencil and
mechanism. The final distance that of the mechanism alone.
Fig. 3.
These effects on the diagram are different. That of the pencil causes
172
ON THE THEORY OF THE INDICATOR.
[49
the pencil to be behind its true position, by a quantity which will bear to
the extreme distance, a ratio equal to the sine of the inclination of the curve
it is describing at the instant, to the atmospheric-line.
The effect of this alone on a rectangular diagram would be to round off
the corners as in Fig. 3.
With an early cut off, the effect would be as shown in Fig. 4.
Fig. 4.
The friction of the mechanism causes the pencil to be behind its true
position by a nearly constant quantity, and hence during expansion and
exhaust the pencil will be too high, and during compression and admission
the pencil will be too low. This is shown in Fig. 5. Its effect on the area
of the diagram is therefore not very great.
Fig. 5.
The magnitude of these effects, taken together, on the area of the
diagram, depends on the construction of the instrument and on pencil-
pressure. From numerous experiments with Richards' and Thomson's
indicators, it was found that a comparatively slight alteration of pencil-
pressure from that just sufficient to mark the diagram, would cause an
excess of 0'5 Ib. during expansion, and an equal fall during compression.
While if pencil-pressure were made sufficient to prevent serious oscillations
when n=15, the mean acting pressure was affected by as much as 1'5 Ib.
Thus it would appear possible to make a difference of as much as 5 per cent,
in a locomotive in mid-gear by pencil- friction.
The conclusions, then, as regards the motion of the pencil, are, that the
general effects of inertia and friction are both to increase the size of the
diagram ; that so long as the speeds are such that n is not greater than 15,
49] ON THE THEORY OF THE INDICATOR. 173
the effect of inertia is less than 1 per cent., but that if n is less than 30,
oscillations will show themselves unless the pencil-friction He Increased.
They may, by this, be kept down till w = 15, but not farther, and then the
necessary friction will affect the area of the diagram about 5 per cent. A
speed, therefore, which makes w = 15 is about the limiting speed at which
diagrams can be taken accurate to 5 per cent., while for the diagrams to be
sensibly accurate and free from oscillation the speeds must not be greater
than will make n = 30.
These speeds for Richards' indicators are given in Table II.
TABLE II.
N
x-
e
20
30
40
50
60
70
80
90
100
SECTION III. — DISTURBANCES ON THE DRUM.
These are the disturbances (3), (4), (5), section (1). They arise from the
elasticity of the cord and mechanism connecting the drum with the piston of
the engine. In order to express them definitely —
I is the indicated length of the diagram in inches ;
y the yielding of the mechanism in inches per Ib. of the tension ;
/ the moment of inertia of the drum.
(3) The Inertia of the Drum. — If the obliquity of the connecting-rod of
the engine be disregarded, and x be put for the distance OP (Fig. 1), the
force arising from inertia is proportional to N*x and the disturbance arising
from this cause will be yIN-x. And as x will be positive or negative ac-
cording as P is to the right or left of 0, the diagram will be uniformly
elongated.
The effect of the obliquity of the connecting-rod would be to increase
?i=30
n=15
69
138
85
170
99
198
105
210
120
240
130
260
139
278
147
294
155
310
174 ON THE THEORY OF THE INDICATOR. [49
this elongation at the back-end and diminish it at the front, increasing the
area of the back-end diagram, and diminishing that of the front somewhat,
but it is small unless the connecting rod is very short.
(4) The effect of the varying Stiffness of the Spring. — Let q be the
difference of tension of the spring at the extreme ends of the diagram.
Then the disturbance of the point P will be
I
This effect is therefore opposite to that of (3), and the joint effect will be
and since IN2 will be zero at small speeds, and it increases as the square of
the speed, when the speed is low the diagram will be qy too short, but as the
speed increases this shortening will diminish until at some speed INZ = j ,
I
and for higher speeds the diagram will be elongated. With the Richards'
indicator, the critical speed appears to be 150 = ^. In most diagrams these
effects are apparent, but, except when the connecting-rod is short, they do
not affect the indicated pressure.
(5) The effect of the Friction of the Drum. — Let F be the tension on the
string necessary to overcome the friction of the drum in either direction.
Then during the forward stroke the string will be stretched from this
cause yF, and during the backward stroke it will be shortened yF. The
effect will be to place the drum always behind its true position by yF. This
is shown in Fig. 6.
AC&, &c. represent the positions of the crank on its circle, as explained
in reference to Fig. 1; but in this case CiC2, &c. are chosen so as to correspond
with the equidistant positions of the piston. Projecting dc2 with the con-
necting-rod as radius on to the atmospheric-line the points are obtained in
which, for a true diagram, the pencil would be when the crank was in the
positions C&, &c., but owing to the cause under consideration, as the crank
moves from A towards B, the pencil will be (at the points <T ) at a distance
Fy behind its true position, and from B to A (at the points £) Fy behind
its true position.
When the crank arrives at A from B the pencil will not, as it should,
arrive at A, but at the point (marked ^ A) distant Fy towards B. This is
the end of the indicated stroke, and here the drum will remain until the
piston has reversed its position (with regard to & A), that is, until the crank
49]
ON THE THEORY OF THE INDICATOR.
175
has reached A'; hence, as the crank moves from A to A', the drum will be
stationary, and then move off distant Fy behinds its true posTlioli, which
Fig. 6.
distance it will maintain until the crank reaches B, when the drum will
again rest (at 9 B) until the crank has reached B', when it will again start
towards A distant Fy behind its true position.
The effect of this disturbance on a diagram is very great.
In the first place, it must be noticed that, supposing y the same, i.e., the
length of cord used the same, the effect will be the same on both diagrams.
In starting from either end the drum does not move until the engine-piston
has moved through a distance Fy, and the crank has moved through AA'
or BB', so that, however the pencil of the indicator may have been moved,
in this interval it will merely describe a vertical line (a very common feature
of diagrams). For the rest of the motion the drum will move at a constant
distance behind its true position, so that the two halves of the diagram will
be of the right shape, but wrongly placed with regard to each other. If,
then, the pressure at the ends of the true diagram rose and fell instan-
taneously, so that the extreme ends are vertical, as shown by the line ACBD
in Fig. 7, the indicated diagram A'CBD' would be obtained from the true
diagram by simply giving a horizontal shift (as in Fig. 7) AA' = 2Fy to the
lower half of the diagram-line ADB.
The apparent cut-off is then shortened by
AA' = 2yF (6).
The diagram is shortened by 2yF.
176 ON THE THEORY OF THE INDICATOR.
The area is diminished by
[49
and putting im = area j- .
A A'
Fig. 7.
The effect/ on pm, or pm = im +/, is given by
.(7).
It is thus seen that / increases with the expansion and compression, and
is zero when these are zero.
This effect of the friction of the drum appears to be so important, and
to have been so entirely unperceived, that it may be well to introduce a
short discussion of the circumstances on which it depends, and on its effects.
The circumstances are the elasticity of the cord and the friction of the
drum, and the important question is, how far these exist in the ordinary
indicators ? In answer to this, it may be said that the diagrams, which led
to the discovery of this effect, were taken with an indicator which had been
in constant use for several years. It was in apparently perfect condition,
and the diagrams did not differ essentially from those which had been
previously taken. The cord was one which was supplied by the maker.
The manner of the discovery was as follows: For years the Author had
pursued in the class the method of testing the vibrations of the indicator-
pencil by projecting them on to the crank-circle, as shown in Fig. 1, and he
had all along noticed that the first oscillation fell short, and shorter in the
back-diagram than the front. The cause of this was not obvious, as there
seemed to be several possible explanations, and it was partly with a view to
determine this cause that Mr Brightmore's investigation was commenced.
A slight error in the reducing-rod, which had a fixed centre and a slot in
which a stud in the slide-block worked, was altered at Mr Brightmore's
49] ON THE THEORY OF THE INDICATOR. 177
suggestion. This, however, did not get rid of the effect. A new cord
obtained from the makers was substituted for the old one, and theTeffect was
found to be much enhanced, the new cord being more elastic than the old
one. This reduced it to the stretching of the cord, but it was only after
carefully working out the effect of the inertia of the drum, and it was seen
this effect was to lengthen, not shorten, the first oscillation at the back-end,
that it occurred to the Author to look to the friction. The indicator was
then taken to pieces, cleaned and oiled ; then the effect was much reduced.
Several new wires and cords were used which gave less effects, and eventually
the steel wire was adopted by Mr Brightmore as the best. The test supplied
by the oscillations could only be applied to diagrams taken at high speeds,
and the test furnished by the effect upon area was vague. What was
wanted was an independent means of determining the simultaneous positions
of the drum and the engine-piston. As the best method of meeting this,
it was decided to arrange an electric-circuit through the pencil to the drum,
with sufficient electromotive force to prick the paper, making the engine-
piston close this circuit at eleven definite equidistant points in its motion
backwards and forwards. After some difficulty this was successfully carried
out by Mr Brightmore and Mr Foster. In this way the stretching of the
cord during the backward and forward strokes was definitely ascertained
by Mr Brightmore. Taking the smallest results obtained with a cord, it
appeai-s from these experiments that the least difference of stretching was
to make
2Fy = 0'05Cm inches ........................... (8),
where C is the length of the cord in feet ; so that there is obtained from
i M| nation (7)
, . . . 0-05(7
This equation gives the value of f or pm - im for any diagram in terms of
the length of the cord, on the assumption that the stretching is the same
per foot of cord. The length of cord is generally 1'5 times the stroke for the
front-end, and 2*7 times for the back-end, or 2'1 for both, hence putting S
for the stroke in feet
f-a-i n Q'0758
« *"• 7:l'/ + 0075S '
for the front-end,
, . . 0-1358
/-fc-^-^i+ongB
for the back-end, or
... . ., 0-1058
as the mean.
o. R. ii. 12
178
ON THE THEORY OF THE INDICATOR.
[49
In the College engine, Avith 3 cwt. on the brake, at a speed of one hundred
and seven revolutions,
S= 1-5;
1= 5-0;
*\ - ta = 30-0 ;
im = 23-0.
From (12)
/=0-24;
or 4- — O'Ol.
In a locomotive-diagram, Fig. 8, published in Richards' Indicator, by
Porter,
8= 2;
1= 4;
t, - i3 = 105 ;
*»»= 40;
/= 3-25;
i = 0-08.
In the case of a condensing-engine $ = 3'5, cutting-off at apparently
£-0-2;
^m
and in the case of a compound-engine expanding ten times
f
i- = O'lO.
49]
ON THE THEORY OF THE INDICATOR.
179
These would seem to be the smallest results that can have occurred in
ordinary practice. The conclusion, however, that hitherto the normal indi-
cated power from engines has been from 10 to 20 per cent, too small is one
which must be received with hesitation, or must wait for verification. Yet
it may be pointed out that there are not wanting independent evidences of
such an effect. There are features common to most diagrams which are
shown in this investigation to be due solely to this effect.
(i) In diagrams taken from engines at high speeds the admission-line
would not but for this effect be vertical. It would show a certain amount
of detail, and the first oscillation would not have a sharp top. They would
be as shown in Fig. 9, whereas they commonly are as in Fig. 10.
Fig. 9.
Fig. 10.
(ii) It is commonly found that the expansion-line is above the true
expansion-line for the steam allowing for clearance. This fact has been
much commented upon, and is sometimes assumed to indicate leaking valves,
and sometimes a large amount of evaporation from the jacket, either of
which circumstances may explain some rise of the expansion-line towards
the end of the stroke, but it is difficult to see how they can explain the rise
from cut-off which is usually observed. Now this apparent rise in the curve
of expansion is exactly what would result if the apparent cut-off were too
early, and this is the result of the effect that has been considered. The
author has tried several diagrams, and he finds that, correcting the cut-off
by formula 6, the expansion-line comes out very close indeed to the true
curve.
(iii) In making these comparisons the explanation of another feature
of diagrams became apparent. When the two diagrams are traced on the
same card there is sometimes seen a want of symmetry about them, and
almost invariably when this is the case the cut-off is shorter on the back
than on the front-diagram. This would be the result of the friction of the
12—2
180
ON THE THEORY OF THE INDICATOR.
[49
drum, supposing the cord for the back-diagram longer than that for the front.
Where this is. the case the relative lengths of the cord are about 1 to 1'8.
These observations are all illustrated in Fig. 11, which represents a
facsimile diagram from Richards' Indicator.
Fig. 11.
To test this diagram a tracing was taken, and reversed so that the front-
diagram was superimposed on the back. It was then observed —
(a) That the diagrams were of different lengths, and the difference was
about the same as the difference in cut-off.
(6) That notwithstanding the apparent cut-off in the back-diagram is to
that in the front in the ratio of 2 to 3, the expansion-line of the back-
diagram was exactly the same shape as that of the front.
(c) That if the diagrams were restored by formula 8, supposing the
lengths of cords used to have been 5 feet and 9 feet, the diagrams became
exactly similar, and, allowing 2 per cent, clearance, the expansion-line comes
to be the true expansion-line for that cut-off. This rearrangement is shown
in the dotted lines in Fig. 11, the mean pressure from which is 14 per cent,
larger than from the original diagrams.
Such instances as these seem to sufficiently establish a primd facie case
against the confidence which appears to be at present placed in the accuracy
of indicator-diagrams. But, in conclusion, the author would state that he
should be very disappointed if anything in this investigation should have
the effect of diminishing reliance on the indicator itself. He would have
the instrument treated as other instruments have been treated, and instead
of its results being, assumed accurate, he would have it the object of careful
study and experimental investigation, so that the limits of its wonderful
perfection may be exactly known, and that reliance placed on it which such
knowledge must afford.
49 A.
EXPERIMENTS ON THE STEAM-ENGINE INDICATOR.
By ARTHUR WILLIAM BRIGHTMORE, B.Sc., Stud. Inst. C.E.,
Late Berkeley Fellow in Owens College, Manchester.
THE object of these experiments was to ascertain definitely to what extent
certain disturbing causes, which exist in the indicator, affect the diagram.
These disturbing causes are : —
1st. The necessary inaccuracy of the indicator springs, when cold or hot.
2nd. The effect of the inertia of the piston and parallel- motion bars on
the area.
3rd. The effect of the oscillations of the spring on the diagram, and the
extent to which these may be reduced without sensibly altering its area.
4th. The effect produced by the stretching of the indicator-cord. To
get rid, as far as possible, of the error due to this cause, in the experiments
relating to the second and third causes, a thin steel wire (B. W. G. 22) was
used instead of a cord.
The following is a description of the apparatus employed : —
INDICATOR.
The indicator was an ordinary Richards indicator, made by Elliott Bros.,
London ; having Watt's parallel-motion for magnifying the deflection of the
spring. Springs by different makers were used.
ENGINE.
The engine employed was the one which is used for the Owens College
workshop. It was not chosen on account of any particular adaptability for
the purpose; in fact, in some of the experiments, although it fulfilled the
182
EXPERIMENTS ON THE STEAM-ENGINE INDICATOR.
[49 A
requirements, the results were not so marked as they would have been, had
the point of cut-off been earlier ; but it was necessary in the experiments
to have complete control over the engine, and to be able to run it with the
brake on only, and the College engine presented these facilities.
It is a non-condensing engine, with 9-inch cylinder, 18 inches length of
stroke, and a fly-wheel 16 feet in circumference. The point of cut-off is
towards the end of the stroke. It works up to a boiler-pressure of about
47 Ibs. on the square inch, and to a speed of about 150 revolutions per
minute.
REDUCING-MECHANISM.
In order to give the paper-drum a reduced motion of the piston, the wire
employed to rotate the drum was attached to a rod, one end of which turned
on a pin in the cross-head, and the other end worked in a slot, fixed vertically
over the middle position of the cross-head pin, as shown in Fig. 12.
Fig. 12. MECUANISM FOB REDUCING THE MOTION or THE PISTON.
By this method of reducing the motion of the piston of the engine, the
only error that comes in is due to the slight change of inclination of the wire.
BRAKE.
The work done was measured by the friction-brake used in the class
experiments at Owens College.
49 A]
EXPERIMENTS ON THE STEAM-ENGINE INDICATOR.
183
It consists of small flat blocks of wood threaded on a cat-gut rope, and
is passed round the fly-wheel. To one end of this rope a board, of which
the other extremity rests on the ground, is fastened ; and the load is placed
on the board close to its attachment to the brake. The other end of the
brake is attached to a spring-balance, which measures the tension on it ;
the arrangement is shown in Fig. 13.
Fig. 13. FBICTION-BHAKE.
Thus, the rate of work was obtained by multiplying the difference of the
tensions at the two ends by 16 feet (the circumference of the fly-wheel).
SPEED-INDICATOR (Fie. 14).
This was also a class instrument. It consists of a small paddle-wheel
fixed on a vertical axis, in a small circular box containing coloured liquid.
Fig. 14. SPEED-lNDlCATOK.
Near the bottom of this box, an upright glass tube is inserted. The paddle-
wheel was rotated by a cord, driven from a pulley on the main shaft, and
passing round a pulley fixed on the same vertical spindle with it.
184
EXPERIMENTS ON THE STEAM-ENGINE INDICATOR.
[49 A
The rotation of the paddle causes the liquid to rise in the tube to a
height dependent on the speed of the engine.
Thus the scale was graduated by running the engine at constant speeds
and counting.
INDICATOR-SPRINGS (Fio. 15).
Before commencing the experiments, it Avas necessary to test the accuracy
of the indicator-springs.
To do this, the indicator was rigidly fixed in a vertical position, and
pressure was applied to the centre of the indicator-piston by means of a rod,
pressed upwards by one end of a long beam, balanced on a knife-edge ; the
weight being hung on the other end of the beam.
Fig. 15. APPARATUS FOB TESTING THE SPKINGS.
The deflection of the springs was measured by Professor Reynolds' small
cathetometer, used in his experiments on "Thermal Transpiration," and
fully described in the Philosophical Transactions of the Royal Society,
Part II. 1879.
It consists of a microscope carried by a vertical sliding-piece moved by a
very accurate screw with fifty threads to the inch, and is capable of measuring
to fo.ooo incn- Thus, by continually adjusting the screw, so that some well-
defined mark on the piston-rod lay on the horizontal cross-hair, and noting
the reading for each particular weight, the deflections under the various
pressures were arrived at.
To prevent the piston of the indicator sticking in a wrong position, owing
to friction, the frame to which the indicator was attached was tapped with a
light hammer each time a fresh weight was added.
49 A] EXPERIMENTS ON THE STEAM-ENGINE INDICATOR. 185
Table I. gives the results of these experiments for five springs, at the
ordinary temperature.
The next thing was to see what effect an increase of temperature would
have on the springs. Now, the temperature of the indicator-spring never
rises above 212° Fahrenheit, owing to its being open to the atmosphere, and
moisture always being present in the indicator. Hence the springs were
surrounded with steam at 212° Fahrenheit, by passing it through a hole in
the cap of the indicator ; of course the steam was at first all condensed ; but
by waiting until steam issued from another hole in the cap, the temperature
was maintained uniformly at 212° during the experiments, which were con-
ducted as in the previous cases. The result of these experiments for the
same five springs is given in Table II.
Tables I. and II. show the uniformity of the increase of the deflection
with a constant addition to the pressure on the spring. They also prove
that the deflection of a spring is greater, under the same weight, the higher
the temperature ; hence the necessity of setting indicator-springs when hot,
i.e., when at the temperature of boiling water. It appears also from these
Tables, that in the case of the springs experimented upon, the deflection
under a given weight at 212° Fahrenheit is about 3 per cent, greater than at
the ordinary temperature ; therefore a diagram, taken with a spring which is
perfectly correct when cold, will be 3 per cent, too large.
This is shown more clearly in Table III., which gives the mean deflection
of springs under 1 Ib. when cold and when hot, as calculated from Tables I.
and II., and the deflection under 1 Ib. as calculated from the number marked
on the spring. The percentage error in the fifth column is the difference
between columns three and four, and is allowed for in all the following calcu-
lations. It will be noticed from this Table, that in one case only did this
error amount to 2 per cent.
TABLE I. — DEFLECTION OF SPRINGS, WHEN COLD.
Strength
of Spring,
Ibs. to the
Inch.
20
32
32*
50
80
c •-> £
Deflec-
tion
under lib.
.2 § '~
'.Q
o> a I"H
«"1
3
|| |j
S d^
«"!
3
to 6
.a
<i> a T~<
qU o IN
Reading
of Micro-
meter.
S o rH
3
Weight in
Scale Pan.
0-6829
0-4887
0-4888
0-4832
0-4791
0-0246
0-0156
0-0157
0-0099
0-0060
1
0-7075
0-5043
0-5045
0-4931
0-4860
0-0242
0-0156
0-0154
0-0104
0-0061
2
0-7317
0-5199
0-5199
0-5035
0-4921
0-0245
0-0154
0-0154
0-0101
0-0063
3
0-7562
0-5353
0-5353
0-5136
0-4984
0-0243
0-0153
0-0152
0-0101
0-0062
4
0-7805
0-5506
0-5505
0-5237
0-5046
0-0240
0-0156
0-0153
o-oioo
0-0062
5
0-8045
0-5662
0-5658
0-5337
0-5108
0-0240
0-0154
0-0152
o-oioo
0-0062
6
0-8285
0-5816
0-5810
0-5437
0-5170
0-0245
0-0153
0-0153
0-0095
0-0062
7
0-8530
0-5969
0-5963
0-5532
0-5232
0-0250
0-0157
0-0156
0-0099
0-0062
8
0-8780
0-6126
0-6119
0-5631
0-5294
0-0250
0-0157
OO155
0-0098
0-0062
9
0-9030
0-6283
0-6274
0-5729
0-5356
0-0248
0-0157
0-0153
0-0100
0-0061
10
0-9278
0-6440
0-6427
0-5829
0-5417
0-0249
0-0155
0-0157
0-0101
0-0060
11
0-9527
0-6595
0-6584
0-5930
0-5477
0-0247
0-0155
0-0154
0-0099
0-0060
12
0-9774
0-6750
0-6738
0-6029
0-5537
0-0246
0-0158
0-0160
0-0099
0-0059
13
1-0020
0-6908
0-6898
0-6127
0-5596
0-0244
0-0156
0-0156
o-oioo
0-0060
14
1-0264
0-7064
0-7054
0-6227
0-5656
0-0243
0-0158
0-0155
o-oioo
0-0061
15
1-0507
0-7222
0-7209
0-6327
0-5717
0-0246
0-0156
0-0161
o-oioo
0-0059
16
1-0753
0-7378
0-7370
0-6427
0-5776
0-0238
0-0157
0-0159
o-oioo
0-0061
17
1-0991
0-7535
0-7529
0-6527
0-5837
0-0251
0-0156
0-0154
o-oioo
0-0059
18
1-1242
0-7691
0-7683
0-6627
0-5896
0-0244
0-0150
0-0151
o-oioo
0-0062
19
1-1486
0-7841
0-7834
0-6727
0-5958
0-0242
0-0152
0-0154
o-oioo
0-0072
20
1-1728
0-7993
0-7988
0-6827
0-6030
0-0152
0-0156
0-0102
0-0062
21
...
0-8145
0-8144
0-6929
0-6092
...
0-0154
0-0155
0-0101
0-0062
22
0-8299
0-8299
0-7030
0-6154
0-0156
0-0153
0-0103
0-0061
23
...
0-8455
0-8452
0-7133
0-6215
0-0151
0-0155
o-oioo
0-0063
24
...
0-8606
0-8607
0-7233
0-6278
0-0152
0-0160
0-0101
0-0062
25
...
0-8768
0-8767
0-7334
0-6340
0-0099
0-0062
26
...
0-7433
0-6402
...
0-0102
0-0066
27
...
0-8535
0-6468
* Different maker.
TABLE II.— DEFLECTION OF SPRINGS, WHEN HOT.
Strength
of Spring,
20
32
32* 50
80
Ibs. to the
Inch.
60 6 .
.£>
ec 6 .
£
60 0 .
.0
eco
.a
2f2 •
£
.2 o £
t) __ ,— i
? B y
® ^ rH
*"" pi *"*
8 o""
S 8 b
8 c^
.H b H
8 a'"'
Weight in
g s "s
0) —
* .2 QJ
|l|
|||
41 S
-J M
|g|
* ° s
3) '^3 ^
!§!
S *
Scale Pan.
* -g
3
* o C
3
a
w's
3
rt^
§
0-6872
0-4941
0-4937
0-4862
0-5953
01)245
0-0154
0-0162
OO104
0-0063
1
0-7117
0-5095
0-5099
0-4966
0-6016
()-()-2~)[
0-0160
0-0157
OO108
0-0065
2
0-7368
0-5255
0-5256
0-5074
0-6081
0-0251
0-0158
0-0157
0-0103
0-0064
3
0-7619
0-5413
0-5413
0-5177
0-6145
0-0249
0-0159
0-0156
0-0103
0-0062
4
0-7868
0-5572
0-5569
0-5280
0-6207
0-0250
0-0157
0-0157
0-0103
0-0063
5
0-8118
0-5729
0-5726
0-5383
0-6270
0-0251
0-0162
0-0156
0-0103
0-0063
6
0-8369
0-5891
0-5882
0-5486
0-6333
0-0253
0-0156
0-0157
0-0098
0-0063
7
0-8622
0-6047
0-6039
0-5584
0-6396
0-0258
0-0162
0-0164
0-0105
0-0064
8
0-8880
0-6209
0-6203
0-5689
0-6460
0-0258
0-0163
0-0159
0-0104
0-0066
9
0-9138
0-6372
0-6362
0-5793
0-6526
0-0256
0-0160
0-0160
0-0099
0-0062
10
0-9394
0-6532
0-6522
0-5892
0-6588
0-0259
0-0163
0-0158
0-0102
0-0062
11
0-9653
0-6695
0-6680
0-5994
0-6650
0-0258
0-0160
0-0162
0-0104
0-0064
12
0-9911
0-6855
0-6842
0-6098
0-6714
0-0250
0-0162
0-0158
0-0098
OO066
13
1-0161
0-7017
0-6700
0-6196
0-6780
0-0250
0-0159
0-0162
0-0106
0-0065
14
1-0411
0-7176
0-7162
0-6302
0-6845
0-0246
0-0163
0-0163
0-0105
0-0062
L5
1-0657
0-7339
0-7325
0-6407
0-6907
OO259
0-0164
0-0161
0-0101
0-0063
16
1-0916
0-7503
0-7486
0-6508
0-6970
0-0258
(t)
0-0161
0-0104
0-0067
17
M174
0-7689
0-7647
0-6612
0-7037
()-<\-2:r2
0-0160
0-0158
0-0105
0-0064
18
1-1426
0-7849
0-7805
0-6717
0-7101
0-0242
0-0159
0-0157
0-0105
0-0061
19
1-1668
0-8008
0-7962
0-6822
0-7162
• • .
0-0159
0-0160
0-0106
0-0065
20
...
0-8167
0-8122
0-6928
0-7227
0-0159
OO157
0-0102
0-0062
21
0-8326
0-8279
0-7030
0-7289
0-0163
0-0170
0-0105
0-0063
22
...
0-8489
0-8449
0-7135
0-7352
• • •
0-0162
0-0156
0-0064
n
0-8(55 1
0-8605
0-7416
...
0-0159
0-0160
• . .
00063
24
...
0-8810
0-8765
0-7479
• • .
0-0162
0-0164
...
0-0063
25
0-8972
0-8929
0-7542
» • •
0-0062
26
...
...
0-7602
Different maker.
t Condensed steam let out of cylinder.
188
EXPERIMENTS ON THE STEAM-ENGINE INDICATOR.
[49 A
EFFECT OF INERTIA OF THE MOVING-PARTS ON THE AREA OF THE DIAGRAM.
Having ascertained the errors in the springs, the next question was to
find how far the effect of inertia tends to alter the area of the diagram
before the oscillations appear. To do this, diagrams were taken at various
speeds and with several springs. In Table IV. the efficiences, i.e., the ratios
of the brake-pressures to the mean diagram-pressures, are given at the
various speeds, instead of the mean pressures as calculated from the
diagrams, on account of the difficulty of keeping the load on the brake
exactly constant.
TABLE III. — MEAN DEFLECTIONS OF SPRINGS UNDER 1 Ib.
Spring.
Experimental
Deflection, cold.
Experimental
Deflection, hot.
Deflection from
Mark on Spring.
Percentage
Error.
Inch.
Inch.
Inch.
20
0-0245
0-02525
0-02523
0-08
32
0-0155
0-01600
0-01580
1-25
32*
0-0155
0-01595
0-01580
0-95
50
o-oioo
0-01030
0-01009
2-08
80
0-0062
0-00636
0-00630
0-94
* Different maker.
Now if the inertia affects the areas of the diagrams, the areas of the
diagrams, and hence the mean diagram- pressures, will vary directly with the
velocity, and inversely as the stiffness of the spring (the weight on the brake
being constant) ; i.e., the efficiencies will vary directly with the stiffness of
the spring and with the inverse of the velocity. However, an examination
of the Table shows no appreciable increase of the efficiency with greater
stiffness of the spring, and no more decrease, as the velocity increases, than
would be accounted for by the greater friction.
Table IV. is not filled in for the 20 and 32 springs at the higher speeds,
because the oscillations begin to come in.
The inference is, " that in a given engine, when the ratio of the speed to
the stiffness of the spring, used to indicate it, is not so great as to cause
oscillations to appear in the diagram, the area is not appreciably affected by
the momentum of the moving parts." This seems natural, for, after the
initial disturbance on the admission of the steam to the cylinder, the motion
49 A]
EXPERIMENTS ON THE STEAM-ENGINE INDICATOR.
189
of the spring is gradual, and hence its deflection would correspond to the
pressure on it.
TABLE IV.
Speed.
Efficiencies.
Spring.
20
32
50
80
44
0-94
...
0-95
...
0-945
68
0-93
0-94
0-93
...
0-933
84
...
0-93
0-93
0-93
0-930
107
0-93
0-94
0-93
0-933
127
...
0-93
0-92
0-925
OSCILLATIONS.
When the ratio of the speed of the engine to the stiffness of the spring,
used to indicate it, exceeds a certain value, which is different for different
engines, oscillations appear in the diagram.
The equation which gives the time of oscillation of the spring, modified
by the parallel-motion bars (Fig. 16), devised by Professor Reynolds, is, taking
the axis of x vertically upwards : —
W
(1),
190 EXPERIMENTS ON THE STEAM-ENGINE INDICATOR. [49 A
k-
where W ' = W + (w + w2) — + 16^ ,
CL
+ (w + w.2)- + ^vl],
ct /
and e = the stiffness of spring.
W = weight of piston + £ weight of spring.
w = weight of rod AD (Fig. 16).
w1 = weight of rod DF.
w2 = weight of rod FH.
P = whole pressure of steam on the piston.
a = AB.
b = AC= GH = distance of centres of gravity of rods AD, FH from
A and H respectively.
k = radius of gyration of AD, FH, about A and H respectively.
W is, in fact, the weight which would have to oscillate at B to be
equivalent to the moving-parts, and the expression P — Q represents the
force which would have to be applied at B, if the parts referred to were
removed, to be equivalent to them.
Equation (1) is of the well-known form for finding the time of a complete
oscillation (T), and then is obtained in the ordinary way—
g.e
Or calling N the number of oscillations per minute —
30 Ig7e_
~ 7T V W"
It will be noticed that in equation (1), the rotation of the rod DF, which
is very slight, is neglected, as also is the friction of the instrument.
In the case of the indicator employed, the values of the above constants
were —
W* = 0-10529 Ib. TfS2 = 0-10954 Ib.
w = 0-00957 Ib.
w, = 0-01037 Ib.
w, = 0-00866 Ib.
a = 0'75 inch.
b = 1 inch.
A* = 1-83.
49 A] EXPERIMENTS ON THE STEAM-ENGINE INDICATOR. 191
Whence from the above Wn = 0'33063 Ib.
^32=0-33488 Ib.
;ui(l from preceding experiments £20 = 475.
e:,2 = 750.
NOTE. — The suffixes 20 and 32 refer to the springs marked 20 and 32
respectively.
Thus N*> = 2050.
^2 = 2560.
It will be noticed on substituting for W, that the rod DF has as much
influence in causing the oscillations to come in as all the other moving parts
together.
To verify these results, diagrams were taken with weak springs, in order
to bring in oscillations. It must be understood that the diagrams in this
Paper are not intended as specimens of good diagrams, but are merely to
illustrate the various points considered.
The time of oscillation of the indicator-springs may be approximately
obtained from such diagrams in the following manner: — first, project the
crests and hollows of the oscillations vertically down on to the atmospheric-
line ; next, with a radius equal to the length of the connecting-rod (reduced
to the same scale as the length of the diagram), and centre on the atmo-
spheric-line produced, project the points so obtained upon a circle described
on the atmospheric-line with the length of diagram as the diameter ; then
the arcs of the circle intercepted between alternate intersections represent
the angle turned through by the crank during the time of a complete
oscillation of the spring. Hence, assuming that the crank-shaft rotates
uniformly, these arcs would represent the time of a complete oscillation.
There are several reasons why the number of oscillations per minute
so obtained should not quite equal the number as obtained above from
theory. Firstly, the neglect of the rotation of the bar DF, and of the
friction in the equation, would make a slight difference ; but the most
important reason is the gradual decrease of pressure in the cylinder of the
engine, consequent upon the motion of the piston and initial condensation.
This diminution of pressure causes the crests to lie behind, and the hollows
to be in advance of their true position (Figs. 17 to 25), by an amount varying
with the rate of decrease. Supposing for the moment the lag to be equal in
amount for each crest, the projection of it (the lag) upon the crank-circle will
include a greater arc towards the ends than in the middle of the diagram ;
thus, other things being the same, causing the time of oscillation to appear
192
EXPERIMENTS ON THE STEAM-ENGINE INDICATOR.
[49 A
too great at one end of the diagram, and too small at the other end of the
diagram. However, this tendency is counteracted, at least during the first
half of the stroke (and it is during this period chiefly that the time of oscilla-
tion is measured), by the retardation of the velocity of oscillation, and
consequently the greater effect of the reduction of pressure in causing the
crests to lag as the stroke progresses. That the velocity of oscillation
decreases with the distance from the point of admission is seen by integrat-
ing equation (1), where —
dx
dt
qe
qe
where c = £ the distance of a hollow from the atmospheric-line.
Now 2 f — — c j is equal to the range of oscillation, as may be seen by
again integrating equation (1), and in the case of the diagrams referred to,
the range of oscillation, and hence from above, the velocity of oscillation of
the spring diminished as the stroke advances, which is almost self-evident,
for the time of oscillation is independent of the range, so that if the range be
reduced the velocity must be reduced also.
From equation (2) it is also seen that, other things being the same, the
number of oscillations in a diagram increases with the stiffness of the spring,
hence the counteracting effect, just referred to, would be less marked as the
stiffness of the spring used is increased, so that for this reason the number of
oscillations per minute as obtained from a diagram would be nearer the
truth the weaker the spring.
Again, the number of oscillations per minute will probably be nearer the
truth the greater the speed of the engine ; for the number of oscillations in
Fig. 17. Front-end diagram taken with 20 spring at 141 revolutions.
a diagram is smaller the greater the speed of the engine, because the time of
oscillation of the spring is independent of the speed of the engine, and
49 A]
EXPERIMENTS ON THE STEAM-ENGINE INDICATOR.
193
hence the ratio of the velocity of oscillation to the rate of reduction of
pressure is less the higher the speed of the engine, hence the counteracting
effect referred to is greater. These two points are illustrated in the diagrams,
Figs. 17 to 25, and the accompanying Table V.
Fig. 18. Front-end diagram taken with 20 spring at 127 revolutions.
Fig. 19. Front-end diagram taken with 20 spring at 107 revolutions.
Fig. 20. Back-end diagram taken with 20 spring at 144 revolutions,
o. it. ii. 13
194 EXPERIMENTS ON THE STEAM-ENGINE INDICATOR. [49 A
Fig. 21. Back-end diagram taken with 20 spring at 127 revolutions.
Fig. 22. Front-end diagram taken with 32 spring at 144 revolutions.
Fig. 23. Front-end diagram taken with 32 spring at 127 revolutions.
49 A]
EXPERIMENTS ON THE STEAM-ENGINE INDICATOR.
195
Fig. 24. Back-end diagram taken with 32 spring at 144 revolutions.
Fig. 25. Back-end diagram taken with 32 spring at 127 revolutions.
TABLE V.
Speed.
Revolutions per
minute.
End.
Sprin».
Number of Oscillations.
Difference
per cent.
From Diagram.
From Formula.
144 (Fig. 17)
Front
20
1,950
2,050
5-0
127 (Fig. 18)
»
20
1,920
6'5
107 (Fig. 191
»
20
1,883
8-5
144 (Fig. 20)
Back
20
1,950
5-0
127 (Fig. 21)
»
20
1,930
6-0
144 (Fig. 22)
Front
32
2,370
2,560
7-5
127 (Fig. 23)
32
2,300
10-0
144 (Fig. 24)
Back
32
2,300
10-0
127 (Fig. 25)
>»
32
2,300
lO'O
In calculating the oscillations from the diagrams a mean value was taken.
The distance to which the oscillations extend depends on the range of
13—2
196
EXPERIMENTS ON THE STEAM-ENGINE INDICATOR.
[49 A
the first one, and on the friction of the pencil. The range of the first oscil-
lation is great if the period of a semi-oscillation nearly coincides with the
time the steam takes to attain its maximum pressure on admission ; this
happens when the engine is running fast. It is small when the time of
attaining the greatest pressure of steam and the time of a semi-oscillation
are not nearly equal. Thus, when the steam is wire-drawn on entering the
cylinder of an engine, that engine would give a better diagram at high
speeds than if this were not the case.
Again, if the steam be throttled on entering the indicator, the time of
the steam attaining its maximum pressure in the indicator-cylinder will be
lengthened; hence the extent of the first oscillation will be reduced, and
therefore the oscillations in the diagram will be reduced ; but the diagram
so obtained does not give a correct idea of the work done, but is too small
in proportion to the amount of throttling.
The effect of the friction of the pencil in lessening the extent of the
oscillations varies with the pressure on the pencil. When the oscillations
are thus reduced by pressing the pencil on the paper an indefiniteness is
introduced into the results, owing to the pencil sticking either too high or
too low, and the results cannot be relied on.
To illustrate this point diagrams were taken under the same conditions,
of which the results are given in Table VI.
In the case of the weaker springs, 20 Ibs. and 32 Ibs., the pencil was
pressed on the diagram-paper so as to reduce the oscillations. Diagrams
were taken with stiffer springs, in which oscillations do not perceptibly enter,
to check the results so obtained.
TABLE VI. — FRONT-END EFFICIENCIES.
Speed.
20 Spring (pencil
pressed).
32 Spring (pencil
pressed).
50 Spring.
80 Spring.
69
0-932
0-927
0-959
0-958
87
0-931
0-942
0-954
0-954
108
0-918
0-907
0-955
0-954
Mean efficiencies
0-927
0-925
0-956
0-955
Table VI. shows that in those experiments in which the pencil was pressed
on to the paper the results are too small by more than 3 per cent. No
49 Aj EXPERIMENTS ON THE STEAM-ENGINE INDICATOR. 197
doubt if the engine had cut-off earlier, and been working with a higher
pressure of steam, the results would have been still more discordant.
Probably the most accurate method of arriving at the mean pressure
when the oscillations extend a good way into the diagram, at least when
the cut-off occurs late in the stroke as in the present case, is to draw a
line midway between the crests and hollows, and to the value for the mean
pressure obtained by taking this line add an amount, which in the case of
indicators similar to the one employed in these experiments is 0'35 Ib.
To see the reason for this, referring back to equation (1), and integrating
it twice —
Iw
Substituting in this t = IT A/ (time of half oscillation)
20
x = — - c,
e
i.e., Q is the arithmetical mean of ex and ec.
Substituting in this expression the value for Q, and taking the area of
the indicator piston as 0'5 square inch, the following value for the intensity
of pressure (p) is obtained : —
p = e (x + c) + 2 ( W+ (w -f wa) - + 4wj ) .
\ & /
Hence if a line midway between the crests and hollows be taken as repre-
senting the pressure, the mean pressure so obtained will be too small by the
amount of the second term on the right, which for the indicator employed
= 0'35 Ib. This would be negligible for any considerable pressure.
It was found that with the indicator used, a diagram tolerably free from
oscillations could be taken from the engine up to a speed of about 90 revolu-
tions per minute, with a spring of 20 Ibs. to the inch. Hence, since the
time the steam takes to attain its maximum pressure in the cylinder varies
with the speed of the engine (in different engines it would also vary with
the arrangement of the slide-valve), it might be expected to obtain a
diagram tolerably free from oscillations at a speed of from 400 to 500 revo-
lutions per minute, with an indicator having a parallel-motion in which the
rod corresponding to DF is absent, and in which the other moving-parts are
as light again as in the present case. This would be the case with an
indicator of smaller diameter, in which a much stronger spring could be
used for the same weight. For much higher speeds than this, unless the
relative time occupied in attaining the maximum pressure increased with
198
EXPERIMENTS ON THE STEAM-ENGINE INDICATOR.
[49
the speed, it would appear that the diagrams would be affected to a great
but unknown extent by the oscillations of the spring.
VITIATION OF THE DIAGRAM BY THE STRETCHING OF THE
INDICATOR-CORD.
The effect of the stretching of the cord varies greatly with the shape of
the diagram, and with the state of lubrication of the paper drum. Owing
to the late cut-off, the engine employed in the experiments was not well
suited for showing this effect. However, in some experiments, when the
paper-drum wanted oiling, the diagram given with the cord was more than
7 per cent, smaller than that given with the steel wire. The effect is in all
cases to reduce the area, though not necessarily to reduce the mean pressure
calculated from it.
To ascertain if the diagrams from the engine in question would show
much difference when taken with cord and with wire, the experiments
summarised in Table VII. were made. The lengths and efficiencies given
are the mean of the front- and back-end diagrams.
TABLE VII.
Speed.
Wire.
String.
Length.
Efficiency.
Length.
Efficiency.
68
Inches.
5-11
0-93
Inches.
4-78
0-94
84
5-11
0-93
4-80
0-94
107
5-13
0-94
4-80
0-94
127
5-12
0-93
4-80
0-97
Although the efficiency as calculated from the two sets of diagrams is
inconsiderable, yet the difference in their lengths points to a large difference
in their areas.
The difference in the tension of the indicator-cord at various parts of
the stroke may be shown by considering the equation of motion of the
indicator-drum.
This equation during the outward stroke is
dt*
= Ta- J\L -
49 A] EXPERIMENTS ON THE STEAM-ENGINE INDICATOR. 199
where / = the moment of inertia of the drum about its axis.
T = the tension in the cord.
a = radius of drum.
Mg = moment of resistance of the drum-spring about the axis of drum.
Jiff = moment of friction about the same line.
Hence r-s(' 3? +*• +
ri-R
~r , the angular acceleration of the drum about its axis, is a maximum
to begin with, and continues to decrease during the stroke, becoming zero
near the middle of the stroke.
Ms is constant during the stroke.
Mf is a maximum on starting, then suddenly decreases and then varies
directly with some power of the velocity, increasing therefore until about
the middle of the stroke, and then diminishing.
Thus it is evident that during the outward stroke the tension T is a
maximum to begin with, decreases rapidly about the middle of the stroke,
and more slowly towards the end.
At the end of the stroke the friction suddenly changes sign, thus causing
a sudden diminution in the tension at the commencement of the inward
stroke ; afterwards the tension increases rapidly about the middle of the
stroke, and more slowly towards the end.
Hence it might be expected that that part of a diagram taken during
the outward stroke would be shortened to commence with, then slightly
stretched, and slightly shortened at the end; and that that part taken
during the inward stroke, would be first shortened, then lengthened a little,
and slightly shortened towards the end, almost as in the case of the outward
stroke.
To show that this actually takes place, an arrangement was devised by
Professor Reynolds, the object of which was to prick holes in the diagram
corresponding to eleven equidistant positions of the piston. For this purpose
a Grove battery (D Fig. 26) of five cells, in conjunction with a Ruhmkorrf
coil, was used. But in order to get the holes pricked in their proper
positions, instead of the ordinary arrangement for making and breaking
contact, the following plan was adopted, the hammer of the coil being held
back. The wire from one pole of the battery was connected with one of the
binding-screws (//) of the primary coil as usual, but the wire from the other
pole of the battery was connected with the engine. A wire from the other
200
EXPERIMENTS ON THE STEAM-ENGINE INDICATOR.
[4-9 A
binding-screw (G) of the primary coil was attached to the contact-breaker
(B). This consisted of a smooth piece of wood, into which eleven pieces of
wire were inserted at equal distances, and filed level with the wood, the
Fig. 26. ELECTRICAL APPARATUS FOR SHOWING THE DISTORTION OF A DIAGRAM BY THE
INDICATOR-COKD.
distance between the first and the last wires being the length of the stroke
of the engine. The contact-breaker was fixed on the lower slide bar, so that
the central wire should be at the middle of the stroke, and so that a pointer
(A), which was secured to the cross-head, should slide on the smooth piece
of wood. Hence every time the pointer crossed a wire on the contact-breaker
the circuit of the primary current was complete, and a spark of the induced
current passed through the diagram-paper. To bring this about one wire
of the induced current was connected with the metallic drum (E), and the
other to a cup of mercury (F), into which the metallic pencil dipped, thus
completing the circuit of the induced current when the pencil touched the
paper.
In the diagrams, Figs. 27 to 34, which were taken in this manner, the
position of the pricked holes, corresponding to the eleven equidistant posi-
tions of the piston, are indicated by small circles. The relative positions of
these circles show which parts of the diagrams are lengthened, and which
are shortened. An examination shows that the effect is not merely to shorten
Fig. 27. Front-end pricked diagram taken with wire at 107 revolutions.
Fig. 28. Back-end pricked diagram taken with wire at 107 revolutions.
49 A]
EXPERIMENTS ON THE STEAM-ENGINE INDICATOR.
Fig. 29. Front-end pricked diagram taken with string at 107 revolutions.
o o e — PO— o
Fig. 30. Bcick-end pricked diagram taken with string at 107 revolutions.
Fig. 31. Front-end pricked diagram taken with wire at 127 revolutions.
201
-e o-
<P OO
Fig. 32. Back-end pricked diagram taken with wire at 127 revolutions.
Fig. 33. Front-end pricked diagram taken with string at 127 revolutions.
Fig. 34. Back-end pricked diagram taken with string at 127 revolutions.
the ends and lengthen the middle of the diagrams, but also to distort them,
i.e., to cause corresponding points in their upper and lower parts not to lie
in the same vertical line. The amount of this distortion is shown by the
distance between corresponding points on the atmospheric-line. It will
also be noticed that even in the diagrams taken with wire instead of with
cord this distortion is not altogether absent. The indefiniteness in the
stretching of the cord is shown by some of the points being marked twice.
202 EXPERIMENTS ON THE STEAM-ENGINE INDICATOR. [49 A
In high-speed diagrams of short length these effects would cause a marked
modification in their form when taken with cord.
At high speeds, when the spring of the drum is not stiff enough to keep
the cord tight near the centre of the stroke, and the velocity is greatest, a
shortening of the middle portion of the diagram, taken during the inward
stroke, and a lengthening of the end, would result.
These considerations show that in indicators intended to take diagrams
from engines running at high speeds, the drum, as well as all the other
moving-parts, should be as light as possible.
50.
ON THE DILATANCY OF MEDIA COMPOSED OF RIGID
PARTICLES IN CONTACT. WITH EXPERIMENTAL ILLUS-
TRATIONS*.
[From the "Philosophical Magazine " for December, 1885.]
IDEAL rigid particles have been used in almost all attempts to build
fundamental dynamical hypotheses of matter: these particles have generally
been supposed smooth.
Actual media, composed of approximately rigid particles, exist in the
shape of sand, shingle, grain, and piles of shot ; all which media are influenced
by friction between the particles.
The dynamical properties of media, composed of ideal smooth particles
in a high state of agitation, have formed the subject of very long and
successful investigations, resulting in the dynamical theory of fluids.
Also, the limiting conditions of equilibrium of such media as sand, have
been made the subject of theoretical treatment by the aid of certain
assumptions.
These investigations, however, by no means constitute a complete theory
of granular masses ; nor does it appear that any attempts have been made
to investigate the dynamical properties of a medium consisting of smooth
haul particles, held in contact by forces transmitted through the medium.
It has sometimes been assumed that such a medium would possess the
properties of a liquid, although in the molecular hypothesis of liquids now
accepted, the particles are assumed to be in a high state of motion, holding
each other apart by collisions ; such motion being rendered necessary to
account for the property of diffusion.
* This Paper was read before Section A of the British Association at the Aberdeen Meeting,
September 10, 1885, and again before Section B, at the request of the Section, September 15.
204 ON THE DILATANCY OF MEDIA [50
Without attempting anything like a complete dynamical theory, which
will require a large development of mathematics, I would point out the
existence of a singular fundamental property of such granular media, which
is not possessed by known fluids or solids. On perceiving something which
resembles nothing within the limits of one's knowledge, a name is a matter
of great difficulty. I have called this unique property of granular masses
"dilatancy," because the property consists in a definite change of bulk,
consequent on a definite change of shape or distortional strain, any dis-
turbance whatever causing a change of volume and generally dilatation.
In the case of fluids, volume and shape are perfectly independent ; and
although in practice it is often difficult to alter the shape of an elastic body
without altering its volume, yet the properties of dilatation and distortion
are essentially distinct, and are so considered in the theory of elasticity.
In fact there are very few solid bodies which are to any extent dilatable
at all.
With granular media, the grains being sensibly hard, the case is, according
to the results I have obtained, entirely different. So long as the grains are
held in mutual equilibrium by stresses transmitted through the mass, every
change of relative position of the grains is attended by a consequent change
of volume ; and if in any way the volume be fixed, then all change of shape
is prevented.
In speaking of a granular medium, it is assumed to be in such a condition
that the position of any internal particle becomes fixed, when the positions
of the surrounding particles are fixed.
This condition is very generally fulfilled, but not always where there is
friction ; without friction it would be always fulfilled.
From this assumption it at once follows, that no grain in the interior
can change its position in the mass by passing between the contiguous
grains without disturbing these ; hence, whatever alterations the medium
may undergo, the same particle will always be in the same neigh-
bourhood.
If, then, the medium is subject to an internal strain, the shapes of the
internal groups of molecules will all be altered, the shape of each elementary
group being determined by the shape of the surrounding particles. This
will be rendered most intelligible by considering instances ; that of equal
spheres is the most general, and presents least difficulty.
A group of such spheres being arranged in such a manner that, if the
external spheres are fixed, the internal ones cannot move, any distortion of
the boundaries will cause an alteration of the mean density, depending on
the distortion and the arrangement of the spheres. For example : —
50] COMPOSED OF RIGID PARTICLES IN CONTACT. 205
If arranged as a pile of shot as in Fig. 2, which is an arrangement of
tetrahedra and octahedra, the density of the media is -^ g» taking the
density of the sphere as unity.
7T •
If arranged in a cubical formation, as in Fig. 1, the density is ^ , or \/2 times
less than in the former case.
Fig. 1. Fig. 2.
These arrangements are both controlled by the bounding spheres ; and in
either case the distortion necessitates a change of volume.
Either of these forms can be changed into the other by changing the
shape of the bounding surface.
In both these cases the structure of the group is crystalline, but that is
on account of the plane boundaries.
Practically, when the boundaries are not plane, or when the grains are of
various sizes or shapes, such media consist of more or less crystalline groups
having their axes in different directions, so that their mean condition is
amorphous.
The dilatation consequent on any distortion for a crystalline group may
be definitely expressed. When the mean condition is amorphous, it becomes
difficult to ascertain definitely what the relations between distortion and
dilatation are. But if, when at maximum density, the mean condition is not
only amorphous but isotropic, a natural assumption seems to be, that any
small contraction from the condition of maximum density in one direction,
means an equal extension in two others at right angles.
As such a contraction in one direction continues, the condition of the
medium ceases to be isotropic, and the relation changes until dilatation
ceases. Then a minimum density is reached ; after this, further contraction
in the same direction causes a contraction of volume, which continues until
206 ON THE DILATANCY OF MEDIA [50
a maximum density is reached. Such a relation between the contraction in
one direction, and the consequent dilatation, would be expressed by
1 / . „ a.
6—1 = el \ I sin2 — ;
V e1
e being the coefficient of dilatation, a that of contraction, and el the maximum
dilatation ; the positive root only to be taken.
The amorphous condition of minimum volume is a very stable condition ;
but there would be a direct relation between the strains and stresses in any
other condition if the particles were frictionless and rigid.
If the particles were rigid, the medium would be absolutely without
resilience, and hence the only energy of which it would be susceptible would
be kinetic energy ; so that, supposing the motion slow, the work done upon
any group in distorting it would be zero. Thus, supposing a contraction in
one direction and expansion at right angles, then if px be the stress in the
direction of contraction, and py, pz the stress at right angles, a being the
contraction, b arid c expansions,
pxa + pyb + pzc = Q;
or, supposing b = c, p,, = pz,
pxa + py (a + c) = 0.
With friction the relation will be different; the friction always opposes strain,
i.e. tends to give stability.
It is a very difficult question to say exactly what part friction plays ; for
although we may perhaps still assume without error,
py _ 1 — sin <f>
px I + sin $ '
where tf> is the angle of repose, we cannot assume that tan <f> has any relation
to the actual friction between the molecules.
The extreme value of 0 is a matter of arrangement ; as in the case of
shot, which would pile equally well although without friction.
Supposing the grains rigid, the relations between distortion and dilata-
tion are independent of friction ; that is to say, the same distortion of any
bounding surfaces must mean the same internal distortion whatever the
friction may be.
The only possible effect of friction would be to render the grains stable
under circumstances under which they would not otherwise be stable ; and
hence we might, with friction, be able to bring about an alteration of the
boundaries other than the alteration possible without friction ; and thus we
50] COMPOSED OF RIGID PARTICLES IN CONTACT. 207
might possibly obtain a dilatation due to friction. How far this is the case
can be best ascertained by experiment.
In the case of a granular medium, friction may always be relaxed by
relieving the mass of stress, and any stability due to this cause would be
shown by shaking the mass when in a condition of no stress.
But before applying this test, it is necessary to make perfectly sure that
during the shaking the boundary spheres do not change position.
Another test of the effect of friction is, by comparing the relative
dilatation and distortion with different degrees of friction. If the dilatation
were in any sense a consequence of friction, it would be greater when the
coefficient of friction between the spheres was greater. Where the granular
mass is bounded by solid surfaces, the friction of the grains against these
surfaces will considerably modify the results.
The problem presented by frictionless balls is much simpler than that
presented in the case of friction. In the former case the theoretical problem
may be attacked with some hope of success. With friction the property is
most easily studied by experiment.
As a matter of fact, if we take means to measure the volume of a mass
of solid grains more or less approximately spheres, the property of dilatancy
is evident enough, and its effects are very striking, affording an explanation
of many well-known phenomena.
If we have in a canvas bag any hard grains or balls, so long as the bag is
not nearly full it will change its shape as it is moved about ; but when the
sack is approximately full, a small change of shape causes it to become
perfectly hard. There is perhaps nothing surprising in this, even apart from
familiarity ; because an inextensible sack has a rigid shape when extended
to the full, any deformation diminishing its capacity, so that contents which
did not fill the sack at its greatest extension fill it when deformed.
On careful consideration, however, many curious questions present them-
selves.
If, instead of a canvas bag, we have an extremely flexible bag of india-
rubber, this envelope, when filled with heavy spheres (No. 6 shot), imposes
no sensible restraint on their distortion ; standing on the table it takes
nearly the form of a heap of shot. This is apparently accounted for by the
fact that the capacity of the bag does not diminish as it is deformed. In
this condition it really shows us less of the qualities of its granular contents
than the canvas bag. But as it is impervious to fluid, it will enable me to
measure exactly the volume of its contents.
Filling up the interstices between the shot with water, so that the bag is
208 ON THE DILATANCY OF MEDIA [50
quite full of water and shot, no bubble of air in it, and carefully closing the
mouth, I now find that the bag has become absolutely rigid in whatever form
it happened to be when closed.
It is clear that the envelope now imposes no distortional constraint on the
shot within it, nor does the water. What, then, converts the heap of loose
shot into an absolutely rigid body ? Clearly the limit which is imposed on
the volume by the pressure of the atmosphere.
So long as the arrangement of the shot is such that there is enough
water to fill the interstices, the shot are free, but any arrangement which
requires more room, is absolutely prevented by the pressure of the
atmosphere.
If there is an excess of water in the bag when the shot are in their
maximum density, the bag will change its shape quite freely for a limited
extent, but then becomes instantly rigid, supporting 56 Ib. without further
change. By connecting the bag with a graduated vessel of water, so that the
quantity which flows in and out can be measured, the bag again becomes
susceptible of any amount of distortion.
Getting the bag into a spherical form, and its contents at maximum
density, and then squeezing it between two planes, the moment the squeezing
begins the water begins to flow in, and flows in at a diminishing rate until
it ceases to draw more water.
The material in the bag is in a condition of minimum density under the
circumstances. This does not mean that all the parts are in a condition of
minimum density, because the distortion is not the same in all the parts ;
but some parts have passed through the condition of maximum, while others
have not reached it, so that on further distortion the dilatations of the latter
balance the contractions of the former. If we continue to squeeze, water
begins to flow out until about half as much has run out as came in ; then
again it begins to flow in. We cannot by squeezing get it back into a con-
dition of uniform maximum density, because the strain is not homogeneous.
This is just what would occur if the shot were frictionless ; so that it is not
surprising to find that, using oil instead of water, or, better (on account of the
india-rubber), a strong solution of soap and water, which greatly diminishes
the friction, the results are not altered.
On measuring the quantities of water, we find that the greatest quantity
drawn in is about 10 per cent, of the volume of the bag ; this is about one-
third of the difference between the volumes of the shot at minimum and
maximum density.
—= : 1, or 30 per cent, of the latter.
v 2
50] COMPOSED OF RIGID PARTICLES IN CONTACT. 209
On easing the bag it might be supposed that the shot would return to their
initial condition. But that does not follow : the elasticity of form of
the bag is so slight compared with its elasticity of volume, that resti-
tution will only take place as long as it is accompanied with contraction
of volume.
So long as the point of maximum volume has not been reached, approxi-
mate restitution follows quite as nearly as could be expected, considering
that friction opposes restitution. But when the squeezing has been carried
past the point of maximum volume, then restitution requires expansion ; and
this the elasticity of shape is not equal to accomplish, so that the bag retains
its flattened condition. This experiment has been varied in a great variety
of ways.
The very finest quartz sand, or glass balls f inch in diameter, all give the
same results. Sand is, on the whole, the most convenient material, and its
extreme fineness reduces any effect of the squeezing of the india-rubber
between the interstices of the balls at the boundaries ; which effect is very
apparent with the balloon bags, and shot as large as No. 6.
A well-marked phenomenon receives its explanation at once from the
existence of dilatancy in sand. When the falling tide leaves the sand firm,
as the foot falls on it the sand whitens, or appears momentarily to dry round
the foot. When this happens the sand is full of water, the surface of which
is kept up to that of the sand by capillary attraction ; the pressure of the
foot causing dilatation of the sand, more water is required, which has to be
obtained either by depressing the level of the surface against the capillary
attraction, or by drawing water through the interstices of the surrounding
sand. This latter requires time to accomplish, so that for the moment the
capillary forces are overcome ; the surface of the water is lowered below that
of the sand, leaving the latter white or dryer until a sufficient supply has
been obtained from below, when the surface rises and wets the sand again.
On raising the foot it is generally seen that the sand under the foot and
around becomes momentarily wet ; this is because, on the distorting forces
being removed, the sand again contracts, and the excess of water finds
momentary relief at the surface.
Leaving out of account the effect of friction between the balls and the
envelope, the results obtained with actual balls, as regards the relation
between distortion and dilatation, appear to be the same as would follow if
the balls were smooth.
The friction at the boundaries is not important as long as the strain over
the boundaries is homogeneous, and particularly if the balls indent them-
selves into the boundaries, as they do in the case of india-rubber. But with
o. K. ii. 14
210 ON THE DILATANCY OF MEDIA [50
a plane surface, the balls at the boundaries are in another condition from the
balls within. The layer of balls at the surface can only vary its density from
2/V3 to 1. This means that the layer of balls at a surface can slide between
that surface and the adjacent layer, causing much less dilatation than would
be caused by the sliding of an internal layer within the mass. Hence,
where two parts of the mass are connected by such a surface, certain con-
ditions of strain of the boundaries may be accommodated by a continuous
stream of balls adjacent to the surface. This fact made itself evident in two
very different experiments.
In order to examine the formation which the shot went through, an
ordinary glass funnel was filled with shot and oil, and held vertical while
more shot were forced up the spout of the funnel. It was expected that the
shot in the funnel would rise as a body, expanding laterally so as to keep
the funnel full. This seems to have been the effect at the commencement of
the experiment ; but after a small quantity had passed up it appeared,
looking at the side of the funnel, that the shot were rising much too fast, for
which, on looking into the top of the funnel, the reason became apparent.
A sheet of shot adjacent to the funnel was rising steadily all round, leaving
the interior shot at the same level with only a slight disturbance.
In another experiment one india-rubber ball was filled with sand and
water ; at the centre of this ball was another much smaller ball, communi-
cating through the sides of the outer envelope by means of a glass pipe with
an hydraulic pump. It was expected that, on expanding the interior ball by
water, the sand in the outer ball would dilate, expanding the outer ball and
drawing more water into the intervening sand. This it did, but not to the
extent expected. It was then observed that the outer envelope, instead of
expanding, generally bulged in the immediate neighbourhood of the point
where the glass tube passed through it ; showing that this tube acted as a
conductor for the sand from the immediate neighbourhood of the interior
ball to the outer envelope, just as the glass sides of the funnel had acted for
the shot.
As regards any results which may be expected to follow from the recog-
nition of this property of dilatancy, —
In a practical point of view, it will place the theory of earth-pressures
on a true foundation. But inasmuch as the present theory is founded
on the angle of repose, which is certainly not altered by the recognition of
dilatancy, its effect will be mainly to show the real reason for the angle
of repose.
The greatest results are likely to follow in philosophy, and it was with
a view to these results that the investigation was undertaken.
50] COMPOSED OF RIGID PARTICLES IN CONTACT. 211
The recognition of this property of dilatancy places a hitherto^ unrecog-
nized mechanical contrivance at the command of those who would explain
the fundamental arrangement of the universe, and one which, so far as I have
been able to look into it, seems to promise great things, besides possessing
the inherent advantage of extreme simplicity.
Hitherto no medium has ever been suggested which would cause a
statical force of attraction between two bodies at a distance. Such attraction
would be caused by granular media in virtue of this dilatancy and stress.
More than this, when two bodies in a granular medium under stress are near
together, the effect of dilatancy is to cause forces between the bodies, in very
striking accordance with those necessary to explain coherence of matter.
Suppose an outer envelope of sufficiently large extent, at first not abso-
lutely rigid, filled with granular media, at its maximum density. Suppose
one of the grains of the media commences to grow into a larger sphere ; as
it grows, the surrounding medium will be pushed outwards radially from the
centre of the expanding sphere. Considering spherical envelopes following
the grains of the medium, these will expand as the grains move outwards.
This fixes the distortion of the medium, which must be contraction along
the radii, and expansion along all tangents.
The consequent amount of dilatation depends on the relation of distortion
and dilatation, and on the arrangement of the grains in the medium. At
first the entire medium will undergo dilatation, which will diminish as the
distance from the centre increases. As the expansion goes on, the medium
immediately adjacent to the sphere will first arrive at a condition of minimum
density; and for further expansion this will be returning to a maximum
density, while that a little further away will have reached a minimum. The
effect of continued growth will therefore be, to institute concentric undula-
tions of density from maximum to minimum density, which will move
outwards ; so that after considerable growth, the sphere will be surrounded
with a series of envelopes of alternately maximum and minimum density,
the medium at a great distance being at maximum density. At a definite
distance from the centre of the sphere not more than
where R is the radius of the sphere, the density will be a minimum, and
between this and the sphere there may be a number of alternations,
depending on the relative diameters of the grains and the spheres.
The distance between these alternations will diminish rapidly as the
sphere is approached. The distance of the next maximum is I'ZR, the
next minimum is given by T09.R, and the next maximum T06.R.
14—2
212
ON THE DILATANCY OF MEDIA
[50
The general condition of the medium around a sphere which has expanded
in the medium, is shown in Fig. 3, which has been arrived at on the sup-
position that the sphere is large compared with the grains.
Curve, vf DenJ?
Fig. 3.
From a radius about 1'4<R outwards the density gradually increases,
reaching a maximum density at infinity ; and at all distances greater than
I'8R the law is expressed by
J?? = JL
dr ~ rn '
where n has some value greater than 3, depending on the structure of the
medium.
Within the distance 1'4>R the variation is periodic, with a rapidly
diminishing period. In this condition, supposing the medium of unlimited
extent and the sphere smooth, the sphere may move without causing further
expansion, merely changing the position of the distortion in the medium ;
for the grains, slipping over the sphere, would come back to their original
positions. It thus appears that smooth bodies would move without resistance,
if the relation between the size of the grains and bodies is such, that the
energy due to the relative motion of the grains in immediate proximity may
50] COMPOSED OF RIGID PARTICLES IN CONTACT. 213
be neglected. The kinetic energy of the motion of the medium would be
proportional to the volume of the ball, multiplied by the density of the
medium, and the square of the velocity.
But the momentum might be infinite, supposing the medium infinite in
extent, in which case a single sphere would be held rigidly fixed.
If we suppose two balls to expand instead of one, and suppose the dis-
tortion of the medium for one ball to be the same as if the other were
not there, the result will be a compound distortion. Since, however, the
dilatation does not bear a linear relation to the distortion, the dilatation
resulting from the compound distortion will not be the sum of the dilatations
for the separate distortions, unless we neglect the squares and products of
the distortions as small.
Supposing the bodies so far apart that one or other of the separate
distortions caused at any point is small, then, retaining squares and products,
it appears that the resultant dilatation at any point will be less than the
sum of the separate dilatations, by quantities which are proportional to the
products of the separate distortions.
The integrals of these terms through the space bounded by spheres of
radii R and L, are expressed by finite terms, and terms inversely propor-
tional to L, which latter vanish if L is infinite. Thus, while the total separate
dilatations are infinite, the compound dilatations differ from the sum of the
separate by finite terms, and these are functions of the product of the
volumes, and the reciprocal of the distance.
Assuming stress in the medium, the difference in the value of these
finite terms for two relative positions of the bodies, multiplied by the
stresses, represents an amount of work which must be done by the bodies
on the medium in moving from one position to another.
To get rid of the difficulty of infinite extent of medium, if for the moment
we assume the envelope sufficiently large and imposing a normal pressure
upon the medium, then, since the work done will be proportional to the
dilatation, the force between the bodies will be proportional to the rate at
which this dilatation varies with the distance between them.
The force between the bodies would depend on the character of the
elasticity, as well as on the dilatation.
It is not necessary to assume the outer envelope elastic ; this may be
absolutely rigid, and one or both the balls elastic.
In such case the two balls are connected by a definite kinematic relation.
As they approach they must expand, doing work which is spent in producing
214
ON THE DILATANCY OF MEDIA
[50
energy of motion ; as they recede, the kinetic energy is spent in the work
of compressing the balls.
As already stated, the momentum of the infinite medium for a single
ball in finite motion may be infinite, and proportional to the product of
the volume of the ball by the velocity; but with two balls moving in
opposite directions, with velocities inversely as the masses, the momentum
of the system is zero. Therefore such motion may be the only motion
possible in a medium of infinite extent.
When the distance between the balls is of the same order as their
dimensions, the law of attraction changes with the law of the compound
dilatations, and becomes periodic, corresponding to the undulations of density
surrounding the balls. Thus, before actual contact was reached, the balls
would suffer alternate repulsion and attraction, with positions of equilibrium
more or less stable between, as shown in Figs. 4 and 5.
Fig. 4.
Fig. 5.
We have thus a possible explanation of the cohesion and chemical
combination of molecules, which I think is far more in accordance with
actual experience than anything hitherto suggested.
It was the observation of these envelopes of maximum and minimum
density, which led me to look more fully into the property of dilatancy.
The assumed elasticity of the surrounding envelope, or of the balls, has
only been introduced to make the argument clear.
The medium itself may be supposed to possess kinetic elasticity arising
from internal distortional motion, such as would arise from the transmission
of waves, in which the motion of the medium is in the plane of their fronts.
The fitness of a dilatant medium to transmit such waves is only less
striking than its property of causing attraction, because in the first respect
it is not unique.
50] COMPOSED OF RIGID PARTICLES IN CONTACT. 215
But, as far as I can see, such transmission is not possible in a medium
composed of uniform grains. If, however, we have comparatively large grains
uniformly interspersed, then such transmission becomes possible. If, notwith-
standing the large grains, the medium is at maximum density, the large
grains will not be free to move without causing further dilatation ; and it
seems that the medium would transmit distortional vibrations, in which the
distortions of the two sets of grains are opposite.
Such waves, although the motion would be essentially in the plane of the
wave, would cause dilatation, just as waves in a chain cause contraction in
the reach of the chain. They would in fact impart elasticity to the medium,
exactly as, in the case of a slack chain having its ends fixed but otherwise
not subject to forces, any lateral motion imparted to the chain will cause
tension, proportional to the energy of disturbance divided by the slackness
or free length of chain.
Distortional waves therefore, travelling through dilatant material which
does not quite occupy the space in which it is confined when at maximum
density, would render the medium uniformly elastic to distortion, but not in
the same degree to compression or extension. The tension caused by such
waves would depend on the gross energy of motion of the waves, divided
by the total dilatation from maximum density consequent on the wave-
motion. All such waves, whatever might be their length, would therefore
move with the same velocity.
If, when rendered elastic by such waves, the medium were thrown into a
state of distortion by some external cause, this would diminish the possible
dilatation caused by the waves. Thus work would have to be done on the
medium in producing the external distortion, which would be spent in in-
creasing the energy of the waves. For instance, the separation of two bodies
in such a medium, which, as already shown, would increase the statical
distortion, would increase the energy of the waves and vice versd.
As far as the integrations have been carried for this condition of elasticity,
it appears, with a certain arrangement of large and small grains, that the
forces between the bodies would be proportional to the product of the
volumes divided by the square of the distance ; i.e. that the state of stress
of the medium may be the same as Maxwell has shown must exist in the
ether to account for gravity. We have thus an instance of a medium,
transmitting waves similar to heat-waves, and causing force between bodies
similar to the forces of gravitation and cohesion, in such a manner as to
constitute a conservative system. More than this, by the separation of the
two sets of grains, there would result phenomena similar to those resulting
from the separation of the two electricities. The observed conducting power
of a continuous surface for the grains of a medium, closely resembles the
216 ON THE DILATANCY OF MEDIA, ETC. [50
conduction of electricity. And such a composite medium would be suscep-
tible of a state in which the arrangement of the two sets of grains were
thrown into opposite distortions, which state, so far as it has yet been
examined, appears to coincide with the state of a medium necessary to
explain electrodynamic and magnetic phenomena according to Maxwell's
theory.
In this short sketch of the results which it appears to me may follow
from the recognition of the property of dilatancy, I have not attempted to
follow the exact reasoning even so far as I have carried it.
In the preliminary acceptance of a theory, the mind must be guided
rather by a general view of its adaptability, than by its definite accordance
with some out of many observed facts. And as it seems, after a preliminary
investigation, that in space filled with discrete particles, endowed with
rigidity, smoothness, and inertia, the property of dilatancy would cause
amongst other bodies, not only one property, but all the fundamental proper-
ties of matter, I have, in pointing out the existence of dilatancy, ventured
to call attention to this dilatant or kinematic theory of ether, without waiting
for the completion of the definite integrations, which must take long, although
it is by these that the fitness of the hypotheses must be eventually tested.
51.
EXPERIMENTS SHOWING DILATANCY, A PROPERTY OF
GRANULAR MATERIAL, POSSIBLY CONNECTED WITH
GRAVITATION.
[From the " Proceedings of the Royal Institution of Great Britain."]
(Read February 12, 1886.)
IN commencing this discourse, the author said, My principal object
to-night is to show you certain experiments which I have ventured to think
would interest you on account of their novelty, and of their paradoxical
character. It is not, however, solely or chiefly on account of their being
curious that I venture to call your attention to them. Let them have been
never so striking, you would not have been troubled with them, had it not
been that they afford evidence of a fact of real importance in mechanical
philosophy.
This newly recognised property of granular masses, which I have called
dilatancy, will, it may be hoped, be rendered intelligible by the experiments,
but it was not by these experiments that it was discovered.
This discovery, if I may so call it, was the result of an attempt to
conceive the mechanical properties a medium must possess, in order that it
might fulfil the functions of an all-pervading ether — not only in transmitting
waves of light, and refusing to transmit waves like those of sound, but
in causing the force of gravitation between distant bodies, and actions of
cohesion, elasticity, and friction between adjacent molecules, together with
the electric and magnetic properties of matter, and at the same time allowing
the free motion of bodies.
It will be well known to those who attend the lectures in this room, that
although a vast increase has been achieved in knowledge of the actions called
218 EXPERIMENTS SHOWING DILATANCY, A PROPERTY OF [51
the physical properties of matter, we have as yet no satisfactory explanation
as to the prima causa of these actions themselves ; that to explain the trans-
mission of light and heat, it has been found necessary to assume space filled
with material possessing the properties of an elastic jelly, the existence
of which, though it accounts for the transmission of light, has hitherto
seemed inconsistent with the free motion of matter, and failed to afford the
slightest reason for the gravitation, cohesion, and other physical properties of
matter. To explain these, other forms of ether have been invented, as in
the corpuscular theory and the celebrated hypothesis of La Sage, the im-
possibilities of which hypotheses have been finally proved by the late
Professor Maxwell, to whom we owe so much of our definite knowledge of
the fundamental physics. Maxwell insisted on the fact, that even if each of
the physical properties could be explained by a special ether, it would not
advance philosophy, as each of these ethers would require another ether
to explain its existence, ad infinitum. Maxwell clearly contemplated the
existence of one medium, but it was a medium which would cause not one
but all the physical properties of matter. His writings are full of definite
investigations as to what the mechanical properties of this ether must be, to
account for the laws of gravitation, electricity, magnetism, and the trans-
mission of light, and he has proved very clear and definite properties,
although, as he distinctly states, he was unable to conceive a mechanism
which should possess these properties.
As the result of a long-continued effort to conceive a mechanical system
possessing the properties assigned by Maxwell, and further, which would
account for the cohesion of the molecules of matter, it became apparent that
the simplest conceivable medium — a mass of rigid granules in contact with
each other — would answer not one but all the known requirements, provided
the shape and mutual fit of the grains were such, that while the grains
rigidly preserved their shape, the medium should possess the apparently
paradoxical, or anti-sponge property, of swelling in bulk as its shape was
altered.
I may here remark, that if ether is atomic or granular, that it should be
a mass of grains holding each other in position by contact, like the grains in
the sack of corn, is one of only two possible conceptions ; the other being
that of La Sage, or the corpuscular theory that the grains are free like
bullets, moving in space in all directions.
Nor, in spite of its paradoxical sound, is there any great difficulty of con-
ceiving the swelling in bulk. When the grains are in contact, it appears at
once that the mechanical properties of the medium must be to some extent
affected by the shape and fit of the grains. And having arrived at the con-
clusion, that in order to act the part of ether, this shape and fit must be such
•51] GRANULAR MATERIAL, POSSIBLY CONNECTED WITH GRAVITATION. 219
that the mass could not change its shape, without changing its volume or
space occupied, the next thing was to see what possible shape could be
given to the grains, so that while these rigidly preserved their shape, the
medium might possess this property of dilatancy.
It was obvious that the grains must so interlock, that when any change
of shape of the mass occurred, the interstices between the grains should
increase. This would be possessed by grains shaped to fit into each other's
interstices in one particular arrangement.
In an ordinary mass of brickwork or masonry well bonded without mortar,
the blocks fit so as to have no interstices ; but if the pile be in any way
distorted, interstices appear, which shows that the space occupied by the
entire mass has increased. (Shown by a model.)
At first it appeared that there must be something special and systematic,
as in the brick wall, in the fit of the grain of ether, but subsequent con-
sideration revealed the striking fact, that a medium composed of grains, of
any possible shape, possessed this property of dilatancy, so long as one im-
portant condition was satisfied.
This condition is, that the medium should be continuous, infinite in
extent, or that the grains at the boundary should be so held as to prevent a
rearrangement commencing. All that is wanted is a mass of hard smooth
grains, each grain being held by the adjacent grains, and the grains on the
outside prevented from rearranging.
Smooth hard spheres arranged as an ordinary pile of shot are in their
closest order, the interstices occupying a space about one-third that occupied
by the spheres themselves. By forcing the outside shot so as to give the
pile a different shape, the inside spheres are forced by those on the outside,
and the interstices increase. Thus by shaping the outside of the pile, the
interstices may be increased to any extent, until they occupy about nine-
tenths of the volume of the spheres : this is the most open formation.
A further change of shape in the same direction causes a contraction of the
interstices, until a minimum volume is reached, and then again an expansion,
and so on. The point to be realised is, that in any of these arrangements, if
the whole of the spheres on the outside of the group are fixed, those inside
will be fixed also. (Shown by a model.)
An interior portion of a mass of smooth hard spheres therefore cannot
have its shape changed by the surrounding spheres, without altering the
room it occupies, and the same is true for any granular mass, whatever be
the shape of the grains.
Considering the generality of this conclusion, the non-discovery of this
property as existing in tangible matter, requires a word of explanation.
220 EXPERIMENTS SHOWING DILATANCY, A PROPERTY OF [51
The physical properties of elasticity, adhesion, and friction, so far render
the molecules of ordinary matter incapable of behaving as a system of parts
with the sole property of keeping their shape, and so prevent evidence of
dilatancy in solids and fluids. This is quite consistent with dilatancy in the
ether, for the properties of elasticity, cohesion, and friction, in tangible
matter, are due to the presence of the ether, so that it would be illogical
for the elementary atoms of the ether to possess these properties.
This, although a sufficient reason why dilatancy has not been recognised
as a property of solid and fluid matter, does not explain its non-existence in
masses of solid, hard, free grains, as of corn, shot, and sand. To understand
why it has not been observed in these, it must be remembered that, to
ordinary observation, these present only an outside appearance, and that the
condition essential for dilatancy, that the outside grains should not be free to
rearrange, is seldom fulfilled. Also these granular forms of matter, though
commonplace, have not been the subjects of physical research, and hence
such evidence as they do afford has escaped detection.
Once, however, having recognised dilatancy as a universal property of
granular masses, it was obvious that if evidence of it was to be sought from
tangible matter, it must be sought in what have hitherto been the most
commonplace and least interesting arrangements. That an important
geometrical and mechanical property of a material system should have been
hidden for thousands of years, even in sand and corn, is such a striking
thought, that it required no little faith in mechanical principles to undertake
the search for it, and although finding nothing but what was strictly in
accordance with the conclusions previously arrived at, the evidence obtained
of this long-hidden property was as much a matter of visual surprise to the
lecturer, as it can be to any of the audience.
To render the dilatancy of a granular mass evident, it was necessary to
accomplish two things: (1) the outside grains must be controlled so that they
could not rearrange, and this without preventing change of shape and bulk
of the mass; (2) the changes of bulk or volume of the mass, or of the
interstices between the grains, must be rendered evident by some method of
measurement which did not depend on the shape of the mass.
A very simple means — a thin india-rubber envelope or boundary — answered
both these purposes to perfection. The thin india-rubber closed over the
outside grains sufficiently to prevent their change of position, and the
impervious character of the bag allowed of a continuous measure of the
volume of the contents, by measuring the quantity of air or water necessary
to fill the interstices.
Taking an india-rubber bag which will hold six pints of water, without
stretching, and having only a small tubular aperture, getting it quite dry,
51] GRANULAR MATERIAL, POSSIBLY CONNECTED WITH GRAVITATION. 221
and putting into it six pints of dry sea sand, such as will run_in an hour-
glass, sharp river sand, dry corn, shot or glass marbles, it presents no very
striking appearance, but all the same when filled with any of these materials,
it cannot have its form changed, as by squeezing between two boards,
without changing its volume. These changes of volume are not sufficient to
be noticeable while the squeezing is going on, but they may be rendered
apparent. It is sufficient to do this with the bag full of clean dry Calais
sand, such as is used in an hour-glass.
The tube from the bag is connected with a mercurial pressure-gauge, so
that the bag is closed by the mercury.
The actual volume occupied by the quartz grains is four and a half pints.
The remaining space, one and a half pints, is occupied by the interstices
between the grains in their closest order ; these interstices are full of air, so
that three-quarters of the bag are occupied by quartz, and one-quarter by
air. Since the bag is closed, and no more air can get in, if interstices are
increased from one pint and a half to two pints, the air must expand, and
its pressure will fall from that of the atmosphere to three-quarters of an
atmosphere. As soon as squeezing begins, the mercury rises on the side
connected with the bag, and steadily rises as the bag flattens, until it has
risen seven inches, showing that the bag has increased in capacity by half
a pint, or one-twelfth of its initial capacity.
That by squeezing a porous mass like sand we should diminish the
pressure of the air in the pores is paradoxical, and shows the anti-sponginess
of the granular material ; had there been a sponge in the bag, the pressure
of the air would have increased with the squeezing.
This experiment has been mainly introduced to prevent a possible im-
pression that the fluid filling the interstices has anything to do with the
dilatation besides measuring it.
Water affords a more definite measure of volume than air.
Taking a small india-rubber bottle with a glass neck full of shot and
water, so that the water stands well into the neck. If instead of shot the
bag were full of water, or had anything of the nature of a sponge in it,
when the bag was squeezed the water would be forced up the neck. With
the shot the opposite result is obtained ; as I squeeze the bag, the water
decidedly shrinks in the neck.
This experiment, which you see is on a very small scale, was not designed
to show to an audience ; it was the original experiment which was made
for my own satisfaction, when the idea of dilatancy first presented itself.
The result, but for the knowledge of dilatancy, would appear paradoxical,
not to say magical. When we squeeze a sponge between two planes, water
222 EXPERIMENTS SHOWING DILATANCY, A PROPERTY OF [51
is squeezed out ; when we squeeze sand, shot, or granular material, water
is drawn in.
Taking a larger apparatus, a bag which holds six pints of sand, the
interstices of which are full of water without any air — the glass neck being
graduated so as to measure the water drawn in. On squeezing the bag
with a large pair of pincers, a pint of water is drawn from the neck into
the bag. This is the maximum dilatation ; the grains of sand are now in
the most open order into which they can be brought by this squeezing ;
further squeezing causes them to take closer order, the interstices diminish,
and the water runs out into the vessel, arid for still further squeezing is
drawn back again, showing that as the change of form continues, the
medium passes through maximum and minimum dilatations.
This experiment may be repeated with granules of any size or shape,
provided they are hard, and shows the universality of dilatancy.
Although not more definite, perhaps more striking evidence of dilatancy
is afforded by the means which the non-expansibility of water affords of
limiting the volume of the bag. An impervious bag full of sand and water
without air cannot have its contents enlarged without creating a vacuum
inside it — the interstices of the sand are therefore strictly limited to the
volume of the water inside it, unless forces are brought to bear sufficient
to overcome the pressure of the atmosphere and create a vacuum. Since
then, owing to this property of dilatancy, the shape of a granular mass at
its greatest density cannot change without enlarging the interstices, if we
prevent this enlargement by closing the bag we prevent change of shape.
Taking the same bag, the sand being at its closest order — and closing
the neck so that it cannot draw more water. A severe pinch is put on
the bag, but it does not change its shape at all ; the shape cannot alter
without enlarging the interstices, which cannot enlarge without drawing
more water, and this is prevented. To show that there is an effort to enlarge
going on, it is only necessary to open a communication with a pressure-
gauge, as in the experiment with air. The mercury rises on the side of the
bag, showing when the pinch is hardest (about 200 Ibs. on the planes) that
the pressure in the bag is less by 27 inches of mercury than the pressure
of the atmosphere ; a little more squeezing and there is a vacuum in the
bag. Without a knowledge of the property of dilatancy such a method
of producing a vacuum would sound somewhat paradoxical. Opening the
neck to allow the entrance of water, the bag at once yields to a slight
pressure, changing shape, but this change at once stops when the supply
is cut off, preventing further dilatation.
In these experiments neither the thickness of the bag, nor the character
of the fluid, has anything to do with the dilatation of the contents,
51] GRANULAR MATERIAL, POSSIBLY CONNECTED WITH GRAVITATION. 223
considered as forming an interior group of a continuous medium, the bag
merely controlling the outside members as they would be controlled by
surrounding grains, and the fluid merely measuring or limiting the volume
of the interstices.
It has, however, been absence of such control of the outside grains, and
such means of measuring the volume of the interstices, that has prevented
the dilatancy revealing itself as a general mechanical property of granular
material ; as a mechanical property, because dilatancy has long been known
to those who buy and sell corn. It is seldom left for the philosopher to
discover anything which has a direct influence on pecuniary interests ; and
when corn was bought and sold by measure, it was in the interest of the
vendor to make the interstices as large as possible, and of the vendee to
make them as small ; of the vendor to make the corn lie as lightly as
possible, and of the vendee to get it as dense as possible. These interests
are obvious ; but the methods of getting corn dense and light are paradoxical
when compared with the methods for other material. If we want to get
any elastic material light we shake it up, as a pillow or a feather bed, or
a basket of dried fruit ; to get these dense we squeeze them into the
measure. With corn it is the reverse ; it is no good squeezing it to get
it dense ; if we try to press it into the measure we make it light — to get it
dense we must shake it — which, owing to the surface of the measure being
free, causes a rearrangement in which the grains take the closest order.
At the present day the measure for corn has been replaced by the scales,
but years ago corn was bought and sold by measure only, and measuring
was then an art which is still preserved. It is understood that the corn is
to be measured light, and the method employed is now seen to have made
use of the property of dilatancy. The measure is filled over full and the
top struck with a round pin called the strake or strickle. The universal art
is to put the strake end on into the measure before commencing to fill it.
Then when heaped full, to pull the strake gently out and strike the top;
if now the measure be shaken it will be seen that it is only nine-tenths full.
Sand presents many striking phenomena well known but not hitherto
explained, which are now seen to be simply evidence of dilatancy.
Every one who walks on the strand must have been painfully struck
with the difference in the firmness and softness of the sand at different
times ; letting alone when it is quite dry and loose. At one time it will be
so firm and hard that you may walk with high heels without leaving a
footprint ; while at others, although the sand is not dry, one sinks in so as to
make walking painful. Had you noticed you would have found that the
sand is firm as the tide falls, and becomes soft again after it has been left
dry for some hours. The reason for this difference is exactly the same as
224 EXPERIMENTS SHOWING DILATANCY, A PROPERTY OF [51
that of the closed bags with water and air in the interstices of the sand.
The tide leaves the sand, though apparently dry on the surface, with all
its interstices perfectly full of water, which is kept up to the surface of the
sand by capillary attraction ; at the same time the water is percolating
through the sand from the sands above, where the capillary action is not
sufficient to hold the water. When the foot falls on this water-saturated
sand, it tends to change its shape, but it cannot do this without enlarging
the interstices — without drawing in more water. This is a work of time, so
that the foot is gone again before the sand has yielded. If you stand still,
you will find that your feet sink more or less, and that when you move, the
sand becomes wet all round the space you stood on, which is the excess of
water you have drawn in, set free by the sand regaining its densest form.
One phenomenon attending walking on firm sand is very striking ; as the
foot falls, the sand all round appears to shoot white or dry momentarily, soon
becoming dark again. This is the suction into the enlarging interstices
below the foot, which for the moment depresses the capillary surface of the
water below that of the sand.
After the tide has left the sand for a sufficient time, the greater part of
the water has run out of the interstices, leaving them full of air, wrhich by
expanding allows the interstices to enlarge, and the foot to sink in far
enough to make walking unpleasant.
If we walk on sand under water, it is always more or less soft, for the
interstices can enlarge, drawing in water from above.
The firmness of the sand is thus seen to be due to the interstices being
full of water, and to the capillary action or surface tension of the water at
the surface of the sand. This capillary action will hold the water up in the
sand for some inches or feet, according to the fineness of the sand. This
is shown by a somewhat striking experiment. If sand running in a stream
from a small hole in the bottom of a vessel, as in an hour-glass, fall into
a vessel containing a slight depth of water, the sand at first forms an island,
which rises above the water. The sand which then falls on the top of this
island is dry as it falls, but capillary action draws up the water which fills
the interstices and gives the sand coherence. The island grows vertically,
very fast, and assumes the form of a column, sometimes with branches like
a tree or a fern, some inches or even a foot high. The strength of these
consists in the surface tension of the water preventing air from being drawn
in to enlarge the interstices, which therefore cannot change shape ; it is
therefore another evidence of dilatancy.
By substituting an impervious envelope for the surface of water, firmness
of sand saturated with water may be rendered very striking.
51] GRANULAR MATERIAL, POSSIBLY CONNECTED WITH GRAVITATION. 225
Thin india-rubber balloons, which may be easily expanded with the mouth,
afford an almost transparent envelope.
Taking one containing about six pints of sand and water, closed without
air, there being more water than will fill the interstices at the densest, but
not enough to allow them to enlarge to the full extent. When standing on
the table, the elasticity of the envelope gives it a rounded shape. The sand
has settled down to the bottom, and the excess of water appears above
the sand, the surface of which is free. The bag may be squeezed and its
shape altered, apparently as though it had no firmness, but this is only
so long as the surface is free. But taking it between two vertical plates and
squeezing, at first it submits, apparently without resistance, when all at once
it comes to a dead stop. Turning it on to its side, a 56-lb. weight produces
no further alteration of shape ; but on removing the weight, the bag at once
returns to its almost rounded shape.
Putting the bag now between two vertical plates, and slightly shaking
while squeezing, so as to keep the sand at its densest, while it still has a free
surface, it can be pressed out until it is a broad flat plate. It is still soft as
long as it is squeezed, but the moment the pressure is removed, the elasticity
of the bag tends to draw it back to its rounded form, changing its shape,
enlarging the interstices, and absorbing the excess of water; this is soon gone,
and the bag remains a flat cake with peculiar properties. To pressures on its
sides it at once yields, such pressures having nothing to overcome but the
elasticity of the bag, for change of shape in that direction causes the sand
to contract. To radial pressures on its rim, however, it is perfectly rigid,
as such pressures tend further to dilate the sand ; when placed on its edge,
it bears one cwt. without flinching.
If, however, while supporting the weight it is pressed sufficiently on the
sides, all strength vanishes, and it is again a rounded bag of loose sand and
water.
By shaking the bag into a mould, it can be made to take any shape ;
then, by drawing off the excess of water and closing the bag, the sand
becomes perfectly rigid, and will not change its shape without the envelope
be torn ; no amount of shaking will effect a change. In this way bricks can
be made of sand or fine shot full of water and the thinnest india-rubber
envelope, which will stand as much pressure as ordinary bricks without
change of shape ; also permanent casts of figures may be taken.
I have now shown, as fully as time will allow, the experiments which
afford evidence of the existence of the property of dilatancy, and how it
explains natural phenomena hitherto but little noticed.
Beyond affording evidence of the existence of the property dilatancy,
o. R. ii. 15
226 EXPERIMENTS SHOWING DILATANCY, A PROPERTY OF [51
these experiments have no direct connection with gravitation or the physical
properties of matter.
These properties cannot be deduced by direct experiment on granular
material, for the simple reason that the grains of the medium which con-
stitutes the ether must be free from friction, while the grains with which
we work are subject to friction. These properties can only be deduced
by mathematical reasoning, into which I will not drag you to-night. I
will merely point out two or three facts, which may serve to convey an
idea of how dilatancy should have such a bearing on the foundation of the
universe.
If you look at this diagram, you see it represents a ball surrounded by a
continuous mass of grain, the density of the grains being indicated by the
depth of colour. If that ball were to grow in volume, it would have to push
out the medium on all sides, and in that way it would distort the groups of
grains, or change their form, causing the interstices to increase ; those nearer
the ball would be distorted more than those further away. Then the inter-
stices of these would grow the most rapidly, and those adjacent to the ball
would first come to their openest order for further growth ; these would
contract somewhat, those a little further away would reach the openest order,
and if the process of growth steadily continued, we should have a series of
undulations of density, commencing at the ball and moving outwards ; the
first of these waves of open order would not, however, get beyond half
the diameter of the ball away. The diagram represents the interstices that
would result, if a single grain of the material had grown to the size of the ball,
pushing the medium out before it. It is not necessary that the ball should
have grown, to produce this result; however the ball were originally placed,
if it were moved away from its original place, it would assume this arrange-
ment, and with this arrangement it would be free to move. Now, although
I cannot attempt to enter upon the relation between the density of the
medium, and the force of attraction between two bodies in it, I may call your
attention to this fact, that the dilatation as calculated, varies exactly as the
force of gravitation, inversely as the square of the distance from an infinite
distance till close to the ball, and then goes through several undulations,
corresponding exactly to the variations in the attraction of bodies necessary
to explain the elasticity and cohesion of molecules. As is shown in the
other diagrams, these undulations in density, which may be experimentally
produced, not only appear to afford a clear explanation of cohesion, but are
the only suggestion of an explanation ever made. And further, similar
undulations have been found necessary to explain one of the phenomena
of light. My reason for calling your attention to them was partly an
experiment, which, although not the most striking, is. the most advanced
experiment in the direction of dilatancy.
51] GRANULAR MATERIAL, POSSIBLY CONNECTED WITH GRAVITATION. 227
The apparatus is that represented in the diagram; the medium is con-
tained in the large elastic bag ; in the middle of this bag is a small hollow
elastic ball, which can be expanded by water forced in through a tube
passing through the medium and outside ball ; the quantity of water which
passes in is measured by a mercury gauge, the water being forced in by the
pressure of the mercury. The medium between the two balls is sand and
water, and is connected with a gauge, the water drawn from which measures
the dilatation.
The full pressure of 30 inches is on the interior ball, but produces no
expansion, because the medium outside cannot dilate, as the supply of water
is now cut off; opening the tap to admit water to the outer ball, it at once
draws water. It has now drawn 3 oz. ; in the meantime the mercury has
fallen, showing that an ounce and a half was admitted to the interior ball,
the expansion of which drew the water into the outer envelope. This
experiment is not striking, but it is definite, and enables us to measure the
dilatation consequent on a given distortion.
It is impossible for me to go further into this explanation, so I will merely
state that the ability of the grains of a medium to slide over a smooth surface
has been experimentally shown to produce phenomena closely resembling the
conduction of electricity, to complete which it is only necessary to construct
the medium of two different sorts of grains, different in size or different
in shape, the separation of which would afford the two electricities, and be a
simple way out of the difficulty hitherto found in explaining the non-exhaus-
tibility of the electricity in a body. Hitherto the two electric fluids have
been supposed to reside together in the matter of the machine, which, how-
ever much has been withdrawn, has never shown signs of exhaustion. In the
dilatant hypothesis, these electricities are the two constituents of the ether
which the machine separates, and it is worth noticing that the ordinary
electrical machine resembles in all essential particulars the machines used by
seedsmen for separating two kinds of seed, trefoil and rye-grass, which grow
together : as long as there is a supply of the mixture, the machine is never
exhausted.
This dilatant hypothesis of ether is very promising, although it cannot be
put forward as proved until it has been worked out in detail, which will take
long. In the meantime it is put forward mainly to excite interest in the
property of dilatancy, to the discovery of which it has led. This property,
now that it has once been recognised, is quite independent of any hypothesis,
and offers a new field for philosophical and mathematical research quite
independent of the ether.
15—2
52.
ON THE THEORY OF LUBRICATION AND ITS APPLICATION
TO MR BEAUCHAMP TOWER'S EXPERIMENTS, INCLUD-
ING AN EXPERIMENTAL DETERMINATION OF THE VIS-
COSITY OF OLIVE OIL.
[From the "Philosophical Transactions of the Royal Society," Part I., 1886.]
SECTION I. — INTRODUCTORY.
1. LUBRICATION, or the action of oils and other viscous fluids to diminish
friction and wear between solid surfaces, does not appear to have hitherto
formed a subject for theoretical treatment. Such treatment may have been
prevented by the obscurity of the physical actions involved, which belong to
a class as yet but little known, namely, the boundary or surface actions of
fluids ; but the absence of such treatment has also been owing to the want of
any general laws discovered by experiment.
The subject is of such fundamental importance in practical mechanics,
and the opportunities for observation are so frequent, that it may well
be a matter of surprise that any general laws should have for so long escaped
detection.
Besides the general experience obtained, the friction of lubricated surfaces
has been the subject of much experimental investigation by able and careful
experimenters. But, although in many cases empirical laws have been
propounded, these fail for the most part to agree with each other and with
the more general experience.
2. The most recent investigation is that of Mr Beauchamp Tower, under-
taken at the instance of the Institution of Mechanical Engineers. Mr Tower's
first report was published November, 1883, and his second report in 1884
(Proc. Inst. Mechanical Engineers}.
52] ON THE THEORY OF LUBRICATION, ETC. 229
In these reports Mr Tower, making no attempt to formulate, states the
results of experiments apparently conducted with extreme care and under
very various and well-chosen circumstances. Those results which were
obtained under the ordinary conditions of lubrication so far agree with
the results of previous investigators as to show a want of any regularity.
But one of the causes of this want of regularity, irregularity in the supply of
the lubricant, appears to have occurred to Mr Tower early in his investiga-
tion, and led him to include amongst his experiments the unusual circum-
stances of surfaces completely immersed in oil. This was very fortunate, for
not only do the results so obtained show a great degree of regularity, but while
making these experiments he was accidentally led to observe a phenomenon
which, taken with the results of his experiments, amounts to a crucial proof
that in these experiments with the oil bath the surfaces were completely and
continuously separated by a film of oil ; this film being maintained by the
motion of the journal, although the pressure in the oil at the crown of the
bearing was shown by actual measurement to be as much as 625 Ibs. per
sq. inch above the pressure in the oil bath.
These results obtained with the oil bath are very important, notwith-
standing that the condition is not common in practice. They show that with
perfect lubrication a definite law of variation of the friction with the pressure
and velocity holds for a particular journal and brass. This strongly implies
that the irregularity previously found was due to imperfect lubrication.
Mr Tower has brought this out: — Substituting for the bath an oily pad,
pressed against the free part of the journal, and making it so slightly greasy
that it was barely perceptible to the touch, he again found considerable
regularity in the results ; these, however, were very different from those with
the bath. Then with intermediate lubrication he obtained intermediate
results, of which he says : — " Indeed, the results, generally speaking, were so
uncertain and irregular that they may be summed up in a few words. The
friction depends on the quantity and uniform distribution of the oil, and may
be anything between the oil bath results and seizing, according to the
perfection or imperfection of the lubrication."
3. On reading Mr Tower's report it occurred to the author as possible
that, in the case of the oil bath, the film of oil might be sufficiently thick for
the unknown boundary actions to disappear, in which case the results would
be deducible from the equations of hydrodynamics. Mr Tower appears to
have considered this, for he remarks that according to the theory of fluid
friction the resistance would be as the square of the velocity, whereas in his
results it does not increase according to this law. Considering how very
general the law of resistance as the square of the velocity is with fluids,
there is nothing remarkable in the assumption of its holding in such a case.
230 ON THE THEORY OF LUBRICATION [52
But the study of the behaviour of fluid in very small channels, and par-
ticularly the recent determination by the author of the critical velocity
at which this law changes from that of the square of the velocity to that
of the simple ratio, shows that with such highly viscous fluids as oils, such
small spaces as those existing between the journal and its bearing, and such
limited velocity as that of the surface of the journal, the resistance would
vary, cceteris paribus, as the velocity. Further, the thickness of the oil film
would not be uniform and might be affected by the velocity, and as the
resistance would vary, cceteris paribus, inversely as the thickness of the film,
the velocity might exert in this way a secondary effect on the resistance ; and,
further still, the resistance would depend on the viscosity of the oil, and this
depends on the temperature. But as Mr Tower had been careful to make all
his experiments in the same series with the journal at a temperature of
90° Fahr., it did not at first appear that there could be any considerable
temperature effect in his results.
4. The application of the hydrodynamical equations to circumstances
similar in so far as they were known to those of Mr Tower's experiments, at
once led to an equation between the variation of pressure over the surface
and the velocity, which equation appeared to explain the existence of the
film of oil at high pressure. This equation was mentioned in a paper read
before Section A. of the British Association at Montreal, 1884. It also
appears from a paragraph in the President's Address (Brit. Assoc. Rep., 1884,
p. 14) that Professor Stokes and Lord Rayleigh had simultaneously arrived at
a similar result. At that time the author had no idea of attempting its
integration. On subsequent consideration, however, it appeared that the
equation might be transformed so as to be approximately integrated, and the
theoretical results thus definitely compared with the experimental.
5. The result of this comparison was to show that with a particular
journal and brass, the mean thickness of the film of oil would be sensibly
constant, and hence, if the viscosity was constant, the resistance would
increase directly as the speed. As this was not in accordance with Mr Tower's
experiments, in which the resistance increased at a much slower rate, it
appeared that either the boundary actions became sensible, or that there
must have been a rise in the temperature of the oil which had escaped the
thermometers used to measure the temperature of the journal.
That there would be some excess of temperature in the oil film, on which
all the work of overcoming the friction is spent, is certain ; and after carefully
considering the means of escape of this heat, it seems probable that there
would be a difference of several degrees between the oil bath and the film
of oil.
52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 231
This increase of temperature would be attended by a diminution of
viscosity, so that, as the resistance and temperature increased with the
velocity, the viscosity would diminish and cause a departure from the simple
ratio.
6. In order to obtain a quantitative estimate of these secondary effects,
it was necessary to know exactly the relation between the viscosity and
temperature of the lubricant used. For this purpose an experimental deter-
mination was made of the viscosity of olive oil at different temperatures as
compared with the known viscosity of water. From the results of these
experiments an empirical formula has been deduced, by means of which
definite expressions have been obtained for the approximate variation of the
viscosity with the speed and load. Taking these variations of viscosity into
account, the results obtained from the hydrodynamical theory are brought
into complete accordance with these experiments of Mr Tower. Thus we
have not only an explanation of the very novel phenomena brought to light
by these experiments, and what appears to be an important verification of
the assumptions on which the theory of hydrodynamics is founded, but we
also find, what is not shown in the experiments, how the various circum-
stances under which the experiments have been made affect the results.
7. Two circumstances particularly are brought out in the theory as
principal circumstances which seem to have hitherto entirely escaped notice,
even that of Mr Tower.
One of these is the difference in the radii of the journal and of the brass
or bearing.
It is well known that the fitting between the journal and its bearing
produces a great effect on the carrying power of the journal, but this fitting
is rather supposed to be a matter of smoothness of surface than a degree of
correspondence in radii. The radius of the bearing must always be as much
larger than that of the journal as is necessary to secure an easy fit ; but
more than this, I think, has never been suggested.
Now it appears from the theory that if viscosity were constant the friction
would be inversely proportional to the difference in radii of the journal and
the bearing, and this although the arc of contact is less than the semicircum-
ference. Taking the temperature into account, it appears, from the com-
parison of the theoretical results with the experimental, that at a temperature
of 70°'5 Fahr. the radius of one of the brasses used was '00077 inch greater
than that of the journal, while at a temperature of 70° Fahr. that of the
other was '00084 inch, or 9 per cent, larger than the first.
These two brasses were probably both bedded to the journal in the same
way, and had neither of them been subjected to any great amount of wear, so
232 ON THE THEORY OF LUBRICATION [52
that there is nothing surprising in their being so nearly the same fit. It
would be extremely interesting to find whether prolonged wear of the brass
tends to preserve or destroy the fit. This does not appear from Mr Tower's
experiments. It does appear, however, that with an increase of temperature
the brass expands more than the journal, and that its radius increases as the
load increases in a very definite manner.
Another circumstance brought out by the theory, and remarked on both
by Lord Rayleigh and the author at Montreal, but not before expected,
is that the point of nearest approach of the journal to the brass is not by any
means in the line of the load, and, what is still more contrary to common
supposition, is on the off* side of the line of load.
This circumstance, the reason for which is rendered perfectly clear by the
conditions of equilibrium, at once accounts for a singular phenomenon
mentioned by Mr Tower, viz., that the journal having been run in one
direction until the initial tendency to heat had entirely disappeared, on being
reversed it immediately began to heat again ; but this effect stopped when
the process had been often repeated. The fact being that running in one
direction the brass had been worn to the journal only on the off side for that
direction, so that when the motion was reversed the new off side was like
a new brass.
7 A. The circumstances which determine the greatest load which a
bearing will carry with complete lubrication, i.e., with the film of oil extend-
ing between brass and journal throughout the entire arc, are definitely
shown in the theory.
The effect of increasing the load beyond a certain small value, being to
cause the brass to approach nearer to the journal at a point H, which moves
Fig. 1.
from A towards 0 as the load increases, and when the load is such that the
least separating distance is about half the difference of radii, the angular
* " On" and "off" sides of the line of load are used by Mr Tower to express respectively the
sides of approach and succession, as B and A in the figure, the arrow indicating the direction of
rotation.
52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 233
position of H is 40° to the off side of 0, the middle of the brass. At this
point the pressure in the oil film is everywhere greater than at A and B, the
extremities of the brass, but when the load further increases the pressure
towards A on the off side becomes smaller or negative. This, when sufficient,
will cause rupture in the oil film, which will then only extend between the
brass and journal over a portion of the whole arc, and a smaller portion as the
load increases. Thus, since the amount of negative pressure which the oil will
bear, depends on circumstances which are uncertain, the limit of the safe load
for complete lubrication is that which causes the least separating distance to
be half the distance of radii of the brass and journal.
The rupture of the oil film does not take place at the point of nearest
approach, and hence the brass may still be entirely separate from the journal,
and could the integrations be effected it would be possible to deal as
definitely with this condition as with that of complete lubrication ; but these
difficulties have limited the actual application of the theory to complete
lubrication. This however by no means requires an oil bath, but merely
sufficient oil on the journal.
What happens when the supply of oil is limited, i.e., insufficient for com-
plete lubrication, cannot be definitely expressed without further integrations;
but sufficient may be seen to show that the brass will still be completely
separated from the journal, although the separating film will not touch the
brass, except over a limited area; but in this case it is easy to show by
general reasoning that in the one extreme, where the supply of oil is limited,
the friction increases directly as the load and is independent of the velocity,
while in the other, where the oil is abundant, the circumstances are those of
the oil bath.
The effect of the limited length of the journal is also apparent in the
equations, as is also the effect of necking the shaft to form the journal,
so that the ends of the brass are against flanges on the shaft.
The theory is perfectly applicable to cases in which the direction of the
load on the bearing varies, as with the crank pin and with the bearings
of the crank shaft of the steam-engine ; but these cases have not been
considered, as there are no definite experiments to compare.
8. Although in the main the present investigation has been directed to
the circumstances of Mr Tower's experiments, viz., a cylindrical journal
revolving in a cylindrical brass, it has, on the one hand, been found necessary
to proceed from the general equations of equilibrium of viscous fluids, and,
on the other hand, to consider somewhat generally the physical property of
viscosity and its dependence on temperature.
234 ON THE THEORY OF LUBRICATION [52
The property of viscosity has been discussed at length in Section II. ;
which section also contains the account of an experimental investigation as
to the viscosity of olive oil.
The general theory deduced from the hydrodynamical equations for
viscous fluids, with the methods of application, is given in Sections IV., V.,
VI, VII., and VIII.
As there are some considerations which cannot be taken into account in
the more general method, which method also tends to render obscure the
more immediate purpose of the investigation, a preliminary discussion of the
problem, illustrated by aid of the graphic method, has been introduced as
Section III. Finally, the definite application of the theory to Mr Tower's
experiments is given in Section IX.
SECTION II. — THE PROPERTIES OF LUBRICANTS.
9. The Definition of Viscosity.
In distinguishing between solid and fluid matter, it is customary to define
fluid as a state of matter incapable of sustaining tangential or shearing stress.
This definition, however, as is well known, is only true as applied to actual
fluids when at rest. The resistance encountered by water and all known
fluids flowing steadily along parallel channels, affords definite proof that
in certain states of motion all actual fluids will sustain shearing stress.
These actual fluids are, therefore, called in the language of mathematics
imperfect or viscous fluids.
In order to obtain the equations of motion of such fluids, it has been
necessary to define clearly the property of viscosity. This definition has
been obtained from the consideration that to cause shearing stress in a body
it is necessary to submit it to forces tending to change its shape. Forces
tending to cause a general motion, whether linear, revolving, uniform expan-
sion, or uniform contraction, call forth no shearing stress.
Using the term distortion to express change of shape, apart from change
of position, uniform expansion, or contraction, the viscosity of a fluid is
defined as the shearing stress caused in the fluid while undergoing distor-
tion, and the shearing stress divided by the rate of distortion is called the
coefficient of viscosity, or, commonly, the viscosity of the fluid.
This is best expressed by considering a mass of fluid bounded by two
parallel planes at a distance a, and supposing the fluid between these planes
52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 235
to be in motion in a direction parallel to these surfaces with^ a_ velocity
which varies uniformly from 0 at one of these surfaces to u at the other.
Then the rate of distortion is
u
a
and the shearing stress on a plane parallel to the motion is expressed by
/* being the coefficient of viscosity or the modulus of the resistance to distor-
tional motion.
10. The Character of Viscosity.
In dealing with ideal fluids, it is of course allowable to consider p as
being zero or having any conceivable value ; but practically, as regards
natural philosophy, the value of any such considerations depends on whether
the calculated behaviour of the ideal fluid is found to agree with the
behaviour of the actual fluids — whether taking a particular fluid, a value of
fj, can be found such that the values of f calculated by equation (1) agree with
the values off determined by experiments for all values of a and u.
In the mathematical theory of viscous fluid, //, is assumed to be constant
for a particular fluid. This supposition is sometimes justified by reference
to some assumed dynamical constitution of fluids ; but apart from such
hypotheses there is no more ground for supposing a constant value for /*
than there is for supposing a particular law of gravitation, in other words,
there is no ground at all. If a particular value of //. is found to bring the
calculated results into agreement with all experimental results, then this
value of /j, defines a property of actual fluids, and of course it has been with
this object that the mathematical theory of ^ has been studied.
The chief question as regards //, is a simple one — within a particular
fluid is fji constant ? In other words, is viscosity a property of a fluid like
inertia which is independent of its motion ? If it is, our equations may be
useful ; if it is not then the introduction of p, into the equations renders
them so complex that it is almost hopeless to expect anything from them.
Another question of scarcely less practical importance relates to the
character of //, near the bounding surfaces of the fluid. If /u, is constant
in the fluid, does it change its value near the boundary of the fluid ? Is
there anything like slipping between the fluid and a solid boundary with
which it is in contact ?
As regards the answers to these questions the present position is some-
what as follows : —
236 ON THE THEORY OF LUBRICATION [52
11. The Two Viscosities.
The general experience that the resistance varies as the square of the
velocity is an absolute proof that /JL is not constant unless a restricted meaning
be given to the definition of viscosity, excluding such part of the resistance
as may be due, in the way explained by Prof. Stokes*, to internal eddies or
cross streams, however insensible these may be, so long as they are not simply
molecular motions.
On the other hand in the definite experiments made by Colomb, and
particularly by Poiseuille, it was found that the resistance was proportional
to the velocity, and therefore that //, was absolutely constant — i.e., independent
of the velocity f.
To meet this discordance it has been supposed that //. varied with the
rate of distortion — i.e., is a function of u/a, but is sensibly constant when
u/a is small | .
To assume this, however, is to neglect Poiseuille's experiments, in which
he found for water the resistance absolutely proportional to the velocity in
a tube '6 mm. diameter up to a velocity of 6 metres per second, which
corresponds to a value of u/a = 20,000.
On the other hand it is found by Darcy § and others in large tubes that
ni
the resistance varies as the square of the velocity for values of - , as low
as 1. Thus in a tube of *6 mm. we have p. constant for all rates of distor-
tion below 20,000, while in a tube of 500 mm. diameter p is a function of
the distortion for all values greater than 1.
It is, therefore, clear that if /i is a function of the distortion it must also
be a function of the dimensions of the channels, and in that case /j, cannot
be considered as a property of the fluid only.
The change in the law of resistance from the simple ratio has, however,
been shown by the author to be due to a change in the character of the
motion of the fluid from that of direct parallel motion to that of sinuous
or eddying motion ||.
* Stokes's Reprint, vol. i., p. 99.
t Paris Mem. Savans Etrang., torn. 9 (1846), p. 434.
I Lamb's Motion of fluids, 1879, Art. 180.
§ Recherches Kxpl. Paris, 1852.
|| "An Experimental Investigation of the circumstances which determine whether the Motion
of Water shall be Direct or Sinuous." Phil. Trans., vol. 174 (1883), p. 935.
52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 237
In the latter case, although the mean motion at any point taken over a
sufficient time is parallel to the pipe, it is made up of a succession of motions
crossing the pipe in different directions.
The question as to whether, in the case of sinuous motion, p is to be
considered as a function of the velocity or not, depends on whether we
regard f as expressing the instantaneous shearing stress at a point, or the
mean over a sufficient time. Whether we regard the symbols in the
equations of motion as expressing the instantaneous motion or the mean
taken over a sufficient time.
If the latter, then /* must be held to include, in addition to the mean
stress, the momentum per second parallel to u carried by the cross streams
in the negative direction across the surface over which / is measured.
If, however, we regard the motion at each instant, then we must restrict
our definition of viscosity by making f the instantaneous value of the
intensity of resistance at a point.
This is a quantity which we have and can have no means of measuring
except under circumstances which secure that f is constant for all points
over a given surface, and for all instants over a given time.
It thus appears that there are two essentially distinct viscosities in fluids.
The one a mechanical viscosity arising from the molar motion of the fluid,
the other a physical property of the fluid. It is worth while to point out
that, although the conditions under which the first of these — the mechanical
viscosity — can exist, depend primarily on the physical viscosity, the actual
magnitudes of these viscosities are independent, or are only connected in a
secondary manner. This is shown by a very striking but little noticed fact.
When the motion of the fluid is such that the resistance is as the square of
the velocity, the magnitude of this resistance is sensibly quite independent
of the character of the fluid in all respects except that of density. Thus,
when in a particular pipe the velocity of oil or treacle is sufficient for the
resistance to vary as the square of the velocity, the resistance is practically
the same as it would be with water at the same velocity, while the physical
viscosity of water is more than one hundred times less.
The answer, then, to the question as to the constancy of /z, may be clearly
given — IJL measures a physical property of the fluid which is independent of
its motion. But in this sense //, is the coefficient of instantaneous resistance
to distortion at a point moving with the fluid.
This restriction is equivalent to restricting the applications of the
equations of motion for a viscous fluid to the cases in which there are no
eddies or sinuosities.
238 ON THE THEORY OF LUBRICATION [52
This, as shown by the author, is the case in parallel channels so long as
the product of the velocity, the width of the channel, and the density of the
fluid divided by /* is less than a certain constant value. In a round tube
this constant is 1400, or
At a temperature of 50°, we have, with a foot as unit of length, for water —
^ = 0-00001428,
P
Dv < '02,
so that if D, the diameter of the channel, be '001 inch, v would have to
be at least 240 feet per second for the resistance to vary other than as
the velocity.
As regards the slipping at the boundaries, Poiseuille's experiments, as
well as those of the author, failed to show a trace of this, although f
reached the value of 0702 Ib. per square inch, so that within this limit it
may be taken as proved that there is no slipping between any solid surface
and water. With other fluids, such as mercury in glass tubes, it is possible
that the case may be different ; but, as regards oils, the probability seems
to be that the limit within which there is no slipping will be much higher
than with water.
12. Experimental Determination of the Value of p for Olive Oil.
Since the value of p for water is known for all moderate temperatures,
in order to obtain the value for oil it is only necessary to ascertain the
relative times taken by the same volumes of oil and water to flow through
the same channel, care being taken to make the channel such that there are
no eddies and that the energy of motion is small compared with the loss
of head.
These times are proportional to
t
P'
where p is the fall of pressure ; therefore the times multiplied by the
respective falls of pressure are proportional to the viscosities.
The arrangement of apparatus used is shown in Fig. 2.
The test tube (A) containing the fluid to be tested was fixed in a beaker
of water, which was heated from below and maintained at any required
temperature.
52]
AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS.
239
A syphon (B), made of glass tube T% inch internal diameter, with the
extremity of its short limb drawn down to capillary size for aT length of
about 6 inches, this six inches being bent up and down so as only to occupy
Fig. 2.
some 2 inches at the bottom of the test tube. The long limb of the syphon
extended to about 2 feet below the mean level of the fluid in the test tube.
Two marks on the test tube at different levels served to show when a
definite volume had been withdrawn.
The syphon used was the same for each set of experiments on oil and
water, so that the pressure urging the fluid through the tube was propor-
tioned to the density of the fluids — that is, it was 1'915 as great for oil
as water, disregarding the effect of the variation of temperature on volume,
which in no case amounted to 1 per cent.
Experiments were first made with water at different temperatures, the
times taken for the water to fall from the first mark to the second being
carefully noted. The syphon was then dried and replaced and oil substituted
for water.
Two sets of similar apparatus were used on different occasions,
different samples of oil being used. In the first set the experiments
on oil were made at temperatures from 95° to 200° Fahr.; in the second
set, from 61° to 120° Fahr. In so far as the temperatures overlapped, the
240
ON THE THEORY OF LUBRICATION
[52
viscosities for the two oils agreed to within 4 per cent., but as the law
of variation of the viscosity seemed to change rapidly at about 140° Fahr.,
only the second set have been recorded. These are shown in Table I.
TABLE I. — VISCOSITY OF OIL COMPARED WITH WATER: 11 April, 1884.
Temperature.
M
Number.
Third.
Time
seconds.
107
experi-
mental.
experi-
mental.
log /j.
calculated.
107
calculated.
Fahrenheit.
Centigrade.
1
Water .
60
15-5
25
1-640
2
»
M
„
M
„
3
j>
M
4
Olive oil
61
16
2040
123-00
5-08990
5-090133
123-06
5
81
1350
81-00
6-90848
6-89807
79-08
6
M
94
1000
60-00
6-77815
6-78290
59-34
7
»
120
555
33-40
6-52375
6-52375
33-40
From Poiseuille's experiments it is found that, measuring viscosity in
pounds on the square inch, for water at a temperature of 61° Fahr.,
/i=10-7x 1-61.
Adopting this value of yu, for the experiments on water at 61° Fahr., the
other experimental values of fi for water at different temperatures, obtained
as being in the ratios of the times, were found to be in very close agreement
with those calculated from Poiseuille's law for the respective temperatures.
This tested the efficiency of the apparatus. It has not been thought necessary
to record any experiment on water except at the temperature of 61° Fahr.
The experimental value of p, for oil are in the ratios of the times multi-
plied by '915, the specific gravity of oil ; these are given as the experimental
values of //- in the table. Another column contains the values of /z for oil,
calculated from an empirical formula fitted to the experimental values.
This formula was found by comparing the logarithms of the experimental
values of \i. It appeared that the differences in these logarithms were
nearly proportional to the differences in the corresponding temperatures,
or that T being temperature in degrees Fahr.,
log ,*!- log ,*, = -0096 (Z'.-r,),
in degrees Centigrade
log K - log ^ = -00535 (T, - rl\) ;
whence since '0096 = '0021 Iog10e,
•00535 = -0123 Iog10e,
52]
AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS.
241
for degrees Fahrenheit
for degrees Centigrade
1 _ — •0221(2', -
1 — g--oi23(r,-r2)
.(2).
This ratio holds well within the experimental accuracy from temperatures
ranging from 61° to 120° Fahr. This is shown in the table, and again in
Fig. 3, in which the ordinates are proportional to log/i, the abscissae being
proportional to the corresponding temperatures.
5.1
5.0
6.9
6.8
6.G
6.5
70
80 90 100
Temperature Fahrenheit
Fig. 3.
110
120
13. The Comparative Values of fi for Different Fluids and
Different Systems of Units.
The values of p, given by different writers for air and water, are ex-
pressed in various units of force and length, so that it is a matter of some
trouble to compare them. To facilitate this for the future comparative
values are here given. Those for water have been deduced from Poiseuille's
formula, for air from Maxwell's formulae, and for olive oil from the experi-
ments recorded in the previous article.
The units of length, mass, and time, being respectively the centimetre,
gramme, and second, in which case the unit of force is the weight of one
980'oth (g) part of a gramme, expressing temperature in degrees Centigrade
by T and putting
o. R. ii.
0-03367932' +0-0002209936772 (3),
16
242 ON THE THEORY OF LUBRICATION [52
for water . . . /a = (M)177931P ]
air. . . . /* = 0-0001879 (1 + 0'00366T)[ (4).
olive oil . . /* = 3'2653e-'01232'* j
With the same unit of length, but g grammes as unit of mass and
1 gramme as unit of force, the values of yu. are for
water .../* = O'OOOOISIP \
air. . . . fi = 0-00000019153 (1 + -0036627) L (5).
olive oil . . p, = 0-0033303e-°123r j
The units of length and mass being the foot and pound and the
temperature in degrees Fahr. for
water . . . /* = 0'0011971P
air. . . . fi = 0-000011788 (1 + -0020274T) (6).
olive oil . . ^ = 0-21943e-022ir
With the same unit of length the unit of mass being g (32'1695) Ibs.
and the unit of force 1 Ib. for
water .../*- 0'000037166P \
air.-. . . /* = 0-00000036645(1 + '002074^)1 (7).
olive oil . . fi = 0-00682 13e-'022ir J
Taking the unit of length 1 inch and the unit of force 1 Ib. for
water . . . p = 0'000000258105P \
air. . . . p = 0-0000000025447 (1 + '0020274T) I (8).
olive oil . . ^ = 0-00004737e->0221JI j
SECTION III. — GENERAL VIEW OF THE ACTION OF LUBRICATION.
14. Tlie case of two nearly Parallel Surfaces separated by a Viscous
Fluid.
Let AB and CD (Fig. 4) be perpendicular sections of the surfaces,
CD being of limited but of great extent compared with the distance h
* For olive oil the values of /* have only been tested between the limits of temperature 16° and
49° C. or 61° and 120° Fahr.
52]
AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS.
243
between the surfaces, both surfaces being of unlimited length in_a direction
perpendicular to the paper.
Fig. 4.
Case 1. Parallel Surfaces in Relative Tangential Motion. — In Fig. 5
the surface CD is supposed fixed, while AB moves to the left with a velo-
city U.
Then by the definition of viscosity (Art. 9) there will be a tangential
resistance
W U
F==fJ'J'
and the tangential motion of the fluid will vary uniformly from U at AB
to zero at CD. Thus if FG (Fig. 5) be taken to represent U, then PN will
represent the velocity in the fluid at P.
Fig. 5.
The slope of the line EG therefore may be taken to represent the
force F, and the direction of the tangential force on either surface is the
same as if EG were in tension. The sloping lines therefore represent
the condition of motion and stress throughout the film (Fig. 5).
Case 2. Parallel Surfaces approaching with no Tangential Motion. —
The fluid has to be squeezed out between the surfaces, and since there is
no motion at the surface, the horizontal velocity outward will be greatest
half-way between the surfaces, nothing at 0 the middle of CD, and greatest
at the ends.
If in a certain state of the motion (shown by dotted line, Fig. 6) the
space between AB and CD be divided into 10 equal parts by vertical lines
(Fig. 6, dotted figure), and these lines be supposed to move with the fluid,
16—2
244
ON THE THEORY OF LUBRICATION
[52
they will shortly after assume the positions of the curved lines (Fig. 6), in
which the areas included between each pair of curved lines is the same as
Fig. 6.
Ill
the dotted figure. In this case, as in Case 1, the distance QP will
represent the motion at any point P, and the slope of the lines will represent
the tangential forces in the fluid as if the lines were stretched elastic strings.
It is at once seen from this that the fluid will be pulled towards the middle
of CD by the viscosity as though by the stretched elastic lines, and hence
that the pressure will be greatest at 0 and fall off towards the ends
C and D, and would be approximately represented by the curve at the
top of the figure.
Case 3. Parallel Surfaces approaching with Tangential Motion. — The
lines representing the motions in Cases 1 and 2 may be superimposed by
adding the distances PQ in Fig. 6 to the distances PN in Fig. 5.
The result will be as shown in Fig. 7, in which the lines represent in
the same way as before the motions and stresses in the fluid where the
surfaces are approaching with tangential motion.
In this case the distribution of pressure over GD is nearly the same
as in Case 2, and the mean tangential force will be the same as in Case 1.
The distribution of the friction over CD will, however, be different. This
is shown by the inclination of the curves at the points where they meet
the surface. Thus on CD the slope is greater on the left and less on the
AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS.
245
right, which shows that the friction will be greater on the left ancL less on
the right than in Case 1. On AB the slope is greater on the right and
less on the left, as is also the friction.
Case 4. Surfaces inclined with Tangential Movement only. — AB is in
motion as in Case 1, and CD is inclined as in Fig. 8.
The effect in this case will be nearly the same as in the compound
movement (Case 3).
For if corresponding to the uniform movement U of AB, the velocity
of the fluid varied uniformly from the surface AB to CD, then the quantity
carried across any section PQ would be
U
PQx
2'
and consequently would be proportional to PQ; but the quantities carried
across all sections must be the same, as the surfaces do not change their
relative distances ; therefore there must be a general outflow from any
vertical sections PQ, P'Q' given by
f (PQ-P'Q'X
This outflow will take place to the right and left of the section of greatest
pressure. Let this be PjQ^ then the flow past any other section PQ is
f (PQ-PM
to the right or left according as PQ is to the right or left of PjQj. Hence
at this section the motion will be one of uniform variation, and to the
right and left the lines showing the motion and friction will be nearly
as in Fig. 7. This is shown in Fig. 9.
This is the explanation of continuous lubrication.
The pressure of the intervening film of fluid would cause a force tending
to separate the surfaces.
246 ON THE THEORY OF LUBRICATION [52
The mean line or resultant of this force would act through some point 0.
This point 0 does not necessarily coincide with P1? the point of maximum
pressure.
Fig. 9.
For equilibrium of the surface AB, 0 will be in the line of the resultant
external force urging the surfaces together, otherwise the surface ACD would
change its inclination.
The resultant pressure must also be equal to the resultant external force
perpendicular to AB (neglecting the obliquity of CD). If the surfaces were
free to approach the pressure would adjust itself to the load, for the nearer
the surfaces the greater would be the friction and consequent pressure for
the same velocity, so that the surfaces would approach until the pressure
balanced the load.
As the distance between the surfaces diminished 0 would change its
position, and therefore, to prevent an alteration of inclination, the surface
CD must be constrained so that it could not turn round.
It is to be noticed that continuous lubrication between plane surfaces
can only take place with continuous motion in one direction, which is the
direction of continuous inclination of the surfaces.
With reciprocating motion, in order that there may be continuous lubri-
cation, the surfaces must be other than plane.
15. Revolving Cylindrical Surface.
When the moving surface AB is cylindrical and revolving about its axis,
the general motion of the film will differ somewhat from what it is with
flat surfaces.
Case 5. Revolving Motion, CD flat and symmetrically placed. — The
surface velocity of AB may be expressed by U as before. The curves of
motion found by the same method as in the previous cases are shown in
Fig. 10.
52]
AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS.
247
The curves to the right of GH, the shortest distance between the surfaces,
will have the same character as those in Fig. 9 to the right of G, at which is
also the shortest distance between the surfaces.
Fig. 10.
On the left of GH the curves will be exactly similar to those on the right,
only drawn the other way about, so that they are concave towards a section
at P2 in a similar position on the left to that occupied by Pj on the right.
This is because a uniformly varying motion would carry a quantity of
fluid proportional to the thickness of the stratum from right to left, and
thus while it would carry more fluid through the sections towards the right
than it would carry across GH, necessitating an outward flow from the
position Pl in both directions, the same motion would carry more fluid away
from sections towards G than it would supply past GH, thus necessitating
an inward flow towards the position P2.
Since G is in the middle of CD these two actions, though opposite, will
be otherwise symmetrical, and
From the convexity of the curves to the section at P2 it appears that this
section would be one of minimum pressure, just as Pt is of maximum. Of
course this is supposing the lubricant under sufficient pressure at C and D
to allow of the pressure falling. The curve of pressure would be similar to
that at the top of Fig. 10, in which C and D are points of equal pressure,
P1HP.2 the singular points in the curve.
Under such conditions the fluid pressure acts to separate the surfaces on
the right, but as the pressure is negative on the left the surfaces will be
248
ON THE THEORY OF LUBRICATION
[52
drawn together. So that the total effect will be to produce a turning
moment on the surface AB.
Case 6. The same as Case 5, except that 0 is not in the middle of CD. —
In this case the curves of motion will be symmetrical on each side of H at
equal distances, as shown in Fig. 11.
Fig. 11.
If C lies between H and P2 the pressure will be altogether positive, as
shown by the curve above Fig. 11 — that is, will tend to separate the surfaces.
16. The Effect of a Limiting Supply of Lubricating Material.
In the cases already considered C and D have been the actual limits of
the upper surface. If the supply of lubricant is limited C and D may be
the extreme points to which the separating film reaches on the upper surface,
which may be unlimited, as in Fig. 12.
Case 7. Supply of Lubricant Limited. — If the surface AB be supposed
to have been covered with a film of oil, the oil adhering to the surface and
moving with it, then the surface CD to have been brought up to a less
distance than that occupied by the film of oil, the oil will accumulate as
it is brought up by the motion of AB, forming a pad between the surfaces,
particularly on the side D.
The thickness of the film as it leaves the side C being reduced until the
whole surface AB is covered with a film of such thinness that as much leaves
at C as is brought up to D, then the condition will be steady.
52]
AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS.
249
Putting b for the thickness of the film of oil outside the pad, the quantity
of oil brought up to D by the motion of this film will be per second
bU,
Fig. 12.
and the quantity which passes the section P1Q1, across which the velocity
varies uniformly, will be
2
Therefore since there is no further accumulation
also, since GP2=GP, (Fig. 10, Case 5)
P2Q2 = 26.
And since the quantity which passes P2Q2 will not be sufficient to occupy
the larger sections on the left, the fluid will not touch the upper surface to
the left of P2. The limit will therefore be at P2, the fluid passing away with
AB in a film of thickness 6.
This is the ordinary case of partial lubrication : AB, the surface of the
journal, is covered with a film of oil ; CD, the surface of the brass or bearing,
is separated from AB by a pad of oil near H, the point of nearest approach.
This pad is under pressure, which is a maximum at Plt and slopes away
to nothing at D and P2, the extremities of the pad, as is shown by the curve
above, Fig. 1 2.
250 ON THE THEORY OF LUBRICATION [52
17. The Relation between Resistance, Load, and Speed for Limited
Lubrication.
In Case 7 a definite quantity of oil must be in the film round the journal,
or in the pad between the surfaces. As the surfaces approach, the pad will
increase and the film diminish, and vice versd. The resistance increases with
the length of the pad, and with the diminution of the distance between
the surfaces. The mean intensity of pressure increases with the length of
the pad, and inversely with the thickness of the film, but not in either case
in the simple ratio. The total pressure, which is equal to the load, increases
with the intensity of pressure and the length of the pad.
The definite expressions of these relations depend on certain integrations,
which have not yet been effected. From the general relations pointed out,
it follows that an increase of load will diminish HG and PiQlt and con-
sequently the thickness of the film round the journal, and will increase the
length of the pad. It will therefore increase the friction.
Thus with a limited supply of oil the friction will increase with the load
in some ratio not precisely determined.
Further, both the friction and the pressure increase in the direct ratio of
the speed, provided the distance between the surfaces and the length of the
pad remains constant ; then, if the load remains constant, the thickness of
the film must increase, and the length of the pad diminish with the speed ;
and both these effects will diminish friction in exactly the same ratio as the
reduction of load diminishes friction.
Thus if with a speed U a load W and friction F a certain thickness of oil
is maintained, the same will be maintained with a speed MU, a load MW,
and the friction will be MF.
How far this increase of friction is to be attributed to the increased
velocity, and how far to the increased load, is not yet shown in the theory
for this case ; but, as has been pointed out, if the load be altered from M W
to W, the velocity remaining the same, the friction will be altered from MF
in the direction of F. Therefore, with the load constant, it does appear
from the theory that the friction will not increase as the first power of the
velocity.
There is nothing therefore in this theory contrary to the experience that,
with very limited lubrication, the friction is proportional to the load and
independent of the velocity, while the theoretical conclusion that the friction,
with any particular load arid speed, will depend on the supply of oil in the
pad, is in strict accordance with Mr Tower's conclusion, and with the general
disagreement of the coefficients of friction in different experiments.
52]
AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS.
251
17 A. The Conditions of Equilibrium with Cylindrical Surfaces.
So far CB has been considered as a flat surface, in which case the equili-
brium of CB requires that it should be so far constrained by external forces
that it cannot either change its direction or move horizontally.
When AB is a portion of a cylindrical surface, having its axis parallel to
that of AB, the only condition of constraint necessary for equilibrium is that
CB shall not turn about its axis. This will appear on consideration of the
following cases : —
Case 8. Surfaces Cylindrical and the Supply of Oil Limited. — Fig. 13
shows the surfaces AB and CD.
Fig. 13.
J is the axis of the journal A B.
I is the axis of the brass CD.
JL is the line in which the load acts.
0 is the point in which JL meets AB.
R = JP.
R + a = IQ.
h = PQ.
h0 = HG.
The condition for the equilibrium of 7 the centre of the brass is that the
resultant of the oil pressure on DC together with friction shall be in the
direction OL, and the magnitude of this resultant shall be equal to the load.
252 ON THE THEORY OF LUBRICATION [52
As regards the magnitude of this resultant, it increases as HG diminishes
to a certain limit, i.e., as the surfaces approach, so that in this respect equili-
brium is obviously secured, and it is only the direction of the resultant
pressure and friction that need be considered.
Since the fluid film is in equilibrium under the forces exerted by the two
opposite surfaces, these forces must be equal and opposite, so that it is only
necessary to consider the forces exerted by AB on the fluid.
From what has been already seen in Cases 6 and 7 it appears that the
resultant line of pressure JM always lies on the right or on side of GH. The
resultant friction clearly acts to the left, so that if JM be taken to represent
the resultant pressure and MN the resultant friction, N is to the left of M
and JN the resultant of pressure and friction is to the left of JM.
Taking LJ to represent the load, then LN will represent the resultant
moving force on GD that is on /. Since H will move in the opposite
direction to /, and since the direction of the resultant pressure moves in the
same direction as H, the effect of a moving force LN on / will be to move N
towards L until they coincide. Thus, as long as JM is within the arc
covered by the brass, a position of equilibrium is possible and the equilibrium
will be stable.
So far the condition of equilibrium shows that H will be on the left or
off side of the line of load, and this holds whether the supply of oil is
abundant or limited ; but while with a very limited supply of oil, i.e., a very
short oil pad, H must always be in the immediate neighbourhood of 0, this
is by no means the case as the length of the oil pad increases.
Case 9. Cylindrical, Surfaces in Oil Bath. — If the supply of oil is
sufficient, the oil film or pad between the surfaces will extend continuously
from the extremities of the brass, unless such extension would cause negative
pressure which might lead to discontinuity. In this case the conditions
of equilibrium determine the position of H.
The conditions of equilibrium are as before —
1. That the horizontal component of the oil pressure on the brass shall
balance the horizontal component of the friction ;
2. That the vertical components of the pressure and friction shall balance
the load.
Taking the surface of the brass, as is usual, to embrace nearly half the
circumference of the journal and, to commence with, supposing the brass to
be unloaded, the movement of H may be traced as the load increases.
When there is no load, the conditions of equilibrium are satisfied if the
52]
AND ITS APPLICATION TO MR B. TOWERS EXPERIMENTS.
253
position of H is such, that the vertical components of pressure and friction
are each zero, and the horizontal components are equal and opposite.
This will be when H is at 0 (Fig. 13); for then, as has been shown, Case 5,
the pressure on the left of H will be negative, and will be exactly equal to
the pressure at corresponding points on the right, so that the vertical com-
ponents left and right balance each other. On the other hand the horizontal
component of the pressure to the left and right will both act on the brass to
the right, and as these will increase as the surfaces approach, the distance JI
must be exactly such that these components balance the resultant friction,
which by symmetry will be horizontal and acting to the left.
It thus appears that when the brass is unloaded its point of nearest
approach will be its middle point. This position, together with the curves of
pressure, are shown in Fig. 14.
Fig. 14.
As the load increases, the positive vertical component on the right of GH
must overbalance the negative component on the left. This requires that H
should be to the left of 0.
It is also necessary that the horizontal components of pressure and
friction should balance.
254 ON THE THEORY OF LUBRICATION [52
These two conditions determine the position of H and the value of JI.
As the load increases it appears from the exact equations (to be discussed
in a subsequent article) that OH reaches a maximum value, which places H
nearly, but not quite, at the left extremity of the brass, but leaves JI still
small as compared with GH.
For a further increase of the load H moves back again towards 0.
In this condition the load has become so great that the friction, which
remains nearly constant, is so small by comparison that it may be neglected,
and the condition of equilibrium is that the horizontal component of the
pressure is zero, and the vertical component equal to the load.
H continues to recede as the load increases. But when H C becomes
greater than HP.2, the pressure between P2 arid C would become negative if
the condition did not break down by discontinuity in the oil, which is sure to
occur when the pressure falls below that of zero, and then the condition
becomes the same as that with a limited supply of oil.
This is important, as it shows that with extreme loads the oil bath comes
to be practically the same as that of a limited supply of oil, and hence that
the extreme load which the brass would carry would be the same in both
cases — as Mr Tower has shown it to be.
In all Mr Tower's experiments with the oil bath it appears that the
conditions were such that as the load increased H was in retreat from C
towards 0, and that, except in the extreme cases, P2 had not come up to C.
Figs. 2, 3, 4, show the exact curves of pressure as calculated by the
exact method to be given, for circumstances corresponding very closely with
one of Mr Tower's experiments, in which he actually measured the pressure
of oil at three points in the film. These measured pressures are shown by
the crosses.
The result of the calculations for this experiment is to show, what could
not indeed be measured, that in Mr Tower's experiment the difference in the
radii of the brass and journal at 70°, and a load of 100 Ibs. per square inch
was :
a = -00077
£#=•000375
(The angle) OJH = 48°.
18. The Wear and Heating of Bearings.
Before the journal starts the effect of the load will have brought the brass
into contact with the journal at 0. At starting the surfaces will be in
52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 255
contact, and the initial friction will be between solid surfaces, causing some
abrasion.
After motion commences the surfaces gradually separate as the velocity
increases, more particularly in the case of the oil bath, in which case at
starting the friction will be much the same as with a limited supply of oil.
As the speed increases according to the load, GH approaches, according
to the supply of oil, to a, and varies but slightly with any further increase of
speed ; so that the resistance becomes more nearly proportional to the speed
and less affected by the load.
When the condition of steady lubrication has been attained, if the
surfaces are completely separated by oil, there should be no wear. But
if there is wear, as it appears from one cause or another there generally is, it
would take place most rapidly where the surfaces are nearest : that is, at GH
on the off side of 0.
Thus while the motion is in one direction the tendency to wear the
surfaces to a fit would be confined to the offside of 0.
This appears to offer a very simple and well-founded explanation of the
important and common circumstance that new surfaces do not behave so
well as old ones ; and of the circumstance, observed by Mr Tower, that in
the case of the oil bath, running the journal in one direction does not prepare
the brass for carrying a load when the journal is run in the opposite direction.
This explanation, however, depends on the effect of misfit in the journal and
brass which has yet to be considered.
Case 10. Approximately cylindrical surfaces of limited length in the
direction of the axis of rotation. Nothing has so far been said of any
possible motion of the fluid perpendicular to the direction of motion and
parallel to the axis of the journal. It having been assumed that the surfaces
were truly cylindrical and of unlimited length in direction of their axes, and
in such case there would be no such flow.
But in practice brasses are necessarily of limited length, so that the oil
can escape from the ends of the brass. Such escape will obviously prevent
the pressure of the film of oil from reaching its full height for some distance
from the ends of the brass and cause it to fall to nothing at the extreme
ends.
This was shown by Mr Tower, who measured the pressure at several
points along the brass in the line through 0, and found it to follow a curve
similar to that, shown in Fig. 15, which corresponds to what might be
expected from escape at the free ends.
256
ON THE THEORY OF LUBRICATION
[52
If the surfaces are not strictly parallel in the directions TU and VW,
the pressure would be greatest in the narrowest parts, causing axial flow
from those into the broader spaces. Hence, if the surfaces were considerably
Journal
Fig. 15.
irregular, the lubricant would, by escaping into broader spaces, allow the
brass to approach and eventually to touch the journal at the narrowest
spaces, and this would be particularly the case near the ends.
As a matter" of fact, the general fit of two new surfaces can only be
approximate ; and how nearf the approximation is, is a matter of the time
and skill spent on preparing, or, as it is called, bedding them. Such bedding
as brasses are subject to would not bring them to a condition in which the
hills and hollows differed by less than a YIJOOO^^ Part °f an incn> so that two
such surfaces touching each other on the hills would have spaces as great as
of an inch between them. This seems a small matter, but not
a
when compared with the mean width of the interval between the brass and
the journal which, as will be subsequently shown, was less than -nfoffth °f
an inch.
It may be assumed, therefore, that such inequalities generally exist in
the surfaces of new brasses and journals. And as the surfaces according
to their material and manner of support yield to pressure the brass will
close on the journal at its ends, where, owing to the escape of oil, there is
no pressure to keep them separate.
rn'] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 257
The section of a new brass and journal taken at GH will therefore be,
if sufficiently magnified, as shown in Fig. 16, the thickness of the film,
Journal
Longitudinal Section
Fig. 16.
instead of being, say, of y^f^ths of an inch, varies from 0 to y^^ ths, and
is less at the ends than at the middle.
In this condition the wear will be at the points of contact, which will be
in the neighbourhood of GH on the off side of 0 (Fig. 13), so that, if the
journal runs in one direction only, the surfaces in the neighbourhood of GH
(on the off side) will be gradually worn to a fit, during which wear the
friction will be great and attended with heating, more or less, according to
the rate of wear and the obstruction to the escape of heat.
So long, however, as the journal runs in one direction only GH will be
on one side (the off side) of 0, and the wear will be altogether or mainly on
this side, according to the distance of H from 0.
In the meantime the brass on the on side is not similarly worn, so that
if the motion of the journal is reversed, and the point H transferred to the
late on side, the wear will have to be gone through again.
That this is the true explanation is confirmed if, as seems from Mr Tower's
report, the heating effect on first reversing the journal was much more
evident in the case of the oil bath.
For when the supply of oil is short, HG will be very small, and H will
be close to 0. So that the wearing area will probably extend to both sides
of 0, and thus the brass be partially, if not altogether, prepared for running
in the opposite direction.
When the supply of oil is complete, however, as has been shown, H is
.10 or 60° from 0, unless the load is in excess, so that the wear in the
neighbourhood of H on the one side of 0 would not extend to a point 100°
or 120° over to the other side.
Even in the case of a perfectly smooth brass, the running of the journal
under a sufficient load in one direction should, supposing some wear, ac-
cording to the theory render the brass less well able to carry the load when
running in the opposite direction. For, as has already appeared, the pressure
between the journal and brass depends on the radius of curvature of the
o. R. ii. 17
258
ON THE THEORY OF LUBRICATION
[52
brass on the on side being greater than that of the journal. If then the
effect of wear is to diminish the radius of the brass on the off side, so that
when the motion is reversed the radius of the new on side is equal to or less
than that of the journal, while the radius of the new off side is greater, the
oil pressure would not rise. And this is the effect of wear ; for as will
be definitely shown, the effect of the oil pressure is to increase the radius of
curvature of the brass, and as the centre of wear is well on the off side, the
effect of sufficient wear will be to bring the radius on this side, while the
pressure is on, more nearly to that of the journal, so that on the pressure
being removed, the brass on this side may resume a radius even less than
that of the journal.
SECTION IV. — THE EQUATIONS OF HYDRODYNAMICS AS APPLIED TO
LUBRICATION.
19. According to the usual method of expressing the stress in a viscous
fluid (which is the same as in an elastic solid)* :
Pzz=~P
Pxy = P,,x = /*
Pyz=pzy = /*
(Y\ — fr\ —
PZX — PXZ —
dv dw
dtK dy dz
du dv dw
du
dv du\]
dx dy 1
dw dv \
+ — -
K.
jly dz )
du dw\
dz ~dx ) ,
(10).
In which the left-hand members are the stresses on planes perpendicular to
the first suffix in directions parallel to the second, the first three being the
normal stresses, the last six the tangential stresses.
* Stokes, " On the Theories of the Internal Friction of Fluids in Motion, and of the Equi-
librium and Motion of Elastic Solids." — Trans. Cambridge Phil. Soc., vol. vni., p. 287. Also
reprint, vol. i., p. 84. Also Lamb's Motion of /•'/<*/</*, p. 219.
52]
AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS.
250
The values of these substituted in the equations of motion
811 v dpxx do,,~ dn,*.^
P s-7 = P-^- + ~ J 1"
ot ax
dy dz
Sv v
^=PY
dx
dpyy
dy
= pZ +
dx dy
dpzy
dz
dz
(11)
Sp _ fdu dv dw\
Bt ^ \dx dy dz )
give the complete equations of motion for the interior of a viscous fluid.
These equations involve terms severally depending on the inertia and the
weight of the fluid, also the variation of stress in the fluid.
In the case of lubrication the spaces between the solid surfaces are so
small compared with
U
that the motion of the fluid is shown to be free from eddies as already
explained (Art. 11). Also that the forces arising from weight and inertia
are altogether small compared with the stresses arising from viscosity.
The equations which result from the substitution from (9) and (10) in
the first 3 of (11) may therefore be simplified by the omission of the inertia
and gravitation terms, which are the terms involving p as a factor.
In the case of oil the remaining terms may still further be simplified by
omitting the terms depending on the compressibility of the fluid.
Also if, as is the case, /z, is nearly constant, the terms involving dp may
be omitted, or considered of secondary importance.
From equations (11) we then have
dp
dx* dy- dz
v\
f)
dp _ ,'d2w d*w <l-ir\
dz ~ ^ \dx* + df + <h- )
„ _ du dv dw
dx dy dz
Again, since in the case of lubrication we always have to do with a film
17—2
.(12).
260 ON THE THEORY OF LUBRICATION [52
of fluid between nearly parallel surfaces, of which the radii of curvature are
large compared with the thickness of the film, we may, without error,
disregard any curvature there may be in the surfaces, and put
x for distances measured on one of the surfaces in the direction of
relative motion,
z for distances measured on the same surface in the direction perpen-
dicular to relative motion,
y for distances measured everywhere at right angles to the surface.
Then, if the surfaces remain in their original direction, since they are
nearly parallel,
v will be small compared with u and w, and the variations of u and w
in the directions x and z are small compared with their variations in
the direction y.
The equations (12) for the interior of the film then become
dp _ d2u
=
, ........................... (13).
dp _ 6?w
dz ^^djf
r._du dv dw
~dx dy dz
Equations (10) become
du
dw
20. The fluid is subject to boundary conditions as regards pressure and
velocity. These are —
(1) At the lubricated surfaces the fluid has the velocity of those
surfaces ;
(2) At the extremities of the surfaces or film the pressure depends on
external conditions.
Thus taking the solid surfaces as y = 0, y = h, and as being limited in the
directions x and z by the curve
52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 261
For boundary conditions
y = Q 11=11, w = 0 v = 0
i TT TT Ml . T7-
y = h u=U, w = 0 v=U1^+Viy (15).
21. Equations (13) may now be integrated, the constants being deter-
mined by the conditions (15).
The second of these equations gives p independent of y, so that the first
and third are directly integrable, whence
rh~J.
h
}• (16).
17 '
dp , ..
w = x- -y- (y — h) y
2 M dz ^
Differentiating these equations with respect to x and z respectively, and
substituting in the last of equations (13)
dv _ 1 r d_ (dj> _ , , { d_\dp_( _/\,ll d_\TT^~y jjy\
Integrating from y = Q to ij = h, and substituting from conditions (15)
d I dp\ d / ,3 dj.
From equations (16) and (14)
(l»).
Putting fxfz for the shearing stresses at the solid on the surfaces in the
directions x arid z respectively, then taking the positive sign when y = h, and
the negative when y — 0
...(19).
Equations (17) and (19) are the general equations of equilibrium for the
lubricant between continuous surfaces at a distance h, where h is any
continuous function of &• and z, and p is constant.
262 ON THE THEORY OF LUBRICATION [o2
22. For the further integration of these equations it is necessary to
know the exact manner in which x and z enter into h, as well as the function
which determines the limit of lubricated surfaces.
These integrations have been effected either completely or approximately
for certain cases, which include the chief case of practical lubrication.
Complete integration has been obtained for the case of two parallel
circular or elliptical surfaces approaching without tangential motion. This
case is interesting from the experiment, treated approximately by Stefan*,
of one surface-plate floating on another in virtue of the separating film of air.
It is introduced here, however, as being the most complete as well as the
simplest case in which to consider the important effect of normal motion
in the action of lubricants. It corresponds with Case 2, Section III.
Complete integration is also obtained for two plane surfaces
7 7 it *'\
h = /ij 1 + m -
\ a/
between the limits at which p = II (the pressure of the atmosphere)
x = 0, x = a,
the surfaces being unlimited in the direction of z. This corresponds with
Case 4, Section III.
For the most important case, that of cylindrical surfaces, approximate
integration has been effected for the case of complete lubrication with the
surfaces unlimited in the direction of z. Case 9, Section III.
SECTION V.— CASES IN WHICH THE EQUATIONS ARE COMPLETELY
INTEGRATED.
23. Two Parallel Plane Surfaces approaching each other, the Surfaces
having Elliptical Boundaries.
Here h is constant over the surfaces, and when
/•*i2 «^2
-, + 3-1, p = tt (20),
or c~
U0, Ul are zero.
* Wien. Sitz. Ber., vol. 69 (1874), p. 713.
~>2] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 263
Equation (17) becomes
d d \ 12ft dh
The solution of which is
Therefore 2</> (t) (— +
and El = Q, &c.
. . n
)
p_n=^^ --— I- + - - ll ^ . ...(24).
h3 a? + c2 (a2 c2 j dt
From equations (19)
_ _ 24ft a-c2 dh\
«"**••*[ (25),
/. _ 24/z a2 dh
*z~ ~hT a2+c2'^ rf7/
supposing surfaces horizontal and the upper surface supported solely by the
pressure of the fluid. The conditions of equilibrium in this case are obvious
by symmetry.
The centre of gravity of the load must be vertically over the centre of the
ellipse. Since by symmetry
n pxdxdz = 0
j
^°'° __J* " f (26).
nfxdxdz=Q
I
no V (l— j)
fzdxdz = 0
B
264
ON THE THEORY OF LUBRICATION
[52
And
ra>/l--
W=l I 'p-Udxdy (27)
0 J 0
3/i7r a3c3 eM
H3" a'2+ c2 ' dt
.(28).
Therefore integrating
*-/^sr»(fi-fi) (29)>
(a2 + c2) If V/*.22 »i /
£ being the time occupied in falling from h-^ to 7<2.
21. Plane Surfaces of unlimited Length and parallel in t/te Direction of z.
The lower surface unlimited in the direction x and moving with a velocity
— U. The upper surface fixed and extending from x = 0 to x = a. This case
corresponds with Case 4, Section III.
The boundary conditions are
— a
> (30)
y = k Ul = 0 V1 = 0
, / O!\
ft = h0 1 + in -
\ aj
p is a function of x only.
And from equation (17), Section IV., by integration
being the value of h when x= xl where the pressure is a maximum.
Integrating with respect to x, and putting p = II at the boundaries
-2- cm
t*1 ~~ o i m {<**)
^ :: : ,.., _*j. (88>
I 1 + W - I 1 + VM
a \ (
'"' f
"i+il
52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 265
or putting W for the load per unit of breadth, TV is a maximum when
m = l'2 approximately and
again, by equation (19)
f, = Pjf (36);
therefore I fxdx = , — — loge (1 + in) (37) ;
and if m — 1'2
^=•6572^ (38).
In order to render the application of equations (35) and (38) clear, a
particular case may be assumed.
Let fi = 10~5,
which is the value for olive oil at a temperature of 70° Fahr., the unit of
length being the inch, and that of force the Ib.
Let U = 60 (inches per sec.)
A, = -0003.
Then from (35), the load in Ibs. per square inch of lubricated surface is
given by
W
— = 1070a2,
a
and from (38), the frictional resistance in Ibs. per square inch is
F_
a
This seems to be about the extreme case of perfect lubrication between
plane metal surfaces having what appears to be about the minimum value
of h,.
266
ON THE THEORY OF LUBRICATION
[52
SECTION VI. — THE INTEGRATION OF THE EQUATIONS FOR THE CASE OF
CYLINDRICAL SURFACES.
25. General Adaptation of the Equations.
Fig. 17 represents a section of two circular cylindrical surfaces at right
angles to the axes ; as in Art. 17.
J is the axis of the journal AB ;
/ is the axis of the brass CD ;
Fig. 17.
JO is the line of action of the load cutting the brass symmetrically, and
R = JP
h = PQ
h0 = HG, the smallest section
JI=ca
^JQ = 0i. PI being the point of maximum pressure.
• ...(39).
">:>] AXl) ITS APPLICATION TO MH B. TOWER'S EXPERIMENTS. 207
Then taking x for distances measured in the direction OA from 0 on the
surface AB, and putting r for the distance of any point from /,
x = R6 ]
J .............................. (40)
y = r - .RJ
(41).
ccT2
Neglecting quantities of the order ,,
Ki
h=a{l + c sin (0 - </>„)} ........................ (42).
For if / be moved up to /, Q moves through a distance ca in the
direction JH.
The boundary conditions are such that
(1) all quantities are independent of z\
(2) f/o is constant, U± and Vl = 0 ;
(3) putting #0 = OJA, 0! = OJB, whence by symmetry 00 - — 0l,
Putting — L for the effect of the external load and — M for the external
moment per unit of length in the direction z, and assuming that there are
no external horizontal forces, the conditions of equilibrium for the brass are
f»i
(psin0-/cos0) dB = 0 (44)
J*»
/•», T.
{pcos0+fsiu0}d0 = ~^ (45)
J e0 -ti
[e M
//•"-I <«>•
Substituting from equations (40), (41), (42) in equations (17) and (19),
Section IV., putting
a/ a
K, = ^ (47)
and remembering the boundary conditions, these equations become on
integration
dp = 6RpU0c{sm(0-<l>0)-siii (</>, -j,,
8-3
208
ON THE THEORY OF LUBRICATION
3/A UQc (sin (0 - <ft0) - sin (fa -
-2
[52
.(49).
{1 +csin(0 — <£)}
26. The Method of Approximate Integration.
The second numbers of equations (48) and (49) may be expanded so that
, sin (6 -<£„) + A cos 2 (0 - <£) + &c.
2n cos 2tf (0 - </>0) + ^2;i+1 sin {(2* + 1) (0 -
sin (0 - </>0) + J53 cos 2 (6> -</>) + &c.
ll cos 2*' (0 - <£) + 52M+1 sin \(2x + 1) (0 -
...(50)
Putting
= sn < —
2)(r+l)r(r- 1)...
.(52)
r=2»
+ 2
+
-"•271+1 ~~ ("" */
(2n + 2) (27
22JI+1
22W+1
r=2n+3
r - 2n - 1
2
+
.(53).
52]
AND ITS APPLICATION TO MR B. TOWERS EXPERIMENTS.
269
,
( r + 2
r=2*>
L (? + )J 7 •
(» )••• 2
r=2
2r
2
cr
cr+1x
- 3 (2rc
r=2<»
[4-3 (2vi + 2)] cMl+1 -
]r .(r— L) ... ^
2r-l
2) c271-1-2 x
2
r=2«o+3 r
CJf*-
.(54).
The coefficients ,40, .4i, &c., B0, Bl, &c., are thus expanded in a series of
ascending powers of c with numerical coefficients which do not converge.
It seems, however, that if c is not greater than -6 the series are themselves
convergent, and it is only necessary to go to the tenth or twelfth term,
to which extent they have been calculated, and are as follows : —
A, = - V5c - 3'75c3 - 6-565C5 - 9"85c7 - 13'51c9 - 1 7'Gc11
- {1 + 3cs + 5-625c4 + 8-7 5cG + 12'225c8 4- 16'2c10} x
A, = 1 + 4-oc2 + 9'375c4 + 15'23c6 + 21-92c" + 29"8c10 + 38'6c12
+ (3c + 7-oc3 + 1313c5 + 19-7c7 + 27'Olc9 + 35'2cn} %
A2 = I'oc + 5c! + 9'85c5 + 15'75c7 + 22'56c9 + 20'24cn
+ |3cs + 7'5c4 + 13-13c6 + 19-7C8 -(- 27'02c10 + 35'2c12} ;
A3-- I'oc2 - 4-7c4 - 9'2c6 - 14-7c8 - 21'45c10
- {2-5c3 + 6-5Gc5 + 1 l'78c7 + 18'03c9 + 25'4cu} %
A, = - 1-25C3 - 3-94c5 - 7'875c7 - 12-88C9 - 18'8cn
- {1-875C4 + 5'2oc6 + 9'85c8 + 15'48c10 + 22'45c12} x
\ r /V
At = -939C4 + 3'07c6 + 6'33c8 + 10'68c10
+ {2-63c5 + 3-94c7 + 776C9 + 12'Gc11] %
(55).
270
ON THE THEORY OF LUBRICATION
0 = I - {2-5c2 + 4125c4 + 5-.312.")C6 + 6"54c8}
- (3c -f 4-5c3 + 5-625c5 + 6'562c7 + 7'63c9}
, = 2c+ 6c3 + 9-75c° + ir3125c7
+ {6c2 + 9c4 + 11-25C6 + 12-5c8} %
2 = 2-5c2 + 5-5c4 + 7-97c" + lO'OGc8
+ (4-5c8 + 7-5c5 + 9'85c7 + 11 -8c9} %
S = - 2c3 - 4-37oc5 - 0 5625c7 - 8'61c9
- {3c4 + 5'625c6 + 7-873c8 + 9'84c10} x
, = - l-375c4 - 3-2c6 - 5'03c8
- (l-875c8 + 3'94c7 + 5'09c9} x
[52
27. The Integration of the Equations.
Integrating equation (50) between the limits 00 and 0
P-PQ _ A ff) fi x
KlC '
— A^ {cos (0 — (j)n) — cos (#0 — <£„)}
+ -JT {sin 2 (0 — <£0) — sin 2 (#0 — <£rt)}
&c. &c.
_ 2M+1 {cos f(2n + 1) (0 - <&„)! - cos [(2w
V^i I I *•
whence putting 6 = ^ by condition (43)
j
0 = ^0^1 - Al sin ^i sin (f>0 + ~ sin 2^ cos 2<£0 - &c.
2
sin [(2n + 1) 0,] sin (2n + 1) ^
Putting
j 1 . /» .
E = Al cos ^i cos <^>o + ~^~ cos 2^i sin 2<£0 + &c.
+ ^^T cos K2w + !) 0il cos ^2w + !> 0o
.(56).
(57),
.(58).
cos 2w^, sin 2n<60
(59),
52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 271
whence from equations (57) and (58)
~ = E + A00 - A, cos (0 - </>„) + 2 sin 2(0- 00)
(60).
Multiplying equation (60) by sin 0 and integrating between 00 and 0l
remembering that 0n = — fll
re, p _ p
-j^-- sin 0d0 = 2A0 (sin 9l — 0l cos 0J
J ea -K-C
/sin 20i sin 00
, - — g— - - 1 sin
A9f . a sin 30 cos 20\
-y (sm0cos20 -- — —
&c. &c.
sin (2n + 2) ^ sin (2n + 1) ^>0 sin <2n0l sin (2?? + 1) ^
2/1 + 2 2rc
AM (sin (2n + 1) Ol cos 2w</> sin (2n + 1)0 cos 2n0|
'2n { 2n - 1 ~~2wTT~ ' j
Multiplying equation (60) by cos 0, and integrating from 00 to 0^
atn
A2/sin'301 . i • A - Oj\ e
•+ -^ I — - — sin 2<£0 — f sin 0 sin 20 ) — &c.
SB \ 0 /
A.2n+l (2// + 1 gin + 2)
•In +
2//' (2w + 1
Multiplying equation (51) by cos# and integrating
[e /cos 0
- y-1>— =
J e -K-2
sn j sn 0 .
sin (j - - + 0l sin
/sin 3^ cos 200 .
+ B.2 ( — g— r + sin 9l cos 20o
- &c.
D (sin (2?i 4- 1 ) #1 cos 2w00 sin ( 2n — 1) 6l cos
2n + l 2n-l J
(sin(2n + 2)g1sin(2ra + l)00 sin 2n^ sin (2?t + 1
-"i+1 | 2n+2 2n
..................... (63)
•272 ON THE THEORY OF LUBRICATION [52
Multiplying equation (51) by sin 9 and integrating
2<91cos(f>0
-
-sing, sin
R (sin (2w + 1) B{ sin 2n(j)0 sin (2?? — 1) Ol sin 2re^»)
i 2?i + 1 "2n — I }
(sin (2» + 2) ^ cos (2n + 1) ^>0 sin ^n9l cos (2?i— 1) <£0]
l ~"~ " "5T
...... (64).
Integrating equation (51)
- T' fy. = 2B0^ - 25j sin ^ sin <^>0 -f 25a sin 2^ cos
./ »„ ^ 2 ^
sn nj cos
2?i
— sin (2/i + 1) 0jsin(2?i + 1) <^>0 (65).
Substituting from equations (61) — (65) in the equations of equilibrium
(44), (45), (46), there results—
From (44)
0 = 2 (KjAo + KzB0) sin 0, - ^K^A^ cos (9,
K,B,) sin <£n - (A^c^a + /T.,5,) ft, sin
+ &c.
c^an ^ sin (2n - 1) 0, cos
~ " -
if r> 1 cos
2n + 1
sin 2n0j sin (2n
- -g^-
K1cAm+1 „ ^ sin (2n + 1) ^ sin (2n
f1cm+1 „ R ^ sn n + sn n + (>„
V 2w + 1 ~ Aa^+1J ' 2rc + 2 "J ...... (°
52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 273
From (45)
- , cos
sn>i-
cos (2n + 1) ^0 (67)
From (46)
If I 97?
?A - 25, sin ^ sin (/>„ + -2 sin 2(9, cos 2</>0
4 •£— sin 2n0l cos 2w<£0
2/Jan^i • /,-»
)
! sin (2w+l)<U ..................... (68).
J
The equations (66), (67), (68), together with (58), which expresses the
boundary conditions as regards pressure, are the integral equations of equi-
librium for the fluid between the brass and journal, and hence for the brass.
The quantities involved in these equations are
R, U, M, L, p, B \ and a, c, <£„, c/^.
If, therefore, the former are given, the latter are determined by the
solution of these equations.
SECTION VII. — SOLUTION OF THE EQUATIONS FOR CYLINDRICAL
SURFACES.
28. c and \/ ^ small compared with Unity.
In this case equations (55) become
A0 = -x B0=l |
A,= 1 B, = 0 [ (69).
A,= 0, &c. £., = 0, &c.J
Equation (58) gives
0 = X0, + sin 01 sin </>0 (70).
o. B. ii. 18
274 ON THE THEORY OF LUBRICATION [52
Equation (66) gives
0 = (2^0% - 2^T2) sin 0, - 2^0%^ cos ^ - ^c bm ' sin fa
sin 00 ............... (71).
Equation (67) gives
L -, /sin 20j
^ ,
osfa ..................... (72)
and equation (68) gives
(73).
Also equation (57)
P — Po = K±c (cos #1 cos fa — j(0 — cos (6 — <f>)} ............ (74).
Eliminating ^ between equations (70) and (71)
K2 2 sin #1
The equations (74) and (75) suffice to determine a, c, fa and <f>0 under
the conditions ./ _ and c small so long as </>0 is not small, in which case the
terms retained in the equations become so small that some that have been
neglected rise into relative importance.
To commence with let
L = 0.
(Cases 5 and 9, Section III.)
Then by (72) cos $0 = 0
-, i /r,^ sin 01
and by (/O) % = __J>
putting for ^ its value sin (fa — fa)
sinfl,
cosfa = — j— ' .............................. (76).
"i
Equation (76) gives two equal values of opposite signs for fa. These
correspond to the positions of Pl and P2) the points of maximum and
minimum pressure.
For the extreme cases
(77).
52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 275
•
From equations (73) and (47)
tym
a=^M ................................. (78>
, f a sin Bl
and f,,m(7o) .- ................. (79).
(0i -- £ - +ism2^j
When L increases
From (72) and (75)
R /sin 20! \ 2 sin fl
tan <£„ = JT, (— — ? - 4 - - — _ - - ...... (80).
Id T
( "i - — 7j —
Hence as L increases tan <f>0 diminishes until the approximation fails.
This, however, does not happen so long as c is small.
As the load increases from zero, equation (80) shows that G moves away
from 0 towards A.
It also appears from equation (70) that % and <£t diminish as </>0 diminishes,
and that fa is positive as long as the equations hold.
To proceed further it is necessary to retain all terms of the first order of
small quantities.
Retaining the first power of c only, equations (55) become
A0 = - 1-oc - x B0 = 1 - 3c%
.(81).
I
Ao = l'5c
From (58)
sin 20,
v(0, + 3csin 0} sin <f>0) = — sin 0, sin <f>0 — r5c0, + I'.V - cos 2<6n...(821
/v \ J O r u V /
From (66)
0 = }— ^K\c (1'oc + %) + 2^ (1 ~ ^c%)} gin 0i
l'5c + Y) 0i cos0,
+ [KlC (1 + 3cX) - 2K2c] ~~1 sin ^
- [KlC (1 + 3cx) + 2JSTjc} ^, sin </>„
s200 ..................... (83).
18—2
276 ON THE THEORY OF LUBRICATION [52
From (67)
L ( sin 2#A
• « = - [Kic (1 + 3c%) ~ 2/Tgc] ( 0! ~ — cos </>„
(J sin 30j sin 2<£0 - sin 0D sin 2<£0) (84).
T
From (68)
M
- 4c sin 0, sin </>0} (85).
In the equations (82) to (85) terms have been retained as far as the
second power of c, but these terms have very unequal values. As ^ and
sin <f>0 diminish c increases, and the products of c% or c sin <£0 may be regarded
as never becoming important and be omitted when multiplied by K^c or 7T2.
Making such omissions and eliminating ^ between (82) and (83)
(,„ / . - sin30A ..sin 20,)
, sin 0! + c -$0, sin 0! - -^— - - 3 (sin 0X - 0X cos 0j) — =—
(86).
sn <0 =
03 f 0X - sm - 2 (sin 0! - 0! cos 00 sin 01
Equation (86) is a quadratic for c in terms of sin (/>0, from which it is clear
that as c increases from zero <£0 goes through a minimum value when
(< i
/f ! , sin 30A , x sin 201 '
f 0j sin 0j -- - — - 3 (sm 0, - 0, cos ?,) — =--
\ o / ^
As the load increases from zero the value of c increases from that of
equation (79) to the positive root of (87). As the load continues to increase
c further increases, but <j>0 again increases, so that, as shown by equation (86),
for values of <£„ greater than the minimum there are two loads, two values of
c, and two values of %.
If 0j is nearly - , c will be of the order \ ^ when <£„ is small, and sin </>0
& V zzt
will be of the order 4c ; so that, so long as \ / —^ is sufficiently small, no
error has been introduced by the neglect of products and squares of these
quantities.
For example
0! - 1-37045 (78° 31' 30" as in Tower's experiments) ...... (88).
52] AND ITS APPLICATION TO Mil 13. TOWER'S EXPERIMENTS. 277
By equation (86)
sin </>„ = 3'934c + T9847 -^ (89).
And by (87) at the minimum value of fa
a
2R
• j / a\
sin fa = o'ol * / -y^
Putting x = sin fa — sin fa equation (82) becomes
sin fa = - -16776c + "5656 ^ . . .(91),
Me
or, when fa is a minimum,
/
sin<f>1 = -682A/^ (92).
Therefore x = 4'928 A/^ (93).
Equations (84) and (85) give
^ = -11753^0..
(95),
J.IT
whence equations (47)
c = '388a4-. ...(96),
Iwl
p_ M
<U7)-
So long then as a -^ is not greater than 0'2, these approximate solutions
are sufficiently applicable to any case.
For greater values of -^ the solution becomes more difficult, as long how-
ever as c is not greater than '5 the solution can be obtained for any particular
value of c.
278 ON THE THEORY OF LUBRICATION
29. Further Approximation to the Solution of the Equations for
particular Values of c.
The process here adopted is to assume a value for c. From equations
(53) and (54) to find
AQ = A0' + A0"X B0 = B0' + B0"X (98),
&c. &c.
where J./, A", J5/, B" are numerical.
These coefficients are then introduced into equations (58) and (66) which
on eliminating % give one equation for </>„.
The complex manner in which fa enters into the equation renders
solution difficult except by trial, in which way values of fa corresponding to
different values of c have been found.
The value of fa substituted in equations (58) or (66) gives % and fa.
The corresponding values of c,, fa and % being thus obtained, a complete
table might be calculated. This, however, has not been done, as there does
not exist sufficient experimental data to render such a table necessary.
What has been done is to obtain fa and fa for c = '5, 6^ having the value
1'3704 as in equation (88) and in Tower's experiments.
The value c = '5 was chosen by a process of trial in order to correspond
with the experiments in which Mr Tower measured the pressure at different
parts of the journal as described in his second report, and as being the
greatest value of c for which complete lubrication is certain.
Putting c = '5, equations (43) and (44) give
A0 = - 1-5351 -2-3018% #0 = - '012 -2-304%
A^= 3-0723 + 3-0721% B,= 2-143 + 2-286%
A2= 1-8647 + 1-5360% B.,= 1139+ '896%
V (99).
A3 = - -8911- -6571% #3=- -455- -316%
At = - -3753- -2582% J54 = - -146- -097%1
As= -1396+ -1343%
Taking 01 = 1-3704 or 78° 31' 30" j
it was found by trial that when [ (10U),
= 48° I
52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. ^79
and K.2 was neglected (under the circumstances K2 was about '0003^),
equation (58) gave
X = - -82295 or - sin 55° 22' 40"\
and equation (66) gave • (101).
X = - -82274 „ - sin 55° 21' 40"
The difference "00021 being in the same direction and about the magnitude
of the effect of neglecting Kz.
This solution was therefore sufficiently accurate, and adopting the value
of <£,-<£„ <£i = 7° 21' 40" (102).
Equations (99) then became
A0= -3587 B0= 1-91
A,= -5449 Bl= 2-263
A,= -6010 J52= -303! ,103)
A3 = --3505 B3 = - -195
A 4 = - -0407 B, = -066
A5 = - -0291 ,
Substituting the values from equations (100), (102) and (103) in
L
equation (67) -g « - r2752JT,e (104),
equation (68) ^2 = - 4'7546#2 (105).
By equation (59) # = --25257 (106).
By equation (60)
P ~ Po = - -25257 + -35870 - '545 cos (0-48°)
+ -3005 sin 2 (0 - 48°) + '1168 cos 3 (0 - 48°)
- "0407 sin 4 (0 - 48°) + '0058 cos 5 (0 - 48°) . . .(107).
From equation (107) values of
P-P*
Kc
have been found for values of 0 differing by 10°, and at certain particular
values of 0 —
0= ± 29° 20' 20" points at which the pressure was measured
0 = — 7° 21' 40" point of maximum pressure
0 = — 76° 38' 40" point of minimum pressure
0 = ± 78° 31' 30" the extremities of the brass
These are given in Table II.
280 ON THE THEORY OF LUBRICATION [52
TABLE II. — The Pressure at Various Points round the Bearing.
0 off side.
Arc radius = 1.
P-Po
0 on side.
Arc r&dius — 1
P-Po
-KlC
-KlC
0 / II
t tt
000
o-o
1-0151
000
1-0151
- 7 21 40
-0-12837
1-0269
-10 0 0
-0-17453
1-0232
10 0 0
0-17453
0-9331
-20 0 0
- 0-34907
0-9412
20 0 0
0-34907
0-8022
-29 20 20
-0-5120
0-7923
29 20 20
0-5120
0-6609
-40 0 0
-0-6981
0-5612
40 0 0
0-6981
0-5003
-50 0 0
-0-8737
0-3349
50 0 0
0-8737
0-3555
-60 0 0
- 1-0472
0-1449
60 0 0 1-0472
0-2249
-70 0 0
-1-2217
0-0293 70 0 0
1-2217
0-1002
-76 38 20
-1-3367
- 0-001
-78 31 20
-1-3704
0-0002
78 31 20
1-3704
0-0002
The Figs. 18, 19, 20 represent the results of Table II.
Fig. n). Fig. iS.
0 01 0'2 0'3 0'4 0'5 0'6 07 0'8 0'9
Oil
In the curve Fig. 18 the ordinates are the pressures, and the abscissae
the arcs corresponding to 6.
52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 281
In the curve Fig. 19 the ordinates are the same plotted -to -abscissae
= R sin d.
In the curve Fig. 20 the horizontal ordinates are the same as the vertical
ordinates in Figs. 18 and 19, and the vertical abscissae are= R cos 0.
These theoretical results will be further discussed in Section IX., where
they will be compared with Mr Tower's experiments.
29 A. c = '5 is the Limit to this Method of Integrating.
In the case considered, in which 6 = 78° 31' 20", Table II., shows that the
pressure towards the extreme off side is just becoming negative. With
greater values of c this negative pressure would increase according to the
theory.
The possibility of this negative pressure would depend on whether or not
the extreme off edge of the brass was completely drowned in the oil bath,
a condition not generally fulfilled, and even then it is doubtful to what
extent the negative pressure would hold, probably not with certainty below
that of the atmosphere.
With an arc of contact anything like that of the case considered it would
be necessary, in order to proceed to larger values of c than 5, that the limits
between which the equations have been integrated would have to be changed
from
to
This integration has not been attempted, partly because it only applies,
in the case of complete lubrication, when the value of c > '5 renders approxi-
mation very laborious, but chiefly because it appears almost obvious that the
value of c, which renders the pressure negative at the off extremity of the
brass is the largest value of c under which lubrication can be considered
certain.
The journal may run with considerably higher values of c, the continuity
of the film being maintained by the pressure of the atmosphere, which would
be most likely to be the case with high speeds. But although the load
which makes c = '5 is not necessarily the limit of carrying power of the
journal, it would seem to be the limit of the safe working load, a conclusion
which, as will appear on considering Mr Tower's experiments, seems to be in
accordance with experience.
This concludes the hydro-mechanical theory of lubrication so far as it has
282 ON THE THEORY OF LUBRICATION [52
been carried in this investigation. There remain, however, physical consider-
ations as to the effect of variations of the speed and load on a and p, which
have to be taken into account before applying the theory.
SECTION VIII. — THE EFFECTS OF HEAT AND ELASTICITY.
30. p> and a are only to be inferred from the experiments.
The equations of the last section give directly the friction, the intensity
of pressure, and the distance between the cylindrical surfaces, when the
velocity, the radii of curvature of the journal and the brass, the length of
the brass, and the manner of loading are known (i.e. when U, R, a, 01} L, and
JM are known); and, further, if M the moment of friction is known, the equa-
tions afford the means of determining a when fj, is known, or //, when a
is known.
The quantities U, R, 01} and L are of a nature to be easily determined in
any experiment or actual case, and M is easily measured in special experi-
ments, but with a and fi it is different.
By no known means can the difference of radii (a) of the journal and its
brass be determined to one ten-thousandth part of an inch, and this would
be necessary in order to obtain a precise value of a. As a matter of fact
even a rough measurement of a is impossible. To determine a, therefore, it
is necessary to know the moment of friction or the distribution of pressure ;
then if the value of yu- be known by experiments such as those described for
olive oil (Section II.), a can be deduced from the equations for any particular
value of p. But although the values of /*, may have been determined for all
temperatures for the particular oil used, and that value chosen which corre-
sponds with the temperature of the oil bath in the experiment, the question
still arises whether the oil bath (or wherever the temperature is measured)
is at the same temperature as the oil film. Considering the thinness of the
film and the rapid conduction of heat by metal surfaces, it seemed at first
sight reasonable to assume that there would be no great difference, but when
on applying the equations to determine the value of a for one of the journals
and brasses used in Mr Tower's experiments, it was found that the different
experiments did not give the same values for a, and that the calculated
values of a increased much faster with the velocity when the load was
constant than with the load when the velocity was constant, it seemed
probable that the temperature of the oil film must have varied in a manner
unperceived, increasing with the velocity and diminishing the viscosity,
which would account for an apparent increase of a.
52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 283
That a should increase with the load was to be expected, considering that
the materials of both journal and brass are elastic, and that the loads range
up to as much as 600 Ibs. per square inch, but there does not appear any
reason why a should increase with the velocity unless there is an increase of
temperature in the metal. If this occurs, the apparent increase of a would
be partly real and partly due to the unappreciated diminution of p owing to
the rise of temperature.
Until some law of this variation of temperature and of the variation of a
with the load is found, the results obtained from the equations, with values
of fj, corresponding to some measured temperature, such as that of the oil
bath or a point in the brass, can only be considered as approximately
applicable to actual results. Even so, however, the degree of approximation
is not very wide as long as the conditions are such that the journal " runs
cool."
But, treated so, the equations fail to show in a satisfactory way what is
one of the most important matters connected with lubrication — the circum-
stances which limit the load which a journal will carry. For, although it
may be assumed that the limit is reached when ca, the shortest distance
between the surfaces, becomes zero or less than a certain value, yet, accord-
ing to the equations, assuming a and p to be constant, the value of c
^increases directly as U if the load be constant; so that the limiting load
should increase with U. But this is not the case, for it seems from experi-
ments that at a certain value of U the limiting load is a maximum if it does
not diminish for a further increase of U.
Although, therefore, the close agreement of the calculated distribution of
the pressure over the bearing with that observed and the approximate
agreement of the calculated values of the friction for different speeds and
loads, such as result when /* and a are considered constant, seem to afford
sufficient verification of the theory, and hence a sufficient insight into the
general action of lubrication, without entering into the difficult and some-
what conjectural subject of the effects of heat and elasticity, yet the
possibility of obtaining definite evidence as to the circumstances which
determine the limits to lubrication, which, not having been experimentally
discovered, are a great desideratum in practice, seemed to render it worth
while making an attempt to find the laws connecting the velocity and load
with a and p.
As neither the temperature of the oil film nor the interval between the
surfaces can be measured, the only plan is to infer the law of the variations
of these quantities for such complete series of experiments as Mr Tower's.
In attempting this, a probable formula with arbitrary constants is first
assumed or deduced from theoretical considerations, and then these constants
284 ON THE THEORY OF LUBRICATION [52
are determined from the experiment and the general agreement tested. In
order to determine the actual circumstances on which the constants depend,
it is important to obtain the formula from theoretical considerations. This
has therefore been done, although these considerations would not be sufficient
to establish the formulae without a close agreement with the experiments.
31. The Effect of the Load and Velocity to alter the Value of the Difference
of Radii of the Brass and Journal, i.e., of a.
The effect of the load is owing to the elasticity of the materials, hence it
is probable that the effect will be proportional to the load L. To express
this put
a=a0 -\-rnL (109).
The effect of the temperature on a is owing to the different coefficients of
expansion of brass and iron. Thus : —
aT-a0=(B'-S')(T-T0)R,
where B' and S' are the respective coefficients of expansion of the bearing
and journal. These for brass and iron are : —
B' = -00001 11
S' = -0000061,
therefore putting T — T0 = Tm (the mean rise of temperature due to friction)
aT =a0(l + -000005 - Tm]
...(110),
jy
putting E for "000005 - .
d0
If, as seems general, a0 is about '0005 inch, then with a 4-inch journal
#=•02 about (Ill),
which is sufficiently large to be important.
32. The Effect of Speed on the Temperature.
Putting —
T0 the temperature of the surroundings and bath,
T! the mean temperature of the oil as it is carried out of the film,
Tm the mean rise in temperature of the film due to friction,
Q the volume of the oil carried through per second,
52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 285
D the density,
S the specific heat,
H the heat generated,
Hl the heat lost by conduction,
C' a coefficient of conduction,
taking the inch, lb., and degree Fahr., as units, and 12 J as mechanical
equivalent of heat.
m _ rn _ -" -"o
~DSQ"'
Putting H-H^ = qTmq .............................. (112)
where q is a constant depending on the relative values of T—T0 and Tm,
also on how far the metal of the journal assists the oil in carrying out heat.
On substituting the values of Hlt H2, Q, it appears
R6
= 12JC'.
There does not appear to be any reason to assume any of the quantities
in the denominator to be functions of the temperature except h^. By
equation (42)
hl=a{l +c sin (fa - <£„)}.
Equation (11 2 A) may thus have the form
Tm = - L - ........................ (113),
~
where A = 3JDSqa0(l +ml) [l+c sin(^ -<f>0)} ............... (114).
This shows that A is a function of the load, increasing as the first power
and diminishing as the second power, but the experiments show the effects
of these terms are small, and A is constant except for extreme loads.
286 ON THE THEORY OF LUBRICATION [52
33. The Formulas for Temperature and Friction, and the Interpretation of
the Constants.
From equation (8), Section II.,
p^^-cw-T.) ^|
[ (115).
0 = -0221 (for olive oil)J
From equations (109) and (110)
a = (a0 + ml){l + E(T-T0)} (116),
or approximately since E (Tl — T0) is small
^ (117).
Whence substituting in the equation which results from (51) c being
small
f=^U (118).
Putting Tx for any particular temperature, p,x, ax corresponding values
j ~ * {J C/ x • ••••••••.•.*•••.....••* (J.-LtJ).
From (113)
f=A{Tm + ET*m}+^Tm (120).
These equations (119) and (120) are independent, and therefore furnish
a check upon each other when the constants are known.
Thus substituting the experimental values of U and/ in (120) a series
of values of Tm are obtained, which when substituted in (119) should give
the same value off.
In these equations the meaning of the constants is as follows : —
C is the rate at which viscosity increases with temperature.
E is the rate at which a increases with temperature, owing to the
different expansion of brass and iron.
A U expresses generally the mechanical equivalent of the heat which
is carried out of the oil film by the motion of the oil and journal
for each degree rise of temperature in the film.
B expresses the mechanical equivalent of the heat conducted away
through the brass and journal.
The respective importance of these two coefficients is easily apparent.
When the velocities are low, but little heat will be carried out, and hence
52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 287
the temperature of the journal depends solely on the value of B. But
when the velocities are high, B becomes insignificant compared with AU,
and it is A alone which keeps the journal cool.
The value of A may to some extent be inferred from the quantities
which enter into it as in equation (114). Thus in the case of Mr Tower's
experiment, since
E = 2 {(9 = 1-37 a = -00075 J = -772
D = 0-033 S= 0-31 (for olive oil)
A = -00630 (121).
It is very difficult to form an estimate of q, but it would seem probable
that it has a value not far from 2 ; and, as will be subsequently shown, in
the case of Mr Tower's experiments, q is about 3*5 or
A =0-0223 (122).
As B expresses the rate at which the heat generated in the oil film is
carried away, by conduction through the oil and the surrounding metal, any
estimate of its value is very difficult. If we could measure the temperature
at the surfaces of the metal, B might be made to depend only on the thick-
ness and conductivity of the oil film. But before heat can escape from the
journal or bearing, it must pass along intricate metal channels formed by
the journal or shaft and its supports ; and, on consideration, it appears that
in ordinary cases the resistance of such channels would be much greater
than the resistance of the oil film itself. For example, in the case of a
railway axle, the heat generated must escape either along the journal to
the nearest wheel, or through the brass and the cast-iron axle box to the
outside surface, so that either way it must traverse at least three or four
inches of iron. This is about the best arranged class of journals for cooling.
In most other cases heat has much further to go before it can escape.
However, in every case B will depend on the surrounding conditions, and
can only be determined by experiment. From the experiments, to be
considered in the next section, it appears that
B=l (about) (123).
But it is to be noticed that Mr Tower has introduced a somewhat
abnormal condition by heating the oil bath above the surrounding tem-
perature. For in this way, letting alone the heat generated by friction,
there must have been a continual flow of heat from the bath along the
journal to the machinery; and, considering the comparatively limited surface
of the journal in contact with the hot oil and the large area of section of
the journal, it appears unlikely that the temperature of the journal was
raised by the bath to anything like the full temperature of the latter,
288 ON THE THEORY OF LUBRICATION [52
a conclusion which is borne out by Mr Tower's experiments with different
temperatures in the bath (Table XII., page 291), which shows that the tem-
perature of the bath produced a much smaller effect on the friction than
would have followed from the known viscosity of oil had the temperature of
the oil film corresponded with the temperature of the bath.
Thus the temperature of the film independent of friction is not the
temperature of the bath or surrounding objects, and as it is unknown until
determined from the experiments, it will be designated as
T
-LX}
and the suffix x used to designate the particular value for T= Tx of all those
quantities which depend on the temperature as
If T be the mean temperature of the film,
T-Tx=Tm .............................. (123 A)
where Tm is the rise of temperature due to the film.
33 A. The Maximum Load the Journal will carry at any Speed.
It has already been pointed out that the carrying power of the journal
is at its greatest when c is between '5 and '6. If, therefore, taking the load
constant, c passes through a minimum value as the velocity increases with a
constant load, then the load which brings c to a constant value will be a
maximum for some particular velocity, and if the particular value of c be
that at which the carrying power is greatest, the carrying power will be
greatest at that particular speed.
The question whether, according to the theory, journals have a maximum
carrying power at any particular speed turns on whether
-jjj (c being constant)
is zero for any value of U.
This admits of an answer if the values for JJL, a, Tx, and equations (119)
L f
and (120) hold, for when c is constant „ ,T, ,~. Tr, and A are constant,
KtU K2U
whence differentiating and substituting it appears that when c is constant
f>
~
52] AND ITS APPLICATION TO MR E. TOWER'S EXPERIMENTS. 289
where U is to be taken positive, and T increases as U increases, -This shows
that ,„, for a constant value of c, changes sign for some value of U if T
continues to increase with U.
Hence, according to the theory, the values of L, which make c constant
as U increases, approach a maximum value as U increases, and since this
value, when c is about '5, represents the carrying power of the journal, this
approaches a maximum as U increases.
SECTION IX. — APPLICATION OF THE EQUATIONS TO MR TOWER'S
EXPERIMENTS.
34. References to Mr Tower's Reports.
From the experiments described in Mr Tower's Reports I. and II., in the
Minutes of the Institution of Mechanical Engineers, 1884, the journal had
a diameter of 4 inches, and the chord of the arc covered by the brass was
3'92 inches, the length of the brass being 6 inches.
The loads on the brass in Ibs., divided by 24, are called the nominal load
per square inch.
The moments of friction in inches and Ibs., divided by 24 x R, are called
the nominal friction per square inch.
These nominal loads, arid nominal frictions, with the number of revolu-
tions from 100 to 450, at which they were taken, are arranged in tables for each
kind of oil used, and also for the same oil at different temperatures of the bath.
All the tables relative to the oil bath in the first report refer to the same
brass and journal. And with this brass, to be here called No. 1, no definite
measurements of the actual pressure were made.
The second report contains the account of the pressure measurements,
but it is to be noticed that these were made with a new brass, here called
No. 2, and that the only friction measurements recorded with this brass are
three made at velocities five times less than the smallest velocity used in the
case of brass No. 1.
It thus happens that while by the application of the foregoing theory to
the friction experiments on brass No. 1 a value is obtained for a, the
difference in radii of brass No. 1 and the journal ; and from the pressure
experiments on brass No. 2 a value is obtained for a in the case of brass
No. 2; since these are different brasses there is no means of checking the one
estimate against the other.
o. R. ii. 19
290
ON THE THEORY OF LUBRICATION
[52
The following tables extracted from Mr Tower's reports are those to
which reference has chiefly to be made.
The first of these extracts is a portion of Table I. in Mr Tower's first
report ; this related to olive oil, but corresponds very closely with the results
for lard oil also, although not quite so close for mineral oil.
From TABLE I.— (Mr Tower's 1st Report, Brass No. 1.) Bath of Olive Oil.
Temperature, 90 deg. Fahr., 4-in. journal, 6-in. long. Chord of arc of contact
= 3-92 in.
O ^ ~Q rd
Nominal friction per square inch of bearing.
ll .**
100 rev.
150 rev.
200 rev.
250 rev.
300 rev.
350 rev.
400 rev.
450 rev.
§j§ a o >>
105 ft.
157 ft.
209 ft.
262 ft.
314 ft.
360 ft.
419 ft.
471 ft.
per mm.
per mm.
per mm.
per mm.
per mm.
per mm.
per mm.
per mm.
520
•416
•520
•624
•675
•728
•779
•883
468
...
•514
•607
•654
•701
•794
•841
•935
415
•498
•580
•622
•705
•787
•870
•995
363
...
•472
•580
•616
•689
•725
•798
•907
310
•464
•526
•588
•650
•680
•742
•835
258
•361
•438
•515
•592
•644
•669
•747
•798
205
•368
•43
•512
•572
•613
•675
•736
•818
153
•351
•458
•535
•611
•672
•718
•764
•871
100
•36
•45
•555
•63
•69
•77
•82
•89
The second extract is Table IX. in the first report ; this shows the effect
of temperature on the friction of the journal with lard oil.
From Mr Tower's 1st Report, Brass No. 1.
TABLE IX. — Bath of Lard Oil. Variation of Friction with Temperature.
Nominal Load 100 Ibs. per square inch.
Nominal friction per square inch of bearing.
Temperature.
Fahr.
100 rev.
150 rev.
200 rev.
250 rev.
300 rev. ! 350 rev.
400 rev.
450 rev.
per mm. per mm.
per mm.
per mm.
per mm. per mm.
per nun.
per mm.
120
•24
•29
•35
•40
•44
•47
•51
•54
110
•26
•32
•39
•44
•50
•55
•59
•64
100
•29
•37
•45
•51
•58
•65
•71
•77
90
•34
•43
•52
•60
•69
•77
•85
•93
80
•4
•52
•63
•73
•83
•93
1-02
1-12
70
•48 -65
•8
•92
1-03
1-15
1-24
1-33
60
•59
•84
1-03
1-19
T30
1-4
1-48
1-56
52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 291
The third extract is from the second report, being Table XIL^-represent-
ing the oil pressure at different parts of the bearing as measured with brass
No. 2.
From Mr Tower's Second Report, Brass No. 2. Heavy Mineral Oil
Nominal Load 333 Ibs. per square inch. Number of revolutions 150 per
minute. Temperature, 90°.
TABLE XII. — Oil pressure at different points of a bearing.
Longitudinal planes on centre off
Pressure per square inch Ib. Ib. Ib.
Transverse plane, middle 370 625 500
Transverse plane, No. 1 355 615 485
Transverse plane, No. 2 310 565 430
The points at which the pressure was measured were at the intersections
of six planes, three parallel vertical longitudinal planes parallel to the axis of
the journal, one through the axis, and the other two, one on the on and the
other on the off side, both of them at a distance of '975 inch from that
through the axis, three transverse planes, one in the middle of the journal,
the other two respectively at a distance of one and two inches on the same
side of that through the middle.
In referring to these experimental results in the subsequent articles,
The nominal load per square inch is expressed by L' ;
The number of revolutions per minute by
The nominal friction by /' ;
The effect of the fall of pressure ajflhe ends of the journal on the
mean pressure is expressed by - , thus —
s
n is a coefficient depending on the way in which the journal fits the shaft.
£'. £
~A» \ (124).
_ £7x60
'1-rrR
s= 1-21
19—2
292 ON THE THEORY OF LUBRICATION [52
35. The Effect of Necking the Journal.
The expression (124) for f assumes that the journal was not necked into
the shaft. From Mr Tower's reports it does not appear whether or not the
brass was fitted into a neck on the shaft ; but since there is no mention of
such necking, the theory is applied on the supposition that there was not.
If there were, the friction at the ends of the brass would increase the
moment of friction. Put b for the depth of the neck and a' for the thickness
of the oil film at the ends, then the moment of resistance of these ends would
be-
Hence if M be the moment of friction of the cylindrical portion of the
journal only
And from equation (97) —
TT ft
.(126).
f,_ , 0.0 -!>
' -360 a 1 TlT
For example, a 5-inch shaft necked down to a 4-inch journal would give
b — '5 inch. Whence, assuming
|
and l
the relative friction of the ends to that of the journal would be 11 '36 to
31-00, or 28 per cent, of the friction; and the values of a, calculated on the
assumption of no necking, would have to be increased in the ratio n — T33.
Even if there is no necking the value of a will probably not be the same
all along the journal, in which case the values of a2 and a in Kl and K2 will be
means, and then the square of the mean will be less than the mean of
the squares, so that n will probably have a value greater than unity, although
there may be no necking of the shaft.
36. A first Approximation to the Difference in the Radii of the
Journal and Brass No. 1.
The recorded temperature in Mr Tower's Table I. is 90° Fahr. Accepting
this, and taking the value of /A, equation (8), Section II.,
Hn^IQ-' x 6-81 (128).
52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 293
By equation (97) —
nu. — nM
a =2757W (129)'
Since R = 2 and U ' = —
oO
HfJL "72/"
~^ = U
= 3-46^ (130).
Whence substituting from equation (128) for nw
a N
- = 10-xl-97^-, (131),
and from the tabular Nos. for L' = 100.
N=100, /' = -36
- = 10~4 x 5'5 (inch)
n 7
.(132).
- = 10-*xlO(inch)
n ' •
These are the extreme cases; for intermediate velocities intermediate
values of - are found.
n
In order to be sure that these are the values of - , which result from the
n
application of the equations, it is necessary (since the approximate equations
only have been used) to see that the squares of c may be neglected.
Substituting from equation (124) in (96)
which for L' = 100, N= 100, gives
and for // = 100, ^=450,
c = '033/i2
c = -065ns
.(133).
So that the approximations hold, and, as already stated in Art. 30, this
considerable increase in the value of a with the load constant suggests that
the temperature of the film was not really 90°. And as this point has been
294 ON THE THEORY OF LUBRICATION [52
considered in the last section, the equations of that section may be at once
used to determine the law of this temperature, after which the values of -
n
may be determined with precision.
37. The Rise in Temperature of the Film owing to Friction.
In order to determine the values of E, A, and B in equations (119) and
(120), by substituting in these equations corresponding values of N and f
for Z/=100, the tabular values of/' were somewhat rectified by plotting
and drawing the curve N, f. These corrected values are in the second row,
Table III.
From these values, and the corresponding values of N, it was then found
by trial that the equations (119) and (120) respectively
. Ue-(C+E)(T-T,) f
ax + mL
E
c = '0221,
are approximately satisfied for values of T — Tx = Tm, if
u, -01345
.(134)
ax + m x 400 n
A = -0223081
# = •0222 I (135).
E = -95914 J
TABLE III. — Rise of Temperature in the Film of Oil caused by Friction,
calculated by Equation (120) from Experiments with a Nominal Load
100 Ibs. (see Table I., Tower, p. 290).
Nominal friction per square inch, as calculated by equation (119) from the rise of temperature.
N
Revolutions per minute . .
100
150
200
250
300
350
400
450
b
»<_, .
•sll
o a
f
Nominal friction f^ble L»
per square inch •] n O1 , ,'
for olive oil 1 Correcte<i
•36
•45
•54
•63
•69
•77
•82
•81
r2 O
0) 5
\ to a curve
•33
•55
•55
•63
•705
•768
•83
•8!
2^V
T-T0
Rise of temperature by equa-
O s II
Fahr.
tion (120)
3-45
5-83
8-13
10-02
11-77
13-26
14-48
15-31
|ft
Nominal friction per square
&
/'i
inch calculated by equation
(119) assuming c small . .
•336
•453
•546
•628
•697
•76
•823
•Si
52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 295
Which seemed to agree very well with the reasoning in Section VIII.
With these values of the constants the values of T ' — Tx were then calculated
from equation (120), and are given in the third row in Table III. These
temperatures were then substituted in equation (119), and the corresponding
values off calculated, these are given in the fourth row, Table III.
The agreement between these calculated values of/' and the experi-
mental values is very close ; and it may be noticed that a very small
variation in any of A, B and E makes a comparatively large difference in
some one or other of the calculated values of/', or, in other words, these
are the only positive values of these quantities which satisfy the equations.
The only difference between the experimental and calculated values of/
which is not explainable as experimental error, is for the lowest speed at
which the experimental value of /' is 0'7 per cent, too large. This is
important as it is in accordance with what might be the result of neglecting
c2, since at that speed c is becoming too large to be neglected, and taking
c'J into account the calculated value of / agrees very closely with the
experimental.
38. The Actual Temperature of the Film.
Having found the approximate values of Tm, the rise of temperature,
owing to friction, it remains to find Tx, the temperature of the film, the rise
due to friction, so that
Tx + Tm = temperature of film.
This is found from Mr Tower's Table IX. (see p. 290).
Putting TH for temperature of the bath,
T0 for temperature of surrounding objects,
and assuming
Tg)+Tm+T. .................. (136).
From equations (119) and (130)
f — nf*°
J 3'46 ' a0 + mL
whence
log/' — 0448 [Z(T. - T,) + r,,J log . + 1<* -t ...(137).
In Table IX. (Tower) the values of /' are given for the same values of
N' and L' corresponding to different values of TB.
Substituting corresponding values of/' and TB in equation (137), and
296 ON THE THEORY OF LUBRICATION [52
subtracting the resulting equations, we have an equation in which the only
unknown quantities are Z and the differences of Tm.
The values of f being known, the values of Tm are obtained from equations
(120), (134), and (135), and substituting these, the equations resulting from
(137) give the values of Z. Thus from Table IX. (Tower)
L' = 100
[7=100
TB=1Q /='48 Tm = 4>-8.
5-9 - 4-8 log -59 - log -48
From (136) Z+ — ^ ^-$- .
10 -0443 x 10 x loge
Therefore ^=35 ................................. (138).
From this value of Z the values of f corresponding to those in Tower's
Table IX. have been calculated and agree well with the experimental values.
The smallest temperature of the oil bath recorded in Tower's Table IX.
is 60° Fahr., therefore it is assumed that this was the normal temperature,
whence
Tx = -35(^-60) + 60 ........................ (139).
Hence it is concluded that the actual temperature of the oil film in all
the experiments with the bath, at a temperature of 90° Fahr., is given by
T=7Q-o + Tm .............................. (140).
By the formula for p, since Tx = 70'5
= -000009974
= •00001 (approximately) .................. (141),
and since, by equation (134), when L' = 100
~
~TOl345
= -0007413n
= -00074w (approximately) ......... (142).
This is the value of ax with a load of 100 Ibs. per square inch.
52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 297
39. The Variation of a- with the Load.
All Mr Tower's experiments, when the loads are moderate and the
velocities high, show a diminution of resistance with an increased load.
Since c increases with the load and the friction increases as c increases,
a and fj. being constant, the diminution of friction with increased loads shows
either that the load increases the temperature of the film and so diminishes
the viscosity, or increases the radius of curvature of the brass as compared
with that of the journal, i.e., increases a.
These effects have been investigated by substituting the experimental
values of/' and L't obtained with the same velocity in equations (119) and
(120).
In this way, from equation (119) the value of m is determined, where,
from equations (117) and (135),
7'o> ........................ (143).
And the equation (120) gives the effects of the load on the value of the
constant A. After trial, however, it appears that the effects of the load
upon the constant A are small so long as the loads are moderate, and that
the diminution of the resistance with the increased load is explained by the
value obtained for m from equation (119). From this equation, taking
LSf'1, L2'f'2, simultaneous values of L' and/', and assuming Tx independent
of the load,
f*
(144)
which gives the value of m.
The slight irregularities in the experiments affect the values of m thus
found to a considerable extent, and a mean has been taken, which is
(145).
Putting Tx= 70-5, T0= 60, ax = '00074, when L' = 100, from equation (143)
a0 -i- ml' = -0005861w.
Therefore a0 = '0004885/1 (146).
The value for a thus obtained is therefore
ci = -0004885/i(l +-002//l)e'oaB(r*+r— ^ (147),
and for the experiments with the bath at 90° Fahr.
a = -0004885;i (1 + -002/7) e«a<r- +«»••)
or a='0006161/i(l + -002AV032*7'"' (148).
298 ON THE THEORY OF LUBRICATION [52
From equation (148) and the values of Tm, Table III., the values of
a
n
for Mr Tower's experiments have been calculated.
Putting p = -00001e-'022ir" (149),
and, substituting in equations (47) from (148) and (149),
66-
(1 + -002Z')2™2
•(150),
- -00005139 (1 + -002/7) e'0
•tti
for the circumstances of Mr Tower's experiments, to which the equations of
Sections VI. and VII. then become applicable.
40. Application of the Equations to the Circumstances of Mr Tower s
Experiments on Brass No. 1, given in Table I., p. 290, to determine c, $0, 0j,
/' and p— p0.
The circumstances are, the unit of length being the inch,
R = Z (inches)
B = 78° 31' 20"
— = -0004885, already deduced equation (14(5)
72*
L = 484£'
T0 = 60, assumed
Tx = 70-5, equation (140)
Tm = tabular values, Table III., the increase with load being neglected t
(151).
For a first Approximation as long as c is small.
Equations (89), (91), (94), and (95) are used to determine c, <f>0, </>,// for
the experiments in Table I. (Tower), these being made with brass No. 1.
Putting as in equation (124)
,, _ nM
J ~ f) 7?2 '
52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 299
equation (94) gives
and by equation (150)
fl>=. 004658
.(152).
From equation (152) the values of/' to a first approximation have been
calculated, using the values of Tm given in Table III. These are given as//
in Table IV. t
TABLE IV.— Olive Oil, Brass No. 1.
Length of the journal 6 inches
Chord of the arc of contact of the brass 3-92 inches.
Radius of the journal 2 inches.
Temperature of the oil bath 90° Fahr.
„ surrounding objects 60° Fahr. (assumed).
Difference in radii of brass and journal at 60° 0-0006 inch (deduced).
Effect of necking or variations in radius to increase friction ...1-25.
// the nominal load in Ibs. per square inch, being the total load divided by 24.
N the number of revolutions per minute.
/' the nominal friction in Ibs. per square inch from Table I. in Mr Tower's first Report
(see Art. 34, p. 290).
(//) the nominal friction calculated by complete approximation for c-5 (see Art. 40,
equation (159) ).
/2' the nominal friction calculated to a second approximation, equation (154).
// the nominal friction calculated to a first approximation, equation (152).
c the ratio of the distance between the centres of the brass and journal to the difference
in the radii, equation (153).
a the difference in the radii of the brass and journal (see equation (157)).
0j the angular distance from the middle of the arc of contact of the point of maximum
pressure, equation (91).
p0-* the angular distance from the middle of the arc of contact of the point of nearest
approach (see equation (89)).
2' the rise in temperature of the film of oil owing to the friction, equation (120).
N.
100
150
200
250
300
350
400
450
Tm Fahr.
3-45
5-83
8-13
10-02
1177
13-26
14-46
15-37
•498
•580
•622
•705
•787
•870
•995
P
•300
(•57)
•360
(•65)
•414
•460
•504
•544
/2'-108
•589
c
•67
•578
•590
•487
•457 -I3U
•413
L =41.)
a
•00154
•00162
•0017
•00178
•00182
•00187
•00191
•00194
<j)l
- 7°0'0"
- 7°0'0"
...
...
d> -*"
-42°0'0"
- 42° 0' 0"
...
...
\ r * 2
TABLE IV. — Olive Oil, Brass No. 1 — continued.
N.
100
150
200
250
300
350
400
450
rmFahr.
3-45
5-83
8-13
10-02
11-77
13-26
14-46
15-37
f
•472
•580
•616
•689
•725
•798
•907
...
(-498)
/2'-920
J\
•233
•314
•378
•434
•482
•526
•573
•616
,
c
•520
•462
•408
•380
•357
•340
•312
~ 1 °
•00151
•00161
•00168
•00171
•00177
•00181
•00183
Ui
...
- 7° 0' 0"
...
...
...
U-l
- 42° 0' 0"
...
...
f
*
•464
•526
•588
•650
•684
•742
•835
(-370)
A '765
•805
•845
A
•249
•336
•404
•464
•517
•582
•610
•660
c
•510
•392
•337
•305
•284
•266
•254
•234
// =310
a
•00138
•00144
•00151
•00158
•00161
•00166
•00172
•00172
0i
- 7° 0' 0"
...
...
...
, IT
0n— —
- 42° 0' 0"
TO 2
/'
•361
•436
•514
•592
•644
•669
•747
•789
/i
•62
•670
•712
•760
•810
X'
•266
•358
•431
•495
•550
•600
•650
•707
c
•377
•287
•242
•224
•200
•195
•180
•171
// = 253
a
•00128
•00135
•00141
•00148
•00151
•00156
•00159
•00161
f/
•368
•430
•512
•572
613
•675
•736
•818
A
•380
•457
•530
•595
•65
•701
•755
•810
/i'
•285
•385
•464
•534
•592
•646
•700
•755
^
c
•255
•195
•165
•152
•L41
•132
•126
•118
./^ == ^vjo
a
•00119
•00126
•00131
•00138
•00240
•00145
•00148
•00135
0i
...
...
- 1°13'0"
- 1° 5'G
^-f
...
...
-60° O'O"
-62° O'O
r/
•351
•458
•535
•611
•672
•718
•764
•871
/ '
/ 9
•352
•458
•530
•601
•665
•717
•778
•840
A'
•307
•414
•498
•574
•638
•695
•753
•813
C
•165
•126
•107
•098
•089
•086
•082
•075
x, =152 •{
a
•00111
•00116
•00122
•00128
•00130
•00134
•00137
•00139
(I).
...
- 1°13'0"
- 1°0'0"
- 0°57'0"
- 0°50'0"
- 0°44'0"
- 0°38'0"
- 0°31'0
<7)|
* ...
-61° O'O"
-65° O'O"
- 27° 20' 0"
-69° O'O"
-69°40'0"
-70°20'0"
-72° O'O
//
•360
•450
•550
•630
•690
•770
•820
•890
A'
•352
•465
•555
•637
•708
•770
•831
•897
A
•336
•463
•546
•628
•697
•760
•823
•890
c
•090
•0691
•0600
•0541
•0492
•0471
•0460
•0420
// =100 <
a
•00101
•00106
•00112
•00116
•00120
•00123
•00125
•00127
0i
- 0°43'0"
- 0°26'0"
- 0°17'0"
- 0°10'0"
- 0° 4'0"
- 0°2'30"
- 0° O'O"
+ 0° 5'0
^ ""
l*o-2
-68°30'0"
-73°20'0"
-75°20'0"
-76°30'0"
-77°20'0"
-77°40'0"
- 0°78'0"
- 78° 40' 0
52] ON THE THEORY OF LUBRICATION AND ITS APPLICATION, ETC. 301
As compared with the experimental values/' given in the Table IV., it is
seen that the agreement holds as long as c is less than -06, after jvhich, as c
increases, the values of// become too small, or while the values of// continue
to diminish as the load and a increase, the experimental values of/' after
diminishing till c is about '1 or '15 begin to increase again. In order to see
how far this law of variation was explained by the theory, it was necessary to
find /' the values of /' to a second approximation, and before this to obtain
the values of c.
Putting, as in equation (124),
L = 4-84Z',
equation (94) gives
c=- 2-059^;
and by equation (150)
2 T '
c = '031 16(1+ -1002 L')2^ e0665^ (153).
Equation (153) gives the values of
c
w2'
s*
To obtain the value of n from the experiments, these values of — are
substituted in the equation for /', retaining the squares of c, which obtained
from (.-filiations (85) and (89), is
//=//(l + 5c2) (154),
whence, substituting the values of— obtained from (153) we have
/•'-/«' + «'(£
Therefore, choosing any experimental values of /', and subtracting the corre-
sponding value of// in Table IV., n is given by —
5
<n:
In this experiment irregularities become important, and it has been
necessary to calculate several values of n in this way and take the mean,
which is
w = l'2o (156).
It has been shown (Art. 35, p. 292) that necking might account for a
302 ON THE THEORY OF LUBRICATION [52
value of n as great as T33, while if there were no necking n would still have
a value in consequence of variations of a along the journal.
Substituting this value of n in equations (148) and (153),
a = -00077 (1 + -002 L') e'022271-' \
L' (157),
c = -0487(1 + -002 L'Y - e'066577™ |
' N
from which equation the values of a and c have been calculated for Table IV.
for all values of L' less than 415 Ibs. These are all Mr Tower's experiments
with olive oil, except those of which Mr Tower has expressed himself doubtful
as to the results.
The values of c as given by equation (94) are onlj" a first approximation,
and are too large, but the error is not large, even when c= "5 only amount-
ing to 8 per cent., as is shown by comparing equation (104) with (95).
With these values of c in the equation (154) the values of /2' have been
calculated for all values of c up to '250. At c~'l2 these values of/2' are
about 5 per cent, larger than the experimental values, but they have been
carried to c = "25 in order to show that the calculated friction follows in its
variations the idiosyncracies of the experimental frictions, falling with the
load to a certain minimum, and then rising again.
These values of f£ carry the comparison of the frictions deduced from the
theory up to loads of 205 Ibs. for all velocities, and up to 363 Ibs. for the
highest velocity. To carry the approximation further, use has been made of
the more complete integrations of the equations for the case of
These are given by equations (104) and (105).
As already stated, comparing (104) with (94) it appears that when c= '5
the approximate values of c in the Table IV. are about 8 per cent, too large ;
that is to say, a value c = '540 in the table would show that the actual value
was c = "5.
Comparing equation (95), from which the values // have been calculated,
with equation (105), it appears that when c = '5 the values of f by (105) are
given by
2-3773
1-37 *1'
This is not, however, quite satisfactory, as that portion of the friction
which is due to necking does not increase with the load. This portion
in fi is
n-l ,
n Jl '
52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 303
So that for c = "5
/'.Mm+^[(>-1>/ .^..,.(158),
aod since n = T25, this gives for c = '5
/' = 1-585/V.
If, therefore, any of the approximate values of c were exactly '540, the
complete value of f would be 1*585 times the value of f^. This does not
happen, the nearest approximate values of c being "578, '520, '520, '510.
Multiplying the corresponding values of f^ by 1*585, the results are as
follows : —
Tabular,
c
Tabular.
A'
1-605 /,'
Experimen-
tal. /
Difference.
•578
•36
•57
•58
•01
•520
•414
•656
•65
-•005
•520
•314
•498
•472
-•026
•510
•249
•394
It thus appears that the approximation is very close, the calculated
values for the first, in which c is greater than *540, being too small, and for
the rest, in which c was smaller than *540, too large, which is exactly what
was to be expected.
These corrected values of // have been introduced in Table IV. in
brackets. As they occur with different loads and different velocities, they
afford a very severe test of the correctness of the conclusions arrived at
as to the variations of A and T with the load and temperature, also as to the
condition expressed by n. Had the values of c and f been completely
calculated as for the case of c=*5, there would have been close agreement
for all the calculated and experimental values of /'.
This close agreement strongly implies, what was hardly to be expected,
namely, that the surfaces, in altering their form under increasing loads,
preserve their circular shape so exactly that the thickness of the oil film is
everywhere approximately
a(l + c sin (0 -</>)).
A still more severe test of this is, however, furnished by the pressure
experiments with brass No. 2 in Mr Tower's second report.
304 ON THE THEORY OF LUBRICATION [52
41. The Velocity of Maximum Carrying Power.
The limits to the carrying powers are not very clearly brought out in
these recorded experiments of Mr Tower, as indeed it was impossible they
should be, as each time the limit is reached the brass and journal require
refitting. But it appears from Table I. and all the similar tables with the
oil bath in Mr Tower's reports, that the limit was not reached in any case in
which the load and velocity were such as to make c less than '5. In many
cases they were such as to make c considerably greater than this, but in such
cases there seems to have been occasional seizing. There seems, however, to
have been one exception to this case, in which the journal was run at
20 revolutions per minute with a nominal load of 443 Ibs. per square inch
with brass No. 2 without seizing, in which case c, as determined either by
the friction or load, becomes nearly '9.
It does not appear that any case is mentioned of seizing having occurred
at high speeds, so that the experiments show no evidence of a maximum
carrying power at a particular velocity.
This is so far in accordance with the conclusions of Art. (33a), for, substi-
tuting the values of ABCE, as determined Art. (37), it appears by equation
(123 B) that the maximum would not be reached until Tm, the rise of tem-
perature due to friction, reached 72° Fahr., which, seeing that at a velocity
of 450 revolutions T,n is less than 17°, implies that the maximum carrying
power would not be reached until the speed was 1500 or 2000 revolutions;
notwithstanding that -jjj (c constant) is very small at 450 revolutions.
This is with the rise of temperature due to legitimate friction with
perfect lubrication. But if, owing to inequalities of the surfaces, there is
excessive friction without corresponding carrying power, i.e., if /?, the effect of
necking, is as large as 3 or 4, which it is with new brasses, then the maxi-
mum carrying power might be reached at comparatively small velocities ;
thus suppose T= 13 when N= 100, U = 21, equation (123 B) gives
^ = 0
dU
or the maximum carrying power would be reached ; all which seems to be in
strict accordance with experience, particularly with new brasses.
42. Application of the Equations to Mr Towers Experiments with Brass
No. 2 to determine the Oil Pressure round the Journal.
The approximate equation (74) is available to determine the pressure at
any part of the journal, i.e., for any value of 6 so long as c is small, but these
52]
AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS.
305
approximations fail for much smaller values of c than for others ; for this
reason, together with the fact that the only case in which the pressure has
been measured c is large, the pressures have only been calculated for c = '5,
in which the approximations have been carried to the extreme extent.
These are obtained directly from equation (107), and the pressures divided
by K,c are given in Table II., Section VI.
The results of Mr Tower's experiments with brass No. 2 are given in
Table XII., Art. 34.
Had the friction been recorded in the experiments in which Mr Tower
measured the pressures with brass No. 2 as with brass No. 1, the values of c
might have been obtained as in the case of brass No. 1. But as this was
not done the value of c for these experiments with brass No. 2 could only be
inferred from the agreement of the relative oil pressures measured in
different parts of the journal, those calculated for the same parts with a
particular value of c. This was a matter of trial, and as it was found that the
agreement was very close when
c = -5,
further attempts were not made.
With the section at the middle of the brass the calculated and experi-
mental results are shown in Table V.
TABLE V. — Comparison of Relative Pressures, calculated by Equation
(107) when c = '5, with the Pressures measured by Mr Tower, see Table XII.,
Art. 34, Brass No. 2.
The values of 0
measured from
middle of arc at
which pressures
were measured
Pressure
measured at
the middle of
the journal.
Table XII.,
Tower
P ~ l>»
calculated.
Table II.
Kelative
values,
experimental
llelative
values,
calculated
-*
-20 20 20
500
•7923
•800
•781
639
000
625
1-0150
1-000
1-000
615
29 20 20
370
•6609
•592
•651
560
This agreement, although not exact, is, considering the nature of the
test, very close. The divergence seems to show that in the experiments
c was somewhat more than '5, but it is doubtful if the agreement would have
been exact, as, owing to the journal having been run in one direction only, it
seems probable that the radius of the brass was probably slightly greatest on
the on side.
o. R. ii. 20
306 ON THE THEORY OF LUBRICATION [52
Deducing the value of K& by dividing the experimental pressure by the
calculated values of -«-— ° the values given in the last column are found.
An alteration in the value of c would but slightly have altered the middle
value of - ,_ in the same direction as the alteration of c ; hence taking
Kc
this value, and making c = '520, as being nearer the real value,
K,c = - 640 (159).
In these experiments N = 150,
77 = 333 (160).
From equation (104) L = — 2'5504 x Kc,
§ = 408 (161),
therefore s = jy?
= 1-21 (162).
To find a K, = - 1230,
by equation (150)
ft«-A»«-'0665rm
#,
whence ax2 = '000001 Q88e~'WG5Tm .............. . ...... (163)
and taking Tm the same as with brass No. 1, and olive oil at JV=180,
i.e., 5-83° Fahr., with brass No. 2, at 70'5° Fahr.
ax =-00086
instead of with brass No. 1
= -0007 7
(164).
The difference in the radii of curvature of the two brasses, the one
deduced from the measured friction, the other deduced from the measured
differences of pressure at different positions round the journal, come out
equal within j^^th part of an inch, and the values of a differing only by
11 per cent. Had the frictions been given with brass No. 2, this agreement
would have afforded an independent comparison of the values of a. As it
is, the only probability of equality in these two brasses arises from the
probability of their having been bedded in the same way.
In deducing the value a for brass No. 2, it has been assumed that the
oil, which was mineral, had the same law of viscosity as the olive oil. Both
these oils were used with brass No. 1, and the results are nearly the same,
52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 307
the mean resistances, as given by Mr Tower, are as 0-623 to 0'6o4, or the
viscosity of the mineral oil being 0'95 that for olive oil ; had this-been taken
into account, the value of ax for brass No. 2 would have been still nearer
that for brass No. 1, being '00084 as against '00077.
As the radii of the two brasses seem to be so near, and as the resistance
was measured for brass No. 1 under circumstances closely resembling those
of the experiment with No. 2, a further test of the exactness of the theory
is furnished by comparing the calculated friction with brass No. 2 with that
measured with brass No. 1, with the same oil, the same speed, and nearly
the same load.
As in equation (158)
-/' = 2-3773^ + 1 -37 (w- l)Kz .................. (165)
Whence, taking account of the values of /* for mineral and olive oils, and
the values of a for brass No. 1 and No. 2 for mineral oil and brass No 2,
Kz has 0'87 1 of the value in equation (150)
•871 x -00346- -
(1 + -002/7) n 6)'
which, when T=5-83°, iV=150, /7 = 337, n = l'25, being substituted in the
equations
#a = - 01665
/' = 446 .................................... (167).
In Mr Tower's Table IV., it appears that with brass No. 1, mineral oil,
^=150 /7 = 310 /'=4-4 77 = 415 /' = -51,
whence interpolating for
L = 337
/' = 4-58 ................................. (168).
This agreement is very close, for taking account of the difference of
radius, the calculated friction for brass No. 2 should have been about '95
of the measured friction with brass No. 2.
In order to show the agreement between the calculated pressures and
those of Mr Tower, the values of ^^- for c = '5 have been plotted, and are
K&
shown in Figs. 18 and 19 (page 280), the crosses indicating the experiments
with brass No. 2, as in Table VII. (Tower).
20—2
308 ON THE THEORY OF LUBRICATION [52
43. Conclusions.
The experiments to which the theory has been definitely applied may be
taken to include all Mr Tower's experiments with the 4-inch journal and oil
bath, in which the number of revolutions per minute was between 100 and
450, and the nominal loads in Ibs. per sq. inch between 100 and 415. The
other experiments with the oil bath were with loads from 415, till the journal
seized at 520, 573, or 625 ; and a set of experiments with brass No. 2 at
20 revolutions per minute. All these experiments were under extreme
conditions, for which, by the theory, c was so great as to render lubrication
incomplete, and preclude the application of the theory without further
integrations.
The theory has, therefore, been tested by experiments throughout the
entire range of circumstances to which the particular integrations under-
taken are applicable. And the results, which in many cases check one
another, are consistent throughout.
The agreement of the experimental results with the particular equations
obtained on the assumption that the brass, as well as the journal, are truly
circular, must be attributed to the same causes as the great regularity
presented by the experimental results themselves.
Fundamental amongst these causes is, as Mr Tower has pointed out, the
perfect supply of lubricant obtained with the oil bath. But scarcely less
important must have been the truth with which the brasses were first
fitted to the journal, the smallness of the subsequent wear, and the variety
of the conditions as to magnitude of load, speed, and direction of motion.
That a brass in continuous use should preserve a circular section with a
constant radius requires either that there should be no wear at all, or that
the wear at any point P should be proportional to sin (90° -POH).
Experience shows that there is wear in ordinary practice, and even in
Mr Tower's experiments there seems to have been some wear. In these
experiments, however, there is every reason to suppose that the wear would
have been approximately proportional to c sin (<£0 -0) = c sin (90° - POH),
because this represents the approach of the brass to the journal within the
mean distance a, for all points, except those at which it is negative; at
these there would be no wear. So long then as the journal ran in one
direction only, the wear would tend to preserve the radius and true circular
form of that portion of the arc from C to F (Fig. 17, p. 266), altering the
radius at F, and enlarging it from F to D. On reversal, however, C and
F change sides, and hence alternate motion in both directions would
52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 309
preserve the radius constant all over the brass. The experience, emphasized
by Mr Tower, that the journal after running for some time in one -direction
would not run at first in the other, strongly bears out this conclusion. Hence
it follows that had the journal been continuously run in one direction, the
condition of lubrication, as shown by the distribution of oil pressure round
the journal, would have been modified, the pressure falling between 0 and
B on the on side of the journal, a conclusion which is borne out by the fact
that in the experiments with brass No. 2, which was run for some time
continuously in one direction, the pressure measured on the on side is
somewhat below that calculated on the assumption of circular form, although
the agreement is close for the other four points (see Fig. 18, page 280).
When the surfaces are completely separated by oil it is difficult to see
what can cause wear. But there is generally metallic contact at starting,
and hence abrasion, which will introduce metallic particles into the oil
(blacken it) ; these particles will be more or less carried round and round,
causing wear and increasing the number of particles and the viscosity of
the oil. Thus the rate of wear would depend on the impurities in the oil,
the values of c, I/a and the velocity of the journal, and hence would render
the greatest velocity at which the maximum load could be carried with a
large value of c small. A conclusion which seems to be confirmed by
Mr Tower's experiments at twenty revolutions per minute.
In cases such as engine bearings, the wear causes the radius of curvature
of the brass continually to increase, and hence a and c must continually
increase with wear. But in order to apply the theory to such cases the
changes in the direction of the load (or E7i and Fj) have to be taken into
account.
That the circumstances of Mr Tower's experiments are not those of
ordinary practice, and hence that the particular equations deduced in order
to apply the theory to these experiments do not apply to ordinary cases,
does not show that the general theory, as given in equations (15), (18), and
(19) could not be applied to ordinary cases were the conditions sufficiently
known.
These experiments of Mr Tower have afforded the means of verifying
the theory for a particular case, and hence have established its truth as
applicable to all cases for which the integrations can be effected.
The circumstances expressed by
^ 77' R' °' ^°' ^l' n' m' G> ^' E' B
which are shown by the theory to be the principal circumstances on which
310 ON THE THEORY OF LUBRICATION, ETC. [52
lubrication depends, although not the same in other cases, will still be the
principal circumstances, and indicate the conditions to be fulfilled in order
to secure good lubrication.
The verification of the equations for viscous fluids, under such extreme
circumstances, affords a severe test of the truth and completeness of the
assumptions on which these equations were founded. While the result of
the whole research is to point to a conclusion (important in Natural
Philosophy) that not only in cases of intentional lubrication, but wherever
hard surfaces under pressure slide over each other without abrasion, they are
separated by a film of some foreign matter, whether perceivable or not.
And that the question as to whether this action can be continuous or not,
turns on whether the action tends to preserve the matter between the
surfaces at the points of pressure, as in the apparently unique case of the
revolving journal, or tends to sweep it to one side, as is the result of all
backwards and forwards rubbing with continuous pressure.
The fact that a little grease will enable almost any surfaces to slide for a
time, has tended doubtless to obscure the action of the revolving journal to
maintain the oil between the surfaces at the point of pressure. And yet,
although only now understood, it is this action that has alone rendered our
machines, and even our carriages possible. The only other self-acting system
of lubrication is that of reciprocating joints with alternate pressure on
and separation (drawing the oil back or a fresh supply) of the surfaces.
This plays an important part in certain machines, as in the steam-engine,
and is as fundamental to animal mechanics as the lubricating action of the
journal is to mechanical contrivances.
53.
ON THE FLOW OF GASES.
[From the " Philosophical Magazine," March, 1886.]
(Read before the " Manchester Literary and Philosophical Society,"
November 17, 1885.)
1. AMONGST the results of Mr Wilde's experiments on the flow of gas,
one. to which attention is particularly called, is that when gas is flowing from
a discharging vessel through an orifice into a receiving vessel, the rate at
which the pressure falls in the discharging vessel is independent of the
pressure in the receiving vessel, until this becomes greater than about five-
tenths the pressure in the discharging vessel. This fact is shown in Tables
IV. and V. in Mr Wilde's paper; thus, the fall of pressure from 135 Ibs.
(9 atmospheres) in the discharging vessel is 5 Ibs. in 7 '5 seconds for pressures
in the receiving vessel, ranging from one half-pound to nearly 5 or 6 atmo-
spheres.
With smaller pressures in the discharging vessel, the times occupied by
the pressure in falling a proportional distance are nearly the same, until the
pressure in the receiving vessel reaches about the same relative height.
What the exact relation between the two pressures is when the change
in rate of flow occurs, is not determined in these experiments. For as the
change comes on slowly, it is at first too small to be appreciable in such
short intervals as 7\5 and 8 seconds. But an examination of Mr Wilde's
Table VI. shows that it lies between '5 and '53.
This very remarkable fact, to which Mr Wilde has recalled attention,
excited considerable interest fifteen or twenty years ago. Graham does not
appear to have noticed it, although On reference to Graham's experiments it
appears that these also show it in the most conclusive manner (see Table IV.,
Phil Trans. 1840, Vol. IV. pp. 573—632; also Reprint, p. 106). These
312 ON THE FLOW OF GASES. [53
experiments also show that the change comes on when the ratio of the
pressures is between '483 and '531.
R. D. Napier appears to have been the first to make the discovery*. He
found, by his own experiments on steam, that the change came on when the
ratio of pressures fell to '5 (see Encyc. Brit. Vol. xn. p. 481). Zeuner,
Fliegner, and Hirn have also investigated the subject.
At the time when Graham wrote, a theory of gaseous motion did not
exist. But after the discovery of the mechanical equivalent of heat and
thermodynamics, a theory became possible, and was given with apparent
mathematical completeness in 185(5. This theory appeared to agree well
with experiments until the particular fact under discussion was discovered.
This fact, however, directly controverts the theory. For on applying the
equations giving the rate of flow through an orifice to such experiments as
Mr Wilde's, it appears that there is a marked disagreement between the
calculated and experimental results. The calculated results are even more
remarkable than the experimental ; for while the experiments only show
that diminishing the pressure in the receiving vessel below a certain limit
does not increase the flow, the equations show that by such diminution of
pressure the flow is actually reduced and eventually stopped altogether.
In one important respect, however, the equations agree with the experi-
ments. This is in the limit at which diminution of pressure in the receiving
vessel ceases to increase the flow, which limit by the equations is reached
when the pressure in the receiving vessel is '527 of the pressure in the
discharging vessel.
The equations referred to are based on the laws of thermodynamics, or
the laws of Boyle, Charles, and that of the mechanical equivalence of heat.
They were investigated by Thomson and Joule (see Proc. Roy. Soc., May
1856), and by Prof. Julius Weisbach (see Civilingenieur, 1856); they were
given by Rankine (articles 637, 637 A, Applied Mechanics), and have since
been adopted in all works on the theory of motion of fluids.
Although discussed by the various writers, the theory appears to have
stood the discussion without having revealed the cause of its failure ; indeed,
Hirn, in a late work, has described the theory as mathematically satisfactory.
Having passed such an ordeal, it was certain that if there were a fault, it
would not be on the surface. But that by diminishing the pressure on the
receiving side of the orifice the flow should be reduced and eventually
* The account of E. D. Napier's experiments is contained in letters in the Engineer, 1867,
vol. xxiii. January 4 and 25. They were made with steam generated in the boiler of a small
screw-steamer and discharged into an iron bucket, the results being calculated from the heat
imparted to a constant volume of water in the bucket in which the steam was condensed.
53] ON THE FLOW OF GASES. 313
stopped, is a conclusion too contrary to common sense to be allowed to pass
when once it is realized; even without the direct experimental evidence
in contradiction, and in consequence of Mr Wilde's experiments, the author
was led to re-examine the theory.
2. On examining the equations, it appears that they contain one assump-
tion which is not part of the laws of thermodynamics, or of the general
theory of fluid motion. And although commonly made and found to agree
with experiments in applying the laws of hydrodynamics, it has no founda-
tion as generally true. To avoid this assumption, it is necessary to perform
for gases integrations of the fundamental equations of fluid motion which
have already been accomplished for liquids. These integrations being
effected, it appears that the assumption above referred to has been the cause
of the discrepancy between the theoretical and experimental results, which
are brought into complete agreement, both as regards the law of discharge
and the actual quantity discharged. The integrations also show certain facts
of general interest as regards the motion of gases.
When gas flows from a reservoir sufficiently large, and initially (before
flow commences) at the same pressure and temperature, then, gas being a
nonconductor of heat when the flow is steady, a first integration of the
equation of motion shows that the energy of equal elementary weights of the
gas is constant. This energy is made up of two parts, the energy of motion
and the intrinsic energy. As the gas acquires energy of motion, it loses
intrinsic energy to exactly the same extent. Hence we have an equation
between the energy of motion, i.&-. the velocity of the gas, and its intrinsic
energy. The laws of thermodynamics afford relations between the pressure,
temperature, density, and intrinsic energy of the gas at any point. Substi-
tuting in the equation of energy, we obtain equations between the velocity
and either pressure, temperature, or density of the gas.
The equation thus obtained between the velocity and pressure is that
given by Thomson and Joule ; this equation holds at all points in the vessel
or the effluent stream. If, then, the pressure at the orifice is known, as well
as the pressure well within the vessel where the gas has no energy of motion,
we have the velocity of gas at the orifice ; and obtaining the density at the
orifice from the thermodynamic relation between density and pressure, we
have the weight discharged per second by multiplying the product of velocity
with density by the effective area of the orifice. This is Thomson and Joule's
equation for the flow through an orifice. And so far the logic is perfect, and
there are no assumptions but those involved in the general theories of
thermodynamics and of fluid motion.
But in order to apply this equation, it is necessary to know the pressure
314 ON THE FLOW OF GASES. [53
at the orifice ; and here comes the assumption that has been tacitly made :
that the pressure at the orifice is the pressure in the receiving vessel at a
distance from the orifice.
3. The origin of this assumption is that it holds, when a denser liquid
like water flows into a light fluid like air, and approximately when water
flows into water.
Taking no account of friction, the equations of hydrodynamics show that
this is the only condition under which the ideal liquid can flow steadily from
a drowned orifice. But they have not been hitherto integrated so far as
to show whether or not this would be the case with an elastic fluid.
In the case of an elastic fluid, the difficulty of integration is enhanced.
But on examination it appears that there is an important circumstance
connected with the steady motion of gases which does not exist in the case of
liquid. This circumstance, which may be inferred from integrations already
effected, determines the pressure at the orifice irrespective of the pressure in
the receiving vessel when this is below a certain point.
4. To understand this circumstance, it is necessary to consider a steady
narrow stream of fluid in which the pressure falls and the velocity increases
continuously in one direction.
Since the stream is steady, equal weights of the fluid must pass each
section in the same time ; or, if u be the velocity, p the density, and A the
area of the stream, the joint product upA is constant all along the stream, so
that
W
A = ,
gpu
where — is the mass of fluid which passes any section per second.
U
In the case of a liquid p is constant, so that the area of the section of the
stream is inversely proportional to the velocity, and therefore the stream
will continuously contract in section in the direction in which the velocity
increases and the pressure falls, as in Fig. 1, also Fig. 2 A.
Fig. 1.
In the case of a gas, however, p diminishes as the velocity increases and
the pressure falls ; so that the area of the section will not be inversely
53]
ON THE FLOW OF GASES.
315
proportional to u, but to u x p, and will contract or increase according to
whether u increases faster or slower than p diminishes.
As already described, the value of pu may be expressed in terms of the
pressure. Making this substitution, it appears that pu increases from zero
as p diminishes from a definite value p^ until p = •5%7p1 ; after this pu
diminishes to zero as p diminishes to zero. A varies inversely as pu, and
therefore diminishes from infinity as p diminishes from pl till p = '52tjpl;
then A has a minimum value and increases to infinity as p diminishes to
zero, as in Fig. 2.
Fig. 2.
The equations contain the definite law of this variation, which, for a
particular fall of pressure, is shown in Fig. 2 A.
Fig. 2 A.
For the present argument it is sufficient to notice that A has a minimum
value when p=-527jt>,; since this fact determines the pressure at the orifice
when the pressure in the receiving vessel is less than '527^, that being the
pressure in the discharging vessel.
316 ON THE FLOW OF GASES. [53
5. If, instead of an orifice in a thin plate, the fluid escaped through a
pipe which gradually contracted to a nozzle, then it would follow at once,
Fig. 3.
from what has been already said, that when pz was less than '52*7 plt the
naiTowest portion of the stream would be at N, for since the stream converges
to N the pressure above N can be nowhere less than '527^ ; and since
emerging into the smaller surrounding pressure p2 the stream would expand
laterally, N would be the minimum breadth on the stream, and hence the
pressure at N would be '527p1. In a broad view we may in the same way
look on an orifice in the wall of a vessel as the neck of a stream. But if we
begin to look into the argument, it is not so clear, on account of the curva-
ture of the paths in which some of the particles approach the orifice.
Since the motion with which the fluid approaches the orifice is steady,
the whole stream, which is bounded all round by the wall, may be considered
to consist of a number of elementary streams, each conveying the same
quantity of fluid. Each of these elementary streams is bounded by the
neighbouring streams, but as the boundaries do not change their position
they may be considered as fixed.
The figure (4) shows approximately the arrangement of such stream.
But for the mathematical difficulty of integrating the equations of motion,
the exact form of these streams might be drawn. We should then be able to
determine exactly the necks of each of these streams. Without complete
integration, however, the process may be carried far enough to show that the
lines bounding the streams are continuous curves which have for asymptotes
on the discharging-vessel side lines radiating from the middle of the orifice
at equal angles, and, further, that these lines ail curve round the nearest
edge of the orifice, and that the curvature of the streams diminishes as the
distance of the stream from the edge increases.
These conclusions would be definitely deducible from the theory of fluid
motion could the integrations be effected, but they are also obvious from the
figure and easily verified experimentally by drawing smoky air through a
small orifice.
From the foregoing conclusions it follows, that if a curve be drawn from
53]
ON THE FLOW OF GASES.
317
A to B, cutting all the streams at right angles, the streams will all be
converging at the points where this line cuts them, hence the jiecks of the
streams will be on the outflow side of this curve. The exact position of
Fig. 4.
these necks is difficult to determine, but they must be nearly as shown in
the figure by cross lines. The sum of the areas of these necks must be less
than the area of the orifice, since, where they are not in the straight line AB
the breadth occupied on this line is greater than that of the neck. The sum
of the areas of the necks may be taken as the effective area of the orifice ;
and, since all the streams have the same velocity at the neck, the ratio which
this aggregate area bears to the area of the orifice may be put equal to K,
a coefficient of contraction.
If the pressure in the vessel on the outflow side of the orifice is less than
•527_/)1, this is the lowest pressure possible at the necks, as has already been
pointed out, and on emerging the streams will expand again, as shown in the
Fig. 4, the pressure falling and the velocity increasing, until the pressure in
the streams is equal to p.2, when in all probability the motion will become
unsteady.
If 7>2 is greater than '527/),, the only possible form of motion requires the
pressure in the necks to be p2, at which point the streams become parallel
until they are broken up by eddying into the surrounding fluid (Fig. 5).
318
ON THE FLOW OF GASES.
[53
6. There is another way of looking at the problem, which is the first
that presented itself to the author.
Fig. 5.
Suppose a parallel stream flowing along a straight tube with a velocity u,
and take a for the velocity with which sound would travel in the same gas
at rest, the velocity with which a wave of sound or any disturbance would
move along the tube in an opposite direction to the gas would be a — u.
If then a = u, no disturbance could flow back along the tube against the
motion of the gas ; so that, however much the pressure might be suddenly
diminished at any point in the tube, it would not affect the pressure at points
on the side from which the fluid is flowing. Thus, suppose the gas to be
steam and this to be suddenly condensed at one point of the tube, the fall of
pressure would move back against the motion, increasing the motion till
u = a, but not further ; just as in the Bunsen's burner the flame cannot flow
back into the tube so long as the velocity of the explosive mixture is greater
than the velocity at which the flame travels in the mixture.
53] ON THE FLOW OF GASES. 319
According to this view, the limit of flow through an orifice should be the
velocity of sound in gas, in the condition as regards pressure, Density, and
temperature, of that in the orifice; and this is precisely what it is found to be
on examining the equations.
7. The following is the definite expression of the foregoing argument.
The adiabatic laws for gas are : p being pressure, p density, r absolute
temperature, and 7 the ratio of the specific heats,
y-l
'1 m
The equation of motion, u being the velocity and x the direction of motion, is
du _ dp
P dx dx'
u? [P dp „
or «•— - ~+G (2>-
Substituting from equations (1),
JP dp _ 7 JPO r_
Jo p 7- 1 /30r0'
7 - l p0 TO ( v^j
^ y^ \p,
Hence along a steady stream, since W is constant, equation (5) gives
a relation that must hold between A and p.
Differentiating A with respect to p and making -^-- zero, it appears
Y-I r-i
.............................. (6),
__
» / 2 V1
or 1: = -
^
P'or air 7 = 1'408.
/. - = -527 ...(8).
320
ON THE FLOW OF GASES.
[53
It thus appears that as long as p falls, the section continuously dimin-
ishes to a minimum value when p = -o2lp1, and then increases again.
Substituting this value of p in equation (3),
-•/
v.
-</,
(y+l)p0\p
.(10),
y-l
y-l
p
Hence by equation (6),
»=^T (12),
which is the velocity of sound in the gas at the absolute temperature r.
It thus appears that the velocity of gas, at the point of minimum area of
a stream along which the pressure falls continuously, is equal to the velocity
of sound in the gas at that point.
8. From the equation of flow (5) it appears that for every value of A
other than its minimum value, there are two possible values of the pressure
which satisfy the equation, one being greater and the other less than
•527pj.
It therefore appears that in a channel having two equal minima values of
section A and C, as in Fig. 6, the flow from A to B may take place in either
Fig. 6.
of two ways when the velocity is such that the pressure at A and B is •527|>1,
i.e. the pressure may either be a maximum or a minimum at G. In this
respect gas differs entirely from a liquid, with which the pressure can only
be a maximum at C.
54.
ON METHODS OF INVESTIGATING THE QUALITIES OF
LIFEBOATS.
[From the " Proceedings of the Manchester Literary and Philosophical
Society," Vol. xxvi.]
(Read December 14, 1886.)
THE lamentable accidents to the St Anne's and Southport lifeboats on
the 9th inst. seem likely to lead to steps being taken to obtain a more
systematic investigation as to the qualities of these boats than has yet been
undertaken.
It seems, therefore, a proper time to direct attention to certain facts and
general considerations, the importance of which have impressed themselves
upon me during many years' investigation.
Before entering upon this, it may be remarked that there is probably no
class of boats, on the design and construction of which more attention and
skill have been spent, than on lifeboats, or of which the qualities are so well
adapted to the circumstances, taken all round. If we compare the results
of the use of these boats with the results obtained in the use of the navies
of this or any other country, it will, without a moment's hesitation, be
admitted that the designers of lifeboats and lifeboat paraphernalia have
arrived much nearer perfection than the designers of war vessels and their
armaments.
That the high standard already obtained by these boats has not been
the result of scientific investigation, or the theoretical application of any
known principles of equilibrium, does not render the method less scientific,
for the base of all science is observation and experiment, and these boats
o. R. ii. 21
322 ON METHODS OF INVESTIGATING THE QUALITIES OF LIFEBOATS. [54
are the result of such a course of direct experiments and experimental
observation as has not been expended on any other modern structure, nor
is this method of arriving at the best form peculiar to lifeboats.
With the exception of the large modern steamers and ironclads, the
peculiar construction of boats of all sizes is the result of a prolonged process
of trial and failure, and that, although certain general principles, connecting
the qualities of ships with their shapes, have been discovered and recognised
during the last thirty years, still, the recognition of these principles has not
resulted in the suggestion of any considerable improvement to be effected
in what were before high class vessels, such as yachts and fast sailing vessels,
but rather have confirmed the form previously arrived at in these as the
best, and led to their being copied in larger vessels.
The discovery and recognition of principles have undoubtedly been of
immense service in improving the types of our large modern vessels. But
this is mainly because with large ships there is not the same opportunity
for trial and failure as with the small, the number is so much smaller, and
experiments are so much slower and more costly ; but the main reason
is, that the circumstances which call out the highest qualities of the large
vessels become so extremely rare. There is no doubt that many large
vessels pass through their lives without meeting weather which tests their
sea-going qualities in the way in which those of a fishing boat are tested
many times every winter. It was, therefore, an immense step in the way
to study the resistance qualities of large ships, when the late Mr Fronde
brought into practice the rules connecting the resistance of the full-sized
vessel with that of an exact model to scale.
By means of a tank 200 feet long, and models on scales of 1 to 50, or
1 to 20, the resistance and rolling qualities of all Her Majesty's ships have
since been verified before they are constructed. And the same is now done
by manufacturers of mercantile vessels, like Mr Denny, who have tanks of
their own. The qualities of ships thus tested were originally limited to
those of resistance and of rolling, and so far as I know, no extension has
taken place; for although in 1876 it was pointed out by the author before
Section 9 of the British Association, that by constructing models of our war
ships on a scale large enough to enable them to be used as launches, say
1 to 16, and supplying these launches with power as the cube of their
dimensions — then the manoeuvering qualities would be similar if conducted
on scales proportional to their lengths, the time occupied by the launches
in executing a particular evolution, as compared with that occupied by the
ships, being as the square root of their lengths. So that with such models
the officers and seamen could be instructed in the handling of their ships
without cost or risk. This has not been done. The Admiralty replying,
54] ON METHODS OF INVESTIGATING THE QUALITIES OF LIFEBOATS. 323
so far as they did reply, that their officers were continually experimenting
with the launches — disregarding the fact that the launches in ijse_were in
no sense models of the ships, and were supplied with power five or six
times too great in proportion — thus ignoring the point of the suggestion,
namely, that the experience gained by the models might be applicable to
the ships, which with their present launches it is not, and only tends to
mislead those who attempt a comparison.
Since making this suggestion, I have been much engaged in experiments
with water, which have enabled me to extend this law of similarity, until
I find it is possible now to lay down the conditions under which to test
the seaworthy qualities of a vessel from those of its model.
Certain conditions have to be observed, but, in general, it may be
asserted that provided the models are to scale, that the height and length
of the waves are to the same scale, the velocity of the wind being as the
square root of the scale, or in other words, the corresponding depressions
of the barometer in the same scale as the models — the behaviour of the
model would be similar to that of the boat.
Thus, the behaviour of a model three feet long in waves two feet high,
and with a wind twenty miles an hour, would correspond with that of a
boat twenty-seven feet long in waves eighteen feet high, and a velocity of
the wind sixty miles an hour.
The main object of this communication is to point out that this similarity
in the behaviour of models and larger boats under circumstances as regards
the stress of weather, corresponding in scale to that of the models and boats,
affords an opportunity of testing the seaworthy qualities of the lifeboats in
a degree that they cannot otherwise be tested. For, although the size of the
boats does not preclude the possibility of their qualities being actually
tested under any circumstances of sufficiently common occurrence to afford
opportunities, yet the circumstances which call for the highest qualities in
these boats, and in which the boats are most needed, are of extremely rare
occurrence ; this appears at once, when it is considered that it is years since
anything approaching such a storm as wrecked the two boats has been
experienced, and that in order to submit any modified boat to a similar
test, it may be years before there will be another chance, even if it could
be made available when it did come. To make satisfactory tests on the
full-sized boats, command is wanted of the extreme circumstances, and this
cannot be had ; while on the other hand, to test the same qualities in their
models, these extreme circumstances, modified to scale, are all that is wanted,
and these are of such common occurrence as to afford ample opportunity,
even if they cannot be commanded by artificial means.
21—2
324 ON METHODS OF INVESTIGATING THE QUALITIES OF LIFEBOATS. [54
If the qualities to be tested involved the handling of the boats, then
the models must be large enough to carry a crew ; that is to say, they
would have to be small lifeboats. Even with such, much experience can be
and has been gained, which could not be obtained with larger boats, for bad
weather for the smaller is only moderate for the larger, and is of compara-
tively common occurrence compared with that which affords a similar test
for the larger boats.
It is, however, the self-righting qualities of these boats that is for the
moment in question ; this requires no crew, or at most a dummy crew, so
that there is no limit to the smallness of the models, except what arises
from the conditions of dynamical similarity, and these would admit of
models as small as two or three feet.
It may be well to say one word as to the powers of self-righting, and the
question as to how far these powers may be affected by the wind and waves.
I do not know that it has ever been suggested that wind and wave have any
such effect. But it is equally certain, that there is no d priori reason why
they should not, and. short of actual experience, it cannot be said that any
boat which would right itself in calm water would do so equally well in any
storm that might blow. On the other hand there are reasons why wind and
waves must, individually and collectively, affect the stability of an upturned
boat.
In the first place, the wind will keep such a boat broadside on, which
will be in the trough of the sea raised by the wind, although the swell may,
of course, be running in another direction. The wind, acting on the bottom,
will further drive the boat broadside on through the water. This horizontal
thrust of the wind, acting on the part of the boat above water, and balanced
by the resistance of the water on the submerged portion, will tend to right
the boat by turning her keel to leeward, and so far it would seem that the
wind would help to right her, but owing to the shape of the bottom of the
boat when broadside on, there will be a vertical force resulting from the
wind as well as the horizontal, and this vertical force will bear down that
side of the boat toward the wind, and this effect will be enhanced by the
weight of the waves breaking on this side of the boat tending to right
her by turning her keel to windward, or in direct opposition to the horizontal
effect ; and more than this, the vertical effect of the wind and waves to turn
the keel to windward will be greatest when the windward side of the boat's
bottom has some definite inclination to the horizontal, while the horizontal
effect to turn the keel to leeward will continually increase as the keel turns
to windward, so that it is possible that in a particular wind and sea there
may be a position of very stable equilibrium, towards which, if the keel
is to leeward, the vertical effect of the wind and the waves predominating
54] ON METHODS OF INVESTIGATING THE QUALITIES OF LIFEBOATS. 325
over the horizontal effect, will bring it back, and vice versa ; if the keel is
turned to windward, the horizontal effect predominating will alsa tend to
bring it back.
The fact that two boats were found stranded bottom upwards, with part
of their crews underneath, and that one of these is known to have upset in
comparatively deep water, and to have remained in that position during a
long time while drifting into shallow water, seems altogether inconsistent
with the supposition that these upturned boats were in their normal
condition of instability, as when in calm water. For although in a calm
sea the effect of three or four men hanging on to each side of the boat
might prevent the initial motion of turning, before the weight of the iron
keel and ballast obtained sufficient leverage to lift the weight of the men
and so keep the boat stable, this could hardly be the case in a rough sea,
when the waves would be continually altering the balance of the boat.
These are questions which can only be set at rest by experiments, and
the method of models thus affords a means of testing the righting qualities
of these boats under circumstances as severe or more severe than any to
which they will ever be subjected, and this without waiting and without
danger; while with full-sized boats such tests are impossible, for even
should an extreme storm occur opportunely for making the experiment,
the danger involved with full-sized boats would preclude the possibility
of their being undertaken. It is this last consideration which has led to
these suggestions, and not the idea that the experiments on models would
be more satisfactory ; while the fact that the experiments on models could
be made at much smaller cost, is too small a matter to be considered, when,
as in this case, the lives of some of the most heroic of our fellow countrymen,
and the sentiments of the entire nation, are involved.
55.
ON CERTAIN LAWS RELATING TO THE REGIME OF RIVERS
AND ESTUARIES, AND ON THE POSSIBILITY OF EXPERI-
MENTS ON A SMALL SCALE.
[From the " Report of the British Association," 1887.]
1. THE object of this communication is to bring before Section G certain
results and conclusions with respect to the action of water to arrange loose
granular material over which it may be flowing. These results and con-
clusions were in the first instance arrived at during a long-continued in-
vestigation, undertaken with a view to bring the general theory of hydro-
dynamics into accord with experience, rather than with any special reference
to the subject in hand, but have since been to some extent made the subject
of special investigation.
2. A systematic study of the regime of rivers naturally divides itself
under three heads, which may be stated as follows : —
(1) The more general facts observed as regards the regimen of the
beds.
(2) The movements of sand consistent with these observed facts.
(3) The necessary actions of the water to produce these movements
in the material of the beds.
Observed facts. — Amongst the most general facts to be observed as to
the arrangement of the material forming the beds of estuaries are —
(1) The general stability or steadiness of these beds, so far as is shown
by their outline or figure, while, at the same time, as is shown by the
obliteration of all footprints and markings casually placed upon them, also by
the ripple mark, the material at the surface of these beds is being continually
shifted.
55] ON CERTAIN LAWS RELATING TO RIVERS AND ESTUARIES, ETC. 327
(2) The almost absolute steadiness in figure of some of these beds.
(3) The gradual changes in the position and form of -others — the
growth or accumulation of sand-banks *in some places, and the wasting of
banks or removal of sand in others.
Movement of sand. — As regards the movement of sand consistent with
these changes, in the first place the movement, whatever it may be, is one
of the surface, and not one in bulk ; and in the next place such movement
of the surface must be continually going on, whether it produces any
change in the figure of the banks or not. The invariable obliteration of
footprints and marks which may have been left on the sand at low water, as
well as the ripple marks, are absolute evidence of a general disturbance of
the surface, and it requires but little observation to show that the disturbance
is of the character of a drift of sand, in whatever direction the water may
be moving.
Uniform drift. — Where the outline of the banks is not altered, this
drift or motion of the sand must be uniform, as much sand being deposited
at each point as is removed from that point. Although there may be a
general flow of the sand in some direction, if the drift is uniform this
movement will not alter the figure of the bed, which, like the balance in
another kind of bank, does not depend on the rate of deposit and with-
drawal, but on the excess of one of these over the other. The gradual
accumulation or diminution of sand at any point is clearly not due to a
simple action of deposit or removal, as it is always attended with the same
evidence of the drifting of the surface, and is clearly the result of a difference
in the quantities of sand deposited or removed by the drift.
Movement of water. — The manner in which a current of water acts on
the granular material forming the bed of the current has been the subject
of an investigation by various experimenters. It has been found that the
primary action is not so much to drag the grains along the bottom, but
to pick them up, hold them in a kind of eddying suspension, at a greater
or less height above the bed, for a certain distance and then drop them,
so that, when the water is drifting the sand, there is a layer of water adjacent
to the bottom, of a greater or less thickness, charged to a greater or less
extent with sand. The faster the current and the finer the sand the greater
will be the thickness of the charged layer, as well as the denser is the charge
in the layer.
A certain definite velocity, according to the size and weight of the grains,
is required before the water will raise the grains from the bottom, and for all
velocities above the minimum necessary to raise the sand the suspended
charge increases with the velocity, and the rate of drift or the quantity
328 ON CERTAIN LAWS RELATING TO RIVERS AND ESTUARIES, [55
of sand which passes a particular section increases much faster than the
velocity. Attempts have been made, with greater or less success, to deter-
mine exact laws connecting the minimum velocities at which the sand
begins to drift, with the weight of the grains and other circumstances ;
also to determine the exact law of rate of increase of the drift with the
velocity.
For my present purpose, however, it is not necessary to enter upon
such considerations.
From the facts already mentioned, it will appear that the effect of a
uniform current of water over a uniform bed of sand will not be to raise
or lower the bed ; for, as the charge of sand in the water remains uniform,
it must drop as many particles as it raises everywhere on the bed. This is
the action of the water in causing a uniform drift.
It is also evident that, if the charge in the water as it comes to any
particular place is less than the full charge due to its velocity, it will pick
up from that place more sand than it drops, and so increase its charge
at the expense of the bed, which will there be scoured or lowered. And
conversely, if the water as it arrives at any place is overcharged, it will
relieve itself by depositing more than it picks up, and so raise or silt up
the bed.
As regards the circumstances which can cause the water to be charged to
a greater or less extent than that which it would just maintain with such
velocity as it has, the most important are —
(1) An increasing or diminishing velocity. When the water is moving
in a stream from a point where the velocity is less to one where it is greater,
the velocity of the actual water as it moves along is increasing, as will also
be its normal charge of sand ; hence it must be continually picking up more
than it deposits. And conversely, when moving from a point of greater
velocity to one of less, its normal charge will be continually diminishing
through deposits on the bed.
(2) Another circumstance which affects the charge of sand with which
the water may arrive at a particular point is a variation in the character of
the bed. If, for instance, water flows from a rocky bed on to sand, it
may arrive on the sand without charge, and immediately charges itself at
the expense of the bed. Or again, where water flows from a sandy bottom on
to a clean or grassy rocky bottom, it gradually loses its charge, silting up the
bottom.
The direction in which the sand is moved by the water is sensibly in the
direction in which the water which holds the charge is moving. But, as was
first pointed out by Dr James Thomson as affording an explanation of the
55] AND ON THE POSSIBILITY OF EXPERIMENTS ON A SMALL SCALE. 329
generally observed fact that the beds of rivers are scoured on their convex
sides and silted on their concave, the layers of water adjacent tn the bed do
not always move in the general direction of the stream. There are often
steady cross currents at the bottom, as in the case mentioned, though such
cross currents do not exist except under circumstances which may be
readily distinguished. The most important of these is that pointed out by
Dr Thomson — curvature in the general direction of the stream, in which
case the centrifugal force of the more rapidly moving water above over-
balances that of the water retarded by the bottom, and forces the latter back
towards the centre of the curve.
This action is universal, where even the lateral boundaries are such as to
require the water to move in curved streams ; the drift at the bottom does
not follow the general direction of the stream, but sets towards the centre of
the curve.
The result of the foregoing consideration is to lead to the conclusion that
the regime of each part of the bed as to maintenance in steady condition,
lowering or raising it any time, depends solely on the character of the motion
of the water, which if straight and uniform, neither acquiring nor losing
velocity, causes a uniform drift in the direction of the stream, which main-
tains the condition steady. If losing velocity, causes a depositing drift and
raises the bed ; if gaining velocity, causes a scouring drift and lowers the
bed ; while if curved, the direction of the drift is diverted towards the centre
of the curve, with its attendant effect to lower the convex side and raise the
concave side of the bed. This conclusion seems to be of the utmost im-
portance in dealing with this subject. For if it is correct, not only can the
character of the action going on at the bed be inferred from the observed
motion of the water, and vice versa, but since, according to this conclusion,
the character of the action is independent of the magnitude or velocity of the
stream, the results will be the same on a small scale as on a large one,
provided only that the character of the motion of the water is the same
at all points. In this latter respect this conclusion affords an explanation of
a fact that cannot fail to have struck every one who has observed the sand-
beds of the streams running over sands which have been left by the tide,
viz., what an almost exact resemblance they bear to each other, whether
having the size of a moderate river or of the smallest rivulet.
On the large scale of actual estuaries we can only test the conclusion by
actual observation, but on a small scale we can experimentalise in whatever
condition of motion we want to test, and readily observe the effects pro-
duced ; a possibility of which great use has been made in this investigation,
and which will be again referred to.
As applied to a non-tidal river, in which the direction of the motion
330 ON CERTAIN LAWS RELATING TO RIVERS AND ESTUARIES, [55
is always the same, the foregoing conclusion would lead us to expect that
the regime would be steady except at the bends, the sources, and the mouth,
which is exactly what is observed, so that the conclusion so far agrees with
experience. The most striking feature about rivers is the way they wriggle
about in the alluvial valleys ; a phenomenon pointed to by Lyeli as one of
those causes still in progress which had produced the present conditions of
the valleys, and which, as already stated, was explained by Dr Thomson.
From the source of the river, as the rain-water acquires the velocity, it
charges itself with deposit, which charge it maintains with continual taxes
and drawbacks until it reaches the ocean or lake, when its water in again
losing its velocity deposits its charge, continually carrying forward the bar
and extending its delta.
In non-tidal rivers, whether large or small, fast or slow, the characters of
these actions are invariable, however much they may differ in intensity. The
case of tidal estuaries is, however, by no means so simple. Here we have
not, as in a river, a continuous progression of the same character of action at
the same point. On the contrary, at every point the action is changed
twice a day. For the change in the tidal current does not merely change or
reverse the direction of the sand-drift at each part of the bed, but it changes
and often reverses the character of this drift, changing what has been a
scouring drift during the ebb-tide into a depositing drift during the flood ;
so that the question as to whether the regime is stable, depositing, or
scouring is not simply a question as to whether the current at this point
is uniform, accelerated, or retarded, but whether the action of the ebb to
cause, say, scour is equal to, less than, or greater than the action of the flood
to cause deposit.
As there is no likelihood that the resultant effect as regards the general
regime of two opposing influences will resemble what would have been the
simple effect of either of the influences acting alone, this dual control affords
abundant reason why the configuration of the beds of these tidal estuaries
should differ in character from the configuration of the sand-beds of continuous
streams.
There is, however, another and an equally important difference between
the general motion of the water in rivers and tidal estuaries.
The function of the estuary is by no means that of a simple channel to
conduct the tidal water up and down. It equally discharges the function of
a reservoir or basin, to be filled and emptied by each tide.
In consequence of this action as a reservoir, the directions of the motions
of the water during flood and ebb, and particularly towards the top of the
flood and commencement of the ebb, are generally very different from what
55] AND ON THE POSSIBILITY OF EXPERIMENTS ON A SMALL SCALE. 331
they would be were the estuary acting the simple part of a channel conduct-
ing the water from one place to another.
When a vessel is filled by a stream entering on one side, the forward
motion of the water is stopped before reaching the opposite side. But if, as
is always the case, the motion which the water has on entering is more than
sufficient to carry it as far as is necessary, the remaining momentum is spent
in setting up eddies, or a general circulation in the water, so that when the
vessel is full the water within it is not by any means at rest, but may be
circulating round or have any other motion. If, then, the water is allowed
to flow out, the initial motion will not be a steady movement towards the
outlet from all parts of the vessel, but those portions of the water which are
moving towards the outlet will have their motion accelerated, while those
which are moving in the opposite direction will have first to be stopped
before they begin to approach the outlet. And thus the ebb will begin
earlier at some points in the vessel than at others.
It was the observation of such an effect as this in one of our largest
estuaries that first directed my attention to the subject of this paper.
Having investigated this point sufficiently for my own satisfaction nothing
further was done until 1885, when my attention was directed to the inner
estuary of the Mersey.
This estuary may be described as a crescent-shaped shallow pan, eleven
miles long by three broad, lying north-west and south-east, having its upper
horn pointing east and its lower horn north ; the northern horn, being
prolonged for five miles into a narrow deep channel, runs north to the outer
estuary or sandy bay of the sea. One of the most marked features presented
by the configuration of the bed of this inner estuary is the invariable prefer-
ence of the low-tide channels for the concave or Lancashire side ; whereas,
were the estuary acting merely the part of a river, whether during flood or
ebb, it would be expected to follow the usual law, and have the deepest
water on the convex or Cheshire side.
That this prevalence of the deepest water on the concave side must be
the result of the momentum left in the water by the flood at once seemed
to me probable ; for if the bottom were level or deepest on the Lancashire
side the effect of the curved shape would be to cause the flood entering at
the northern horn to follow the south-eastern or Cheshire shore, and the
momentum of this water would tend to carry it round the head of the
estuary and back along the Lancashire side ; would, in fact, tend to set up a
circulation before the top of the flood was reached; so that on the Lancashire
side the water would be moving down the estuary before the ebb commenced;
whence, considering that the flood tends to raise the bottom and the ebb to
332 ON CERTAIN LAWS RELATING TO RIVERS AND ESTUARIES, [55
lower it (for the reasons already pointed out), it seems that the stronger flood
on the Cheshire side would raise this side, while the stronger ebb on the
Lancashire side would lower this. This is supposing the bottom to be level.
In order to verify these conclusions a vessel was constructed having a flat
bottom and a vertical boundary of the same shape as the high-tide line of the
inner estuary from the rock to the same distance above Runcorn. The
horizontal scale was 2" to a mile, and the vertical scale 1 inch to 80 feet,
A shallow tin pan was hinged on to the otherwise open channel at the
rock, by raising and lowering which, when full of water, the motion of the
tide could be produced throughout the model through the narrows ; the true
form of the bed of the channel was given to the model by means of paraffin.
And in order to obtain approximately the proportional depth in the inner
estuary, sand was placed level on the bottom so that the high-tide depth was
reduced to the equivalent of about twenty feet. The idea in making this
model was not so much to obtain a shifting of the sand, as to show the
circulation of the water as resulting from the flood tide with a level bottom.
In the first instance the tide pan was raised and lowered by hand, but as at
the first trial it became evident that the model was not only going to show
the expected circulation, but was also capable of showing, by the change in
the position of the sand, the effect of this circulation on the configuration of
the estuary and other important effects, it was arranged that the model
should be worked from a continuously running shaft. The working of the
model by hand at once showed that there was only one period of working at
which the motion of the water in the model would imitate the motions of the
actual tide in the Mersey, which period was found to be about forty seconds ;
a result that might have been foreseen from the theory of wave motions,
since the scale of velocities varies as the square roots of the scales of wave
heights, so that the velocities in the model which would correspond to the
velocities in the channel would be as the square roots of the vertical scales —
about 3^— and the ratios of the periods would be the ratio of horizontal
scales divided by this ratio of velocities, or
33 J.
31800 " 950 '
Hence, taking 11 '25 hours or 40,700 seconds as the tidal period, the period of
the model
40700
= -jpr:- = 42 seconds (about).
you
This period was adopted for working the model from the shaft.
It was then found that the circulation at the top of the flood, which was
55] AND ON THE POSSIBILITY OF EXPERIMENTS ON A SMALL SCALE. 333
very evident while the bottom was flat, caused a general rise of the sand on
the Cheshire side and lowering on the Lancashire, which went <m for about
2,000 tides. That during this time, owing to the^ increase of flood up the
Lancashire side and the diminution of that on the Cheshire side which
followed from the deepening of the one and the shoaling of the other, the
circulation steadily diminished until its character was so changed that it
could no longer be called a general circulation, and that after this, although
there were further changes in detail going on in the estuary, the two sides
maintained a steady condition as regards depth for low tides.
During this time banks were formed and low-tide channels, which
resembled in all the principal features those actually in the Mersey ; the
eastern bank, with the deep sloynes on the Cheshire side, the Devil's Bank
and the Garston Channel, the Ellesmere Channel and the deep water in
Dungeon Bay and at Dingle Point — all these were very marked in character
and closely approximate in scale.
And, what is as important, the causes of these as well as all minor
features could be distinctly seen in the model.
The eastern and Devil's Bank are seen during the process of their forma-
tion to be simply an internal bar formed by carrying the sand brought down
by the ebb out of the narrows and sloyne, until debouching into the broad
estuary; its velocity is so far diminished that it can no longer carry its
charge, just as happens at the mouth of every river. The peculiar configura-
tion of these banks is explained by the existence of two lines of eddies from
about half-tide to the top of the flood : the first of these is caused by the
sharp corner at Dingle, and lies between Dingle and Garston, the eddies
having their centres over the Devil's Bank ; and the second, caused by the
divergence of the Cheshire Bank towards Eastham, having the lines of
centres over the Eastham Bank. These eddies, which during the most
rapid part of the flood only effect a diminution of the velocity of the flood,
cause, as the velocity slackens toward the top of the flood, back water to set
in along both shores, which back waters, starting the ebb, cause this to
be strongest over the Garstou and Eastham Channels, which are thus kept
open.
The lateral configuration of the shores at Dungeon Bay and at Ellesmere
is seen to cause back waters to exist in these bays during the whole of the
flood iu the latter, and from one to two hours before the top of the flood in
the former, which fully accounts for the deep water at these points. The
existence of these back . waters in the actual channel has been verified.
There are many other circumstances brought to light by this model, which it
is impossible for me here to notice without unduly extending the length of
this paper, if, indeed, I have not already done so. I will therefore only
334 ON CERTAIN LAWS RELATING TO RIVERS AND ESTUARIES, [55
remark that a second start was made with the sand Hat in this second model,
and that the result obtained was the same as regards the general features of
the estuary. So interesting were these results that it was decided to try a
larger scale. A model, having a horizontal scale of 6 inches to a mile, and a
vertical scale of 33 feet to an inch, was therefore made, and the tide produced
as before. The calculated period of this model is 80 seconds, and experiment
bears this out, any variation leading to some tidal phenomena, such as bonos
or standing waves, which are not observed in the estuary.
The disadvantage of the larger model is the time occupied — a little more
than a minute a tide — which means about 300 tides a day, or 2,000 tides a
week. On one occasion the model was kept going for 6,000 tides, and a
survey was then made of the state of the sand. And this will be seen to
present a remarkable resemblance in the general features to the charts of the
Mersey, of which three — 1861, 1871, 1881 — are shown; in fact the survey
from the model presents as great a resemblance to any one of these as they
do to each other.
It is impossible for me to enter upon all the points of agreement.
Taking into account that in both the estuary and the model there are always
changes going on within certain limits, and these changes do affect the
currents to a certain extent, it is not to be supposed that there will be exact
agreement between the currents at all points and at all states of the tides on
the model and estuary. Still there is a general agreement, and in the few
verifications I have made I have found that the current found in the model
at a particular point and state of tide is also to be found in the estuary.
In one respect the great difference between the model and the estuary
calls for remark : this is the much greater depth of the model as compared
with its length and breadth. The vertical scale being 33 feet to an inch,
and the horizontal scale 880 feet to an inch, so that the vertical heights are
nearly twenty-seven times greater than the horizontal distances, such a
difference is necessary to get any results at all with such small scale models;
and it is only natural to suppose that it would materially affect the action.
As a matter of fact, however, it does not seem to do so. And, further, it
would seem that, notwithstanding the general resemblance on the regime of
the beds of large and small streams running over sand, there is in these
a similar difference in vertical scale, the smaller streams not only having a
greater slope, but also having greater depth as compared with their breadth
and steeper banks. So far as the theory of hydrodynamics will apply, it
seems that in the model the effects of the momentum of the water would be
greater, as compared with the bottom resistances, than in the estuary, and
I think that they are. But the effects of momentum in the estuary greatly
preponderate on the resistances, as shown by the fact that the tide at the top
55] AND ON THE POSSIBILITY OF EXPERIMENTS ON A SMALL SCALE. 335
of the flood rises some 2 to 3 feet higher at high spring tides than it does at
the rock ; nor does it do much more than this in the model. In the model
it certainly seems that the general regime is determined by the momentum
effects, and from the almost exact resemblance which this regime bears to
that of the estuary, it would seem that, although the momentum effects may
be diminished by the greater resistance on the bottom, they are still the
prevailing influence in determining the configuration of the banks. Further
investigation will doubtless explain this, and also determine the best propor-
tional depths. From my present experience in constructing another model,
I should adopt a somewhat greater exaggeration of the vertical scale. In the
meantime I have called attention to these results, because this method of
experimenting seems to afford a ready means of investigating and determin-
ing beforehand the effects of any proposed estuary or harbour works; a means
which, after what I have seen, I should feel it madness to neglect before
entering upon any costly undertaking.
I have only to say that, as it was not practical to exhibit the model
to the Section, I have had it working in the new engineering laboratory
at the college. Unfortunately it could not be started before Monday, and
it will not yet have run more than 1,000 tides, since the sand was put in
flat, so that it is not probable that the regime is yet quite stable ; still the
principal features have come out*.
* For continuation see papers 57, 58, and 59.
56.
ON THE TRIPLE-EXPANSION ENGINES AND ENGINE-TRIALS
AT THE WHITWORTH ENGINEERING LABORATORY,
OWENS COLLEGE, MANCHESTER.
[From the "Proceedings of the Institution of Civil Engineers," 1889 — 90.]
(Read December 10, 1889.)
IN designing steam-engines to take their place amongst the appliances
of an engineering laboratory, at the present stage of the development of
these institutions, many considerations present themselves.
The primary purpose of the engines is to afford the students opportunities
of practice in making the various measurements involved in steam-engine-
trials, and to afford them an insight into the action of steam in the engine,
as well as of the mechanical actions ; also to render them familiar with good
examples in steam-engine design.
Another purpose, however, which it is very desirable such engines should
serve, is that of supplying a means of research by which knowledge of the
steam-engine may be extended. A systematic and experimental investigation
of the steam-engine involves two sets of conditions which, unless it be in a
laboratory, can hardly exist together, namely, the time arid attention of the
scientific investigator, and the assistance of a considerable number of trained
observers. In the engineering laboratory these conditions, should exist ;
the first being supplied by the permanent staff, and the second by the
students as their training advances.
The making and repeating of the individual observations involved in a
scientific engine-trial, as well as reducing the results, demands an amount
of patience and perseverance which is severe on one so young and in-
experienced as a student ; but the importance and reality which the research
56]
ON TRIPLE-EXPANSION ENGINES AND ENGINE-TRIALS.
337
adds to all the detail of the work, as well as the complete attention and
overlooking which it ensures from those responsible, constitute _very great
advantages.
Having regard to these two purposes, the Committee, Mr John Rams-
bottom, Mr John Robinson, and the Author, appointed by the Council of
Owens College to select, amongst other appliances, the steam-engines best
adapted for the special purposes of the laboratory, decided that the engines,
while as far as possible representing in their principal members the most
approved existing practice in steam-engine construction, should be specially
designed to afford the utmost facilities for experiments on the use of steam
throughout the entire range, and, if possible, beyond the limits hitherto
accomplished in practice.
As best meeting this demand it was decided to have three engines
working on separate brakes1. All engines to be of the inverted-cylinder
type, with the walls and covers separately jacketed with steam at boiler-
pressure, and so arranged that they could be worked with or without steam
in any or all of the jackets. Each engine to work with steam at any
pressure up to 200 Ibs. per square inch, to run at any piston speed up to
1,000 feet per minute, and to have expansion-gear to cut off from zero up
to | of the stroke. One engine to be supplied with air-pump and surface-
condenser, the other two engines to be furnished with alternative exhausts,
either into the atmosphere, or into steam-jacketed receivers supplying steam
to the next engine, each of the receivers also having an alternative supply
of stearn direct from the boiler. The boiler to be of the locomotive type,
having 5 square feet of grate, to be set in a hot chamber with an economizer
and alternative chimney and forced draught, on the closed stoke-hold system.
The condenser to have 200 square feet of cooling surface. The dimensions of
the engines to be somewhat as follow :
Engine
Diameter
of
Cylinder
Stroke
Diameter
of Crank-
Shaft
No. I (high-pressure)
inches
5
inches
10
inches
2f
No. II (intermediate)
8
10
2£
No. Ill (low-pressure)
Air-pump on No. Ill
12
9
15
4i
4
Feed-pump
H
Z
In addition to the brake, each engine was to be furnished with a fly-
* The advantage of having the engines on separate brakes was suggested to the Author by
Mr J. I. Thornycroft, M. Inst. C.E.
O. R. II. 22
338 ON TRIPLE-EXPANSION ENGINES AND ENGINE-TRIALS. [56
wheel, to act as a belt or rope-pulley, weighing about 1,200 Ibs., carried on
a separate shaft with a coupling to the crank-shaft.
The firm of Messrs Mather and Platt, Salford Iron Works, undertook the
preparation of the designs and the construction of special engines and boiler
to meet in all respects the wishes of the Committee, and spared neither
trouble nor expense in carrying out the work. It was entirely owing to the
zeal and liberality of this firm that the College was enabled to meet the
expense of an undertaking involving so much special work.
The design of the engines, shown in Figs. 1 and 2, contains many novelties.
These were not adopted without what appeared to the Committee to be
sufficient reason, as it was unanimously desired to adhere as far as possible
to ordinary types.
As regards the cylinders, pistons, and valves, there are three noticeable
departures ; these were adopted with a view —
1. To ensure the completeness and efficiency of the steam-jackets.
2. To diminish the resistance to the passage of steam as much as
possible.
3. To keep down the clearance.
4. To obtain an adjustable cut-off from zero at any speed.
1. To obtain completeness in jacketing, both ends (or covers) were
jacketed as well as the walls. To ensure efficiency of the jackets steel
liners were used and the covers were domed, so that the surfaces should
free themselves by gravitation from the water resulting from condensation,
the water being drained from the lowest point in the jacket spaces.
2. To diminish the resistance of the passages, these were abnormally
large, the area of the ports being 13 per cent, or =-- the area of the piston,
I "O
and the steam-chests were very large.
3. To diminish clearance, the ports were made straight, and the valves
brought as close as possible to the cylinder, double valves being used. The
pistons were formed to occupy the space in the cylinder, except | inch
clearance at the ends. The result is that in engine No. I the clearance
space shut in by the main valve is 4 per cent, and 1P7 per cent, more by
the rider, and in engines II and III the clearances shut in by the main
valve are 6 per cent, and 2-5 per cent, more by the riders.
4. To obtain an adjustable cut-off, since at the higher speed the engines
were intended to run 400 revolutions per minute, it was practically impossible
to use any form of trip cut-off. Meyer expansion- valves were used on the
backs of the main valves.
56]
ON TRIPLE-EXPANSION ENGINES AND ENGINE-TRIALS.
239
Sca.lv '/•to1.*'
22—2
340 ON TRIPLE-EXPANSION ENGINES AND ENGINE-TRIALS. [56
The engines are exceptionally strong, being all of them designed to work
safely with a pressure of 200 Ibs. on the square inch, so that the effect of
expansion in one cylinder might be compared with compound or triple
expansion.
The frames of the engines are of a somewhat novel form, and their
purpose may not be immediately apparent. It will be seen, however, that
the front cover is cast with a kind of entablature or box, connected with
the base-plate by four wrought-iron columns placed symmetrically as regards
the piston-rod. The function of these columns is to withstand the vertical
forces arising from the steam-pressures on the cylinder covers, and to
maintain the axis of the cylinder vertical against any forces ; they are not
calculated to maintain a horizontal position against lateral forces such as
might arise from the action of the slide-block. To meet such lateral forces
the base-plate is prolonged upwards in the form of a strong box standard, the
upper portion forming the slide-bars, which at the top encircle the piston-
rod and pass within, but not touching the box cast on the cylinder cover.
Through the sides of this box are four horizontal set-screws, which grip the
top of the standard, and so transmit any lateral force directly to the standard,
as well as admitting of the adjustment necessary to maintain the cylinder
co-axial with the slide-bars.
In this way the vertical forces are taken symmetrically, and cause no
distortion of the engine. The cylinder is held very rigidly by the four
columns, and the horizontal forces arising from the pressures of steam in
the pipe, and particularly from the expansion and contraction of the pipes
under a variation of temperature of more than 300°, are taken by the cast-
iron standard. And, what led more than anything else to this design, all
distortion arising from heat is avoided. The heat-connection between the
cylinder cover at 400° is cut, except for the four columns which are heated
symmetrically and the four set-pins which conduct very little heat to the
slide-bars.
The result appears very satisfactory, the engines running with the slide-
bars cool at 400 revolutions per minute, doing 100 H.-P. with great steadiness.
The somewhat peculiar general arrangement of the engines, Figs. 3, 4,
seems to require a word of explanation. Vertical engines were adopted on
account of the much greater accessibility they afford to all the parts ; also
because they allow of the water from the steam-jackets being drained back
into the boiler with a less difference of level between the floors of the
boiler-house and the engine-room.
The crank-shafts of the engines were raised 3 feet above the floor in
order to allow of the floor being kept level and to admit of pulleys 5 feet
in diameter; also because 3 feet is a convenient height for working the
56]
ON TRIPLE-EXPANSION ENGINES AND ENGINE-TRIALS.
341
brakes, oiling and adjusting the gearing. The most noticeable feature in
the arrangement of the engines — the distance between them — -.was neces-
sitated by the alternative shaft connections which it was decided to give
them, and particularly by the room required for the belt and rope-gearing,
and for working the three separate brakes.
The complete shaft consists of seven separate shafts on separate bearings,
which can be connected into a single shaft by six special coupling-boxes.
The shaft immediately on the right of each engine carries a brake, and these
brake-shafts of the two smaller engines carry 11-inch belt-pulleys, 5 feet in
diameter, weighing 11 cwt., while the brake-shaft for the low-pressure engine
carries two 15-inch pulleys, 3 feet in diameter, weighing 9 cwt., one for a
belt and one for ropes. These pulleys act as fly-wheels when the engines
are working separately; and, in addition to these, there is between the
brake-shaft of the intermediate engine and the crank-shaft of the low-
pressure engine an intermediate shaft carrying a 12-inch rope-pulley, 5 feet
in diameter, weighing 12 cwt., which may be used as an auxiliary fly-wheel
on this engine.
When the crank-shafts are working coupled, as a single shaft, at more
than 200 revolutions per minute, these larger wheels must be removed from
the shafts.
A first-motion shaft, 16 feet distant and 12 feet high, carries pulleys 3 feet
in diameter corresponding to those on the engine-shafts, so that the engines
342
ON TRIPLE- EXPANSION ENGINES AND ENGINE-TRIALS.
[56
can be geared conjointly or separately on to the first-motion, and this again
geared on to one of the brakes, by which means the efficiency of the gearing
may well be tested.
Scale/ '/e
Fig. 5.
Fig. 6.
The coupling-boxes, Figs. 5 and 6, on the main shaft, are intended to
serve two purposes. (1) To afford a ready means of connecting or discon-
necting the several shafts. (2) To allow of any side-play which may arise
from the proximity and number of the bearings.
To serve these purposes it was necessary to have a special flexible
coupling, which led to the design of a modified form of Oldham's coupling,
with an intermediate disk, to which the flanges on the shafts are separately
connected, each with two parallel drag-links at equal distances on each side
of the shaft. The drag-links, which connect one shaft with the disk, being
at right angles to those which connect the disk to the other shaft, so that
the shafts are perfectly free to play laterally. The links are held by pins
screwed into the flanges and disk. To disconnect the shafts all that is
necessary is to remove four of these screws and the two links they hold,
which leaves the shafts free with a considerable interval between them.
These couplings, while very flexible, transmit a perfectly uniform motion
and throw no forces on to the bearings.
The intervals between the engines necessitated by this intermediate
gearing are, 7 feet between No. I and No. II, and 12 feet between No. II
and No. III. These intervals entail no evils in the working of the shaft
except the increased friction arising from the additional weight and number
of the bearings. This friction may be accurately measured and taken into
account in determining the brake H.-P.
56] ON TRIPLE-EXPANSION ENGINES AND ENGINE-TRIALS. 343
The Arrangement of the Intermediate Steam Connections, Figs. 3 and 4.
page 341. — This was adopted in order —
(1) To allow of the engines —
Nos. I, II and III being worked as a triple-expansion condensing engine.
„ II and III being worked as a compound condensing engine.
„ I and II „ „ „ non-condensing engine.
„ III „ „ single condensing engine.
„ I or II „ „ „ non-condensing engine.
(2) To secure that the steam-supply to each engine, under whatever
circumstances it might be working, should be dry without intermediate
drainage, so that the weight of water discharged by the air-pump might
measure the steam admitted to each engine as steam.
(3) To bridge over the intervals between the engines without allowing
the changes of temperature to cause undue stresses in the pipes and the
supports of the engines.
The exhaust -passages from No. I and No. II engines are closed respec-
tively by a 4-inch and a 6-inch steam-valve, while an alternative exhaust-
passage, which may be connected directly with an exhaust-pipe in the floor
or closed by a blank flange, is provided. The steam-valves in the exhaust-
passages open into receivers which supply steam to No. II and No. Ill
engines respectively, which receivers also have alternative connections with
the main steam-pipe, so that each engine can have a separate steam-supply.
The jacketed receivers, which are the intermediate steam -passages between
the engines, are cast-iron pipes 6 and 8 feet long respectively, lined with
wrought-iron pipes 4 inches and 6 inches in diameter, the space between
the pipe and casting constituting the space for the steam at boiler-pressure.
These receiver pipes are connected with the engines which they supply by
S copper pipes of 4 inches and 6 inches diameter respectively, the copper
pipes serving as expansion-joints; the expansion in the 12-foot interval
between No. II and No. Ill engines amounting, with 200 Ibs. of steam in
the jackets, to 0'25 inch.
The arrangement of the steam-pipe which supplies the receivers was
adopted in order that the steam might be dry. This pipe leads from a water-
separator, as a 2^-inch pipe which enters a jacketed receiver No. I, 4 feet
long, lined with a 2^-inch wrought-iron pipe, to prevent condensation of the
steam after leaving the separator. The receiver leads to a point near No. I
engine, and is connected with a casting in which are two steam-valves open-
ing into 2 inch copper pipes which lead to the steam-chest of No. I and the
344
ON TRIPLE-EXPANSION ENGINES AND ENGINE-TRIALS.
[56
receiver between No. I and No II. The other end of the receiver is
connected through a steam-valve with the receiver between No. II and
No. III. In this way, whichever engine is receiving steam from the boiler,
the steam has to traverse a steam-jacketed receiver.
I.rvet of TTntrr m- Jt.nl
Fig. 7.
j
The positions of the boiler and engines, Fig. 7, was adopted to allow not
only of the water from the jackets on the cylinders, steam-chests, and
receivers draining back into the boiler, but also to allow of its doing so when
the pressure of the steam in the separator was 3 Ibs. per square inch below
that in the boiler.
To ensure this, the level of the water in the boiler is kept 6 feet below
the lowest jacket to be drained. The boiler-house, which is separated by a
glass partition from the engine-room, has a floor 5 feet below the engine-
room, and the level of the water in the boiler is 1 foot above the engine-room
floor, the boiler being 20 feet distant horizontally from the engines.
The steam-pipe, 2£ inches in diameter, takes the steam from the top of
the. dome on the boiler and enters the engine-room 2^ feet above the floor;
immediately in the engine-room is a steam-valve ; 2 feet from the wall the
pipe rises vertically 8 feet, then turns horizontally for 10 feet, and then
again turns down vertically until it enters the separator. At a height of
10 feet there is a branch 2 inches in diameter, without a valve, which
supplies all the jackets with steam at the pressure of the boiler less the
56] ON TRIPLE-EXPANSION ENGINES AND ENGINE-TRIALS. 345
resistance of the pipe, which is always less than | Ib. on the square inch.
The main pipe then enters the water- separator through a redvicing-valve
which lowers the pressure 2 Ibs. ; below this reducing-valve is the steam-pipe
leading to the receivers, and below this again the steam-drain from the
jackets enters the separator, and 3 feet below this the water drains from the
jackets. The separator now descends as a vertical pipe 1£ inch in diameter
to the floor, and then proceeds horizontally until it joins the feed-pipe from
the economizer just before entering the boiler, having a back valve and
a stop-valve, and also a blow-off valve.
The separator for 3 feet at its upper end consists of a vertical cast-iron
cylinder 6 inches in diameter; it is then reduced to a l|-inch pipe. Com-
municating with the separator at its top, and at a point 1 foot from the
engine-room floor, is a water-gauge of ordinary construction except that the
tube is 6 feet long. This gauge shows the level of the water in the separator.
When the engines are standing with the blow-off shut, the water remains at
the bottom of the gauge. Any water from the jackets drains back into the
boiler. If the blow-off is opened the pressure in the separator falls and the
water rises to balance the excess of pressure in the boiler, which is shown by
the water-gauge ; steam is drawn through the jackets as it cannot pass the
reducing-valve until the pressure has fallen 2 Ibs. below the boiler ; in this
way the engines are heated.
When the engines are running they draw steam out of the separator
below the reducing-valve, and hence all the steam is drawn through the
jackets until the resistance in the passages reaches 2 Ibs. on the square inch ;
the water in the gauge shows the level at which it stands in the separator.
When the pressure in the separator is 2 Ibs. below that of the boiler, the
water in the separator stands about 5 feet above the floor, which is just the
bottom of the 6-inch cylinder ; the water then as it enters the separator
gravitates to the boiler. If, however, the stop-valve at the bottom of the
separator is closed, the water is collected in the 6-inch cylinder, and, as its
level is shown on the gauge, this furnishes a ready means of measuring the
condensation from jackets and radiation, which measurements may be checked
by draining off the water through the blow-off.
In this way the total condensation from jackets and radiation is deter-
mined, and, on consideration, it will appear that herein is an exact measure
of all the heat supplied from the boiler over and above that which leaves the
engines as steam. It will also be seen that the separator ensures complete
water drainage of the jackets and a draught of steam through the jackets
and jacket-pipes.
The arrangement of jacket-pipes and drains, which is very complex, was
necessary in order that the walls, back and front covers, steam-chest covers,
346 ON TRIPLE- EXPANSION ENGINES AND ENGINE-TRIALS. [56
and receiver-covers for each engine might be separately jacketed, and drained
both of steam and water. In all there are fifteen separate jackets.
To ensure an equal passage of steam through all these jackets, it would
have been desirable, had it been practicable, to supply them in series, so that
the steam should pass from one to the other ; but this, for obvious reasons,
was impracticable, and it was necessary to so arrange the pipes that the
head of steam to cause circulation through each jacket should be nearly
equal.
This is accomplished by carrying the distributing- pipe, 1^- inch in diameter,
throughout the entire length of the engines, as high as practicable. Also
the steam-collecting drain, 1^ inch in diameter, and the water-collecting
drain, 1 inch in diameter, and arranging them so that there might be a fall
all the way in the direction in which the steam was moving. A branch from
the steam-pipe with a valve supplies each receiver-jacket on the top, and
a drain from the bottom of each receiver-jacket branches into two, one branch
falling to the water-drain, and the other rising to the steam-drain, these
branches being f-inch and ^-inch in diameter.
Each engine has a branch from the distributing-pipe and from each of the
drains, which can be closed by valves. The branches from the two drains
unite into one drain before branching to the jackets. Then from the distri-
buting branch on each engine are four branches leading respectively to the
four jackets on the engines, and in the same way four drains from the
four jackets unite in the one branch from the drain. The jacket-pipes
are of copper with iron screwed joints, except the unions, valves, and
flanged-joints to the covers, which are of brass. The system is extremely
complex, but nothing short of this would suffice for the special purpose
of these engines. There are twelve steam-valves, thirty flange connec-
tions, and more than forty unions, and about one hundred elbows, tees,
and running-joints. The use of running joints was a mistake ; they were
adopted for simplification, but they should have been unions, it being
found very difficult to make the back nuts stand. They were first tried
with red-lead and hemp in the ordinary way ; this stood a pressure of
200 Ibs. per square inch for about two days. The couplings were then faced,
and nothing but a little putty was used, but these failed. Then another
method was tried which has answered well, and the whole system has been
working practically tight.
The Covering of Cylinders, &c. — The temperature of the steam-jackets,
about 400° Fahrenheit, rendered the covering of the steam-pipes and
cylinders a matter of first importance, not only to prevent loss of heat by
radiation, but to render it possible to operate near the engines. In the first
instance, the cylinders and receivers were surrounded with 2 inches of glass-
56] ON TRIPLE-EXPANSION ENGINES AND ENGINE-TRIALS. 347
wool, and lagged with 2 inches of baywood, but the glass-wool, being found
to create gritty dust, was removed, and an inner lagging of_ soft pine
substituted for it. The steam-chest covers and the water-separator were
also lagged in the same way ; while all the steam-pipes, except the copper
expansion-pipes and jacket-pipes, which could not be brought under cover
of the wood lagging, have been covered with 2 inches asbestos cement.
The Surface-condenser is of the torpedo-boat type of thin copper, 14 inches
in diameter, and 4 feet long. It has about 160 square feet of heating-surface,
and receives the steam by an 8-inch exhaust-pipe from the 12-inch engine.
The Air-pump, working by side levers from the slide-block of the 12-inch
engine, is 9 inches in diameter, with a 4^-inch stroke, with foot-valve, piston-
valve, and cover- valve, and is designed to work up to 400 revolutions per
minute.
The condenser and air-pump are conveniently placed on a bracket on the
standard of the 12-inch engine, which also forms a stage for indicating the
engine. This stage is 5 feet from the floor, which gives sufficient but not too
much room for conveniently measuring the water from the hot- well, and the
condensing water.
The Feed-pump. — This was adopted in order to maintain a regular feed
in the boiler, as well as to enable the water from the hot-well to be returned
to the boiler. It is worked from the rocking-shaft of the air-pump levers ; it
has a plunger l£ inch in diameter with a 2-inch stroke, and draws water from
a feed-tank 3 feet below it, discharging into a feed-pipe, which, together
with the economizer or water-heater, leads through 70 feet of l|-inch pipe to
the boiler. The inertia of this column of water becomes very considerable
when the speed is as great as 400 revolutions per minute, and this, together
with the 200 Ibs. pressure, seemed to render it doubtful whether the pump
would answer. However, by means of a special device, a cushion of air or
steam was provided about 4 feet from the pump, and by another device the
pump was made to start itself, notwithstanding the 3-feet draw, so that the
pump works silently and without trouble up to 400 revolutions.
The Governors. — For the special investigations into the action of steam,
governors were unnecessary. The load on the engines being constant, the
cuts-off fixed, and the supply of steam regular, small variations of speed
would be of no moment ; while any alteration of the pressures of steam or
cut-off by the governors would only confuse the trials; besides which, the
problem of governing engines working in conjunction as regards steam, but
on separate brakes, was altogether a new one. At the same time, as a matter
of safety, the complexity of the system, the number and inexperience of the
observers engaged at any time on the engines, the extreme circumstances as
348 ON TRIPLE-EXPANSION ENGINES AND ENGINE-TRIALS. [56
regards the steam-pressure and speed under which the engines were designed
to work, rendered it imperative that the engines should be so far governed,
that under no circumstances could the speeds exceed a safe limit, which, with
the 5-foot cast-iron fly-wheels on the shafts, would be about 600 revolutions
per minute.
To meet both these considerations, what seemed to be necessary was
a safety-governor, which, while it would interfere in no way with the passage
of steam at speeds below the limit, would with the utmost certainty cut off
steam at some definite speed before the limit was reached.
To ensure certainty of action, it was necessary that the governor should
be permanently geared to the engine, and not merely engaged by a belt.
And to secure rapidity of action when once the limit of speed was reached,
it was desirable that there should be as little room as possible for steam
between the governing- valve and the piston; in other words, that the
governor should close the expansion-valve.
The Meyer expansion- valves, which had been selected as peculiarly
suitable for the purposes of these engines, actuated as they are by screws
of such moderate pitch that it requires five or six turns to close the valves,
are not susceptible of being opened and closed by the direct force of
governor-balls. It therefore became necessary to adopt some form of engage-
ment-governor which, instead of acting on the valve, should act on a clutch
which engaged the crank-shaft of the engine with the valve-spindle when the
limit of speed was reached. The clutch here adopted is the Author's spiral
steel band-clutch. This clutch, which requires almost an insensible force to
engage it, is absolutely certain in its hold.
In order to operate on the valve-spindle it was necessary to use two pair
of bevel-wheels, which could not be made less than 4 inches and 6 inches in
diameter. To throw this train of wheels suddenly into gear with a shaft
making 400 revolutions per minute seemed a doubtful proceeding, but such
is the softness of action of the clutch, although there is no slipping, that
there is neither noise nor shock. The engagement is silent and instan-
taneous, so that unless special attention is directed to it the movement of the
10-inch hand-wheel will probably escape notice. The clutch is as good
in disengagement as in engagement, and will release the shaft before it has
turned more than 5° or 10°.
Although the main object of these governors was that of a safety-
governor, opportunity was taken to so design them that they should, if
required, open the valve as the speed fell, as well as close it as it rose,
arrangements being made to prevent hunting. The governors so obtained
are extremely efficient, and afford an excellent means of studying the
56]
ON TRIPLE-EXPANSION ENGINES AND ENGINE-TRIALS.
349
action of governors. During the steam trials, however, they are simply
set to act as safety governors, which they have done to perfection, never
having been out of action, or having allowed the speed of the engine to
exceed the limit to which they are set.
Fig. 8.
The boiler (Fig. 8) is of the locomotive type with iron tubes and
fire-box, the shell being of steel -^ inch thick. The tubes are 2 inches in
external diameter and 8 feet long, giving 160 square feet of tube surface.
The fire-box is -fa inch thick, 2 feet 3 inches by 2 feet 4 inches, 4 feet high,
giving 42 square feet of heating-surface.
The area of the grate as used is not more than 4 square feet.
The boiler is furnished with a dome, from the top of which the steam-
pipe descends and passes out at the side.
350 ON TRIPLE-EXPANSION ENGINES AND ENGINE-TRIALS. [56
The feed enters the boiler just below the water-level and in front of the
fire-box.
There is an iron smoke-box at the end of the boiler from which there are
several passages for the gases. The usual passage is beneath the barrel
of the boiler, 3 feet broad and 6 inches deep, and about 6 feet long, proceed-
ing at a slight inclination downwards towards the fire-box; across this passage
the feed-pipe ranges backwards and forwards, and a series of scrapers are
worked to keep the pipes clean. The pipes cross forty times, and give about
50 square feet of heating-surface, 40 square feet of which is kept clean by the
scrapers. In this arrangement the water ascends in the opposite direction to
that in which the gases descend. The gases, after emerging from the water-
heater, descend into a flue leading to the chimney, which is 100 feet high,
and takes the gases from other furnaces, affording generally about f inch
draught.
The boiler and water-heater are enclosed in a brick chamber arched over.
This chamber is 6 feet wide and 9 feet high, extending from the front of the
fire-box to the end of the smoke-box.
At the fire-box end a second chamber is built 6 feet by 6 feet and 8 feet
high. This, by shutting a door, becomes a closed stoke-hold, into which a fan
can be used to force air at any pressure up to 2 inches of water.
In this chamber is an injector, a feed-tank, and water-supply, a window
looking at the safety-valves, and a window into the engine-room, also a
tumbling-hopper for admitting coal.
There are two 1-inch dead- weight safety-valves on the boiler, loaded to
200 Ibs. on Schaffer and Budenberg's gauges, i.e., 400 inches of mercury,
as well as the usual fittings.
THE MEASURING APPLIANCES.
These, in respect of the brake-dynamometers, the indicating gear, the
gauge for jacket-water, and the tumbling-bay and tank for the condensing
water, are of a permanent character. Provision is also made for measuring
the temperature of the gases in the smoke-box as they emerge from the
tubes, and in the flue as they leave the water-heater, and for measuring the
temperature of the feed before passing the pump, as it enters the boiler after
passing the water- heater.
The condensing water is drawn from an iron tank 20 feet by 10 feet by
10 feet, about 116 feet above the engine-room floor. A permanent mercurial
gauge in the engine-room always shows the level of water in this tank.
56] ON TRIPLE-EXPANSION ENGINES AND ENGINE-TRIALS. 351
The great head, although, of course, a waste of power, is of advantage in
securing regularity of flow. The water after leaving the condenser enters
a cast-iron tank, 4 feet by 18 inches by 18 inches, from which it issues over
a tumbling-bay 4 inches wide ; in the tank are bafflers and a float, with
a scale graduated to show in Ibs. per minute the quantity running over
the bay. The water is then caught in a second receiving tank and conducted
to an underground concrete tank 20 by 9 feet by 11 feet, the level of water
in which is shown in the engine-room by a water-gauge, and also indicated
outside by a float. This tank, which has been accurately measured, affords
a very exact means of checking the indications of the float in the tumbling-
bay.
The upper tank holds 12,000 gallons of water, which can be passed
through the condenser before the tank is empty. When the upper tank is
empty, if more water is required the quadruple centrifugal pump is set
in motion, which raises the water at the rate of 10,OQO gallons an hour from
the lower to the upper tank ; but it is seldom necessary to resort to this.
The temperature of the condensing water is measured by a thermometer
in the pipe leading to the condenser, and after leaving the condenser by
a thermometer in the float-tank.
The water from the hot-well flows into an oil-separating tank, from which
it overflows on opening a cock, and is caught in a 100-lb. tip-can after
Mr Bryan Donkin's pattern, from which it may be tipped into the feed-tank,
so that the feed and hot-well discharge is measured at one operation.
The condenser is furnished with a mercurial gauge, which shows the
absolute pressure in the condenser; also by a Bourdon vacuum-gauge, and
the temperature of the discharge from the hot-well is measured by a ther-
mometer in the hot-well. The water, resulting from radiation and jacket
condensation, is measured in the water-separator.
The pressures in the receivers are shown by Bourdon gauges, graduated
to Ibs., which, on the authority of Messrs Schaffer and Budenberg, means
2 inches of mercury — a fact which it is important to know in comparing
these pressures with the indicated pressures.
Each engine is provided with a counter for recording the revolutions.
The Indicating Gear (Fig. 1). — The indicator cocks have a clear ^-inch
way into the cylinder, the cock being placed at the end of a stiff brass tube
screwed horizontally into the cylinder, and reaching through the 4 inches of
lagging. The cock itself forms an elbow, to allow the indicator to have
a vertical position.
The cocks from the back, and from the front of each cylinder are in the
same vertical line, so that the indicators stand vertically over each other in
352
ON TRIPLE-EXPANSION ENGINES AND ENGINE-TRIALS.
[56
a convenient position to receive the motion for the drums. This is obtained,
in the 12-inch engine from the air-pump levers, and in the other engines
from levers specially connected by a link with the slide-blocks.
Fig. 9.
In all cases the indicators are some feet above the levers, and while the
motion of the levers is vertical, that of the drums is horizontal. The connec-
tion of the drum with the lever could be made by a simple cord or wire
passing over the roller on the indicator drum down to the lever ; but con-
56] ON TRIPLE- EXPANSION ENGINES AND ENGINE-TRIALS. 353
sidering that the chief function of the engines was to be regularly indicated,
and this by inexperienced hands, and that the speeds would sometimes be
such that the ordinary method of hooking up would be impracticable, some
more convenient and permanent arrangement seemed desirable. The Author
was thus led to a device which, from its simplicity and convenience, par-
ticularly in the matter of hooking up, as well as its effect in diminishing
errors arising from the stiffness and stretching of the cord, seems likely to be
generally useful.
This method consists of a f-inch pin with a head in the side of the lever,
a light brass plate £-inch thick, with a button-hole to permit its passing
over the head of the pin, and, when pulled up against the pin, allowing of
considerable wear. To this brass is attached a steel wire 19 B.W.G., long
enough to reach beyond the furthest indicator, that on the back of the
cylinder, the wire being held up by a spiral wire spring of such length
and stiffness that it will stretch 6 inches under a force of 25 Ibs. without
causing undue stress in the wire.
The wire connecting the lever with the spring passes the indicators, and
is furnished in convenient positions with two buttons for hooking on the
cords of each of the indicators. This is effected by having a light forked
hook attached to the end of the cord, which has only to be pulled beyond
the button, and one limb of the fork placed on each side of the wire and
then let go, when the spring of the drum pulls the hook up against the
button. Thus hooking up can be accomplished with facility and certainty
at whatever speed the indicator is running. The length of the cord is
reduced to a minimum at both ends of the cylinder.
In these engines, where the pistons of the indicators have a motion
parallel to that of the pistons of the engines, the cord has to turn a right
angle between the drum and the hook. This might be effected by the
rollers on the indicator ; but as they are usually very small and not adapted
for wear, two clips are made to pinch on to the indicator cocks on the
cylinder. The clips have circular sockets in line with the motion of the
piston of the engine with a set-screw; through these passes a ^-inch steel
rod, long enough to carry an adjustable arm to hold the end of the spring,
and two adjustable rollers 2 inches in diameter for the cords to pass over.
The Hydraulic Brake Dynamometers (Figs. 10 to 14). — These are a very
important feature of the system. They are the result of a special investi-
gation as to the possibilities afforded by hydraulic brakes, undertaken by
the Author during the time when the engines were under the consideration
of the Committee and before anything was decided.
Having had a great deal of experience with almost every conceivable
o. R. ii. 23
354
ON TRIPLE-EXPANSION ENGINES AND ENGINE-TRIALS.
[56
form of friction brake, the Author had arrived at the conclusion that,
although it is possible to construct such brakes to work with almost any
Fig. 10.
Fig. 11.
degree of accuracy, certain inconveniences and drawbacks attend their use,
56]
ON TRIPLE-EXPANSION ENGINES AND ENGINE-TRIALS.
355
which in all cases leave much to be desired, particularly where, as in a case
like this, work on the brake is the sole object of the engines.
Fig. 12.
Fig. 13.
(1) Such brakes require constant observation and watching.
(2) A single engine cannot be started without relieving the load.
(3) Such brakes are cumbersome and are not easily adapted to measure
greatly different powers.
(4) Any particular brake cannot without considerable pulling about, such
as altogether removing the brake and brake-wheel, be rendered altogether
nugatory. It was desirable : —
23—2
356
ON TRIPLE-EXPANSION ENGINES AMD ENGINE-TRIALS.
[56
1. That the brakes should be certain in their action without any atten-
tion while the engines were running.
Fig. 14.
2. That they should leave the engines free to start, and then take up
their load without attention.
3. That they should be put on and off by a simple operation.
4. That when turned off they should offer no sensible resistance to the
engines.
5. That they should be capable of being so adjusted as to impose any
particular resistance, from zero to the greatest, at any speed at which it was
desired to run the engines.
6. That the resistance of the brake, when once adjusted, should be
independent of the speed of the engine.
7. That the necessary size and structure of the brakes should not be
such as to incommode or hamper the engines.
8. That the resistance of the brake should admit of absolute determin-
ation from a single observation.
56] ON TRIPLE-EXPANSION ENGINES AND ENGINE-TRIALS. 357
Of these attributes 1 and 2 belong to all fluid resistance, such as that of
the screws of steam-ships or centrifugal pumps, in which cases the-resistance,
varying as the square of the speed, is zero when the engines start.
If the casing of a centrifugal pump, or the tank in which a paddle or
screw works, be suspended on the crank-shaft, making a complete balance
when the shaft is at rest, then, when the shaft is in motion, the moment
of resistance on the shaft will be exactly equal to the moment to turn the
casing round the shaft. This can be readily and absolutely measured by
suspending weights at a definite horizontal distance from the shaft. The
first published account of this form of brake having been made use of for
dynamometric measurement was by Hirn*, in his investigation for the
verification of Joule's mechanical equivalent of heat, and was subsequently
adopted by Joule in his second determination.
In neither of these cases, to the Author's knowledge, was there any
attempt to vary the resistance at a constant speed.
Having occasion to use a dynamometer for measuring the resistance on
the shaft of a multiple steam-turbine at speeds of 12,000 revolutions per
minute, which was engaging his attention in 1876, the Author made use
of a brake, having a centrifugal pump suspended on the shaft and working
into itself. The resistance, or head against which the pump was working,
was regulated by a valve between the exit and inlet passages, that is,
in the external circuit made by the water. This was brought before the
Mechanical Section of the British Association in 1877. At the same
meeting, Mr William Froude gave an account of his hydraulic brake, for
measuring the power of large engines, in which the resistance was regulated
on the same principle as that adopted by the Author, namely, by adjusting
diaphragms or sluices in the passages between the revolving wheel and the
casing. In other respects Mr Froude's brake differed essentially from any
of those previously used, being designed to obtain a maximum resistance
with a given sized wheel. For this purpose Mr Froude invented an internal
arrangement which affords a resistance out of all comparison with any other
form.
Since great resistance, admitting of small brakes, was of extreme import-
ance for these engines, the first step in the special investigation was the
construction of a model Froude's brake with a 4-inch wheel; the object
of which was to ascertain how far the sluices would act in maintaining a
constant resistance at any particular speed, and what was the minimum
resistance when the sluices were closed.
With this brake it was found that the minimum resistance was about
* Theorie mecanique de la Chaleur, 2nd edition, 1865, p. 65.
358 ON TRIPLE-EXPANSION ENGINES AND ENGINE-TRIALS. [56
0'08 of the maximum ; a hardly satisfactory range, considering it was desired
to run the engines at a constant load at from 100 to 400 revolutions per
minute, the maximum resistance of the brake ranging from 1 to 16, so that
the minimum at 400 would be 26 per cent, greater than the maximum at
100 revolutions, apart from the fact that closing the sluices would not render
the brake nugatory.
This, however, was of small importance compared with another fact
revealed by these experiments. When the speed of the brake-wheel
exceeded a certain small limit, determined by the head of water under
which it was working, the maximum resistance gradually fell off in a
surprising and somewhat irregular manner. This falling off was found to
be owing to the brake partially emptying itself of water, due to the air
from the water gradually accumulating in the centre of the vortex — a fact
which, if not dealt with, threatened to render such brakes useless for the
purpose of these engines.
The argument was simple : in a vortex the pressure at the centre is less
than the pressure at the outside. The pressure at the outside in these
brakes is determined by the atmosphere, and the small head under which
they are working ; and the outside forms' a closed surface. The pressure
at the centre will therefore, at different speeds, fall below the pressure of
the atmosphere. Air will be drawn from the water and accumulated in the
centre, occupying the space of the water and diminishing the resistance ;
and, owing to various causes, the action will be irregular. This would be
prevented if passages could be carried through the outside to the axis of
the vortex, carrying a supply of water at or above the pressure of the
atmosphere, so as to prevent the pressure at this point falling below that of
the atmosphere. This was accomplished by perforating the vanes of the
wheel, and supplying water through the perforations. It also appeared
that, by having similar perforations in the casing open to the atmosphere,
the pressure at the centre of the vortex could be rendered constant, whatever
the supply of water and speed of the wheel ; so that it would then be
possible to run the brake partially full, and regulate the resistance, from
nothing to the maximum, without the sluices. These conclusions having
been verified on a model, it was decided to arrange the engines with the
shafts in line, with three brakes on the shafts ; and the brakes, with 18-inch
wheels, were designed according to the resistance given by the model. The
brakes promised all the attributes desirable, except that of running with a
constant load under varying speeds. This matter was considered during their
construction, and an automatic arrangement was devised acting on cocks
regulating the supply and exit of the water to arid from the brake necessary
to keep it cool, the lifting of the lever opening the exit and closing the
supply, so as to diminish the quantity in the brake, and vice versa.
56] ON TRIFLE-EXPANSION ENGINES AND ENGINE-TRIALS. 359
The danger of such an arrangement hunting was carefully considered,
and precautions were taken. The brakes were constructed by Messrs- Mather
and Platt at the same time with the engines, and the engines started with
the brakes and automatic gear complete. During the twelve months they
have been running the brakes have demanded and received no attention
whatever. They are easily tested for balance. They have neither fixed nor
spring attachment, except the bearing on the shaft. They are loaded on a
4-foot lever, with 2-inch play between the stops. When the speed of the
engines reaches about 20 revolutions per minute, the levers rise (whatever
load they have on), and, though always in slight motion, they do not vary
£-inch until the engines stop ; during the run the load on the brakes may
be altered at will, without any other adjustment.
THE ENGINE TRIALS.
Before commencing the trials, the object to which they were to be
directed, and the manner in which they should be conducted, were carefully
considered, and it was decided : —
1. That the purpose of the trials should be the elucidation of the general
laws of the action of steam in the steam-engine, and the more general circum-
stances on which these laws depend.
2. That, from the commencement, the trials should be systematic ;
certain definite conditions being aimed at, and the trials under each set
of conditions continued until consistent results should be obtained, showing
how far the conditions had been achieved.
3. That there should be no casual nor unrecorded trials, but that all
trials should be considered of the same degree of importance.
4. That observations should be noted and reduced on special forms
according to a definite system, to be carefully preserved for future reference;
and that a synopsis of the mean results of each trial should be entered
forthwith in a special record for ready comparison.
The trials have all so far been conducted as part of the regular work
of the laboratory, under the superintendence of the Author, Mr Foster
(assistant in the laboratory) having general charge of the appliances, and
the fireman (Mr Joseph Hall) firing and driving the engines. The detailed
observations were taken and reduced by students (about fourteen on each
trial) under the supervision of Mr Mackinnon, demonstrator of the laboratory.
Diagrams are taken every half-hour simultaneously from the six ends by
six students, who have charge of their respective indicators for the trial.
360 ON TRIPLE-EXPANSION ENGINES AND ENGINE-TRIALS. [56
The same students also reduce the diagrams in the intervals. The three
counters are read every ten minutes by three students, who have respectively
charge of the counters and running of the three engines, calculating the
brake H.-P. as the trial proceeds, and noting any circumstance connected
with the resistance or running of the engine.
One student has charge of the 100 Ib. tip-can, which measures the water
from the hot- well ; and another has charge of the condensing water, noting
the temperature and quantity given by the float every ten minutes. Another
student measures the rate of discharge from the jackets every half-hour. A
student watches the coal-weighing and firing. A student takes the tem-
peratures of the hot- well and feed before and after passing the economizer,
and the temperature of the air in the smoke-box and flue before and after
passing the economizer. Each student reduces his observations as he pro-
ceeds, so that within a few minutes of the end of the trial the reduction is
completed.
The results are then examined by Mr Mackinnon, checked and entered
in the permanent record, the original diagrams and notes of each trial being
carefully preserved.
Two series of trials have been conducted, the one by regular students
between 9.30 A.M. and 5.30 P.M. The other by evening students between
6.30 P.M. and 9 P.M., one of each series being made every week.
In the day trials the fire is lighted the first thing in the morning, and
steam is got up quietly. As the steam rises it is blown freely through the
jackets to heat the engines. If the trial is to be made with jackets, the
blowing through all the jackets is continued until the boiler-pressure reaches
200 Ibs. on the gauges. Should the trial be without jackets, the jacket-
covers on the low-pressure engine are closed when the pressure has reached
about 40 Ibs., and the air-cock is opened ; those on the intermediate cylinder
when the pressure reaches about 80 Ibs., and those on the high-pressure
cylinder at 200 Ibs. In all cases the engines are started, and are allowed
to run just as required for the trial for one hour. The engines are then
stopped fifteen minutes before the trial, the fire is drawn, and the readings
of the counters and level of the water in the boiler and tanks are taken ;
14 Ibs. of wood and 14 Ibs. of coal are allowed for the waste of relighting,
starting, and stopping. The run then commences ; the coal is weighed out
in charges of 100 Ibs., each charge being shot from the scale-pan into the
hopper in the firing-chamber, and completely consumed before the next
weighing is admitted.
The boiler is fed continuously by the feed-pump, either from the water
from the hot-well or, in some trials, from the water from the condenser.
56] ON TRIPLE- EXPANSION ENGINES AND ENGINE-TRIALS. 361
The runs have generally been for six hours, except when forced draught is
used, in which case they are about four hours.
After the last coal has been put on the fire, the engines are run as long
as steam can be kept up, care being taken to bring the level of the water in
the boiler at stopping exactly to that at starting, any difference being allowed
for as 15 Ibs. for each -fa inch.
The ashes which fall through the bars are burned during the trial, and
the ashes after the trial are generally weighed, but no account is taken of
them, nor of any fuel that may be left in the grate.
This was adopted, after trying several systems, as being workable and
very definite ; nor does it appear, on comparing the results from the long
with those of the short trials, that the one has any sensible advantage over
the other. During the experiment the regulator is fully open, and a definite
quantity of water run through the condenser. The engines, therefore, take
all the steam the boilers will produce, the load on the brakes just balancing
the pressure of steam, so that the speed is regulated by the rate at which
steam is made in the boiler, that is, by the draught-gauge. As it was
intended that the scope of these trials should include as far as possible all
conditions under which steam may be used, there was no particular reason
for commencing with one set of conditions rather than another, except such
as arose from convenience, and out of consideration for the engines them-
selves. The fact that the engines were new, and wanted running to bring
the bearings into order, as well as the number of students to be employed,
led to the first series of trials being made with triple expansion and full
pressures of steam.
THE RESULTS OF THE TRIALS.
The trials commenced in March 1888, and were continued at the rate
of two a week till June ; in all twenty trials were made and recorded, the
engines being then complete with the exception of lagging.
These early trials with 200 Ibs. pressure triple expansion, with and
without steam-jackets, and various degrees of expansion, gave very definite
results. But they also revealed the fact that the linings of the cylinders
leaked at pressures above 170 Ibs. per square inch, and that the joints in
the jacket-pipes could not be made to hold. They also showed that, not-
withstanding the precautions taken, the jackets were liable to fall off in
efficiency. The effect of the leaks was not great on the general economy
of the engines, and might easily have passed unnoticed but for the rigour
of the tests to which they were subjected.
362 OX TRIPLE-EXPANSION ENGINES AND ENGINE-TRIALS. [56
At 250 revolutions per minute the thermal efficiency of the engine with
jackets was
Heat equivalent of indicated work per minute n-T7-
Heat discharged + heat equivalent of indicated work
Coal per H.-P = T48 Ib.
The leaks, however, tended to confuse the diagrams, and opportunity was
taken of the long vacation, during which the trials were discontinued, to
reset the linings of the cylinders. The lagging of the engines was completed
as far as it was thought desirable.
The trials were continued in October, when the linings proved to be
perfectly tight, a,nd although at first the jacket-pipes leaked occasionally, the
leakage was not of any sensible magnitude. The jackets were, however, still
found liable to fall off in effect at low speeds. The trial with the jackets was
therefore repeated many times, small alterations being made in the jacket-
pipes, until consistent results were obtained with speeds of 250 revolutions
per minute, giving thermal efficiency, calculated as before, 0'2(), coal per
indicated H.-P., 1'33 Ib. Corresponding trials without the jackets were then
made, followed by trials at higher and lower speeds with and without the
jackets. These furnish a complete series of trials of triple-expansion engines
working with about 200 Ibs. boiler pressure, at piston speeds from 250 to
1000 feet per minute.
Appendix, Table I, shows the mean results as recorded for three trials at
different speeds with and without jackets. Only one trial at each speed is
given, though several trials have been recorded, the results not differing by
1 per cent.
Lines 4 to 29 contain the mean results from the engines.
„ 30 to 42 „ the heat discharged from the engines.
„ 43 to 48 „ „ received by the engines.
„ 49 to 59 „ „ received from the furnace.
„ 60 to 76 „ the general relations between the coal, heat, water
and power.
It will be noticed that the three engines do not run at the same speed in
the same trial. This is a matter of great importance, and shows the ad-
vantage of having for such trials as these the engines working on separate
brakes.
The cut-off in each cylinder regulates the fall of pressure in that cylinder,
but the pressure in the receiver into which it discharges is determined so as
to equalize the steam received, and the steam drains off into the next
engine.
56] ON TRIPLE-EXPANSION ENGINES AND ENGINE-TRIALS. 363
If, then, the shafts are coupled, there can be only one ratio of expansion,
which will make the terminal pressures in the cylinders correspond with the
pressures in the receivers. But when the shafts are free the engines adjust
themselves so that they pass the same quantity of steam, and the cuts-off
are easily arranged to bring the terminal pressure into accordance with the
pressure in the receivers. Thus, with these three separate engines, the full
economic advantage of all degrees of expansion can be obtained. To do this
with coupled engines would require a different ratio of cylinder volumes for
each degree of expansion, these trials showing distinctly what should be the
cylinder volume for each degree with coupled engines.
The Checking of the Results. — The system rendered possible by the use of
a surface-condenser, of accurately measuring the water which has passed
through the engines, as well as the heat discharged from the condenser, and
the feed-water, gives a certainty to the results of the trials not otherwise to
be obtained. There will be always a loss between the water supplied to the
feed-pump and that received by the engines ; hence, unless the loss is
definitely known, the actual water received by the engines can only be
surmised.
In the first forty of these trials the water discharged from the engines,
after being measured, has been returned to the boiler, the deficiency being
carefully ascertained ; and in no case where this has been done has the
deficiency amounted to less than £-lb. per minute, although there were
no visible or perceivable leaks of any sort from joints or glands, and the
boiler, when tested with water-pressure before and after the experiment,
has shown no leak. Great pains have been taken to find where this
water went, but without success, though it certainly did not go through
the engines.
The importance of this point in determining the action of steam in
the cylinder is fundamental. It is only by knowing the quantity of water
passing through the engines that it is possible to compare the actual diagrams
with a theoretical diagram ; and the difference between the feed and the
hot-wrell discharge would in these engines generally amount to from 5 to
10 per cent., and would vitiate any such comparison. As it is, all com-
parisons have been made from the water discharged from the hot-well.
Since each Ib. of dry saturated steam condensed would give up about
1000 thermal-units to the condensing water, the measures of water from the
hot- well and heat from the condenser keep a useful running check upon each
other. It is found that the heat measured (in 1000 thermal units) is about
4 per cent, greater than the water measured in Ibs. when the jackets are on,
and from 1 to 2 per cent, less when the jackets are off.
An exact calculation, as to the heat discharged per Ib. of water, must
364 ON TRIPLE-EXPANSION ENGINES AND ENGINE-TRIALS. [56
involve certain assumptions, of the accuracy of which a careful comparison
with the measured heat affords a valuable test. Such a comparison is shown
in Appendix, Table II.
For the trials with the jackets on, the calculations are made on the
assumption that the steam is released as dry saturated steam, and carries
with it into the condenser the heat of evaporation at release pressure from
the temperature of the hot-well, less the external work of evaporation and
plus the work done by the piston in discharging the exhaust. This expressed
in quantities from Professor Rankine's Table is
H,-h3-(P.2-P3)V2
772
In this calculation no account is taken of the additional heat received by
the steam, during its passage from the cylinder into the condenser, from the
hot walls of the passages.
For the trials without jackets, the calculations are made on the assumption
that the steam is admitted into the low-pressure cylinder as dry saturated
steam, carrying into the cylinder the total heat of evaporation from the
temperature of the condenser at the temperature of admission, and that it
carries this heat, less the heat equivalent of the indicated work done in this
cylinder per Ib. of steam, into the condenser, which, expressed in Professor
Rankine's quantities, is
H^- hs _ (I. H.-P.) x 427
772 Ibs. per minute from the hot-well '
This calculation, therefore, takes no account of the heat that must be lost
by the steam in supplying the heat to be radiated from the exterior of the
cylinder.
Since important actions are not taken into account in these calculations,
the resulting quantities cannot be considered an absolute check upon the
observed quantities; they constitute, however, a valuable relative check.
Thus in Trials 44, 33, 56 (with jackets) the observed discharges of heat
are greater than the calculated by amounts which diminish slightly as the
speed increases. These differences, about 5 per cent, of the total heat dis-
charged, which will be the subject of further remark, reveal no inconsistency
in the observed results, which so far check each other. On the other hand,
in the trials 41, 35, 40 (without jackets), while the observed discharges (for
trials 35 and 40) are from 1 to 2 per cent, below those calculated, allowing a
margin for external radiation, the observed discharge for trial 41 is about
5 per cent, larger than the calculated, an inconsistency which shows error of
observation somewhere. Table II does not supply sufficient evidence to
56] ON TRIPLE-EXPANSION ENGINES AND ENGINE-TRIALS. 365
locate the error, but this is found in Table I in the quantities given under
the head radiation (line 41).
This radiation is obtained as the balance of the total heat received from
the boiler (in the water from the hot-well as dry steam, and in the jacket
water), and the total heat discharged as heat and work ; hence any error in
measuring the heat discharged, or the water from the hot-well, would affect
the apparent radiation. Since all the trials without jackets are made under
approximately the same radiating conditions, and these conditions are such
as would cause slightly less radiation than the trials with jackets, the actual
radiation in the trials without jackets must have been nearly the same, and
somewhat less than in the trials with jackets. In Table I the radiation for
trial 41 is 503 thermal units per minute, 897 for 35, 1170 for 40, and 1266
for the trials with jackets, so that the radiation in trial 41 is clearly some
500 thermal units per minute too small. This might be due to an error
either in the hot-well discharge or in the heat discharge ; but as the former
would affect the heat per Ib. of coal (line 62), and so bring this trial out
of accord with the others, it seems that the error is in the heat dis-
charged.
The correction that would bring the observed heat discharged in
Table II, trial 41, into accord with the others is 60 thermal units per Ib.,
or 460 thermal units per minute, which heat, transferred to the radiation,
would bring this to 963, or nearly the mean of that for trials 35 and 40.
This shows the completeness of the check throughout these results.
The Radiation. — The slight differences which are shown in this quantity,
Table I, line 41, for all the trials with jackets, may have been due to
differences of temperature in the engine-room. The mean radiation with
200 Ibs. steam in the jackets is 1266 thermal units per minute, and the
mean radiation in the trials with the cylinder jackets shut off (omitting 41)
is 1037, the difference being 229, with or without jackets, at a pressure
of 200 Ibs. per square inch. This is exclusive of radiation from the boiler.
The Heat Abstracted during Exhaust. — That during the exhaust the
water in the cylinder, which has resulted from condensation, is re-evaporated
by heat from the walls is well established, and it has been often suggested
that the steam leaving the cylinder may be somewhat superheated by the
hot walls of the passages. The excess of the observed heat discharged over
that calculated in Appendix, Table II, might be explained by the second of
these causes, but not by the first, since the diagrams all show that the steam
was in the condition of dry saturated steam at release ; besides which, the
calculated heat takes account of all the heat it could so possess. To account
for this difference, which amounts to 5 per cent, of the total heat discharged,
by supposing the steam superheated would be to suppose the temperature of
366 ON TRIPLE-EXPANSION ENGINES AND ENGINE-TRIALS. [56
the steam raised from 70° to 100° above the temperature of the condenser.
Considering that the temperature of the steam in the jackets was 250°
higher than that in the condenser, there would be nothing apparently
impossible in thus superheating the steam while passing through the ports
and exhaust passage heated by the jackets. Such a rise of temperature
would, however, be apparent in the exhaust pipe if sought for ; and as
thermometers showed that the temperature of this did not rise at any time
to more than 140° Fahrenheit, which temperature corresponded with the
pressure of steam in the condenser, it is evident that this heat did not go to
raise the temperature of the effluent steam. The fact that the difference
varies so little with the speed of the engine suggests that this absorption of
heat is consequent, in some way, on the mechanical action to which the steam
is subject during exhaust, in a similar manner to that in which the heat
supplied by the jackets to the cylinder is consequent on the expansion, and
this appears to be the case.
The steam in the cylinder at release expands down to the pressure of the
condenser. The expansion takes place partly in the cylinder, partly in the
passages, and will be attended by liquefaction similar to that which results
from ordinary expansion. The liquid, thus formed, may be re-evaporated,
from the hot walls of the cylinder and the passages, without raising the
temperature of the steam above that of the condenser. This expansion is
from the volume (per Ib.) at release to the volume (per Ib.) at the pressure
in the condenser, and the amount of heat for re-evaporation can be de-
finitely estimated. In trials 44, 35, 56 respectively, this heat amounts to
84, 87, 71 thermal units per Ib. of steam. Some considerable portion of
the heat would be supplied from the work done by the steam against the
resistance in the passages, which would be directly reconverted into heat;
but the greater portion would have to be obtained from the surfaces, or else
the steam would enter the exhaust in a supersaturated condition. The
excesses of the observed heat over the calculated, Appendix, Table II, are 64,
29, 31 thermal units per Ib., being well within the heat necessary to re-
evaporate the water, after making allowance for the friction of the passages.
This heat, it is to be noticed, is acquired by the steam from the walls after
the steam has done its work in the cylinder, and must be supplied either by
the jackets or by the condensation in the steam-chest, ports, and cylinder.
It therefore represents heat which passes direct through the engine, without
effecting any work, and is a loss of between 3 and 6 per cent, of the theoretical
efficiency of the steam.
The Diagrams have been taken with six Crosby indicators, and with
springs as low as the speeds and pressures would admit.
The reduction is effected by measuring ten breadths, the pressure and
56] <)N TRIPLE- EXPANSION ENGINES AND ENGINE-TRIALS. 867
back-pressure from the atmospheric line, and then the effective pressure, so
that the results check, and may be directly used to obtain a mean diagram.
These results have been several times checked by a planimeter, without
establishing any sensible difference. As regards the diagrams themselves,
every precaution has been taken to ensure accuracy, and there is no reason to
suppose that there are any prevailing errors of 1 per cent., although errors of
the instruments, and, indeed, of all indicators, when subjected to certain par-
ticular tests, are much greater than this. The check afforded by the brake-
power, although it would not reveal a prevailing error of 2 or 3 per cent.,
has this important effect, that it does away with any possible bias that
might result from enthusiasm to obtain high indicated power, for by so
doing the effect would be to lower the mechanical efficiency of the engine.
It is, however, the consistent agreement of the curves of expansion, as
indicated, with the theoretical curve for the weight of absolute steam shown
by the water discharged to have passed through the engine, that gives the
greatest confidence in the indicated results.
The Reduction of the Diagrams to a mean Compound Diagram. — Con-
sidering the important place which must be occupied by mean compound
diagrams, in comparing the results of the various trials in such an extended
investigation of the steam-engine, it was necessary that some system of
reduction should be adhered to, and the choice of this system was a matter
of the first importance. There was one peculiarity about the working of
these engines which necessitated a departure from any methods previously
adopted, namely, the unequal speeds of the three engines. This fact had
great influence in determining the system adopted. Except as affected by
this, the methods of reduction did not differ from one or other of the plans
usually followed.
The reduction of the twenty-four diagrams, taken during a trial from each
engine, is effected by finding the means of each of the twenty measured
distances from the atmospheric line, which are then reduced to a common
scale, 10 Ibs. to an inch. These ordinates are then plotted, so as to project
the diagram to a length determined, as will be subsequently described. The
volumes of clearance, 4 per cent, on engine I, and 6 per cent, on engines II
and III, valve-clearance l'6o per cent, on engine I, and 2-o on engines II and
III, are then added to obtain the line of zero volume. Thus, a compound
diagram is obtained showing the relation of volumes and pressures of the
whole steam in each of the cylinders. To reduce this diagram, to show the re-
lation of volume and of pressure of the steam discharged from the cylinder, an
ideal compression -line is drawn through the point of the actual compression-
curve which corresponds to the closing of the exhaust. Horizontal lines are
next drawn across the diagram, cutting the expansion-curve, the compression-
line, and the ideal line, and each of these horizontal lines is set back until
368 ON TRIPLE-EXPANSION ENGINES AND ENGINE-TRIALS. [56
the point which was the ideal compression-curve reaches the line of zero
volume. Then the positions taken by the points from the expansion-line
and the actual compression-line show the volume of steam in the cylinder
over and above the volume of that which is shut in at exhaust. All this
reduction may be done arithmetically, or by plotting. The result is that,
while the area of the diagram has not been altered, the actual expansion and
compression-line for the steam passing through the engine is obtained ;
Rankine's curve of saturation for the weight of steam discharged is then
drawn. On account of the varying difference between the speeds of these
engines, the lengths for the compound diagram could not be obtained by
simply projecting the lengths of the separate diagrams, so that they should
be proportional to the effective volumes of the several cylinders. It was
necessary to project them so that they should be proportional to the products
of the effective volumes of each engine multiplied by its revolutions per
minute. Slight as this necessary modification may appear, it does away with
the idea of a relation between the area of a diagram and the size of the
engine, which, once got rid of, leaves it apparent that the separate diagrams
express nothing but the relation which holds between the pressures and
volumes of a certain quantity of steam, which quantity may be changed by
altering the scale of length of the diagrams. Having once realized this, the
advantage becomes apparent, in instituting comparisons between a number
of engine trials, of taking the common scale of length for the diagrams to be
such that they all express the relation between the volume and the pressure
of the common unit (1 Ib.) adopted for the weight of steam. This common
scale is readily obtained by dividing the product of effective volumes, multi-
plied by revolutions, by the weight of steam passing through the engines
per minute, and taking the result as the length of the diagram in any
uniform scale ; ^-inch to the cubic foot has been that adopted for the first
reduction in these trials, the pressures being plotted to 10 Ibs. to an inch.
The diagrams, Fig. 15, p. 370. are such mean diagrams, showing the
Ibs. per square inch pressure and cubic feet volume for each Ib. of steam
passing through the engines, also Rankine's curv£ for saturated steam to the
same scale. In these diagrams : —
The extreme length of the diagram =
The distance from the line of zero]
volume to the expansion or I
the effective volume swept by the
piston for each Ib. of steam
through the engines.
the volume of the steam in the
cylinder at that pressure, less
compression-curve at any par- [ the steam shut in at com'
ticular pressure ) pression per Ib. of steam through
V the eneine.
56]
ON TRIPLE-EXPANSION ENGINES AND ENGINE-TRIALS.
369
The area enclosed in the diagram = effective work per Ib. of steam.
The distance to the right between "
the compression-Hue and that
C I
ot no volume measures
The distance between the expan-
sion-line and the saturation-
curve
The ratio of the horizontal dis-
tances from the line of zero
volume to the curve at cut-off
and release..
the volume of initial steam per Ib.
of steam rendered non-effective
by clearance.
rthe volume of steam per Ib. of steam
through the engines absent on
account of condensation, priming
and leakage.
= the effective ratio of expansion.
The clearness and simplicity of the comparison which these diagrams
institute between the areas actually occupied, and those which would have
been occupied had the steam been saturated, renders it possible, as well as
desirable, to state exactly in what relation the areas stand as regards the
theory and economy of the engine.
The area enclosed between the limits of pressure and volume by the line
of zero volume, the line of condenser pressure, and the saturated curve,
expresses in foot-lbs., the greatest possible amount of heat that can be
converted into work, through the agency of 1 Ib. of steam maintained in a
state of saturation between these limits. The areas included in the measured
diagrams represent the heat which has been so converted by the agency of
each Ib. through the engines, and the various intervening areas represent loss
in conversion.
These are facts which it is important to bear in mind in dealing with
jacketed engines, in which 1 Ib. of steam through the engines does not
represent a certain quantity of heat, which will be the same whether it is
realized or not. For such engines it is impossible to make the diagrams
represent the comparative efficiencies actual and theoretical. With un-
jacketed engines, the case is different, as the Ib. of steam represents, at
a particular pressure, a definite quantity of heat through the engine, how-
ever much of it is converted, and if a special adiabatic line be substituted
for the saturated line, the relation of areas will be the relation of efficiencies.
In the present case, however, it seemed better to treat all the diagrams in
the same way, and to make a separate comparison of the efficiencies with the
highest theoretical efficiency between the same limits. With the unjacketed
as well as with the jacketed trials, the theoretical efficiency has been
calculated as for saturated steam. This comparison for all the trials is
given in Appendix, Table III.
o. R. TI.
24
370 ON TRIPLE-EXPANSION ENGINES AND ENGINE-TRIALS. [56
TRIAL 44.
O ' jb SO 3O 4O
ib io :(o 40
0 T " I " "0." " I " "gL," " I " ' 'al,
F,-rt to Hie Lb. from «.c lint-Well: _
•*>•,_ ' ib z» ' so
r-i,6.V Prri Ic ftit ZH,. Iron, tl,r Hot-Jim
Fig. 15.
ofi] ON TRIPLE-EXPANSION ENGINES AND ENGINE -TRIALS. 37l
THE CONDENSATION, PRIMING AND LEAKAGE OF STEAM IN_ THE
CYLINDERS, AS SHOWN IN THE DIAGRAMS.
There are two quantities which it is almost impossible to separate by the
inherent evidence of the diagrams.
The missing quantity, to use Mr Willans' expression, which is here shown
by the horizontal breadth of the black band, may equally well arise from the
steam having escaped by the piston, or having been temporarily converted
into water.
This much, however, is evident from the diagrams, that with steam in the
jackets, in whatever manner the steam has vanished in the high-pressure
engine, it has all reappeared before the end of the stroke in the inter-
mediate engine, and though some of it has disappeared at the cut-off in the
low-pressure cylinder, it has reappeared again before the end of the stroke.
Hence it seems that there is no escape of steam by the pistons of these two
engines.
The question remains, however, as to whether steam has not escaped
by the pistons of the high- pressure engine, and through the valves, during
expansion into the cylinders of the intermediate and low-pressure engines.
Certain differences in the diagrams taken from No. II engine, when
\vorking with different cuts-off, suggested that the rider valves were held
somewhat off the back of the main valve by the spindle, so that they leaked
steam until the pressure in the cylinder was sufficiently lower than that
in the steam -chest to spring the spindle and force the valves home. It
became particularly evident in the fifty-fifth trial, and then the cover was
removed and the conclusion verified. This source of error was put right,
and the fifty-sixth trial, as compared with the earlier ones, shows what has
been the effect of leakage in these, namely, the breadth of the black band
towards the tops of the diagram from No. II engine.
When the covers were last put on, in August, 1888, the cylinders and
valve-faces were all in equally good condition, and there has been no leak
from the jackets, while the engines were standing with full pressure in
the jackets. The regulators opening into the intermediate receivers were
made tight in August, 1888, and were not again opened till after the forty-
sixth trial. There was then occasion to open them, and as the engines were
standing preparatory to starting the fifty-sixth trial, it was seen that steam
was leaking into No. II receiver, probably about £-lb. per minute; as the
valve was found to be shut, there was nothing to be done, so the trial was
run ; and, as was to be expected, the diagrams from No. I engine show what,
considering the circumstances, is an unusually large black band.
24—2
372 ON TRIPLE-EXPANSION ENGINES AND ENGINE-TRIALS. [56
In the absence of definite evidence of leakage, the Author concludes that
the missing quantity shown by the black band is everywhere due to con-
densation.
It is not the intention in this Paper to endeavour to establish a complete
theory of cylinder-condensation. Though it may be well to state that, before
designing the engines, the theory was carefully considered and formulated,
leaving only the arbitrary constants to be determined from the experiments.
For anything like a complete determination of these constants, the experi-
ments have not sufficiently advanced ; but this is not necessary to show that
in the case of a series of cylinders, all jacketed up to boiler-pressure, the law
of condensation would be precisely that which is shown in the diagrams.
Whenever the bounding surfaces are colder than the steam adjacent to
them condensation occurs. To prevent condensation it is therefore necessary
to maintain all parts of the cylinder surfaces, and port passage surfaces,
at a temperature at least as high as that of the initial steam.
To do this, in the case of expansion, it is not sufficient (as seems to be
commonly assumed) to keep the outside of the metal constituting the walls
and covers, merely at the temperature of the initial steam. That, of course,
would be sufficient if there were no condensation other than what results
from the temperature of the surfaces.
Forty years ago no such other cause of condensation was known. It was
revealed, however, by the discoverers, Rankine and Clausius, in 1849, that
the expansion of steam reduces its temperature below that corresponding to
saturation unless some of the steam is condensed. The manner of action
of this supersaturation, caused by expansion, in absorbing heat from the walls
of the cylinder maintained at a higher temperature than the steam, does not
appear to have been yet ascertained with any degree of certainty ; but it
is certain that steam in this state of supersaturation does absorb heat with
immense rapidity, when the walls are at a higher temperature than the
expanded steam. Also the amount of heat necessary to prevent supersatura-
tion is definitely known, though it is, perhaps, well to recall the fact that it
is not, even approximately, the heat equivalent of the work done by the
steam during expansion.
If the walls of the cylinders are maintained at the temperature of the
initial steam, the expanding steam will absorb heat. This heat must pass
through the walls ; and as heat only flows through metal down the gradient
of temperature, the temperature on the outside must be greater than that on
the inside. Hence it follows that either the steam in the jackets must be
hotter than the initial temperature of the steam in the cylinder, or the mean
temperature of the internal surface of the cylinder will be below that of the
initial steam, in which case there will be cylinder-condensation.
5b'] ON TlilPLE-EXPANSION ENGINES AND ENGINE-TRIALS. 373
How important this degradation of temperature through the walls is, will,
perhaps, best be rendered apparent by stating an actual case.
In expanding 1 Ib. of steam from a pressure of 203 Ibs. to a pressure
of 79 '3 Ibs., the heat per Ib. necessary to prevent supersaturation is
551 T.U.
or about 5 per cent, of the total heat in the initial steam In a cylinder
passing 600 Ibs. of steam per hour, to prevent supersaturation there should
pass through the walls of the cylinder
33,060 T.U.
Now the jacketed surface of the cylinder of the H.-P. engine is less than
1*5 square foot, and the thickness of the metal is more than 0*4 inch. Hence
the heat would have to flow through this thickness of metal at a rate of
•2'2,OQO T.U. per square foot per hour.
From the known laws of conductivity of iron, this would require a difference
of temperature of 38° Fahrenheit.
Thus it appears that, to prevent supersaturation, the temperature of the
steam in the jackets of No. I engine must be 38° higher than the mean
temperature of the internal steam ; or, in other words, that the mean tem-
perature of the internal surfaces will be 38° lower than that of the initial
steam, which is at the same temperature as that in the jackets.
What amount of surface-condensation this difference of temperature
would cause may be, to some extent, inferred by comparison with the differ-
ence between the mean temperature of the surfaces and that of the initial
steam when the jackets are empty. Here the initial temperature is about
383°, and that of the exhaust, 302° ; the mean, 342° ; difference of mean and
initial, 41°; so that in this engine the mean temperature of the walls would
only be affected to the extent of about 3° Fahrenheit by the jackets, suppos-
ing the whole of the heat to prevent supersaturation were supplied by the
jackets. But this would not be quite the case, as some heat is obtained from
the difference in the heat given up and absorbed by the cylinder-con-
densation ; and there is no proof that the steam may not be discharged
with a certain degree of supersaturation.
However, the reasoning leads to the conclusion that, with steam at initial
pressure in them, the jackets would produce a comparatively small difference
on the cylinder-condensation in this engine when passing 10 Ibs. of steam
per minute.
In No. II, the intermediate engine, the case is different. Here the heat
which has to flow into the cylinder through the walls is nearly the same as in
374 ON TRIFLE-EXPANSION ENGINES AND ENGINE-TRIALS. [56
No. I ; but the surfaces are double as large and of the same thickness, so
that the fall of temperature would be about one-half, or 20°. The tempera-
ture of the steam in the jackets is 81° above that of the initial steam, and
the internal walls would still be 60° above the initial temperature. Hence
there should be no condensation on those surfaces which are jacketed. Still
there are in this engine, although much less than is usual in jacketed
engines, portions of the surfaces which are not, so to speak, jacketed, mainly
the surface of the ports and of the piston ; and though these derive heat
from the jackets, it is through a much greater thickness of metal, and hence
would require a much greater difference of temperature to prevent condensa-
tion. Thus, even with the jackets at a temperature of 60° above that of the
steam, there should probably be some initial condensation.
In No. Ill engine the jackets have a temperature of 140° above the
steam, hence the initial condensation should probably be much less than
in No. II
The diagrams (Fig. 15, p. 370) show that this is the case. They exhibit
a little condensation, which seems to increase from cut-off until the expansion
reaches about V5 or 2, and it then diminishes to zero. The increase after
cut-off may be owing to the inertia of the indicator piston depressing the
curve, as the springs used have always been as weak as possible on account
of the low pressure.
They also demonstrate conclusively, with such jacketing as there is
in these cylinders, that a temperature of 140° in the jackets above the initial
temperature is sufficient to prevent sensible cylinder-condensation with as
much as 720 Ibs. of water per hour passing through the cylinders.
The diagrams for the trials 41, 35, 40, show the condensation when
the jackets are empty. These three diagrams are from trials as nearly
as practicable corresponding in power with those with the jackets on. They
are reduced to show the volume per Ib. of water through each engine, and
the outside curve is the saturation-curve for 1 Ib. of steam ; the horizontal
breadth of the black band, therefore, represents volume of steam missing.
This includes the volume missing on account of the condensation resulting
from expansion in each cylinder as well as on account of cylinder-condensa-
tion. It is to be noticed, however, that the steam probably entered each
steam-chest dry, so that the only water in excess of cylinder- condensation is
that resulting from expansion in that cylinder. This would be represented
by a curve draAvn from the points in the saturation-curve corresponding
in pressure to the points of cut-off, and gradually diverging inwards from the
saturation-curve, until at release the horizontal divergence should be about
5 per cent, of the horizontal breadth of the white diagram at that pressure.
50] OX TKII'LE-EXI'AXSION ENGINES AND ENGINE-TRIALS. 375
The great excess of condensation in the intermediate cylinder over the
high-pressure, and in the low-pressure cylinder over the intermediate, is very
apparent. This fully explains the difference in the relative speeds of the
engines with and without the jackets already mentioned, the speed of No. Ill
compared with No. I being as 1*5 with jackets to 1 without jackets.
The distributions of condensation are very similar in the three cylinders.
The ratios which the steam condensed bears to the steam passing through
the engines at cut-off, middle stroke, and release, are shown in Appendix,
Table IV.
The testing of the boiler was carried only so far as was necessary to check
the results of the trials. No chemical tests were taken of the air or coal.
The coal used was Nixon's Navigation mixture, weighed as it came from
the heap in the boiler-house. In most of the trials the feed was carefully
measured, with the result, already mentioned, that it was from 5 to 10 per
cent, greater than the discharge from the hot-well.
Taking the feed, these trials show that the boiler generally evaporated
10'4 Ibs. of water per Ib. of coal with the pressure 195 Ibs. and the feed at
130°. This, if all the water were evaporated, would give 11,350 units of heat
per Ib. of coal.
The temperature at which the gases left the boiler was 500°, and after
passing the water heater 250°, the rise of temperature in the water-heater
being about 100°.
The source of the loss of water was not discoverable, so that it was not
possible to determine whether it escaped as water or steam ; and until this
point could be determined it was impossible to say from observations on the
boiler what the quantity of heat obtained in the boiler might be. The
results in Table I are therefore confined to the steam received by the
engines.
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Date of the trial
Time of trial
Lbs. jacket-water per hour returned to the feed-j^
Degrees Fahrenheit temperature of the boiler
Degrees mean temperature observed after mixtui
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56]
ON TRIPLE-EXPANSION ENGINES AND ENGINE-TRIALS.
379
TABLE II.
Jackets at Boiler-Pressure
Jackets Empty
Number of the trial
44
33
56
41
35
40
Thermal units j Calculated
from the con- 1
denser per Ibt-j Measured
1,011
1,075
1,014
1,043
1,011
1,042
1,014
1,065
1,009
1,001
990
978
the hot- well... I Differences
-64
-29
-31
-51
8
12
TABLE III. — RELATIVE AREAS OF DIAGRAMS PER LB. OF STEAM THROUGH
THE ENGINES, AND THERMAL EFFICIENCIES OF ENGINES.
Number of trial
44
33
56
41
35
40
1
2
3
Theoretical area, ft. & 11 >.
.Mc.isured area „
Percentage of theoretical}
area £
238,645
188,096
79-0
233,545
192,067
82-0
228,420
192,000
84-0
235,500
127,545
54-0
233,000
139,546
60-0
221,860
144,350
65-0
4
5
6
7
8
!)
10
Theoretical efficiency, p.c.
Measured efficiency, p.c.
Percentage of theoretical}
efficiency 5
23-3
18-5
79-4
23-2
19-2
82-6
22-7
19-4
85-4
23-3
13-8
59-2
23-2
15-3
65-9
22-4
15-5
69-4
TABLE IV. — CONDENSATION WITHOUT JACKETS.
Number
of the
Trial
Revolutions
per
Minute
Ratio
of
Expansion
Proportion of Total Steam
condensed at
Cut-off
Mid-Stroke
Release
(
41
146
27
0-40
0-39
0-30
Engine No. I ... 1
35
40
229
322
23
2-0
0-29
0-22
0-27
0-21
0-22
0-17
(
41
127
2-4
0-41
0-345
0-29
Engine No. II... ^
S5
40
215
320
2-4
2-2
0-38
0-30
0-34
0-27
0-26
0-14
(
41
109
2-7
0-51
0-48
0-37
Engine No. lit... I
86
40
184
276
3-05
2-6
0-48
0-32
0-47
0-36
0-33
0-23
57.
REPORT OF THE COMMITTEE APPOINTED TO INVESTIGATE
THE ACTION OF WAVES AND CURRENTS ON THE
BEDS AND FORESHORES OF ESTUARIES BY MEANS
OF WORKING MODELS.
[From the " British Association Report," 1889.]
THE Committee held its first meeting in the Central Institution of the
City and Guilds of London Institute. It was then resolved that the
Committee should avail itself of the permission of the Council of the
Owens College, and conduct its experiments in the Whitworth Engineering
Laboratory.
At the suggestion of Prof. Reynolds it was arranged that the first
experiments should be directed to determine in what respects, and to what
extent, the distribution of sand in the beds of model estuaries of similar
lateral configuration is affected by the horizontal and vertical dimensions,
and the relation which these bear to one another and to the tide period so
as to place the laws of similarity on which the practical applications of the
method depend, on as firm an experimental basis as possible.
It was suggested provisionally that two working tanks should be con-
structed, one as large as the circumstances of the laboratory would admit,
and one of half the linear dimensions of the larger tank. Prof. Reynolds
was empowered to appoint an assistant to make the necessary observations.
At a second Committee, held at Owens College, Manchesoer, the models
constructed were examined, and it was arranged that Prof. Reynolds should
draw up a report on the results so far obtained.
57]
ON THE ACTION OF WAVES AND CURRENTS.
381
On Model Estuaries. By Professor Osborne Reynolds,
Having carefully considered and sketched out designs for the tanks and
appliances in accordance with the resolutions of the Committee on February
6, I obtained the assistance of Mr H. Bamford, B.Sc., from Easter up to the
date of the meeting at Newcastle. The working drawings for the appliances
were commenced immediately after Easter, and the work put in hand, the
experiments being commenced in each tank as soon as it was ready.
The General Design of the Appliances. — A great object in designing the
tanks was to make the most of the facilities in the Whitworth Engineering
Laboratory, Owens College, in respect of a continuously running shaft, a
supply of town's water and wastes, also a water supply (13,000 gallons) from
a storage tank at 116 feet above the floor of the laboratory, and a discharge
into a similar tank below the floor, with pumping power to raise the water
back when required, also floor space.
The available floor space, although very conveniently placed with respect
to the water and power, was strictly limited by resisting structures to 10 feet
wide and 22 feet long. This admitted of an extreme length for the larger
tank of 16 feet, and an extreme width of 4' 8", leaving 2' 6" for the width
of the smaller tank, the remainder of the space being the least possible that
would admit of access to all parts of the tanks. The internal dimensions
of the tanks as designed are : —
TANK A.
Length
Breadth
Height
Fixed rectangular tray having one end open
From laboratory Moor of the tray . . ...
11'IOJ"
3' 9£"
2' 6"
Sides and end above the bottom
9"
Tide generator, one end open
3' 10£"
3' 9A"
Sides at open end
9"
At closed end
1'7"
TANK B. Half the dimensions of A.
The proportions of the tide-generators and fixed pans were determined,
so that in tank A the greatest rise of tide over the whole tank should be 2";
which was double the tide used in my previous experiments, and that con-
sistently with this the generators should be as short as possible. This tide
in tank A required that the generator should displace 10 cubic feet, and as
382 ON THE ACTION OF WAVES AND CURRENTS. [57
the greatest rise and fall that could be conveniently obtained for the end of
the generator was 16", giving a mean rise of 8", the area required was
15 square feet.
A period of 30 seconds was adopted for tank A as the shortest period
likely to be required, and the gearing arranged accordingly. With this
period, and a 2" tide, the horizontal scale would be 1 in 20,000 of that of a
tank with a 30- foot tide, and a period of 12 hours 20 minutes. So that
the 12-foot pan would represent 45 miles.
Provision was made for the production of waves with periods ^th the
tidal period.
Provision was also made for the introduction of land water into the tank
at any point that might be required ; also for scumming the water by an
adjustable weir, which would serve to keep the level of low water constant,
water being supplied into the generator when no land water was required.
The drawings (fig. 1, p. 401) show the tanks and apparatus as they have
been constructed. The pans and tide generators are of pine-boards fastened
with screws. The former rest in a fixed cradle formed by six legs with cross-
bearers, bottom ties, and braces. The floor boards of the pan are screwed to
the cross-bearers, but are left free to expand, the joints being made with
marine glue, after the manner of the decks of ships. The sides are screwed to
the floor only ; they receive lateral support against the pressure of the water
from the prolongations of the legs upwards. The pans are lined with calico
saturated with marine glue, and put down with hot irons, then covered with
a coat of paraffin. The pans of the generators are constructed in the same
way as the others, only instead of the cross-bearers being attached to legs,
they are suspended from two side levers, which are supported on cast-iron
knife-edges resting in cast-iron grooves on the top of the legs at the end
of the pan. These knife-edges are at the exact level of the top of the floor
of the pan, and in line with the joint in the floor between the pan and the
generator, so that there is no opening and closing of this joint. This joint
is, however, covered with indiarubber, which extends up the sides, and by
stretching allows for the opening and closing of these joints.
In tank A these side levers extend 4 feet along the sides of the pan,
beyond the joint, and to their ends is attached a large box for holding
balance weights. These weights are considerably below the knife-edges,
and consequently their moment diminishes as the box descends, i.e. as the
tide rises, but this diminution by no means compensates the diminution of
the water in the generator.
If, therefore, sufficient weight were put into the box to balance the
generator when the tide is low, it would much overbalance it when the tide
is high. To meet this the weights in the box are used mainly to balance
:~>7] ON THK ACTION OF WAVES AND CURRENTS. 383
the dead weight of the generator, which requires about 300 Ibs., and a
varying balance is arranged for the water.
This varying balance consists, in tank A, of a cast-iron cylinder of 500 Ibs.
weight, suspended by links from the side levers across and under the tank.
The cylinder is also suspended by two links from the frame, and this second
suspension is so arranged that when the generator is down the links from
the levers are vertical, and when the generator is up they are horizontal.
In this way a varying balance is obtained, which as far as possible effects a
complete balance in all particulars. In tank B, arrangements which have
the same effect have been carried out in a somewhat different manner, which
will be clear from the drawings.
The glass covering for tank A consists of eight glazed frames, each
having two panes of sheet glass 3' 10" x 10", with |" bearing on the frame
all round ; the external dimensions of the frames are 4' x 2', so that they
are easily handled. The glass is let in flush with the top of the wood, and
each pane is fixed by four small brass clips screwed to the frame. In this
way, except for the clips, the top of the glass cover over the pan presents a
level surface. The frames over the tide generator are connected with those
over the pan by a hinge joint, made of two strips of pine hinged to each
other and to the frames.
A somewhat similar arrangement exists in tank B, except that there are
only four frames each with a single pane 2' x 2'. In both tanks the glass
frames are fastened by screws to the sides, which screws have to be taken
out before the frames can be removed.
The gearing, which is arranged to be driven either from a small water-
engine or the running shafting, is shown in the drawings.
The crank is adjustable so as to give any required tide up to the
maximum. In tank A, the pulley driven by the belt from the motor or
shaft makes 700 revolutions for one of the crank, and has a fly-wheel which
considerably helps the motor over any little irregularities in the balance.
The gearing in tank B is driven either direct from the motor or from a
pulley on the second shaft in the gearing of A, in which way a fixed relation
in speed is obtained when the tanks are working together. The motor was
obtained from Alderman Bailey, Albion Works, Salford ; it is a double-
acting oscillating water-engine with a £" piston and 4" stroke. The available
pressure of water is 50 Ibs. steady pressure ; the consumption is about
1 gallon per 100 revolutions. At the highest speed, 2 tides a minute, the
motor only makes about 200 revolutions per minute, so that the 13,000
gallons will keep it going over three days, and has done so from Saturday
till Tuesday, Monday being Bank Holiday. It has run day and night and
384 ON THE ACTION OF WAVES AND CURRENTS. [57
Sunday, since starting in June, without once stopping, making over
12,000,000 revolutions, and is none the worse. If it used the full pressure
it would, when run at 100 revolutions, do about '044 horse-power. Owing
to the careful balance of the tanks and the use of spur instead of worm
gearing, the work required is not more than '008 horse-power, so that five-
sixths of the pressure is spent in overcoming the fluid resistance, which,
increasing as the square of the speed, affords a very important means of
regulating the speed, which, indeed, is thus rendered very regular.
Surveying Appliances. — Since the configuration of the sand produced
under different circumstances can only be compared by means of records
such as charts or sections, the practicability of the investigation depended
on the finding of some means by which the sections or contour-lines on
the sand could be rapidly and accurately surveyed.
The floor of the estuary was made flat and carefully levelled, so that the
depth of sand at any point could be at once ascertained by sinking a fine
scale through it to the bottom ; and for this purpose scales were constructed
of strips of sheet brass -01' broad and '01" thick. On these the alternate
•01' were painted white, and the intermediate spaces in the first 0' 1 were
painted red, in the second O''l black, and so on, the scales being then
varnished with paraffin.
These scales would stand in the sand edgeways to the current, and so be
made into permanent sand-gauges, which could be read periodically without
removing the glass or stopping the tide. For tank B the scales were half
the size of those for tank A.
The resistance which a few such thin obstructions offered to the water
would be very small, but if the gauges were numerous the resistance would
be a serious matter, so that a more general method of taking a final survey
was necessary.
The ease and simplicity with which the contour-line could be found when
the tides were not running, by adjusting the level of the still water and
observing its boundary on the sand, reduced the question of making a
contour survey to the providing of the means—
1. Of adjusting the level of still water to any required height.
2. Of rapidly and accurately determining the horizontal position of
points on the edge of the water.
The tide-gauge, shown in the drawing on the top of the tank, which
would stand on the glass which gave a level surface, answered well to
determine the level of the water.
For the purpose of surveying the contours a system of horizontal survey-
57] ON THE ACTION OF WAVES AND CURRENTS. 385
lines were set out in the covering frames, consisting of black thread stretched
immediately beneath the glass in the frames. The lines are 6" apart ; those
parallel with sides are called lines, and those at right angles sections. The
first section is 3" from the end of the tank* and the lines are so placed
that one of them bisects the tank.
These survey-lines divide the entire surface of the fixed tray into equal
squares. They are, however, 11" from the bottom and about 8 from the
sand ; besides, they are six inches apart, so that to make accurate use of
them for surveying the sand it was necessary to use some means of projecting
a point vertically up to the level of the glass and scale its distance from a
line and a section. This is accomplished by a little instrument, which may
be called a projector, shown on the top of tank A.
It has a foot which consists of two scales placed at right angles, so that
the zero- lines on both, if produced, would meet in a point about half an inch
from the edge of the scale. About this point there is a hole through the
foot, with cross-wires so placed that they intersect in the point of intersection
of the zero-lines. Vertically above this is a horizontal plate with a pin-hole,
so that, when placed on the horizontal glass, any point below, seen through
the pin-hole on the cross-wires, is vertically below the intersection of the
zero-line of the scales ; and hence if these scales are parallel to the lines
and sections, the distances of the point from these are read at once on the
scales. This method of surveying lends itself readily to plotting on section
paper. This may be done directly, the glass cover of the tank serving for
a table ; each point may be plotted as it is observed ; and in this way
Mr Bamford is now able to survey and plot a complete contour-line in from
fifteen to thirty minutes, and requires only about five hours to make a
complete survey plotting the charts.
One very great desideratum has been a graphic recording tide-gauge. So
much depends on the manner of rise and fall of the tide that it does not
seem sufficient to know that it is produced by a simple harmonic motion ;
the curve should be recorded for each experiment at different parts of the
tank. The want of means and time have prevented any attempt to supply
such a gauge.
Curves have been obtained for most of the experiments by means of the
simple tide-gauge. The crank-wheel being divided into sixteen equal arcs,
one observer observes the wheel and another the gauge. When a particular
number comes to the index the observer at the wheel calls, and the other
observer reads the gauge, and then shifts the sliding pointer to the point at
which the tide-index was, so that on the next revolution, when the call
* This somewhat awkward arrangement was necessary on account of the wood in the frames.
o. K. ii. 25
386 ON THE ACTION OF WAVES AND CURRENTS. [57
comes again, he can observe if the pointer coincides exactly with the index
or requires adjustment. Having brought about coincidence, he then proceeds
to the next number. In this way it takes about half an hour to read the
curve. Time, however, is not the only objection, a greater one being that
any irregularities in the motion of the wheel do not appear in the curve.
The motion of the wheel has been as far as possible checked by the clock,
but still there is room for important errors, which a chronograph would
obviate.
The Selection of Sand. — Sir James Douglas having informed me that
clean shell sand could be obtained, and having sent me samples which, from
the tests to which I subjected them, seemed to be quite as readily moved
by the water as the finest Calais sand, I asked him to procure a quantity —
fifteen bushels of Huna Bay shell sand — and in the meantime I procured a
similar quantity of Calais sand, so that I might be prepared with whichever
showed itself best in actual experiments.
Selection of the Experiments. — It having been decided that in the first
instance the purpose of the experiments should be the comparison of the
distributions of sand produced under particular lateral configurations, and
with different relations between the vertical and horizontal scales in the
same model, and with similar relations in these scales in the two models,
the only matters left for selection in starting these experiments were the
scales and particular configurations.
There was apparently no reason for attempting the very difficult operation
of modelling any actual estuary, and, setting this aside, the question of choice
mainly turned on whether it was best to begin with complex or simple
circumstances. There was considerable temptation to commence with complex,
i.e. boldly irregular boundaries, so that the influence of the boundaries might
predominate over such other influences as exist ; in which case the influence
of the boundaries would be tested by the similarity of the distributions
produced with different ratios of horizontal and vertical scales. On the
other hand, however, it appeared that as the main object of these researches
is to differentiate and examine the various circumstances which influence
the distribution of the sand, it was desirable, in starting, to simplify as
much as possible all the circumstances directly under control, and so afford
an opportunity for other more occult causes to reveal themselves through
their effects, and to determine the laws of similarity of these effects.
The simplest of all circumstances would be that of no lateral boundaries
whatever — a straight foreshore of unlimited length with a shelving sandy
beach, up and down which the tide runs until it has brought the beach
to a state of equilibrium.
57] ON THE ACTION OF WAVES AND CURRENTS. 387
This being an impossibility, the nearest approach to it is that of a beach
or estuary with vertical lateral boundaries parallel to the direction- of flow
of the tide. And the broader such estuary is in proportion to its length
the less would be the effect of the lateral boundaries. The effect of the
tide running straight up and down such an estuary might tend to shift
the sand up or down according to the slope at each point, and the period
and height of the tide, or until some definite relation between these three
quantities was attained. If such a relation exists, its elucidation would seem
to be fundamental to a full understanding of the regime of estuaries.
Further, there was the very important question how far such a tidal
action would leave the bed beach-like, with uniform slope and straight
contours, or groove it with low-water channels as in the mouths of estuaries,
i.e. whether a parallel estuary without land water, having a uniform slope
and straight contours, would be stable under the action of a tide of which
the general motion was straight up and down.
Considering that the new rectangular tanks with their clean paraffined
sides were admirably adapted for such experiments, and that any internal
modelling would have required further time, which was already very short,
if a report was to be presented at the Newcastle meeting, it was decided to
commence with a series of experiments on the general slope and configura-
tion of the sand with parallel vertical sides, after making some preliminary
experiments while the tank A was having a preliminary run to test the
working of the motor.
Following is an abstract account of these experiments and the results
obtained. It has not been thought desirable to introduce into this report
a complete copy of the note-book. The initial conditions of each experiment
are given, together with the date, the number of tides run, and the mean
period of the tide; also notes made during the running on circumstances
which are likely to have affected the general results. The final results are
contained in the charts (or plans as they are headed), the longitudinal and
cross sections, which have been taken from the charts, and the diagram of
mean slope obtained from the areas of contours. These are all appended
to the report.
Preliinlintri/ Experiments with Balls. — Little balls of paraffin the size of
IK ;is were prepared, colouring matter having been first mixed with the
paraffin to distinguish the balls, and to so load some that they would just
sink while others floated. Then, before the motor was started, the water
being quite still, the balls were placed in rows across the tank at definite
distances down the tank, and from the centre line — one set of balls on the
bottom and another set floating above. The motor was then started, and
the change in position of the balls noted.
25—2
388 ON THE ACTION OF WAVES AND CURRENTS. [57
It was supposed that the floating balls would move with the water and
show by any change of their mean position if there was any circulation in
the water. This was what they did when the surface of the water was
perfectly clean, but the slightest scum very greatly diminished the range
of the motion of the floating balls. This matter of scum, if it can be so
called, when it is entirely unperceivable by the eye, is very important in
these model experiments ; for, however slight it is, it tends to prevent the
horizontal motion of the immediate surface, and indirectly to modify the
internal motion of the water ; the only test of perfect freedom from surface
impurity is that small drops caused by a splash falling on the surface float
along. When the surface was in such a state the floating paraffin balls
oscillated up and down with the water, and kept the position for many
oscillations both up and down and across the channel.
The sinking balls are subject to the constant resistance of the bottom,
so that their motion is not equal or proportional to the motion of the water
— for a sufficiently slow motion of the water the ball would not move ; so
that if the ebb were just not sufficient to move the ball, and the flood were
stronger, the ball would be moved up each tide, or vice versa, and the same
resultant motion would follow, even though the ball might be moved some-
what by both ebb and flood ; the strongest would carry the ball farthest.
In this way they furnished a very delicate test as to the symmetry of the
tides and the sufficiency of the balancing.
Experiments on the Movement and Equilibrium of Sand in a Tide Way.
Series 1. — Tide running in a uniform rectangular pan with vertical sides
and end, and a level bottom for the sand to rest on.
Experiment 1 (tank A), commenced June 22. — Three cubic inches of
Calais sand was placed in a heap on section 17 and line lr and 3 cubic
inches of Huna Bay shell sand similarly on section 17 and line lr, the tank
being otherwise clean and empty. Then, with low water O083 of a foot
and high water '23' from the bottom, the tide was set running with a period
of 55 sees. After 3000 tides, the white sand spread upwards from section
16'5 to 12 7 in 7 ripples, having just painted the bottom 1 grain thick down
to section 22.
The Huna Bay shell sand spread upwards from section 18 to 14'25 in
4 ripples, also having painted the floor down.
Experiment 2 (tank A), commenced June 24. — Calais sand was introduced
as a uniform bank across the channel.
Experiment 3 (tank A).— The Calais sand was arranged exactly as for
Experiment 2.
57] ON THE ACTION OF WAVES AND CURRENTS. 389
Experiment 4 (tank A). — A repetition of Experiment 3, observations being
directed more closely to the motion of the sand after starting.
Experiment 5 (tank A), July 5 (Plans III. to VI.). — Calais sand passed
through a sieve into the tank, in which there was sufficient water to cover
the sand until there was enough sand to fill the tank from the upper end to
section 18, to a depth of 0 25 foot, terminating in a natural slope from
section 18 to the floor. Then the sand, which was in excess at the upper
end, was carefully levelled by means of a wooden float guided on the sides
of the tank, and having its straight edge completely across the tank 0'25
foot from the bottom. The scummer was then adjusted to keep the low
water at 0'181 from the floor, and the crank adjusted to give a rise of 0'166
over the whole tank. The actual rise at starting, owing to the sand above
low water, was 0*2' over the whole surface.
The tank was then started, and ran 12,607 tides at a period of 53 sees.,
when the speed was increased to 50 sees, and continued for 3589 tides ;
then, as the condition seemed very steady, the survey for Plan I. was made.
At the starting of the tank careful note was taken of the progressive
appearances, and during the interval of running, which occupied from July 5
to July 15, sand gauges were introduced and read daily, as well as other
notes of progress made. The periods of rise and fall of the tide were
ckecked, and a curve taken which showed the rise and fall at the generator
to be symmetrical and nearly harmonic.
The sand was found to descend down the tank towards the generator in
a steadily diminishing manner, while at the same time it rose towards the
head of the tank at a steadily diminishing rate, until both these changes
ceased to be observable. The configuration of the surface also changed at
a steadily diminishing rate. The chief features in this configuration were
the banks, which gradually formed at the head of the tank in a very sym-
metrical form, and then extended down the tank past low-water level, losing
their symmetry as the low-water channels between them began to take
effect. These banks and channels appear clearly in the plan. The minor
features were numerous minor channels caused by the water running off the
banks. These covered the surface with very beautiful detail, which, however,
it is quite impossible to record. Also ripple bars across the channel below
low water.
After the first survey was taken the tank was started again July 17 at
a somewhat slower period, viz., 6C'7 seconds, and ran for 7815 tides, when
the second survey was made. The daily observation taken as before showed
considerable changes of detail — so much so that it was a matter of surprise
to find that Plan II. corresponded almost exactly with Plan I., the only
390 ON THE ACTION OF WAVES AND CURRENTS. [57
difference being slight divergences and deepenings of the depressions and
raising of the banks.
The tank was again started on July 25 at a period of G0'6, to keep the
sand in motion until the agitator for producing waves could be put in action.
This was accomplished after 7780 more tides, from which no considerable
change was observed. The agitator made 200 beats in the tide generator
for every tide. At first the agitating bar was straight, 3' 6" long, with a
section 6" broad and 1|-" deep. This caused parallel waves '06' high, '8 long.
The effect of the waves was at once apparent in the destruction of all the
beautiful tracery on the banks, which soon presented a smooth washed-out
appearance. After 4000 tides a V-shaped agitator, as shown in the drawing,
was substituted for the first. This sent oblique waves of much the same
size as before. The waves were kept going during the day and stopped
at night ; and after 6000 further tides the tank was stopped to survey for
Plan III. This shows that at low water the waves had to some extent
levelled the sand ; they had also washed off the ridges of the banks and
filled the narrower channels ; yet on the whole the depressions are deeper,
and at the head of the estuary there is a marked change in the arrangement
of the left side.
The model was (August 8) set to run at 33" with the wave agitator
going during the day. On the 19th it was found that so much additional
sand had come down into the generator as to disturb the balance so as to
require 100 Ibs. additional weight to equalise the period, this was added,
and again on Monday the 1 2th it was found that more sand had come
down, requiring 50 Ibs. more weight to re-establish the balance. On the
13th the sand was removed from the generator and the balance restored
by removing the 150 Ibs. It had also been found that from some cause
the speed fell otf considerably ; at one time the speed was 70". The cause
of this was not at first perceived, but somehow the speed was restored,
though it was subsequently found that the belt on the motor was slipping ;
this having been put right, the running at 33" was continued till 12,705
tides had been run since this speed was commenced. Survey IV. was then
taken. It was found that the mean period over the interval had been 43'2"
instead of 33". It being uncertain how far the irregular speed and disturbed
balance had affected the results, the model was started again August 16 to
run at 33" with a mean speed of 33", and after it had run 17,289 additional
tides a partial survey was again taken and plotted in dotted lines on the
plan showing Survey IV.
The dotted contours show a slight change, chiefly in the retreat of the
low-water contour up the estuary, and a change in the distribution of the
sand at the head of the estuary. These changes, however, are very slight
57] ON THE ACTION OF WAVES AND CURRENTS. 391
compared with the great difference presented between Plan IV. and all
the previous plans. In Plan IV. the contour '004 lies between" sections
6 and 11, while in Plan III. it lies between 12 and 13, which shows an
increase from 1 to T47 in the general slope. The low-tide channel on the
left has also increased in magnitude and length, extending nearly across the
head of the estuary.
Experiment 6 (tank A), August 28. — In this experiment the initial
conditions aimed at were precisely the same as in Experiment 5.
The sand which had been removed was returned, and all the sand well
washed in the tank and then placed as before, the float having been examined
and straightened on the surface plate.
After the sand was laid the water in the tank was brought as near as
possible to the level of the sand, and was allowed to stand for twelve
hours, when it was found that the sand looked drier on the right side
than on the left. The generator was then raised by turning the pinion
which, in the position the crank was, would raise the generator about
'008 of an inch for one revolution ; this, considering the surface of water
exposed, would raise the water '005 inch, in which way it was found that
the sand was something like '01 of an inch higher on the right all along
the tank.
The model was then started at the same speed as in Experiment 5, and
the development carefully watched. In all respects it appeared to be the
same as in the previous experiment, and the daily observations showed
the same rate of progress; not only did the sand gauges and the descent
of the sand agree, but the surface of the sand presented the same general
appearance. The experiment was stopped after 8686 tides, before it had
reached the stage of the first survey, Experiment 5, so no survey was taken.
Experiment 1 (tank B), August 5. — In starting this experiment it was
intended that the circumstances should be in every respect homologous to
those of Experiment 5 (tank A).
The sand was introduced in the same way, and brought to the same
figure. The tank was started with a period of 36'5 seconds, that of A
having been 53 seconds, which numbers are as the square roots of the
dimensions of the tanks. The progressive appearances accorded identically
with those noted in tank A, except in one apparently minor particular.
And for the first 1200 tides the downward progress of the sand was nearly
the same (a trifle less)
About this stage an appearance presented itself which had not been
noticed in the previous experiment. The arrangement of the sand appeared
392 ON THE ACTION OF WAVES AND CURRENTS. [57
to show a greater rate of downward progression, at the middle of the tank,
towards the generator than at the sides, and this was followed by a somewhat
more rapid descent of the lower edge of the sand, which after 5000 tides
began to accumulate in the generator, from which about seven pounds was
removed.
From this stage the lower end of the tank B differed considerably from
that of tank A in the same stage. At the upper end the appearances were
almost identical, and the reading of the sand gauges agreed well.
As the experiment progressed, the sand, instead of having a nearly
uniform downward slope from the head to the generator, had a uniform
slope down the middle of the tank, with two large banks extending from
section 8 to section 17 on each side, that on the right being longer.
The experiment was continued for 11,013 tides, when it was found that
the water was much too low, owing to misadjustment of the scummer; then
as there was no possibility of saying how long this had been going on, the
experiment was stopped.
Experiment 2 (tank B, Fig. 7, p. 407), August 28. Plan 1. — In this the
conditions of Experiment 1 were repeated, the edge of the float having been
replaned. The results from starting were almost identical with those
observed in Experiment 1. The sand again came down fastest in the
middle, and faster than in tank A. Seven pounds were removed from
the generator, and subsequently the condition of the model as regards the
lateral banks was nearly the same, except that the longer bank was on the
left. The experiment was continued with speeds exactly corresponding to
those of Experiment 5, tank A, until 16,344 tides had been run; then
Plan I. was taken. The tank was then set running again at 35'5 seconds
and continued for 6757 tides, when considerable changes had taken place
towards the lower end of the tank. A partial survey was then made and
recorded, and the experiment stopped.
Experiment 3 (tank A, Fig. 8, p. 408, and tank B, Fig. 9, p. 409), Sept. 2.—
The sand in both tanks was arranged as before, a new float straightened to
a surface plate being used for B, and the level of the sand in both tanks
tested by water, as in Experiment 6 A, which tests showed that the sand in
A was perhaps -01" highest on the left, while in B it was to something like
the same extent highest on the right.
The tanks were coupled, A being driven from the motor and B from A.
Both were set to low tide at starting, and the start made at full speed,
33 seconds tank A. The progressive appearances simultaneously observed
were identical, with the same exception as before noted. Immediately after
starting, the periods of rise and fall of the generator of A were observed,
and the fall being slightly the larger, 25 Ibs. was removed from the balance
57] ON THE ACTION OF WAVES AND CURRENTS. 393
weight, which restored the equality. After 77 tides it was observed that
the sand in A was coming down much faster than in B, and had already
begun to come into the generator; the periods of rise and fall were noted,
and it was found that the rise was 17 seconds and the fall 15 seconds. The
weight was replaced, the tanks stopped, and 56 Ibs. of sand removed from the
generator and lower end of the trough of A which left the end of the sand
the same in both tanks. The tanks were then started, and the rise and fall
in A were equal.
It may be well to remark that though the tank B is driven from A,
the periods do not synchronise, so that the unequal motion caused by
imperfect balance of A eventually affects all stages of the tide in B
equally, while the resistance of B is so small compared with that of A,
that any want of balance hardly affects the motor when driving both tanks.
In starting there would be the same disturbance of balance in both tanks
owing to the slow descent of the water, from the flat sand, but it would be
only that of A that would affect the balance.
After running 1653 tides, tank A, it was seen that the sand had come
into the generators of both tanks, so a stop was made, and all sand below
section 20 again removed from both tanks — 120 Ibs. from A and 12 Ibs.
from B, making altogether 176 Ibs. from A against 12 Ibs. from B.
Considering that 1 Ib. in B is equivalent to 8 Ibs. in A, and that
altogether in A there would be 1100 Ibs., B was left with about 7 per cent,
more sand, in proportion, than A.
The experiment was then continued, the sand coming down in both tanks,
but not so as to get into the generators. The motion of the sand in the two
tanks followed almost exactly the same course, B gradually taking the lead.
In this case there was not the least sign of the middle channel in B, the sand
keeping level across and following the same course as had previously been
observed in Experiments 5 and 6 A.
When B had run 16,570 tides it was stopped for surveying, while A was
allowed to run on to make up the number of tides.
Surveys were then made.
DISCUSSION OF THE RESULTS. OBTAINED.
Since the experiments have been arranged in accordance with the law of
kinetic similarity, followed in rny previous experiments, it may be well to
restate this law before proceeding to discuss the results.
If h be the depth of water in a uniform trough, it is well known that
the velocity of a wave, of which the length L compared with h is great,
394 ON THE ACTION OF WAVES AND CURRENTS. [57
and of which the height is proportional to h, varies as the square root
of h.
For geometrical similarity at any instant the lengths of the troughs
must be proportional to L.
The period of rise and fall, p, will thus be inversely proportional to
Hence for the law of kinetic similarity,
P^ ....................................... (1),
has a constant value for all scales.
This law takes no account of the resistance of the bed, for a first approxi-
mation to which the law would be
(2),
constant, where A and B are constants to be determined by experiments.
Since the comparative periods of the two tanks have been made propor-
tional to the square roots of their dimensions, e.g. the period of tank A,
\/2 times the period of tank B, the bottom resistances produce dynamically
similar results.
In comparing the results obtained with the same values of k in the
same tank with different periods, the bottom resistances would be different,
and this difference should appear in the results unless too small to be
appreciable, in which case the results would compare with the simple
period.
There are four other sources of possible divergence from the simple
dynamic law, which will become larger as the periods become slower and the
tide lower : —
1. The drainage of the sand after the tide has left it supplies the low-
water channels with a constant stream at low water; the velocity of this
stream will depend on the slope and quantity supplied, and supposing the
quantity to be proportional to hL", the depth of the water in the low-
water channels (not the depth of the channels) will be proportional to the
cube root of the slope ;
2. The size of the grains of sand, which require a certain velocity
before they move ;
57] ON THE ACTION OF WAVES AND CURRENTS. 395
3. The fouling of the sand by growth, &c., which increases as the
shifting of the sand diminishes ; and
4. The viscosity of the water, which causes a definite change in the
internal motion of the water when the velocity falls below a point which is
inversely proportional to the dimensions of the channel.
The effect of 1 would be confined to the channels ; 2 and 3 would
tend to diminish the rate of action ; the 4th might seriously alter the
rate of action at different parts of the estuary, and would also affect the
appearance of the sand surface.
The ground so far covered by the experiments has been confined to
one initial arrangement and to one height of tide in each tank, these being
similar. Two periods have been tried in each tank, the relation between the
periods in the different tanks being as the square roots of the dimensions.
Six experiments have been started:
2 in tank A with a period of 53 sees.
1 » » ?. 33 ,,
2 in tank B „ „ 36'5 „
1 „ M „ 23-3 „
Of the two experiments started at 53 seconds in tank A the first was
continued for 12,097 tides, and then for 3589 tides at a period of 50 sees.,
und a survey made (Fig. 3, p. 403). It was then continued 7815 tides at 65'1
sees., and the plan marked, Fig. 4, p. 404; it was then continued 17,750 tides
with intermittent waves at a period of 6(V6 sees., and a survey made (Fig. 5,
p. 405).
It was then continued for 12,705 tides at periods varying from 33 sees.,
having a mean 43, and Plan 4 (Fig. 6, p. 406) made, then continued at a
period of 33'3 sees, with intermittent waves, when it was re-surveyed
(dotted on Plan 4).
The second experiment at 53 sees., tank A, was continued to 8700 tides
with the same results as the first.
Of the two experiments in tank B the first was continued to 11,013 tides
as in A, then stopped. The second was continued to 12058 tides at 36'8secs. ;
then at 4280 tides at 36 sees., and surveyed (Plan 1 B, Fig. 7, p. 407) ; then
continued to 6769 more tides at 36, and again surveyed.
The experiments started at 33 sees., tank A, and 23'3 sees., tank B, were
continued to 16',603 tides and then surveyed (Figs. 8 and 9, pp. 408, 409).
In all these six experiments the manner in which the water commenced
and proceeded to redistribute the sand was essentially the same, the general
appearances of the surface being, with the exception of one or two particulars,
ON THE ACTION OF WAVES AND CURRENTS. [57
the same at the same number of tides up to 1200. After this the two low-
speed experiments in B began to present more noticeable differences from the
other experiments, which continued to present similar appearances at corre-
sponding tides to the end.
It thus appeared : —
1. That the rate of action was proportional to the number of tides ;
2. That the first result of the tide-way was to arrange the sand in a
continuous slope, gradually diminishing from high water to a depth about
equal to the tide below low water ;
3. That the second action was to groove this beach into banks and low-
water channels, which attained certain general proportions (plans 5 and 7 A
and 2 B, and cross sections, Figs. 5, 8 and 7);
4. That the slope arrived at after 16,000 tides was the same at the high
speed in both models working at corresponding periods, \/2 to 1 (sections, Figs.
8 and 9) ;
5. That in both models the steepness of the actual slope increased as the
tidal period diminished (sections, Figs. 5, 8, 7 and 9).
Owing to the grooving of the surface, the exact slopes at the various
speeds cannot be exactly compared. One way of effecting a comparison has
been to take the highest points on each cross section down the slope, and
plot them as a longitudinal section, and in the same way to take the lowest
points and plot them as another. These are shown in the two longitudinal
sections which accompany each plan.
The increase of the slope with the diminution of the tidal period, both as
regards the banks and channels, is thus rendered apparent ; but these sections
do not admit of an accurate comparison.
Some definite and accurate method of comparing these slopes was essential
before any definite conclusions could be arrived at respecting the laws of
similarity. To meet this the areas above the successive contours have been
taken out. These areas respectively divided by the breadth of the plan give
the mean distance of the respective contours from the head of the estuary,
and the heights of these contours plotted to this mean distance give a
definite mean slope of the sand. There are certain minor objections to this
method, but it is eminently definite and practical, and admits of great
accuracy, the areas being readily taken out with a planimeter with very great
accuracy even for the most complicated contours. The slopes thus taken
out are more readily compared if plotted to scales such that the vertical
distances between high and low water are all equal, the horizontal scales
being determined so that the vertical exaggeration is the same in all cases.
57]
ON THE ACTION OF WAVES AND CURRENTS.
397
The slopes thus taken out from 5 A (Figs. 3 and 6), 7 A (Fig. 8), and
from 3 B (Fig. 9) are shown in (1) Fig. 2. They present a great degree of
regularity ; and it is seen at once that the result of corresponding periods
(33 sees, tank A, and 23 sees, tank B, Figs. 8 and 9) agree very closely.
In order to compare the slopes with the conditions of kinetic similarity,
all that is necessary is to reduce the horizontal distances in the inverse ratio
of the periods, when the slopes should become identical. In doing this the
horizontal distances have all been reduced to represent (according to the
kinetic law) a 30-feet tide with the natural period 44,400 seconds, namely, the
ratio of the lengths of the estuaries made equal to the ratio of the periods
multiplied by the square root of the ratio of the heights. The actual rise and
fall of the tide in the models being taken : —
The horizontal and vertical scales for the five experiments as thus reduced
to a 30-feet tide are given in Table I.
TABLE I.
Reference
Period in
Seconds
Horizontal
Scale
Inches to
Mile
Vertical
Scale
Rise of
Tide
V. A, Plan 1 ...(3)
50
f -0000862 )
U in 1 1,600 ;
5-45
f -00587 \
1 1 in 170 /
•176
V. A, Plan 4 ...(6)
33-3
f -000055 }
tl in 18,200)
3-49
f -00533 |
1 1 in 187 )
•161
VII. A, Plan 1 (8)
33-6
( -000056 |
(1 in 17,900)"
3-55
f -0055 )
( 1 in 182 /
•165
II. B, Plan 1 ...(7)
35-4
j -0000431 |
|1 in 23,200)"
2-72
f -00293 \
1 1 in 341 j
•088
III. B, Plan l...(9)
237
( -0000299 )
|l in 33,400)"
1-895
( -00313 1
1 1 in 317 I
•094
Table II. shows the measured height from low water for each of the
contours, together with its mean distance from the contour at the height
which, reduced, is 30 feet above low water. Also the corresponding heights
of the contours of the 30-feet natural tide, and the corresponding mean
distances of the contours measured in miles, from which the curve of reduced
mean slopes shown in (2) Fig. 2, have been plotted.
Considering the character of the investigation, the agreement between
the slopes is quite as close as could be expected, and there is nothing to argue
from the divergences, except that the effect of the bottom resistances has here
been too small to affect the results.
TABLE II.
TANK A
GO
d
Experiment V
Experiment VII
o
fi to
Plan 1
Plan 4
Plan 1
^
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ft*"*
w *
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a o
ft"-§
a M
Feet
Feet
Unit
6 inches
Miles
Feet
Unit
6 inches
Miles
Feet
Unit
6 inches
Miles
*
—
-•69
-•76
—
-•975
-1-65
—
-•93
-1-67
•176
30-
•00
0
30-
0
0
28-05
•25
•42
•146
24-9
•91
1-
24-39
•79
1-355
22-58
•99
1-67
•116
19-8
2-13
2-34
18-68
1-86
3-2
17-11
1-76
2-98
•086
14-62
4-29
4-7
13-0
2-96
5-07
11-64
3-65
6-18
•056
9-52
6-47
7-1
7-46
4-64
7-95
6-17
5-3
8-95
•026
4-43
9-26
10-15
1-87
6-63
11-38
0-70
7-36
12-44
-•004
•68
11-51
12-52
-- 3-74
8-43
14-5
- 4-77
9-07
15-37
-•034
- 5-8
14-58
16-00
- 9-35
10-30
17-8
-10-24
11-00
18-60
-•064
- 10-9 19-41
21-3
-15-
12-17
21-6
-15-71
13-20
22-32
-•094
-16- 21-31
23-4
- 20-8
13-60
23-4
—
—
—
-•124
— —
—
-26-2
15-88
27-3
—
—
—
TANK B
Experiment II
Experiment III
:
Plan 1
Plan 1
Measured
Heights of
Contours
Height
Mean
Horizontal
Horizontal
Height
Mean
Horizontal
Horizontal
shown on (reduced to
Distance
Distances
(reduced to
Distance
Distances
the Plan
a 30-feet
of Contours
reduced
a 30-feet
of Contours
reduced
Tide) of
from the
to a
Tide) of
from the
to a
Contours
Contour at
30-feet
Contours
Contour at
30-feet
from L.W.
30 feet above
Tide
from L.W.
30 feet above
Tide
L.W.
L.W.
Feet
Feet
Unit 3 inches
Miles
Feet
Unit 3 inches
Miles
*
—
-7-
—
—
-1-82
-2-88
•088
30-
0
—
30'
0
0
•073
24-9
•5
—
25-2
•73
1-16
•058
19-8
2'
—
20-5
1-50
2-37
•043
14-62
3-
—
15-7
2-20
3-49
•028
9-52
5'8
—
10-9
3-53
5-6
•013
4-43
9-
—
6-3
5-25
8-32
-•002
- -68
11-8
— _
- 1-36
7-06
11-2
-•017
-5-8
14-3
—
- 3-4
9-24
14-61
- 8-2
10-14
16-1
—
—
—
—
-13-
12-43
19-8
* The distances in this row are the mean of the Contour at 30 feet from the ends of the tanks.
f It having been found that in measuring the heights which are shown on this plan the datum
had been taken -0106 feet below L.W., the mean distances in the table have been obtained by
interpolation between the mean distances as obtained from the areas of the contours on the plan.
57] ON THE ACTION OF WAVES AND CURRENTS. 399
The Length of the Foreshore. — The interval between mean high and low
water, about 12'5 miles according to the kinetic scale for a 30j-feet tide,
cannot readily be compared with any actual case, since there are no sandy
foreshores subject to a 30-feet tide-way except those which are in a sea-way
and subject to longitudinal currents, while in the deep bays and mouths of
estuaries, slopes are cut up with low-water channels besides a want of
regularity in the lateral boundaries. In such bays as Morecambe Bay, Lynn
deeps, Sol way Firth, the mean distance from the shore to the foot of the
sands at low water must be 8 or 10 miles, and even taking this as the actual
length it leaves no great margin for the resistance of the bottom, which would
be 50 or 100 times greater in the actual case than with a model with a
distortion of 50 or 100 times.
The only divergences of importance occur at the top and bottom of the
slopes. That at the bottom of the curve for Experiment 5 A, Plan 1, is
probably owing to the proximity of the generator, as in this plan the survey
was continued to the end of the pan.
Such results, with regard to low-water channels, as have been obtained
from the experiments already made, are not discussed in this report, because
they have been incidental to the immediate purpose of the experiments ;
they have, however, been carefully recorded for future reference. The same
might be said of the manner of action of the water on the sand, were it not
that these experiments have revealed a part taken by one of these actions, the
importance of which does not appear to have been hitherto observed. This
is the action known as rippling of the sand. In these experiments this action
is seen to play an essential part in determining the rate at which the distri-
bution of the sand is effected, while the result of this action — the ripple
marks — forms a most conspicuous feature in the final distribution, as seen on
the plans, as well as at all preceding stages.
The ripple marks on the strands are too well known to need description,
and there is nothing surprising that similar ripple marks should appear in
the beds of the models. But although presenting a very similar appearance,
and being about of the same size, the ripple marks seen in all the plans are
essentially different in their origin and in the position they take in the
regime of the sand in the models from that held by the observed ripple marks
on the shore sands. This last is caused by the alternating currents produced
by the small swell running inshore, while that in the model is produced by
the alternating action of the tide. There may seem nothing remarkable in
this, considering that these currents in magnitude and velocity are not
dissimilar — but if the models are similar to the results obtained in estuaries,
the converse should hold, and the estuaries should be similar to the models.
In which case we are face to face with a very striking conclusion, that in the
400 ON THE ACTION OF WAVES AND CURRENTS. [57
estuaries there should be — call it ripple mark or wave mark, produced by the
action of the tide, similar to that on the models and on a scale proportional
to the height of tide in the estuary. Thus some of the ripples in the models
are from hollow to crest as much as one-fourth the mean rise of the tide, the
distance between them being 12 times their height. This, in an estuary,
would mean 7 or 8 feet high and 80 to 100 feet in distance.
These ripples in the model are almost confined to the surface of the sand
which is below low-water mark, though in places their somewhat eroded ends
protrude up the slope from the low-water channels. The existence of these
ripples very much enhances the effect of the water to shift the sand — this
was noted in the experiments 2 and 3 on the bars, tank A ; on the smooth
walls of the sand the current, which would be about 6 inches a second, did
not drift the sand at all, except close to the ridge, and then there was no
apparent effect till after 1700 tides, when ripples were just beginning, yet
when the ripple once formed in another 1200 tides the top of the bar had
spread to 12 inches.
The ripples also serve to show in which way any shift of the sand is
taking place, as they have a steep side looking in the direction of motion,
and when the slopes are equal it is an indication of equilibrium.
Conclusions. — So far as these experiments have gone they have shown
that similar results as to the general slope and rate of action of the sand
can be obtained by models working according to the kinetic law as low as
tides of
1 inch with a vertical exaggeration of 100,
or 2 inches „ „ 64.
They have not shown, however, that the limit has been reached. Although
the results obtained with a tide of 1 inch with a vertical exaggeration of 64
in tank B presented peculiarities which appeared in two experiments, it is
still open to question whether these might not have been owing to something
in the initial circumstances. This first series is therefore yet incomplete ; it
should include experiments to show the smallest vertical exaggeration at
which similar results can be obtained with tides as small as half an inch
and as large as 2 inches. This would give the law of the limits ; this
would conclude the first series. Then, if the experiments are continued,
another series might be undertaken to determine whether similar effects
can be obtained from land water acting on such slopes as have been already
obtained ; and again, as to the law of slopes and cross sections on V-shaped
estuaries, and then, though this has been already established in my previous
experiment, as to the effects of irregular lateral configuration in the shores*.
* For continuation, see p. 410.
O. R. II.
26
402
403'
.
svvy snayva? ^y? wo s*fnffy vyj
26—2
404
g
s
I
I
405
406
vo mnffy 3yj,
408
I
-jmmo Jtnf) Mays- fnrfj jnnuoy •yip -uo
409
I
• jaoj wja s^vurKop 7/v ?jM]j?rnu Mffj Mopy M yAoqo r/
fxnar}fjp jvnrn Jffi// SJIM> Mays ffUTf jnvfuqy yip ito svjittry
58.
SECOND REPORT OF THE COMMITTEE APPOINTED TO IN-
VESTIGATE THE ACTION OF WAVES AND CURRENTS
ON THE BEDS AND FORESHORES OF ESTUARIES BY
MEANS OF WORKING MODELS.
[From the " British Association Report," 1890.]
THE Committee held a meeting in the City and Guilds of London
Institute and considered the results obtained since the last report and
the proposals of Professor Reynolds for the continuation of the investigation,
which were approved.
At a second meeting, held at the Owens College, Manchester, it was
arranged that Professor Reynolds should draw up a report on the results
obtained.
At a third meeting, held in the committee room, Section G, at Leeds,
the report submitted by Professor Reynolds was considered and adopted.
On Model Estuaries. By Professor Osborne Reynolds, F.R.S., M. Inst. C.E.
§ I. — INTRODUCTION.
1. In accordance with the suggestion in the report read at the Newcastle-
upon-Tyne meeting of the British Association, 1889, the investigation has
been continued with a view (1) to complete the first series of experiments
by determining the smallest vertical exaggeration at which similar results
can be obtained with tides ranging upwards from half an inch in rectangular
estuaries, and so determine the law of the limits ; (2) to determine how far
similar effects can be obtained with land water acting on such slopes as had
58] ON THE ACTION OF WAVES AND CURRENTS. 411
been already obtained in rectangular estuaries ; and (3) to investigate the
character and similarity of the results which may be obtained with Y-shaped
estuaries.
2. The two models, subject to such modifications as were required for
the various experiments, have been continuously occupied in this investigation,
running, driven by the water motor, at all times when they were not stopped
for surveying or arranging a fresh experiment. They have thus run about
five-sixths of the time day and night. In this way the large model has
worked through in the twelve months 500,000 tides, corresponding to 700
years. These tides have been distributed over ten experiments in numbers
from 32,000 to 100,000. The smaller model has run more tides than the
larger, and these have been distributed over fourteen experiments.
3. The experiments have all been conducted on the same system as is
described in last year's report (see p. 380).
Initially, with two exceptions, the sand has been laid with its surface as
nearly as possible horizontal at the level of half-tide, extending from the
head of the estuary to Section 18, and in the later experiments to Section 17.
The vertical sand gauges, distributed along the middle line of the estuary, .
have been read and recorded once a day. Contour surveys have been made
after the first 16,000 tides, and again after the first 32,000, and in the longer
experiment further surveys have been made ; in all, fifty complete surveys
have been made, and forty-four plans, showing contours at vertical intervals
corresponding to 6 feet on a 30-foot tide, are given in his report.
The general conditions of each experiment, together with the general
results obtained, are given in Table I., p. 436, and a description of each
experiment is given in § IV. p. 423.
The importance of a better means of recording the tide curves was
mentioned in last year's report. Such means have been (see p. 422) obtained
during this year, and automatic tide curves have been taken as nearly as
practical at corresponding numbers of tides during the experiments, these
curves being taken at several definite sections in each tank. Two series
of these curves have been taken in the later experiments, one in which
the paper is moved by a clock, the pencil being moved by a float; the
other in which the paper is moved by the tide generator, by which means
exactly similar motion for the paper is secured at all points of the estuary,
so that differences in the phases of the tide at different parts of the estuary
are brought out. These curves are shown on the plates.
Mr H. Bamford has continued to conduct the experiments, but on account
of the very great amount of detailed work the entire time of a second
412 ON THE ACTION OF WAVES AND CURRENTS. [58
assistant has been occupied. For this the services of Mr J. Heathcott, B.Sc.,
were obtained from October to February, when Mr Heathcott obtained an
appointment in the office of the engineer to the L. & N.W.R. in Manchester.
Mr Greenshields then applied for and obtained the post, and has continued
the work with great patience and zeal.
§ II. — GENERAL RESULTS AND CONCLUSIONS.
4. The Limits to Similarity in Rectangular Estuaries. — In the experi-
ments of last year it was found (1) that as regards
1. Rate of action as measured by the number of tides run ;
2. Manner of action ; and
3. The final condition of equilibrium
with tides of 0176 foot and periods of 50 and 35 seconds the results were
f) V h
similar, according to the hydrokinetic law -, constant ; (2) that, as regards
rate and manner of action, the results obtained with tides of 0'094 foot and
periods 23'7 seconds were similar to those with the tide of 0176; but the
experiment had not proceeded to the final condition of equilibrium.
It was also found that with tides of '088 foot and periods 35*4 seconds,
the results obtained differed in a marked manner from the others as regards
rate and manner of action, so much so as to render the attainment of a final
state of equilibrium impracticable.
These results seemed to indicate that for each rise of tide there exists
some critical period such that for all smaller periods the results would be
similar according to the simple hydrokinetic law, while for larger periods
the results would be dissimilar in a greater or less degree to those obtained
with periods smaller than the critical period. Whether or not the results
obtained with periods greater than the critical periods would present a
general similarity amongst themselves, or even similarity under particular
relations among the conditions, were still open questions.
The experiments, as shown in Table I., Table II., made this year, empha-
tically confirm the conclusions (1) as to the existence for each rise of tide
of a critical period at which the rate and manner of action begin to change,
being similar for all smaller periods ; (2) these experiments also confirm the
general similarity of the final states of equilibrium as regards slopes for
periods smaller than the critical period, as shown in Table II.
The experiments (Experiments IV. and VIII., B) this year, also show that
with tides of 0'094 and 0'097 foot the periods 34'4 and 35'4 seconds are
58] ON THE ACTION OF WAVES AND CURRENTS. 413
greater than the critical periods, although the results show a nearer approach
to similarity, as regards manner and rate of action, than the results -obtained
last year in II. B, with the tide 088 foot and period 35 4 seconds, while the
final conditions of similarity were approximately reached.
With tides of 0'088 foot and periods 69'3 seconds the results in rate and
manner of action are emphatically different from those with less than the
critical period, and with tides of 0'042 foot and periods 5O5 seconds still
greater differences are presented.
On the other hand, it is found (V. B) with tides 0'042 foot and periods
50'5 seconds that if the sand be given a condition corresponding with the
condition of final equilibrium, as if the period were above the critical period
according to the simple hydrokinetic law, this is a state "of equilibrium ; and,
further, that it is not a state of indifference is shown, since on diminishing
the period the sand readily shifted so as to bring it nearer the theoretical
slope for the new period. This shows that the state of equilibrium follows
the simple hydrokinetic law for periods greater as well as less than the
critical period, which is thus shown to be critical only as regards rate and
manner of action in reducing the sand from the initial level state to the
final condition.
The experiments carefully considered suggest that there is some relation
between the rise of tide and critical period. They do not, however, cover
sufficient range to indicate what this relation is with any exactness. The
critical period diminishes with the rise of tide, but much faster than the
simple ratio.
5. Causes of the Change in Manner and Rate of Action. The change in
the action, which sets in at the critical period, is the result of some action,
of which no account is taken in the simple hydrokinetic law. A list of five
such sources of possible divergence from the hydrokinetic law is included in
last year's report (p. 39G), and with a view to obtain an indication of some
relation between the rise of tide and period (or vertical exaggeration, as
compared with the standard tide of 30 feet, by the kinetic law), which
relation would be a criterion of the limiting conditions under which the
simple kinetic law may be taken as approximately accurate, these five
discarded actions were carefully considered.
The fouling of the sand by the water, although it comes in as preventing
further action, cannot take any part in imposing these limits, since it is at
the immediate starting of the experiments that the action is observed to
fail. For the same reason the limits cannot be in any way due to the
drainage from the banks, as these banks have not appeared above water.
Again the limit cannot be due to the size of the grains of sand because
it would then occur at particular velocities, whereas this is not the case.
414 ON THE ACTION OF WAVES AND CURRENTS. [58
The other actions are the bottom resistances and the viscosity of the water,
which causes a definite* change in the internal motion of the water as the
velocity falls below a point which is inversely proportional to the dimensions
of the channel.
That this last source of divergence from the simple kinetic law must
make itself felt at some stage appeared to be certain. But the critical
velocity at which the motion of the water changes from the 'sinuous' or
eddying to the direct is inversely proportional to the depth, and by the
kinetic law the homologous velocities in these experiments are proportional
to the square roots of the depths only ; hence this action would seem to
place a limit, if it were a limit, to the least tide at which the kinetic law
would hold independently of the period, and this is not the case. Observa-
tion of the action of the water above and below the critical periods, however,
confirmed the view that the limit was in some way determined by this
critical condition of the water. For when water is running in an open
channel above the critical velocity, the eddies, of which it is full, create
distortions in the evenness of the surface which distort the reflections,
creating what is called swirl in the appearance of the surface. Now it
was noticed and confirmed by careful observation, that in the cases where
similarity failed, the swirl was absent at the commencement of the experi-
ment, while it was easily apparent, particularly on the ebb, in the other
experiments. Subsequently it appeared that the velocity of the water,
particularly during the latter part of the ebb, which has great effect in the
early stages, might be much affected by the bottom resistances, and hence
not follow exactly the kinetic law.
6. Theoretical Criterion of Similar Action. — The velocities of the water
running uniformly in an open channel, i being the slope of the surface and
m the hydraulic mean depth, is given by
v = A \/im,
where A is constant.
If, then, i is proportional to e (the exaggeration of scale) and m propor-
tional to h, since at the critical velocity v is inversely proportional to h, at
this velocity h*e has a constant value.
The function h3e=C is thus a criterion of the conditions under which
similarity in the rate and manner of action of the water on the sand ceases.
7. The Critical Values of the Criterion for Rectangular Tanks. — Taking
* Reynolds on the Two Manners of Motion of Water, Phil. Trans., 1883, pt. iii. (see page 51).
58] ON THE ACTION OF WAVES AND CURRENTS. 415
h to represent the rise of tide in feet, and e to be the vertical exaggeration
as compared with a 30-foot natural tide by the simple hydrokinetic law,
the values of this criterion have been calculated for each of the experiments
and are given in Table I.
Experiments I. and II., B, First Report, C'=0'046, showed marked slug-
gishness and local action; IV., B, (7 = 0'058 and VIII., B, (7 = 0-064 showed
less, but still a certain amount of sluggishness and local action*, while in
III., B, (7=0-083, the rate of action was good and the action similar to the
experiments with values for C higher than 0'087*, whence it would seem
that the critical value of the criterion is about 0*087, and it may provisionally
be assumed that (7=0-09 indicated the limits of the conditions of similar
action *.
8. The value of the Criterion for V-shaped Estuaries. — This critical
value of C deduced from the experiments in rectangular tanks appears to
correspond very well with the results of the experiments in the V-shaped
estuaries. In the experiments Table I. with V-shaped estuaries in the
small tank, the value of (7 is in no case far from the critical value '09 on
either side. In Experiment IX., B, however, the value of C at starting
was only 0'046 as in I., B, and in consequence of the observed sluggishness
and local character of the action in the lower estuary, the rise of tide was
increased from 0'088 to O'll, which remedied the action and raised the
criterion to O'lOl, and in Experiments X. and XII., B, and in I., D, the
values are between 0'095 and 0'084. In Experiments II., D, F, and F',
owing to the falling off in the tide in consequence of the addition of the
river, the criterion is as low as 0 073. In these experiments signs of slug-
gishness and local action in the lower estuary were observed at starting,
and the difference in the action of the upper estuary as compared with
Tank E in respect of closing up the tidal river may have been due to the
low value of the criterion.
In the experiments in the large tanks the values of (7 are all well above
the critical value: the nearest are the experiments in Tank E, (7=0'17,
which is only double the critical value, and the action was as quick and
general as in the case where C =• 0'5.
It may be noticed that the range through which the value of C as a
criterion has been tested is small. Had the form of criterion been appre-
hended sooner this might have been somewhat extended, though considerable
adaptation of the apparatus would be required to carry it far.
* In both these experiments, IV. and VIII., B, the mean level of the tide was above the initial
level of the sand, which would naturally increase the value of the criterion.
416 ON THE ACTION OF WAVES AND CURRENTS. [58
9. If C =0-08
With a tide 0-1 ft. the greatest period is 32 sees. and least exaggeration 80.
0-12 ft. „ „ 60 sees. „ „ 47.
0-14 ft. „ „ 102 sees. „ „ 30.
„ 0-2 ft. ., „ 6 mins. 9 sees. „ „ 10.
„ 0-43 ft. „ „ 1 h. 33 m. 48 s. „ „ 1.
From which the size of tanks and length of periods necessary to verify this
law for exaggerations of less than thirty can be seen.
10. The General Distribution of Sand in V-shaped Estuaries. — The
experiments all show that with sufficiently high values of the criterion,
as in the rectangular tanks so in those of symmetrical V-shape, the sand
arrives at a definite general state of equilibrium after a definite number
of tides. This state in the rectangular tanks was a general slope which
corresponded to a definite curve, twelve miles long as reduced by the kinetic
law to a 30-foot tide, between the contours at high and low water in the
generator. This slope was furrowed by 3 or 4 shallow channels at distances
of some two miles, commencing very gradually at the top and dying out at
some distance below low water. In the V-shaped estuaries the state of
equilibrium differs from that in the rectangular tanks in a very systematic
manner; it consists in a main low- water channel commencing at the end
and extending all the way down the V out into the parallel portion of the
tank. If this channel is in the middle it is the only channel, but if, as is
as often as not the case, it takes one side of the estuary, then at the lower
end there is on the other side a second channel starting at some distance
down the estuary. The height of the banks above the bottom of the
main low-water channel towards the lower end of the V is much greater
than in the rectangular estuaries. No general method of comparing the
general slope or distribution of the sand in the V-shaped estuaries has
been suggested other than that of comparing the contoured plans and the
longitudinal section taken down the highest banks and lowest channels,
together with the cross sections which have been plotted on the plans.
These are very similar for the similar tanks and corresponding periods.
They show that the slope in the channels down to low water is nearly
the same as in the rectangular tanks, the level of low water being reached
at distances from the head of the estuary a little greater than in the rect-
angular tank, and a little greater in the long V than in the short. Below
low water the slope in the channels is less than in the rectangular estuaries,
which is, doubtless, a consequence of lateral spreading. The slope of the
banks is much less than in the rectangular tanks, and these extend from
two to three times as far from the top of the estuary, according to the
angle of the V.
The range of observations on V-shaped estuaries has necessarily been
5<S] OX THE ACTION OF WAVES AND CURRENTS. 417
limited, and time has not sufficed to duly consider all the results obtained,
but the following conclusions may be drawn :
(1) In similar shaped V-estuaries configurations similar according to the
simple hydrokinetic law are obtained irrespective of scale, provided the
criterion of similarity has a value greater than its critical value. (2) That
the general character is that of a main channel and high banks. (3) That
the estuaries are longer in a degree depending on the fineness of the V than
rectangular estuaries with corresponding tides, while the low- water contour
reaches to nearly the same distance from the top of the estuary.
11. In the experiments with a long (fifty miles) tidal river increasing
in width downwards slowly until it discharges into the top of the V -shaped
estuary the character of the estuary is entirely changed. The time occupied
by the tide getting up the river and returning causes this water to run
down the estuary while the tide is low, and necessitates a certain depth
of water at low water, which causes the channel to be much deeper at
the head of the estuary. In its effect on the lower estuary the experiments
with the tidal river are decisive, but as regards the action of silting up the
river further investigation is required, both to establish the similarity in
the models and to ascertain the ultimate state of equilibrium.
It may, however, be noticed that the general conditions of the experiments
in Tank E do not differ greatly from the conditions of some actual estuary,
as, for instance, the Seine. This estuary is some thirty miles long before it
contracts to a tidal river which extends fifty miles further up. In the
model the tidal river reduced to a 30-foot tide is forty-nine miles long and
the V extends down twenty-eight miles further, while the results in the
model show about the same depth of water in the channel down the estuary
as existed in the Seine before the training walls were put in.
12. The Effects of Land Water. — These come out clearly in the experi-
ments, which show that the stream of land water running down the sand,
although always carrying sand down, does not tend to deepen its channel,
since at every point it brings as much sand as it carries away. If it comes
into the estuary pure, it carries sand from the point of its introduction and
deposits it when it gets to deep water, somewhat deepening the estuary at
the top and raising it below, which effect is limited by the influence the
diminished slope has to cause the flood to bring up more sand than the ebb
carries down. The principal effect of the land water is that running in narrow
channels at low water, which are continually cutting on their concave sides,
it keeps cutting down the banks, preventing the occurrence of hard high
banks and fixed channels. When the quantities of land water are small as
compared with the tidal capacity of the tank, its direct action on the regime
o. R. ii. 27
418 ON THE ACTION OF WAVES AND CURRENTS. [58
of the estuary is small. But that it may have an indirect action of great
importance in connection with a tidal river is clearly shown. In the upper
and contracted end of a tidal river the land water may well be sufficient to
keep it open to the tide, whereas otherwise it would silt up. This was
clearly the effect in the experiments E, 1 and 2, and by keeping the narrow
river open the full tidal effect of this was secured on the sand at the top
of the estuary, causing a great increase of depth. The effects of large
quantities of land water, such as occur in floods, have not yet been investi-
gated.
13. Deposit of the Land Water in the Tidal River. — One incident con-
nected with the land water in the tidal river is worth recording, although
not directly connected with the purpose of the investigation.
The land water, one quart a minute, was brought from the town's mains
in lead pipes. It is very soft, bright water, and was introduced at the top of
the estuary. This went on for about three weeks. At the commencement
the sand was all pure white, and remained so throughout the experiment
except in the tidal river. At the top of the river a dark deposit, which
washes backwards and forwards with the tide, began to show itself after
commencing the experiment, gradually increasing in quantity and extending
in distance. At the end of the experiment the sand was quite invisible
from a black deposit at the head of the river and for 5 or 6 feet down ; this,
then, gradually shaded off to a distance of 12 feet. Nor was it only a
deposit, for the water was turbid at the top of the river and gradually
purified downwards.
On the other hand, in the precisely similar experiment, without land
water the sand remained white and the water clear right up to the top of
the river. This seems to suggest that these experiments might be useful
to those interested in river pollution.
14. The International Congress on Inland Navigation. — During the
Fourth International Congress on Inland Navigation, held in Manchester
at the end of July, the members were invited to see the experiments then
in progress, the subject being one which was occupying the attention of the
Congress. Advantage of the invitation was taken by many engineers, and
especially by the French engineers. M. Mengin, engineer in chief for the
Seine, stated in a paper* read at the Congress that in consequence of the
paper (read by the author before Section G at Manchester) the engineers
interested had advised the Government to stop the improvement works on
the Seine until a model having a horizontal scale of 1 in 3000 was con-
structed, and the effect of the various improvements proposed investigated
* International Congress on Inland Navigation, 1890.
~><S] <>V THE ACTION OF WAVES AND CURRENTS. 419
in the model, the model being then nearly ready, but the experiments had
not commenced. M. Mengin paid several visits to the laboratory and carefully
examined the apparatus and experiments, for which all facilities were placed
at his disposal.
15. Recommendations fur further Experiments. — Although the immediate
objects proposed for investigation this year have been fairly accomplished,
there remain several general points on which further information is very
important, besides the further verification of the criterion of similarity and
the determination of the final conditions of equilibrium with tidal rivers,
already mentioned. It seems very desirable to determine the effect of tides
in the generators diverging from the simple harmonic tides so far used,
simple harmonic tides being the exception at the mouths of actual estuaries.
It would also be desirable before concluding these experiments that they
should include the comparative effects of tides varying from spring to neap.
§ III. — MODIFICATIONS OF THE APPARATUS.
16. General Working of the Apparatus. — The apparatus has worked
perfectly in all respects except that of the driving cord connecting the
water motor with the gearing. For this cord hemp was first used, as it
was liable to be wet. This hemp cord wore out with inconvenient rapidity.
A continuous cord made of soft indiarubber was then tried, and, after several
attempts, has been made to answer well. The only other failure was the
small pinion, which was fairly worn out, and had to be replaced.
17. Extensions. — For carrying out the experiments on the V-shaped
estuaries the original tanks had to be increased in length. To do this it
was necessary to remove temporarily part of the glass partition dividing
the engine room of the laboratory, in which the tanks are placed, from the
testing room. This being done, the tanks were then extended, as shown
(Fig. 1, page 439), the first extension being an addition of a trough 6 feet
long and 2 feet wide to Tank A, and a similar extension of half the size to
Tank B, the new tanks being thence called C and D.
18. Extensions for Tidal Rivers. — The second extension consisted of a
trough 19 feet long and a foot wide to the end of C, the new tank being
thence called E. The corresponding extension to D was not at first made
in the same way, because to do so would require the removal not only of a
panel of the glass partition, but also of a fixed bench, which was a much
more serious matter, or else the extension would have closed up an important
passage. The extension was therefore made, as shown in Figs. 46 and 47,
27—2
420 ON THE ACTION OF WAVES AND CURRENTS. [58
page 479, which admitted of the tidal river being the corresponding length
to that in E, but required a bend of 180°, which was effected by two sharp
corners. This tank was thence called F'. This was the best that could be
done during the time the students were in the laboratory. It was not certain
that the corners would produce any sensible effect, whereas if the results
obtained in F' were not similar to those in E no time would have been lost,
since the straight extension could not be made till the end of June. As
the results in F' were not similar to those in E in a way which might
be explained by the bends, as soon as possible the straight extension was
made similar to E, and the tank called F.
All these tanks were constructed in the same manner as the original
tanks, and covered with glass at the same level as A and B, under which
glass survey lines, conforming to those on A and B, were set out.
19. The Numbering of the Cross Section. — The extension of the tanks
raised the question as to how the new cross sections should be numbered :
the numbering of A and B ran from the ends of the tanks, and it seemed
best to run the numbers in C and D from the ends of these tanks, con-
tinuing this new numbering to the generators. On the other hand, as the
long, narrow extensions in E and F were more in the nature of a tidal river
than an estuary, the numbers in these were carried backwards 1, &c.,
from the ends of C and D, in which the cross sections preserved the same
numbers as before.
20. Appliances for Land Water. — The introduction of land water, besides
the extension of the pipes for its introduction, required certain arrangements
for its regular supply in definite quantities. The water was to be taken from
the town's mains. And in first laying down the pipes, it had been anticipated
that it would be sufficient to regulate the supply by cocks against the pressure
in the mains. Fresh water, regulated in this way, had been from the first
supplied in small quantities into the generators, to ensure the level being
kept properly. The experience thus gained showed that it was impossible
to obtain even approximate regularity in this way, as the nearly closed cocks
always got choked even within twenty-four hours.
To meet this it was arranged to supply the water through thin-lipped
circular orifices under a small but constant head of water, which head can
be regulated to the quantity required. The head of water in the tank from
which the orifices discharge is regulated by a ball cock, which only differs
from an ordinary ball cock in that the ball is not fastened directly on to
the arm of the cock, but is suspended from it by a rod so arranged that
the distance of the ball below the arm can be adjusted at pleasure. This
arrangement has answered well. The cylinder in which the ball cock works
58] ON THE ACTION OF WAVES AND CURRENTS. 421
is made of sheet copper, with a water gauge in the form of a vertical glass
tube, with a scale behind to show the height of water above the -orifices,
which are made in the bottoms of two lateral projections from the sides of
the cylinder. One of these orifices feeds the large, and the other the small
tank. The streams from the orifices descend freely in the air for about
4 inches, and are then caught in funnels on the tops of lead pipes leading
to the respective tanks. The cylinder is fixed against a wall about 8 feet
above the floor, and conveniently near the tanks. Any obstruction in the
pipes conveying the water to the tanks would be at once shown by the
overflow of the funnel. The orifices are made with areas in proportion to
the quantities to be supplied to their respective tanks. Then the supply
cock connecting the ball cock with the main is fully opened, and the ball is
adjusted till the quantity supplied to one of the tanks is correct. The other
is then measured ; if this is not found correct, one of the holes is slightly
enlarged until the proportions are correct.
This having once been done for an experiment, no further regulation is
required except to test the quantities and wipe the edges of the orifice.
When the tanks are stopped for surveying, the water is shut off from the
main and simply turned on again on restarting.
21. The Tide Gauges. — In the experiments made last year a tide gauge
was used. This gauge consisted of a small tin saucer with a central depression
in its bottom, in which a vertical wire rested, restraining any lateral motion
in the float, the wire being guided vertically by a frame made to stand on
the level surface of the class covers, while the wire passed down between
two of the covers opened for the purpose, the frame carrying a vertical
scale. This gauge was used, both to adjust the levels of the water and to
obtain tide curves, by observing the heights of the tide at definite times, and
then plotting the curves with the heights of the tide as ordinates and the
times as abscissae.
For the earlier experiments this year the same gauge was used for both
purposes, and it has been used all through for the purpose of adjusting the
levels of the water, automatic arrangements being used for drawing the tide
curves.
In devising these automatic arrangements several difficulties presented
themselves, besides those inherent in all chronographic apparatus. Anything
in the nature of standing apparatus was inadmissible, as it would interfere
with the working and adjusting of the tanks. The apparatus must be such
as could be put up and taken down with facility, and hence could not admit
of complicated arrangements. A pencil worked direct by a float with a
drum turning about a vertical axis by a clock, all to stand on the level
glass surface, appeared the most drsirable arrangement. In the first instance,
422 ON THE ACTION OF WAVES AND CURRENTS. [58
a clock driving a detached vertical cylinder with a cord was kindly lent by
Dr Stirling from the Physiological Laboratory of Owens College, and an
arrangement of float and stand was constructed by Mr Bamford. The loan of
this clock was temporary, and experience gained with it led to the purchase
of an ordinary Morse clock from Latimer, Clark, & Co. at comparatively
small cost. A pulley was fitted so that the clock would drive the borrowed
cylinder. This clock did its work quite as well as the more costly instrument.
Its rate of action varied considerably with the resistance of the apparatus
to be driven, so much so that the curves taken at different times from the
same experiment could not be compared by superposition. Still the action
of the clock during the individual observations was sufficiently regular to
give a fairly true tide curve, and it became obvious that it would be
impossible to obtain any independent clock-driven apparatus that would
give absolutely constant speeds such as would admit of the comparison of
the curves taken from different parts of the estuary by direct superposition.
To obtain such comparison it would be necessary to move the paper by the
gearing which moved the generator.
22. Compound Harmonic Tide Curves. — On considering how best this
might be done, it appeared that if the paper had a horizontal motion
corresponding to the rise and fall of the generator while the pencil had a
vertical motion corresponding to the rise and fall of the tide at any point
in the tank, then, if the tide were in the same phase as the generator, the
curve would be a straight line or an ellipse of infinite eccentricity, with
a slope (tan 0) equal to the rise of tide divided by the horizontal motion
imparted to the paper, while any deviation of phase would be shown by
the character of the ellipse or closed curve described by the pencil, and
that to obtain the time-tidal curve from such "curves would be easy by
projecting on to a circle, while for the purpose of comparison, and bringing
out any difference of phase or deviation from the harmonic curves, such
compound harmonic curves would be much more definite than the harmonic
curves. This plan was therefore adopted with the happiest results, for,
although it may take some study to become familiar with the curves, the
obvious differences in these curves taken at different parts of the tanks,
and at the same part at different stages of the progress towards a state
of equilibrium are clearly brought out. The method also shows the similarity
of the curves taken in the two tanks, or in different experiments at the
corresponding places and corresponding numbers of tides run, as well as in
the final states of equilibrium. The tide curves (Fig. 48, page 481) bring
out emphatically the inter-dependence of the character of the tide on the
arrangement of the sand, and the coincidence of a state of equilibrium of
the sand with a particular tide curve at each part of the estuary.
In these experiments the balance of the tanks has been adjusted so as
58] ON THE ACTION OF WAVES AND CURRENTS. 423
to make the time intervals of rise and fall of the generator equal, i.e. to
make the motion of the generator harmonic, so that these compound har-
monic curves are at all parts of the tank comparable with a simple harmonic
motion. But it is important to notice that they are not essentially so, being
merely compai'able with the motion of the generator, so that if the generator
were given a compound harmonic motion, such as that of the tide in the
mouths of most estuaries, these curves would have a different dynamic
significance. These curves would still be valuable as showing the state of
progress and final similarity of the tidal motion at the same parts of the
estuaries, but to bring out their dynamical significance it would be necessary
to substitute a simple harmonic motion with the same period as that of the
generator.
§ IV. — DESCRIPTION OF THE EXPERIMENTS ON THE MOVEMENT OF SAND
IN A TIDEWAY FROM SEPTEMBER 9, 1889, TO SEPTEMBER 1, 1890*.
23. Continuation of Experiments VII., Tank A, and III., B, (see Figs.
4, 5, 6, pages 44-1,...) September 7 to October 11. — These experiments were
in progress at the time of the Newcastle Meeting of the British Association,
and had so far advanced that tracings of the first surveys were exhibited
and included in the First Report. So far as they went, they took an
important place in the conclusions arrived at in that report, showing that
with a vertical exaggeration of 100, the results obtained in the small tank
(B) with rectangular estuaries, without land water, as to rate and general
distribution of the sand, were closely similar to those obtained in A, and
that the mean slopes, reduced to a 30-foot tide, in these experiments agreed
with those obtained in A, with vertical exaggerations of 64. It was desirable
to continue these experiments to see how far a state of equilibrium had been
arrived at. This was accomplished by the assistance of Mr Foster, who
kindly looked after the running of the tanks till the return of the author
and Mr Bamford in October, and thus enabled a month, which would other-
wise have been wasted, to be utilised, in obtaining an experience of the
effect of about 100,000 tides after apparent equilibrium had been obtained
in each tank. Daily records of the counters were taken, and, although there
were several stops, the intervals of running gave the periods very constant.
The plans show but little alteration, except that the sand, particularly
in B, had shifted upwards and accumulated somewhat at the head of the
estuary, leaving the slope the same ; a circumstance which would be ac-
counted for by a difference in the level of the water, and which is also
* In the published report of these experiments it is not thought desirable to give the daily
records of progress in the notebook.
424 ON THE ACTION OF WAVES AND CURRENTS. [58
indicated by the mean slope reduced to a 30-foot tide shown in Figures 2 and
3, page 440. The agreement of the slopes here shown as compared with
the mean slope in the case of Experiment V., A, which has been introduced
in this diagram for the sake of comparison, is quite as great as could be
expected, considering the difficulties of the experiments, and affords very
valuable evidence of the permanence of these slopes when once a condition
of equilibrium has been attained.
In respect of the ripple the two tanks presented a very different appear-
ance, which is clearly shown in the plans and sections. While the ripple in
A was comparatively small and shallow, in B it was larger and deeper than
anything previously noticed ; that this was a symptom of the condition of
B being on the verge of dissimilarity seemed probable, and to test this the
period of B was increased from 23'85 to 26'5 seconds, and it was allowed to
run on 16,000 more tides and again surveyed. Plan 3, page 443, shows the
result; the ripple has increased in breadth though rather diminished in
depth.
24. Experiments to find the Limits to Similarity. Experiment IV., B,
Fig. 7, page 444, October 22 to November 27. — In this the rise of tide was
0'094 foot, and the vertical exaggeration as compared with a 30-foot tide 71.
In Experiments I. and II., B, with a rise of tide 0'088 and a vertical exag-
geration 68, described in the First Report, it had been found that the rate
and manner of distribution of the sand did not correspond with that in the
corresponding experiment in the larger tank, indicating that with an exag-
geration 68 the tide of '088 was somewhat below the limit of similarity.
The determination of these limits being a primary object of the investigation,
it appeared desirable to repeat these experiments with a slightly higher tide.
In IV., B, the character of the action presented the same peculiarities as
previously observed, but in a smaller degree, and the final state, as shown
in the plans and in the curve of slopes (Figs. 2 and 3), is a much nearer
approach to the general law, the conclusion being that in IV., B, the con-
ditions were still below the limit, but nearer than in I. and II., B.
Experiment VIII., A, October 22 to November 14. — This was an experiment
to determine the manner of action with the same horizontal scale as the first
part of Experiment V., A, but half the rise of tide. Experiments I. and II.,
B, with a rise of tide of "088 foot and a period of 36 seconds, being a vertical
exaggeration of 68, had indicated that with this rise of tide a change in the
manner of action had already set in, but it was none the less desirable to see
what would be the character of the action and the final state of equilibrium
well below this limit.
The rise of tide in VIII., A. was 0'088 foot and the mean level 0'138 foot
from the bottom, and the period 70 seconds, the sand being placed level at a
58] ON THE ACTION OF WAVES AND CURRENTS. 425
uniform depth of 1£ inch to Section 18 as in the previous experiments. The
vertical exaggeration would thus be only 34.
The manner of action of the water on the sand was in this case essentially
different from that in any previous experiments even in I. and II., B, although
it presented characteristics which had been indicated in those experiments.
Instead of the sand being in the first instance rippled over the whole surface
a middle depression was formed, extending some way up the estuary, the
bottom and sides of which were rippled ; the rest of the sand soon became
set and yellow. After 16,000 tides a survey was made and the experiment
continued to 24,000, when another partial survey was made, showing very
small alterations, and those nearly confined to the rippled channels. It was,
in fact, clear that the apparent equilibrium was owing to the sand having
become set, and that to proceed till real equilibrium was established would
take an almost indefinite time.
As the setting of the sand, owing to the slow action of the water,
appeared to play such an obstructive part, it seemed possible that better
results could be obtained if the sand could be kept alive with waves.
Accordingly the experiment was stopped, to be repeated with waves.
Experiment IX., Tank A, Plans 1, 2, 3, Figs. 8, 9, 10 (with Intermittent
Waves}, November 16 to January 4. — The conditions were the same as in
Experiment VIII., with the addition of the waves.
This experiment presented the same characteristics as those observed
in VIII., A. The rate of action did not fall off so rapidly or completely
as in VIII., but was mainly confined to the channels; and, although the
experiment was continued to 57,000 tides, the condition of equilibrium was
far from being arrived at, owing to the setting of the sand. After the last
survey a small stream of land water (one pint per minute) was admitted at
the top of the estuary, without any perceivable effect for 1000 tides, where-
upon the experiment was stopped.
Experiment V., B, Plan 1, Fig. 11, p. 448, November 21 to December 2. —
This was the corresponding experiment in B to Experiment VIII. in A,
the rise of tide being one-half inch (-042 foot), and the period 50 seconds,
exaggeration 32. The characteristics were yet more definitely marked,
rippling being entirely absent, and the action being entirely confined to
the space between Sections 14 and 18.
Experiment VI., B, December "> to December 9. — In this experiment the
conditions wi-iv exactly the same as in Experiment V., B, except that the
sand, instead of being laid level, was laid with a slope of 1 in 124, the slope
corresponding to the theoretical condition of equilibrium as in the previous
426 ON THE ACTION OF WAVES AND CURRENTS. [58
experiment. After 6757 tides with a mean period of 601 seconds the sand
was not moved anywhere in the slightest degree.
Experiment VII., B, Plans 1 and 2, Figs. 12 and 13, December 9 to
January 3. — This was a continuation of Experiment VI., with the tidal
period diminished in the ratio 1 to V2 from 50 to 35'35.
The effect of changing the period would be to increase the vertical
exaggeration, so that the slope of 1 in 124 would not be the theoretical
mean slope of equilibrium as previously determined, which would be 1 in
87, so that any sensitiveness to the condition of equilibrium would be shown
by the shifting up of the sand.
This commenced at once and continued until the mean slope was about
1 in 100 above Section 13.
The absolute quiescence of the sand in Experiment VI., B, when laid
with the mean slope of equilibrium corresponding to the period, together
with the increase of the slope with the increase of period in Experiment
VII., B, indicates that, although, as shown in Experiment V., the limiting
conditions under which the water could redistribute the sand from the level
condition had been long passed, the conditions of equilibrium remained the
same ; or, in other words, that for a half-inch tide, with a period of 50 seconds
— i.e., an exaggeration of 32 — with the sand originally distributed according
to the theoretical slope of equilibrium, the sand will be in equilibrium, while
if the sand be laid with a smaller slope the water will shift it, tending to
institute the slope of equilibrium.
25. Rectangular Estuaries with Land Water. Experiments X., A, and
VIII. , B, Figs. 14, 15, 16, and 17, January 7 to March 10. — The conditions in
Tank A were the same as in Experiment V., Plan 1. The sand lay 0'25 foot
deep, height of mean tide 0'256, rise 0'176, tidal period 50'2 seconds. A tin
saucer was placed on the sand under Section 1 in the middle of the estuary,
and a stream of water (one quart per minute, about 1/170 of the tidal
capacity of the estuary per tide) run into the pan.
During the early distribution of the sand the land water produced no
apparent effect, but as the sand approached a condition of equilibrium, the
effect of the fresh water in keeping a channel full of water at low tide, from
the source all down the estuary, was very marked. The effect of this river
in distributing the sand at the top of the estuary was also marked. The
channel did not remain in one place ; it gradually shifted from the middle
towards one or other of the sides, cutting away high sandbanks until it
followed along the end of the tank into the corner, and then flowed back
diagonally into the middle. Then, after some 10,000 tides, a fresh channel
would open out suddenly towards the middle of the estuary, and then
58] OX THE ACTION OF WAVES AND CURRENTS. 427
proceed in the same gradual manner perhaps to the other side. This
happened more than once during the progress of the experiment, which
was carried to 85,000 tides. The different positions of the channels are
apparent in the plans 1, 2, and 3 (Figs. 14 to 19). The comparison of these
plans, and the accompanying sections with Plan 1, Experiment V., in the last
report (Fig. 3, p. 403), shows but slight general effect of the land water — so
slight, indeed, that it might pass almost unnoticed. This shows that the
land water does not alter the greatest height of the banks or the lowest
depth of the channels.
It will be noticed, however, in the plans, that the land water has lowered
the general level of the sand in the middle of the estuary at the top, and
raised it towards low water. This effect comes out in the mean reduced
slopes shown in Figs. 2 and 3, p. 440. From these it appears that the effect
of the land water, by continually ploughing up the banks at the top of the
estuary, has been to disturb the previous state of equilibrium, lowering the
sand near the top, and raising it further down the estuary.
In Experiment VIII., B, the conditions at starting were the same as
those in IV., B, and one quart of land water in 2 '8 minutes was admitted
in the same manner as in X., A, the period being 35'4 seconds. The quantity
of land water per tide was one-fourth the quantity in A, while the capacities
of the estuaries are as 1 to 8, or the percentage of land water in B was
1'8 that of the tidal capacity at starting. After running GOO tides the rise
of tide was increased from 0'094 to 0 097 foot without any alteration in the
period. The experiment was then continued to 91,184 tides (Fig. 19).
The apparent effects of the land water observed were exactly the same
in character as in A, but were decidedly greater on account of the larger
quantity. The curves agree fairly with those in A.
26. Experiments in short V-shaped Estuaries with and without Land
Water. — In the tanks A and B inner vertical partitions were introduced so
as to form the upper end of the tank A into a symmetrical V, of length
6 feet and greatest breadth 4 feet ; while that of tank B was formed in a
similar manner into a V of length 3 feet and breadth 2 feet. The lengths
of the tanks were thus unaltered, the tidal capacity being reduced to three-
quarters of what it was before.
The sand was arranged in a similar manner to that previously adopted,
except that the initial depth of the sand was 4 inches (0'33 foot in A)
instead of 3 inches, and the scummers raised so as to maintain the water
higher in a corresponding degree.
Experiment* XL, A, and X., B, Fiyn. 20 t<> 23, March 18 to April 29. —
In tank A the rise of tide was 176 and the period 47 20. The experiments
were first started without land water. The observed character of the action
423 ON THE ACTION OF WAVES AND CURRENTS. [58
was much the same as with the rectangular estuaries, being more intense
towards the top of the V, and quieter at and below the broad end.
The first attempt in Tank B showed that, owing to the diminished
capacity of the estuaries, the sand would not come down even so well as
in corresponding experiments with rectangular estuaries. This led to the
abandonment of Experiment IX., B, and starting X., with a rise of tide
O'llO, without, however, altering the level of the sand. The experiments
were continued in both tanks without land water until about 40,000 tides
had been run, and Plans 1 and 2 had been taken. These plans show the
similarity of the effects in the two tanks. They also show decidedly the
character of the distribution of the sand in the V-shaped estuary. It will
be seen that the extreme positions of the contours up the estuary are much
the same as in the rectangular estuaries, while the extreme positions down
the estuaries are very much increased. The low-water contours extend from
Section 11 to Section 19, while in Experiment V., A, Plan 1, it extends from
Section 11 to Section 13. The low- water channels are nearly the same
depth at corresponding points all down the estuary in both experiments,
while in the V estuaries the banks extend 6 to 7 miles (reduced to a 30-foot
tide) further down.
After Experiments XL, A, and X., B, had proceeded to about 40,000 tides,
corresponding quantities of land water were introduced at the tops of the
estuaries, one quart in one minute in A, about 1/140 of the tidal capacity;
in B one quart in 5'68 minutes, or about 1/140 of the tidal capacity. The
tanks were then run on for 12,000 tides, and surveys for the plans 3 made
(Figs. 24 and 25). The general effect of this land water, as shown in these
experiments, is, as before, to lower the sand at the tops of the estuaries and
slightly to raise it at the bottom. They were not, however, continued long
enough to show a state of equilibrium. As in the rectangular estuaries, the
detailed effects of the land water were much more observable than those
shown in the surveys. The land water continually ploughed up the sand at
the top of the estuary and kept the banks down, but owing to the narrowness
of the estuary the general effects of this were not so striking as in the
rectangular estuaries.
Experiments XII., A, and XII., B, with Land Water, Figs. 26 to 29,
April 29 to May 19. — These were under conditions precisely similar to
XI., A, and X., B ; XI., B, with land water, was started, but owing to an
accident it was restarted as XII., B.
Both experiments were run about 16,000 tides and then surveyed, and
then run on about 16,000 more tides and surveyed again.
The plans are all very similar, and show but little difference from the
plans 3 with land water in the previous experiments.
58] ON THE ACTION OF WAVES AND CURRENTS. 429
27. Experiments in long V-shaped Estuaries without and with Land
Water in Tanks C and D. — Tank C was formed by extending AJay adding
a rectangular trough to the top, and so as to admit of partitions forming
a V extending from Section 23 (12 A), and D was formed by extending B in
a similar manner. The lengths of the tanks were thus extended 6 feet and
3 feet greater than A and B, while the capacities were the same as the
original capacity of A and B.
The sand in C (A extended) was laid 4 inches deep from the top of the
V to Section 28'5 C (17 -5 A).
The sand in D (B extended) was laid 2£ inches deep from the top of the
V to Section 28'5 D (17'5 B).
Experiments I., C and D, Figs. 30 to 33, May 24 to June 16, without Land
Water. — In C the tide was 0162 foot, and the scummer was placed so that
the mean tide when running was O'OOS foot above the initial level of the
sand ; this was riot observed at the time, being a consequence of the land
water raising the level of low water by the necessity of getting over the
weir.
In D the tide was 0105 foot and the mean tide was '010 foot below the
initial level of the sand. Thus reduced to a 30-foot tide, the initial depth
of the sand was 5 feet higher in G than in B. The experiments were run
for about 16,030 tides and surveyed, then restarted, when the level of water
in C fell owing to a leak in the scummer.
This lowered the sand at the lower end of the estuary, and a partial
survey was made, and then the experiment continued until both tanks had
exceeded 30,000 tides. The results, as shown in the plans, are very much
alike, considering the very considerable differences in the initial quantities
of sand. Owing to the much higher level of the sand in D, the top of the
V was much more silted up in the early part of the experiment, and the
sandbanks were higher towards the bottom of the estuary. Otherwise both
tanks show the same characteristics.
The highest point of the contour low water in the generator is still at
Section 15, while the highest point of the contour at high water in the
generator is at Section 4, so that the distance between the highest points
of these sections was still about 11 miles, while the banks at low water
extended down to Section 26.
Experiments II. , Tanks G and D, with Land Water, Figs. 34 to 37, June
17 to July 8. — The conditions in these experiments were the same as in
Experiments I., Tanks C and D, except that the scummer in D was altered,
until the mean tide level was only '003 foot above the initial height of the
430 ON THE ACTION OF WAVES AND CURRENTS. [58
sand, and in Tank A 002 foot above, while the rise of tide in A was slightly
greater and that in B slightly less.
Surveys were taken at about 16,000 and 32,000 tides respectively ; they
are very similar, and the effects of the land water are, as before, to slightly
raise the lower sand and lower the upper. At low water there was still
water in the channels right up to the top of the estuary, and at high water
there was what would correspond in a 30-foot tide with 10 or 12 feet of
water at the top in the low-water channels.
28. Experiments in long V-shaped Estuaries with straight tidal Rivers
extending up from the top of the V with and without Water in Tanks E, F',
and F. — Tank E was formed by opening out the partition boards in Tank C
at the end of the V to a distance of 4 inches. That portion of the V below
Section 12 remained as in Tank C, the position of the partition boards not
being altered. At a section, 12-5, a small angle was formed, so that while
the boards above the section remained straight their ends stood apart 4 inches
instead of closing up to form a V. Tank C was extended by a trough 19 feet
long, in which partition walls were constructed continuing the partitions in
the lower portion up to a section, 38, above the zero in Tank C ; these were
straight, vertical boards, the distance between them contracting from 4 inches
at the lower end to 1 inch at the end of the river.
Tank F' was formed in a similar manner, except that the upper extension
was bent through two sharp right angles so as to return along the side of
the tank ; and subsequently tank F was formed exactly similar to Tank E
with half the dimensions.
Experiment with Land Water, I. and II., Tanks E and F, Figs. 38 to
47, July 11 to July 31. — In Tank E the sand was laid to a depth of 4 inches,
the same as in C, from the upper end of the river, Section 38 down to
Section 28. The rise of tide was 0*140 foot, and the mean level of the tide
about "016 foot above the level of the sand. The period 49 sees, and water
1 quart a minute, or 1/200 the tidal capacity per tide, was introduced at the
upper end of the river.
In Tank F' the sand was laid similar to that in Tank E, the rise of tide
01 foot, and the mean tide O'OOG foot above the level of the sand. The
period being 30'04, land water, 1/200 the capacity of the estuary, was
introduced at the top of the river.
In starting these experiments the effect of the tidal river was very
marked. After the first tide in Tank E, some depth of water remained
in the river, and a long way down the estuary, at low water, and the tide
came up with a bore increasing in height all the way to the top of the
river, and then returned with a bore to the lower end of the river. The
58] ON THE ACTION OF WAVES AND CURRENTS. 431
bore, as before, soon died out over the greater part of the estuary, as the
sand at the bottom became lower. And the bore gradually died out in
the top of the V. until, as the number of tides approached 16,000, the bore
only began to show at about Section 4, and ran up the river very much
diminished from what it was originally.
Owing to the indraught and outflow of the river, the velocity of the
water and its action on the sand was greater at the top of the V and the
mouth of the river than at any part of the estuary, while for some way up
the river, and all the way down the estuary, there was a large volume of
water running at low water. The top of the river was ninety miles (reduced
to a 30-foot tide) from the bottom of the estuary, and the tide did not
commence to fall at the top of the river until after low water at the mouth,
so that nearly all the tidal water in the river ran over the estuary during
the low water. The delay in the return of the water from the river obviously
played a most important part in the effects produced.
At the bottom of the estuary the sand came down much as usual, but
it did not rise at the head of the estuary. For the first 10,000 tides the
sand was all covered at low water and rippled with active ripples up to the
end of the river, and it seemed as if no banks were going to appear. The
sections of the sand appeared as nearly as possible horizontal. The level
having lowered from the bottom of the estuary up to Section 15, from
Section 15 to Section 3 it was somewhat raised, then from 3 upwards to
7 it was lowered, and thence up to the top of the river it was raised in a
gradual slope. At about 12,000 tides two small banks appeared at low
water, one on each side of the estuary at Section 13. Everything was
perfectly symmetrical so far, but from this time the bank on the right of
the estuary extended downwards, while that on the left extended upwards
and a depression or channel formed between them extending across the
estuary in a diagonal manner. This was the condition when at 16,000 tides
the first survey was made.
As the running continued these banks continued to rise, that on the
right downwards, that on the left upwards, until a distinct channel was
formed from the mouth of the river down to Section 20, as shown in the
second survey at 32,000 tides.
The level of the sand at the mouth of the river altered very little,
diminishing during the first 10,000 tides, and then reassuming its original
height, but the sand passed upwards through the mouth, and gradually
raised the level in the river above, until there was only about 0*02 foot in
the shallowest places at low water (corresponding to 5 feet on a 30-foot
tide) ; this level was first reached at the top of the river and then gradually
extended down to Section 19, which point it had reached at 32,000 tides,
432 ON THE ACTION OF WAVES AND CURRENTS. [58
when the second survey was taken. In this condition the bore still reached
the end of the river, raising the water 0'02 foot (5 feet on the 30-foot tide).
Above Section 19 all motion of the sand had ceased, but below this the sand
was still moving up when' the experiment stopped. The bore still formed at
the mouth, but very much diminished, and was very slowly diminishing.
The final condition of the estuary shows the contour at low water in the
generators extending up to Section 9, and the contour at high water in the
generator to Section 11.
In tank F', with the sharp turns in the river, the action of the sand at
the bottom of the tank was at first sluggish, as in Experiment IV. In the
top of the estuary and river the appearance of things for the first 10,000
tides was much the same as in Tank E, except that the ripple of the sand
did not extend more than half-way up the river, and deep holes were formed
at the bends, banks being formed between them. The bore, however, ran
up to the end of the river until some time after the first survey was taken,
and the tide still rose very slightly when the second survey was made,
though the river was barred by a bank between the bends, by which the
flood just passed in small channels at the sides. The sand had risen in the
top of the estuary until it virtually closed the mouth of the tidal river, and
the condition of the estuary resembled that obtained in Tank D. This
virtually ended the experiment, but opportunity was taken to try the effect
of a larger quantity of land water, which was increased to one quart in two
minutes — i.e. nearly three times — and the experiment continued for 20,000
more tides without any material effect.
In Tank F the action at the lower end of the tank was again sluggish.
At the top of the estuary and in the river the conditions of the sand were
as near as possible similar to those in Tank E, but, as it came out, the
mean level of the water, relative to the level of the sand, was some 5 feet
(reduced to a 30-foot tide) lower in F than in E.
The appearances for the first 16,000 tides were the same as far as was
observed ; the ripple now extended up to the top of the river, and no banks
formed at the mouth. Nevertheless, before the second survey was taken,
the tide ceased to rise above the mouth of the river, proving that the
previous failure to realise the same state in the small tank as in the larger,
had riot been entirely due to the bends in the river. The question remained
whether it might not be owing to the higher level of the sand relative to
the mean level of the tide.
This question brings into prominence a fact observed during all the
experiments, but which had not, previous to the experiments on E and F,
assumed a position of importance. This is the gradual diminution of the
rise of tide owing to the lowering of the sand.
58] ON THE ACTION OF WAVES AND CURRENTS. 433
29. The rise of the tide depends not only upon the rise of the generator,
but also upon the tidal capacity of the tank. This capacity is the product
of the area of the surface at high water multiplied by the rise oFtfde, less
the volume of sand and water above low water in the generator. Now in
starting the experiments with the sand at the level of mean tide, not only
is there much more sand above the level of low water in the generator than
when the final condition of equilibrium is obtained, but the quantity of water
retained on the top of the level sand is considerable, so that the tide rises
considerably higher in the generator at starting than when the condition
of equilibrium is obtained, which excess of rise gradually diminishes as the
sand comes down at the lower end of the estuary.
Although the foot of the sand comes down pretty rapidly at the com-
mencement of the experiment, owing to the surface being rippled, the water
runs off slowly, and it is not till the sand at the end of the estuary has been
raised, and a slope formed, that the water runs down freely at low water, so
that during the early part of the experiment not only is the rise of tide at
the head of the estuary high, but also the low tide and the mean level of the
tide. The result is that the mean level of the water at the head of the
estuary is higher during the early part of the experiment. These changes
in the tide at different parts of the estuary and at different stages of the
tide are well shown by the automatic tide curves, page 481. As the sand is
rising at the top of the estuary, the result of the high water is to raise the
first banks above the level to which the tide finally rises.
As these banks come out and the ripple is washed off, leaving smooth
surfaces and channels, from which the water runs, and clean dry banks, the
mean level as well as the rise of tide falls, leaving the tops of the bank,
which were at first covered, high and dry.
These effects were much greater in Experiments C and D than in A and
B, and still more marked in E, F, and F'. In E, F, F', the plans 1 and 2,
taken at 16,000 and 33,000 tides respectively, show the difference in the
level of the sand at the mouths of the respective rivers. In Tank E the
rise of tide at the mouth of the river was observed to be 0'02 higher at
16,000 than at 30,000 tides, and in Tanks F and F' at 16,000 tides there
was a bore which ran up to the top of the river, while at 33,000 tides the
sand at the mouth was not covered at high water.
It thus seems that the condition of things which follows from starting
with the sand level, and a constant height of low water, is to institute a
distribution of sand at the top of the estuary, corresponding to a state of
equilibrium with a higher tide than that which ultimately prevails ; and
the greater the initial height of the sand relative to the mean level of the
water the greater will be this effect. That this action tends to explain
o. R. ii. 28
434 ON THE ACTION OF WAVES AND CURRENTS. [58
the closing of the mouths of the rivers in Tanks F' and F and not in E is
clear. But it is not clear that this is the sole explanation ; the conditions
in F' and F were not far removed from the limits of similarity obtained
in the rectangular tanks, and it is not clear that these limits may not be
somewhat different in the long estuaries with tidal rivers. This is a
matter which requires further experimental examination, for which there
has not been time.
30. Experiment II. in E and F, Figs. 42 to 45, without Land Water,
August 5 to September 1. — These experiments have been made under the
same conditions as in I. E and F, except for the land water. The general
appearance of the progress of the experiments was nearly the same, and
Plan 1 shows little difference. But as the experiment in E proceeded, it
became clear that the river was going to fill up gradually from the end.
The bore no longer reaches the end at 16,000 tides, while it had ceased to
exist and the tide had ceased to rise at Section 11 in the river at 32,000
tides, the end of the estuary also having filled up, the action in F being
nearly the same. Thus we have evidence similarly shown by both estuaries
that, although the fresh water produces little effect on the condition of
equilibrium of a broad estuary, the existence of a long tidal river above
the estuary does produce a great effect on the level of the low-water
channels in the upper portions of the estuary, and that land water, even
in such small quantities, is effective to keep open a long tidal river emptying
into a sandy estuary or bay*.
* For continuation, see p. 482.
28—2
436
ON THE ACTION OF WAVES AND CURRENTS.
TABLE I. — GENERAL CONDITIONS
Shape
of the
Estu-
ary
Per-
cent-
age of
Land
Water
References
Period
in
seconds
Horizoutal scales
Vertical
scale
1 in.
Rise of
tide in
feet
Vertical
exagger-
ation e.
Ex-
peri-
ment
Tank
Plan
Figure
1 in.
Inches
to a
mile
/
o-o
VII
A
2
4 33-5
17,600
3-58
177
0-170
99-7
j
»
III
B
2
5
23-8
33,600
1-88
327
0-094
102-0
»
»
)>
3
6
23-8
33,600
1-88
327
0-094
102-0
»
IV
»
1
7
34-4
23,300
2-71
327
0-094
71-0
»
IX
A
1
8
69-3
10,500
6-02
333
0-090
31-6
*j
j>
V
B
1
11
50-5
23,600
2-68
720
0-042
32-0
3
»
IX
A
2
9
69-3
12,400
5-08
379
0-080
32-0
bo
»
»
w
3
10
67-3
12,600
5-02
366
0-082
34-5
3
>»
VII
B
1
12
34-0
39,200
1-57
986
0-030
39-0
8
55
))
»
2
13
34-0
39,200
1-57
986
0-030
39-0
PH
0-6
X
A
1
14
50-2
11,500
5-49
171
0-176
67-0
1-2
VIII
B
1
16
35-4
22,000
2-87
309
0-097
71-0
0-6
X
A
2
15
48-6
11,900
5-30
171
0-176
69-0
1-2
VIII
B
2
17
34-5
22,600
2-8
309
0-097
73-0
0-6
X
A
3
18
48-6
11,900
5-30
171
0-176
69-0
v
1-2
VIII
B
3
19
34-5
22,600
2-8
309
0-097
73-0
f
o-o
XI
A
1
20
47-5
12,400
5-10
177
0-170
71-0
TS
»
X
B
1
22
35-4
20,700
3-05
273
0-110
75-8
9
&
»
XI
A
2
21
47-2
12,670
5-01
181
0-166
69-5
4
,d
»
X
B
2
23
35-4
20,700
3-05
273
0-110
75-8
T
0-7
XI
A
3
24
47-2
12,400
5-08
177
0-170
70-0
!> '
0-7
X
B
3
25
34-0
21,800
2-90
280
0-107
78-0
|
0-7
XII
A
1
26
48-2
12,300
5-15
179
0-168
68-4
o
J
0-7
XII
B
1
28
34-2
21,700
2-91
280
0-107
77-6
CO
0-7
XII
A
2
27
47-0
12,700
5-00
182
0-165
69-4
0-7
XII
B
2
29
34-2
21,900
2-88
286
0-105
75-0
—
I
C
1
30
49-8
12,100
5-22
185
0-162
65-4
—
I
D
1
32
35-9
20,900
3-03
285
0-105
73-1
c3
,3
—
I
C
2
31
46-2
13,200
4-78
190
0-158
69-5
o3
—
I
D
2
33
34-4
21,800
2-90
286
0-105
76-0
t> "
06
II
C
1
34
48-4
12,500
5-04
188
0-160
66-8
be
C
0-6
II
D
1
36
34-6
22,200
2-85
300
o-ioo
74-1
3
0-6
II
C
2
35
48-4
12,500
5-04
188
0-160
66-8
^H
0-6
II
D
2
37
34-6
22,200
2-85
300
o-ioo
74-1
t-
s
0-5
I
E
1
38
48-9
13,100
4-82
208
0-143
63-2
^
2
0-5
I
F
1
3!)
30-0
25,800
2-45
313
0-096
82-5
h*i
0-5
I
E
2
40
47-8
13,400
4-70
208
0-143
64-6
1
0-5
I
F
2
41
30-0
24,700
2-56
313
0-096
82-5
H .
o-o
II
E
1
42
47-9
13,500
4-67
214
0-140
63-4
£
»>
II
F
1
43
31-5
25,400
2-49
327
0-091
77-8
•+3
't>
>»
II
E
2
44
47-9
13,600
4-64
217
0-138
62-9
^•
bo
»
II
F
2
45
30-3
26,200
2-41
321
0-093
81-86
C
0-5
I
F'
1
46
30-1
25,500
2-48
300
o-ioo
85-1
&
0-5
I
F'
2
47
30-1
25,700
2-46
305
0-098
84-4
ON THE ACTION OF WAVES AND CURRENTS.
* A ***
437
1
THI
AND RESULTS OF .THE EXPERIMENTS.
.
Criterion
of simi-
larity /
C=(tfe)
/
H.-itfht
/of initial
sand in
feet
Height
of mean
tide in
feet
Number
of tides
from the
start
Action of the water on the sand in forming the bed
at the lower end of the estuary
Manner
Bate
Final state
0-490
0-25
0-265
93,839
General
Normal
0-083
0-125
0-140
99,388
General
Normal
Normal
0-083
0-125
0-140
130,176
—
—
Large ripple
0-058
0-125
0-130 16,344
Nearly normal
Nearly normal
Nearly normal
0-023
0-125
0-1325
13,078
Very partial
Very slow
—
0-002
0-65
0-065
17,919
—
Zero
—
0-016
0-125
0-142
36,776
—
—
—
0-019
0-125
0-141
78,986
—
—
Not reached
0-001)
Slope 1
(0-065
17,424
—
Zero
—
o-ooi /
in 124
1 „
39,727
—
—
Nearly normal
0-25S
0-25
0-256
19,437
Normal
Normal
—
0-064
0-1 25
0-148
18,332
Nearly normal
Nearly normal
—
0-362
0-25
0-256
42,820
—
Normal
Normal
0-066
0-125
0-148
68,861
—
—
—
0-362
0-25
0-256
76,273
See description
—
See description
0-066
0-125
0-148
91,184
See description
—
See description
0-346
0-333
0-337
17,206
Normal
Normal
—
0-101
0-166
0-17!)
17,879
—
Normal
—
0-320
0-333
0-348
39,809
—
—
Normal
0-101
0-166
0-169
40,268
—
—
Normal
0-343
0-333
0-348
60,243
Normal
Normal
Normal
0-095
0-166
0-169
57,024
Normal
Normal
Normal
0-327
0333
0-340
16,538
Normal
Normal
—
0-095
0-166
0-168
15,981
Normal
Normal
—
0-315
0-333
0-343
31,991
Normal
Normal
Normal
0-081
0-166
0-175
35,129
Normal
Normal
Normal
0-278
0-333
0-341
16,943
Normal
Normal
0-084
0-187
0-179
16,383
Nearly normal
Nearly normal
—
0-275
0-333
0-345
30,584
Normal
Normal
Normal
0-088
0-187
0-179
35,344
Nearly normal
Nearly normal
Nearly normal
0-274
0-333
0-344
16,90*
Normal
Normal
—
0-074
0-187
0-190
18,128
Nearly normal
Nearly normal
—
0-274
0-333
0-335
31,127
Normal
Normal
Normal
0-074
0-187
0-190
31,928
Nearly normal
Nearly normal
Nearly normal
0-185
0-333
0-350
16,368
Normal
Normal
0-073
0-187
0-191
16,577
Partial
Sluggish
—
0-189
0-333
0-337
32,635
—
Normal
Normal
0-073
0-187
0-191
32,880
—
—
Ripple large
0-174
0-333
0-349
15,871
Normal
Normal
—
0-060
0-187
0-193
17,184
Partial
Sluggish
—
0-163
0-333
0-349
32,501
—
—
Normal
0-066
0-187
0-192
29,!) 17
—
—
Ripple large
0-085
0-187
0-187
16,577
Partial
Sluggish
—
0-080
0-187
0-187
32,677
—
—
Ripple large
438 TABLE II. — MEAN SLOPES OF THE SAND IN RECTANGULAR TANKS.
TANK A
Measured
Experiment V., Plan 4
Experiment VII., Plan 2
Heights of
Contours
shown on
the Plan
Height (re-
duced to a
30-foot Tide)
of Contours
from L.W.
Mean Horizon-
tal Distance of
Contours from
the Contour
at 30 feet
above L.W.
Horizontal
Distances
reduced to
a 30-foot
Tide
Height (re-
duced to a
30-foot Tide)
of Contours
from L.W.
Mean Horizon-
tal Distance of
Contours from
the Contour
at 30 feet
above L.W.
Horizontal
Distances
reduced to
a 30-foot
Tide
Feet
Feet
Unit 6 inches
Miles
Feet
Unit 6 inches
Miles
1
—
- 0-975
-1-65
—
-1-792
- 3-003
0-176
30-00
o-oo
o-oo
30-000
o-ooo
o-ooo
0-146
24-39
0-79
1-355
24-546
0-647
1-133
0-116
18-68
1-86
3-20
19-092
1-254
2-171
0-086
13-00
2-96
5-07
14-638
2-356
4-085
0-056
7-46
4-64
7-95
9-184
3-724
6-447
0-026
1-87
6-63
11-38
3-730
5-428
9-397
-0-004
- 3-74
8-43
14-50
- 1-724
7-467
12-930
- 0-034
- 9-35
10-30
17-80
- 7-178
9-283
16-070
- 0-064
-15-00
12-17
21-60
- 12-632
11-780
20-400
- 0-094
- 20-80
13-60
23-40
- 18-086
14-003
24-235
-0-124
- 26-20
15-88 •
27-30
—
—
—
Experiment X., Plan 1
Experiment X., Plan 2
Feet
Feet
Unit 6 inches
Miles
Feet
Unit 6 inches
Miles
1
—
- 0-690
-0-774
—
-1-302
-1-167
0-176
30-000
o-ooo
o-ooo
30-000
o-ooo
o-ooo
0-146
24-886
0-741
0-810
24-886
0-665
0-752
0-116
19-772
2-147
2-347
19-772
1-900
2-149
0-086
14-658
4-256
4-652
14-658
3-648
4-124
0-056
9-544
6-916
7-560
9-544
6-631
7-507
0-026
4-430
9-880
10-800
4-430
9-101
10-290
-0-004
- 0-684
11-533
12-606
- 0-684
11-227
12-594
- 0-034
- 5-798
13-737
15-013
—
—
• —
TANK B
Experiment III., Plan 2
Experiment IV., Plan 1
Feet
Feet
Unit 3 inches
Miles
Feet
Unit 3 inches
Miles
1
—
- 3-540
- 5-643
—
- 0-994
-1-124
0-094
30-000
o-ooo
o-ooo
30-000
o-ooo
o-ooo
0-079
25-213
0-760
1-240
25-213
0-665
0-760
0-064
20-426
1-330
2-163
20-426
1-558
1-773
0-049
15-639
2-052
3-340
15-639
2-185
2-487
0-034
10-852
3-249
5-290
10-852
4-142
4-714
0-019
6-065
4-332
7-044
6-065
6-859
7-806
0-004
1-278
6-061
9-854
1-278
9-766
11-120
-0-011
- 3-509
7-828
12-727
- 3-509
12-046
13-710
-0-026
- 8-296
9-291
15-110
- 8-296
—
—
-0-031
-13-083
11-341
18-430
—
—
—
Experiment VIII., Plan 1
Experiment VIII., Plan 2
Feet
Feet
Unit 3 inches
Miles
Feet
Unit 3 inches
Miles
1
—
- 0-595
-0-621
—
-1-925
- 2-062
0-097
30-000
o-ooo
o-ooo
30-000
o-ooo
o-ooo
0-082
25-360
0-608
0-634
25-360
0-988
1-060
0-067
20-720
2-090
2-181
20-720
1-672
1-792
0-052
16-080
3-268
3-410
16-080
2-983
3-197
0-037
11-440
5-224
5-472
11-440
5-168
5-538
0-022
6-800
8-987
9-378
6-800
8-398
9-000
0-007
2-160
11-400
11-896
2-160
11-285
12-100
-0-008
- 2-480
13-148
13-720
- 2-480
13-108
14-050
-0-023
—
—
—
-7-120
14-535
15-570
439
440
JKapnm »f Actual -top* »WSt
Oiagrvmof ibpa miuotltoa,30 feetTUb
! y. ftax * Peried, 33 -3 Jos? R,se cf IMt
a - ft ' 3z-a . „ . oies
Fig. 2.
DtAgroffi tfAf tnal Slept* tfieJi- ax. exG-ggtrution of Z9.
of fte/Hs na&tted to a 30 f<C£l Scte*
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31
59.
THIRD REPORT OF THE COMMITTEE APPOINTED TO INVES-
TIGATE THE ACTION OF WAVES AND CURRENTS ON
THE BEDS AND FORESHORES OF ESTUARIES BY MEANS
OF WORKING MODELS.
[From the "British Association Report," 1891.]
THE Committee held a meeting in the rooms of Mr G. F. Deacon,
32 Victoria Street, Westminster (July 29, 1891), and considered the results
obtained since the last report. Professor Reynolds reported that by the date
of the meeting of the British Association the objects of the investigation
would be accomplished, and suggested that it would not be necessary to
continue the investigation beyond that date or to apply to the Association
for reappointment. These suggestions were adopted, and it was resolved
that the thanks of the Committee be communicated to the Council of the
Owens College for the facilities afforded for conducting the experiments in
the Whitworth Engineering Laboratory.
Having considered the disposal of the apparatus, which has no pecuniary
value, the Committee resolved to recommend the Association to place it at
the disposal of the Owens College.
At a second meeting held in the Committee room of Section G at Cardiff
the report submitted by Professor Reynolds was adopted.
§ I. — INTRODUCTION TO REPORT III.
1. In accordance with the suggestions in the Second Report, read at
the Leeds meeting of the British Association, the investigation has been
continued with a view —
59] ON THE ACTION OF WAVES AND CURRENTS. 4.S3
(1) To obtain further information as to the final condition of equilibrium
with long tidal rivers entering the head of a V-shaped estuary. —
(2) To obtain a more complete verification of the value of the criterion
of similarity.
(3) To investigate the effect of tides in the generator diverging from
simple harmonic tides.
(4) To determine the comparative effect of tides varying from spring to
neap.
Opportunity has also been taken : —
(5) To investigate the effect of prolonging the walls of the river into
the estuary through the bar, which was below low water, with prolongations
reaching up to low water, and others reaching up to half-tide — this being
done in both models, so that the similarity of the effects might be seen ; and
(6) To investigate the effect of rendering the estuaries unsymmetrical
by means of large groins, and so to test the laws of similarity obtained in
the symmetrical estuaries as applied to unsymmetrical estuaries.
2. The two models have been continuously occupied in these investi-
gations, when not stopped for surveying or arranging fresh experiments.
In this way each of the models has run 600,000 tides, corresponding to
840 years. These tides have been distributed over six experiments in the
large tank E, and four in the small tank F, in number from 50,000 to
250,000.
3. The experiments have all been conducted on the same system as
described in the previous reports.
All the experiments but one have been made in tanks E and F, without
further modification ; and in all these land water to the extent of (V5 per
cent, of the tidal capacity per tide has been introduced at the top of the
river.
Initially, the sand has been laid to the level of half- tide from Section 13
up the river to Section 26 down the estuary. The vertical sand gauges
distributed along the middle line of the estuary have been read and
recorded each day. Tide curves have been taken at frequent intervals.
Contour surveys have been made, generally after 16,000 tides, and again
after 32,000 ; while in the longer experiments further surveys have been
made. With the spring and neap tide, the rate of action being much the
slower, intervals between the surveys have been longer. In all 26 complete
surveys have been made, and 20 plans showing contours corresponding to
31—2
484 ON THE ACTION OF WAVES AND CURRENTS. [59
every 6 feet, reduced to a 30-foot tide, together with sections and tide curves
(page 506), are given in this report.
The general conditions of each experiment, together with the general
results obtained, are shown in the table, while a description of each experi-
ment is given in § VI.
The Committee have been fortunate in retaining the services of Mr
Greenshields, who has carried out the experiments, observing and recording
the results, besides executing such modifications as have been required,
designing the compound harmonic gearing for the spring and neap tides,
which has answered excellently.
Mr Bamford has kindly continued his assistance in conducting the inves-
tigations and reducing the results.
§ II. — GENERAL RESULTS AND CONCLUSIONS.
4. The conditions of equilibrium with a long tidal river entering at the
top of a V-shaped estuary. — The experiments in tanks C and E made last
year led to the conclusion stated in Art. 11 of the Second Report: 'that
the effect of a river 50 miles long, when reduced to a 30-foot tide, increasing
gradually in width until it enters the top of a V-shaped estuary, is entirely
to change the character of that estuary. The time occupied by the water in
getting up the river and in returning causes this water to run down the
estuary while the tide is low, and necessitates a certain depth at low water,
which causes the channel to be much deeper at the head of the estuary.
In its effects on the lower estuary the experiments with the tidal river
are decisive, but as regards the action of silting up the river further investi-
gation is required, both to establish similarity in the models, and to ascertain
the ultimate condition of final equilibrium.'
From this year's experiments, III., IV., V., VI., and VII., in tank E, and
V. and VI. in tank F, it appears tJiat if the length of the tidal river, reduced
to a 30 foot tide, is 50 miles ; or taking R for the length of the tidal river in
miles and hfor the rise of tide at the mouth of the estuary in feet, if
R = 8-5 VA
the river will keep open so that the tide will rise to the top, the sand falling
gradually from the top of the river to the level of about mean tide at the
mouth.
That the depth of water in the river and at the top of the estuary increases
rapidly with the length of the river, and when
59] ON THE ACTION OF WAVES AND CURRENTS. 485
the level of the sand at the mouth of the river will be more than h feet below
the level of low water, and tlie bottom will be below low water level for more
than half the length of the river above its mouth.
5. The similarity of the results in the tanks E and F. — The experiments
in the tanks E and F this year confirm those of last year, in showing that
during the early stages of forming the estuary from sand at the level of
mean tide, the action in the river is different in the small tank F from
what it is in the large tank E, although the value of the criterion of simi-
larity h3e* may be but little below 0'09.
It was not found practicable to get the value of the criterion any greater
in tank F, but it was found on diminishing the rise of tide in the large
tank E until the criterion had a value 0'09, that the results were still similar,
although the rate of action and the increase in the size of the ripple in-
dicated that the limit was being approached. That the dissimilarity in
tank F was only the result of a phase in the formation of the estuary was
also definitely shown by the effects of dredging out the sand, which was
above the initial level in the river during the early stages of the Experi-
ments V. and VI., after which the action in tank F resumed the same course
as that in E, and led to the same final condition of equilibrium, showing
by the rate of action and size of ripple that the limit of similarity was
approached.
It thus appears that, with such arrangements as these tanks represent,
there are two possible conditions of final equilibrium.
The one is that which has uniformly been presented by tank E, and in
Experiment V. in tank F after dredging ; namely, the tide rising up to the
top of the river and keeping the sand low in the estuary. The other, that
which was presented in Experiments I., II., III., and IV., in tank F ; namely,
the sand at the top of the estuary rising to high water level, as it would do if
there were no river, choking the mouth of the river, except so far as
necessary to allow the land water to pass, and so preventing any tidal action
from the river.
Which of these two conditions the river will assume during the process of
forming the estuary appears to be a critical matter, decided by whether the
tidal action of the river in lowering the sand at the head of the estuary pre-
dominates over the tendency of the tide in the estuary to raise the sand at
the mouth of the river.
There is a possible condition of instability between the river and the
estuary. The emphatic difference in the action of the long tidal river, and
* h IR the actual rise in feet, e the vertical exaggeration as referred to a 30-foot tide.
486 ON THE ACTION OF WAVES AND CURRENTS. [59
mere tidal capacity at the head of the estuary, in keeping down the sand at
the head of the estuary; and, further, the very great effect which an in-
crease in the length of the river has on the depth of water in the estuary
and in the river are clearly shown*. In Experiments III. and V. in tank E,
an increase of from 50 to 70 miles in the length of the river in V. causing
the depth of water to increase from by 40 to 30 feet all down the river
and estuary, lowering the sand in the lower river and upper estuary from the
level of half-tide to 28 feet below low water. In neither of these experi-
ments was the condition of instability reached, but 50 miles was very near
the limit.
In such a state any diminution of the upper tidal waters of the river, by
shortening the river or by land reclamation, might well have caused the
critical stage to be passed and caused the river to silt up — just as in the
other way the increasing of the tidal capacity high up the river by dredging
in Experiment V., tank F, caused the critical stage of silting up to be passed
and the river to open out. The sand actually removed in this experiment
by dredging was 8 per cent, of the tidal capacity, or 400 million cubic yards,
removed at the rate of 7 million cubic yards a year.
In most navigable rivers two processes have been going on — dredging
and land reclamation — the first tending greatly to improve the rivers and
estuaries, the second to deteriorate them so that any improvement has
been a question of balance. Where the rivers have improved they will
probably continue to improve so long as dredging goes on, but if the dredging
should stop, for example in the Thames, there would in all probability
be a gradual deterioration, possibly ending in the silting up of the tidal
river.
6. The effect of Tides deviating from the simple Harmonic Law. — One
attempt was made to study this question, when it was found that it would
require such modifications in the gearing as were not practicable in the time,
and so it was abandoned.
7. The action of Tides varying from Spiking to Neap. — The rates of action
and conditions of final equilibrium in rectangular tanks, in V-shaped estuaries
with a long tidal river, and in each estuary rendered unsymmetrical by large
groins, have been investigated with tides varying harmonically from spring to
neap, and again to spring in 29 tides. The ratio of these at spring and neap
being 3 to 2 as compared with uniform tides, having the same rise as the
spring tides, also for uniform tides having the same rise as the mean of spring
and neap, the results showing definitely :
* See page 50G in which the sections of the rivers and estuaries in tank C, Experiment II., and
tank E, Experiments III. and IV. are plotted to the same vertical and horizontal scales.
59] ON THE ACTION OF WAVES AND CURRENTS. 487
(1) That the condition of Final Equilibrium in all cases with spring and
neap tides ivas the same as that with uniform tides having the^same rise as
springs, and much greater, essentially different, from that with a uniform tide
//lining a rise equal to the mean rise of spring and neap tides.
(2) That the Rate of Action with the varying tide is much smaller than
that of a uniform tide having the rise of the spring tide. The ratios being
definite, about 2'5 to 1.
(3) That the limits of similarity obtained for all spring tides hold
approximately for tides varying from spring to neap.
8. The effects of prolonging the rivers into the estuaries by walls below
high water. — Experiments V. in tanks E and F having arrived at similar final
conditions of equilibrium (in which the depth of the rivers for some distance
above their mouths was reduced to a 30-foot tide, nearly 30 feet at low water,
while the sand in the estuaries gradually rose from the mouths of the rivers
until it reached to within 12 feet of low water at a distance of 14 miles below
the mouth and then fell again, all the sand being below this level, there being
passes which formed a crooked deep water channel), opportunity was taken
to prolong the banks of the river by walls at first up to low water and
extending through the bar to a distance of 44 miles from the mouths of the
rivers. Then raising these walls to half-tide, and finally carrying the walls
forward slowly in tank E at a rate of half a mile a year (700 tides), and in
tank F dredging from between the walls at a rate of seven million cubic
yards a year (700 tides).
This was done in the first place as a further test of the similarity of the
action in the two tanks, and secondly as affording an interesting study as to
the effect of vertical walls in the direction of the current in the bed of a tide-
way. The effect of these walls at the level of low water and at half-tide
were precisely similar in both tanks ; in neither case did they produce any
sensible effect at all on the level of the sand between them. At the level of
half-tide they caused in both tanks a slight silting up outside the walls and
also a slight silting up in the river above its mouth, which effects were very
much increased when the walls were raised to half-tide. On the walls being
removed in tank E and then gradually carried forward, the silting up behind
the wall and deterioration of the river increased, but there was no improve-
ment in navigable depth between the walls.
The dredging in tank F, so long as it was continued, added about 20 feet
on a 30-foot tide or 10 feet on a 15-foot tide, to the navigable depth between
the walls, but there was the same silting up behind the walls and the same
deterioration in the river.
It thus appears that the similarity of the results in both tanks supports
488 ON THE ACTION OF WAVES AND CURRENTS. [59
the conclusion that vertical walls having the horizontal direction of the current
in a straight tideway and terminating well below high water, produce but little
effect on the distribution of the sand between them, so long as the passage is
freely open at both ends, but that if the passage be blocked at one end they form
a bay in which the sand rises at the head.
9. The effects of the tide in estuaries not symmetrical. — Having so far,
in accordance with the original scheme of this investigation (First Report,
1889, p. 5), simplified the circumstances which influence the distribution of
sand by maintaining the lateral boundaries perfectly symmetrical, and as
nearly rectilinear as practicable, and having found definite laws connecting
the distributions of sand in the beds of the model estuaries with the
period and rise of the tide and the length of the estuary, besides the
laws connecting the period of the tide with the horizontal and vertical
scales under which the models give similar results, there remained two
questions :
(1) How far such discrepancies as appear between the general distri-
butions of sand found in the models, and those observed in actual estuaries,
are attributable to irregularities in the boundaries of the latter ?
(2) How far the influence of these boundaries is subject to the same
laws of similarity as those already obtained ?
The original experiments of the author in models of the Mersey which led
to the appointment of the Committee (see page 326) had to a great extent
answered these questions, showing that similar irregularities in the lateral
boundaries exercise similar and predominating influences on the lateral dis-
tributions of the sand in the models and in the estuaries.
It seemed, however, desirable, so far as time allowed, to confirm these
results of the author's and make this investigation complete in itself, by
carrying out experiments in both models similar to those already carried
out, except that the boundaries should be boldly irregular.
Such experiments also afforded opportunity for studying some general
effects of great importance. The relation between the depths of water and
the rise of tide had come out very definite in the symmetrical experiments,
and it was desirable to see how far these relations would be disturbed by
lateral irregularities. For instance: (1) Would bold irregularities in the
boundaries of the estuary alter the depth of water in the river ? Bold
irregularities in the boundaries, causing the water to take a sinuous course,
would have the effect of virtually narrowing and increasing the length of the
estuary, and by causing eddies would obstruct the passage of the water
to some extent. Lengthening the estuary would tend to increase its depth
59] ON THE ACTION OF WAVES AND CURRENTS. 489
at corresponding points, and obstructing the water would tend to diminish
the tidal action in the river ; at all events, until the estuary bad-increased in
depth.
(2) At the mouth of the estuary the flow of water had so far been
straight up and down, and equal all across the estuary. By rendering the
mouth unsymmetrical, circulation would be set up which would render the
up-currents stronger at one part and the down-currents stronger at another,
an effect which would correspond to some extent to that of tidal currents
across the mouth of the estuary.
(3) The large tidal sand ripples below low water in the model estuaries,
with the flood and ebb taking the same course, constitute a feature which it
is impossible to overlook, yet the existence of corresponding ripples had been
entirely overlooked in actual estuaries, until, when they were looked for, they
were found to exist, having been first seen in the models. The reason that
they were overlooked before is, no doubt, explained by the fact that the
bottom is not visible below low water in actual estuaries ; but this is not all.
In the estuaries, these ripples, where found, have been confined to the
bottoms and sides of the narrow channels between high sand-banks, and
they do not occur on the level sands below low water towards the mouths
of estuaries to anything like the same extent as in the models. By rendering
the estuary unsymmetrical and so causing the ebb and flood to take different
courses, this effect, as explaining the greater prevalence of ripples with sym-
metrical estuaries, would be tested.
These considerations led to the repetition of Experiment V. in tank F, at
first with a single groin extending from the right bank into the middle of the
estuary at the mouth, and subsequently to the introduction of three more
groins from alternate sides of the estuary to the middle, up the estuary, and
then to the introduction of similar groins into tank E, during Experi-
ment VII., with spring and neap tides.
The result of these experiments is to show conclusively :
(1) That the laws of similarity found for symmetrical channels with
uniform tides hold with sinuous channels for uniform or- varying tides.
(2) That the greater uniformity of the depth of sand on cross section.* of
models with symmetrical boundaries than with actual estuaries, does not exist
when the banks are equally irregular.
(3) That the circulation caused by the unequal flow of the tide in model
estuaries tends greatly to take the sand out, and that the natural tendency in an
cufuiiry to xntrp the boundaries so as to increase its sinuosities tends greatly to
the deepening of the channels.
490 ON THE ACTION OF WAVES AND CURRENTS. [59
(4) That in the models with boldly irregular boundaries the tidal ripples
are much less frequent than in the symmetrical models, being confined to places
where there are no cross currents, as in actual estuaries.
10. Conclusion of the Investigation. — It seems that the objects of this
investigation have now been accomplished.
The investigation of the action of tides on the beds of model estuaries has
been found perfectly practicable. Two tanks have been kept running night
and day from June 22, 1889, to August 1891, and have each accomplished
upwards of 1,200,000 tides, representing the experience of 2,000 years.
Such difficulties as protecting the sand from extraneous disturbance and
keeping it free from fouling, regulating the levels of the water, the tidal
periods, the rise of tide, forms of the tide curve and the supply of land
water, observing and recording the results, have all been fairly overcome,
though none of the precautions taken could have been safely dispensed with.
The limits to the conditions under which the results will conform to the
simple hydrokinetic law of similarity have been fairly established ; while
above these limits the applicability of the simple hydrokinetic law to these
experiments has been abundantly verified in models varying in scale from
six inches to a mile to an inch and a half to the mile, and with vertical
exaggerations, as compared with a 30-foot bide, ranging from 60 to 100.
The laws of the distribution of the sand in a tideway under circumstances
of progressing complexity have been determined, and have been verified,
not only by repetitions of the same experiment, but also by producing
similar distributions under different circumstances, which circumstances,
however, conformed to the laws of hydrokinetic similarity. Thus the distri-
butions of sand in simple rectangular estuaries, V-shaped estuaries, and
V-shaped estuaries with a long tidal river, have all been investigated and
found to be definite.
Investigations have also been made, with definite results, of the separate
effects of land water in moderate quantities, and of the length of the tidal
river on the depth of water in the river and estuary, and of the effect of bold
irregularities in the configuration of the lateral boundaries of the estuaries,
also of training walls in deep water. And, lastly, the comparative rates and
ultimate action of uniform tides, and tides varying from spring to neap, have
been determined.
It thus appears that this system of investigation has been tested over
a great portion of the ground it is likely to cover, and that most of the
difficulties that are likely to occur have been met, and the necessary pre-
cautions found.
59]
ON THE ACTION OF WAVES AND CURRENTS.
491
It would seem, therefore, by carefully observing these precautions, the
method may now be applied with confidence to practical problems. -
§ III. — THE APPARATUS.
11. General Working of the Apparatus. — All the apparatus has worked
well, although certain repairs have been rendered necessary by wear; thus,
the motor has required new pins, not much, considering it has made over
200 million revolutions. The knife edges, on which the generator of the
large tank rests, which are of cast-iron, and 2 inches long, and each carry
about 1,000 lb., were found to have, after one million oscillations, worn
down ^ of an inch, until they had become so locked in the Vs as to stop
the motor.
12. The modifications in the Tanks have this year been confined to the
introduction of training walls and groins. These have been made of paper
saturated with solid paraffin (which gradually became warped by the
pressure), sheet zinc, and sheet lead or wood, as was most convenient. In
the last experiment the large tank was modified by taking out the partition
boards and stopping the opening at the end so as to reproduce the original
rectangular tank A.
iR. 1.
492 ON THE ACTION OF WAVES AND CURRENTS. [59
13. Gearing for the Spring and Neap Tides. — This arrangement, de-
signed by Mr Greenshields, accomplished the result very neatly and effect-
ually with a minimum of new appliances. It admits of any degree of
adjustment in the ratio of maximum and minimum tides, and works easily
and well.
On commencing the work with spring and neap tides it was found
essential to have an indicator of the phase of the tide, which would be easily
visible without having to examine the gearing. For this, a counter, having
twenty-nine teeth in the escapement wheel, which carried a long finger over
the face, was constructed by Mr Greenshields, and worked well, proving a
great convenience.
§ IV. — DESCRIPTION OF THE EXPERIMENTS ON THE MOVEMENT OF SAND
IN A TIDEWAY, FROM SEPTEMBER 4, 1890, TO AUGUST 1891.
14. Experiment III., Plan 1, Tanks E and F, Fig. 4, Page 507. — These
experiments were intended as a repetition of Experiments I. C and D. (Second
Report, p. 429), which were only continued to 36,000 tides. The only difference
in the conditions being that, while in Experiment I. the sand was initially laid
up to the top of the river, Section 38, in Experiment III. the sand was only
laid up the river to Section 13. These experiments were carried on during
the vacation, Mr Foster kindly keeping the tanks running and reading the
counters daily. In this way 47,000 tides were run in tank E, and 66,000 in
F, when the surveys for Plan 1 were taken.
These surveys show a rather more advanced state than is shown in
Plan 2, Experiment I., but they present exactly the same characters.
In tank E the sand in the estuary is slightly lower in the longer experiment
than in the shorter, but shows the same distribution. In both experiments
in tank E the level of the sand at the mouth of the river is that of mean
tide, and in both experiments the level of the sand reaches the H.W.L. in the
generator at Section 11, or 13 miles up from the mouth, and in both the
tide continued to rise to the top of the river.
In tank F, also, both experiments show the same general distribution of
sand in the estuary and river. In the estuary the phenomenon, previously
observed, with a low value for the criterion, namely, the large ripple, is more
pronounced in the longer experiment ; but in both experiments the river has
become barred at an early stage, showing that the conditions in F, during
the formation of the estuary, have been below those essential for similarity.
The rise of tide observed at the end of the Experiment III. in both
E and F is below those observed at the earlier stages. In tank E the rise of
59] ON THE ACTION OF WAVES AND CURRENTS. 493
tide with the same rise in the generator has fallen to 0125 foot at 47,000 tides,
though it was 0140 foot at 32,000; and in F it was 0*095 fooLat 66,000
against 0'096 foot at 32,000. This phenomenon, which becomes more
pronounced in some of the later experiments, is accounted for by the im-
proved tideway as the experiment gets older, allowing the estuary to empty
itself more completely. It requires notice, since it renders estimates, such
as the value of the criterion of similarity, based upon the rise of tide,
difficult. The same quantity of water passes up and down the estuary,
but does not effect the same rise of tide at the generator, which falls as the
experiment gets older, while the rise of tide up the estuary increases at the
same time.
15. Experiments on Increased Length of Tidal River. Experiments IV.,
E and F, with Land Water, Figs. 3, 5, 11, pp. 506, 508, 514, October 22 to
November 17, 1890.— The sand laid 0'333 foot in E, and 0187 in F from
Section 13 up the river to Section 26 down the estuary. Mean rise of the
tide, 0*310 in E, 0197 in f. Rise of the generators the same as before,
periods 33'47 in E, 22'21 in F.
The conditions were thus the same as in Experiment III., with the
exception that the tidal periods were reduced in the ratio 1 to V2. As
reduced to a 30-foot tide, this would have the effect of increasing the
horizontal scales in the ratio \/2 to 1. Thus, while in Experiment III.
the estuaries from generator to mouth of tidal river represented about
50 miles, and the rivers 54 miles ; in Experiment IV. the estuaries were 70,
and the rivers 76.
With the same tide at the mouth, the elongation of the estuary would
cause the tide to rise higher at the mouth of the river, but as there was
only the same quantity of water from the generator, the tides with the
longer estuaries were smaller at the generators, which would again
diminish the tides at the mouths of the rivers. The tides observed at
the mouths of the rivers were somewhat higher than in Experiment III.
And this fact must be allowed for in considering the results as representing
the effect of increasing the lengths of the rivers on the distribution of sand.
In tank E the effect was very remarkable. For the first 5,000 tides
the sand rose up the river as far as it was laid, the head of the sand
gradually going forward, and the sand falling at the top of the estuary
and in the mouth of the river. Somewhat the same appearances appeared
in tank F, though it soon became apparent that the advance of the head of
the sand was much slower in F, and also the lowering of the sand at the top
of the estuary. Sand was going up the river, but it accumulated in the
lower reaches.
494 ON THE ACTION OF WAVES AND CURRENTS. [59
In E, at 9,000 tides, there was an almost sudden change ; the sand in the
river was rapidly earned to the top, leaving the lower reaches empty. After
11,000 tides the bottom of the river was swept clean from the mouth to
Section 15 (30 miles), and then a steady downward movement of the sand
went on, all down the estuary, until there was deep water all the way down
from 10 miles below the head of the river. The clearing of the bottom
of the river of sand evidently increased the action of the river, increasing
greatly the rise of tide.
In tank F the result was very different ; instead of the sand shifting
suddenly up the river, the sand reached Section 15, and then barred the
river at Section 11, the river then gradually filling up. At 38,000 tides,
when the second survey was made, the tide was still rising at the top of the
river, and the head of the sand still proceeding forwards. The experiment
was continued to 81,000 tides, and the head of the sand reached Section 19,
the tide still rising at the head very slightly. This shows that the conditions
of similarity were more nearly fulfilled in the river in tank F in this experi-
ment than in III. The values of the criterion, however, given in the table,
are lower in IV. than in III. This is because these values are calculated
from the rise in the generators, which were in these experiments O'llO in
tank E, and 0'081 in F, against 0'125 and 0'095 in Experiments III. With
the same water going out of the generator there must have been higher tides
at the mouths of the rivers in IV., and as the vertical exaggeration in Experi-
ment IV. was \/2 times larger than in I. and III., assuming the rise of tide
in tanks E and F, Experiments III. and IV., to be as in Experiments L, the
values of the criterion in Experiments IV. would be at least 0'261 and 0103.
This is in accordance with the observed results.
It seems therefore that in order to apply the criterion to the conditions
of similarity at the top of a long estuary with a tidal river, the actual rise of
the tide at the mouth of the river should be taken in estimating the value of
the criterion for similarity at these points. It appears, however, that in
every case where the criterion, estimated from the tides in the generator,
exceeded the value '09, the conditions of similarity have been fulfilled, while
in no case has it fallen decidedly below this value without decided symptoms
of dissimilarity having appeared, so that this value for the criterion seems to
be established as a good working rule for the formation of an estuary from
sand at the level of half-tide.
If the bottom of the estuary is modelled the case is different, but the
occurrence of large ripples, in experiments in tank F and in Experiment V.
in tank E, when the value of the criterion fell as low as '08, shows that the
similarity of the ripple depends on the same value of the criterion as the
formation of the estuary.
59] ON THE ACTION OF WAVES AND CURRENTS. 495
16. Experiments with Limiting Value of Criterion. — Experiment V. with
Land Water, Tank E, Figs. 6, 7, and 12, pp. 509...,/?-ow November 20 to
December, 24, 189().: — The conditions of this experiment were designed to
bring the value of the criterion, estimated from the rise of tide in the
generator in the final condition of equilibrium, to 0'09, keeping the horizontal
scale as nearly as possible the same as in IV., and diminishing the rise of
tide so as to increase the proportional depth of sand in the river, and thus
prevent the bottom being swept clean when the final condition was reached.
The length of the crank working the generator in IV. had been 4'437
inches; this was reduced to 377 inches in V., reducing the rise of the tide in
the ratio 0'85. To keep the horizontal scale the same the period 33'3 seconds
was increased to 36 seconds, leaving the product p ^h constant.
This reduced the vertical exaggeration e in the ratio 0'85. Thus the
value of h*e is reduced (0'85)4 or 0'52.
Now the value of the criterion in Experiment IV., just before the bottom
was swept with sand, was greater than 0'18, which, multiplied by 0'52, gives
0-093.
As carried out at the final condition shown in Plan 3, Page 510, the period
was 35'6 seconds, the rise of tide 0107, and the value of the criterion 0'0912.
This low value of the criterion showed itself in the rate of progress of
the experiment. It was 13,000 tides before the sand in the river reached
Section 19, against 4,000 in Experiment IV., and 25,000 against 9,000 in IV.
before reaching the head of the river. In the early stage of the experiment
it seemed doubtful whether the sand was going to bar the river as in
Experiment IV., tank F. Except in rate of action, however, the motion
of the sand followed the same course as in Experiment IV., taking a sudden
shift at about 20,000 tides, and then rapidly lowering the sand at the
head of the estuary. At the mouth of the river the bottom of the tank was
reached after 50,000 tides, but only between the ripple bars, so that it was
not swept clean.
The ripples in this experiment were very much larger than anything
before in tank E, showing that the criterion was approaching its critical
value.
The final condition of the estuary, as shown in Plan 3, after 36,000 tides,
shows conclusively the effect of the upper tidal water in a long river on the
bed of the lower estuary. Below Section 19, 32 miles from the top of the
river, there is no sand above the level of low water in the estuary, and from
this the sand falls uniformly to the mouth of the river, where there is a
depth of water, at low tide, of 30 feet. In the head of the estuary there is
496 ON THE ACTION OF WAVES AND CURRENTS. [59
a bar the top of which is only 12 feet below low water ; this is at Section 9,
or 18 miles below the mouth of the river ; below this point the sand
gradually falls to the generator.
Comparing this with the results in Experiments I. and III., where the
reduced length of the river is only some 50 miles, but in which the rise of
tide at the mouth of the river was somewhat greater, the effect of the
extra 20 miles length in the river is seen to have improved the general
and navigable depth of the river and estuary, from the top of the river to
a distance of 40 miles down the estuary, by from 40 to 30 feet.
17. The effects of dredging in the river, Experiment V., in Tank F,from
November 19 to December 23, 1890, Plan 3, Page 510. The initial conditions
of this experiment were the same as those of Experiment IV. in tank F,
except that the mean level of the tide was raised to O'OIG above the initial
level of the sand, and the period was increased from 22 to 23'3 seconds.
The experiment was undertaken with the intention of ascertaining (1)
whether raising the mean level of the tide above the initial level of the
sand, without altering the rise of tide, would prevent the river becoming
barred ; and, supposing this did not succeed, (2) to ascertain whether, if
the bar, which had hitherto formed in the river during the early stages of
the experiments in tank F, were kept down by dredging out the sand as it
rose above the initial level, the later stages would follow the same course as
in tank E.
The results were remarkable, and bring out the critical character of the
conditions at the mouth of the river.
The experiment was allowed to run 30,000 tides, during which the
progress of the sand was much more rapid than in IV., reaching Section 19
in 6,000 tides, as against 36,000 in Experiment IV. and 13,000 in Experiment
V, E., and reaching Section 23 in 16,000. At this point it stuck, and the
sand accumulated at the head of the estuary and in the river, which became
barred at Section 19, on reaching 30,000 tides.
It thus appears that lowering the initial level of the sand produced an
effect on the first action very nearly equal to increasing the rise of tide
by double the amount, but that as the sand distributed itself this effect
passed off.
At 30,000 tides the bar in the river was dredged down to the initial level
of the sand, and this level was maintained by daily dredging till 70,000
tides had been run, 0'08 cubic foot of sand in all being removed.
At this stage the sand in the river suddenly shifted up to the top as in
Experiments IV. and V., E. The sand at the mouth of the river and top of
59] ON THE ACTION OF WAVES AND CURRENTS. 497
the estuary falling until the bottom appeared, dredging was discontinued.
At 95,000 tides the final condition had been reached, which _yvas almost
identical over the whole estuary with that of Experiment V. E after 60,000
tides, as shown in Plan 3, Experiment V., E and F.
The instability of the condition which may prevail at the mouth of a
river is thus clearly shown, as well as the useful effect of improving the
tideway by dredging in the upper reaches in the river. In three experi-
ments in tank F, I., III., and IV., the river became completely barred, and
the estuary became a bay with a stream of land water entering at its
top; in Experiment V. the bar again formed, but on being kept down,
by dredging, to the level of half-tide, till the sand had fallen at the head
of the estuary, the river at length prevailed, and the sand was washed out
till there was 30 feet of water at low tide.
The time occupied and amount of sand removed, in producing this effect,
were considerable. The tidal capacity of the river and estuary is 1 cubic
foot; this reduced to a 30-foot tide is 21,700 million cubic yards, or on
a 15-foot tide is 5,422 million. The amount of dredging, 0'08 cubic foot in
all, represents 1,743 million cubic yards on a 30-foot tide, or 437 millions on
a 15-foot tide. This was distributed over 40,000 tides, or sixty years, so
that even with the 15-foot tide it would represent 7 million cubic yards a
year.
After the dredging the rise of tide fell from '081 to '073 foot, which
would result from the lowering of the sand which was above low water.
18. Experiments with Training Walls. Experiment V. (continued} with
Training Walls, Tanks E and F, from January 7 to February 20, 1891,
Plan 4, Page 516. — Having arrived at similar final conditions of equilibrium
in tanks E and F, in which the sand was entirely below low water from
Section 19 up the rivers (32 miles from the top of the river) to the
generators, and iti which there were bars in the estuary below the mouths
of the rivers, reducing the depth of water at low tide from 28 feet in the
river to a minimum of 12 on the top of the bars, it seemed an opportunity
not to be lost for testing the similarity of the effect in the two tanks of
prolonging the rivers by training walls through the bars.
With this view, walls of thick paper saturated with paraffin, pushed
vertically into the sand and extending up to low water, were run out from
the end of the river, preserving the same divergence as the walls of the
river to Section 22, or 40 miles on a 30-foot tide, the tanks being stopped
for the purpose.
These walls produced no apparent effect whatever on the depth of sand
between the walls, during 20,000 or 30,000 tides. They were then replaced
o. R. ii. 32
498 ON THE ACTION OF WAVES AND CURRENTS. [59
at the upper end by walls of sheet zinc extending to half-tide, which did
produce an apparent effect, inasmuch as the sand accumulated outside the
walls, forming an apparent channel within ; also the sand rose in the river,
doing away with the appearance of a bar. These effects were similar in
both models after 40,000 tides had been run.
The old walls were removed in both tanks and replaced by walls com-
mencing at f tide at the mouths of the rivers, and falling during the first 4
or 5 miles to half-tide, at which they were continued to Section 22.
In tank E the walls were advanced gradually from the mouth of the river
at a rate of about half a mile in 700 tides (a year). The result of this is
shown in Plan 4, Page 516, tank E. There is no improvement in the navig-
able depth of the river.
In tank F the walls were put in and then the tops of the ripple bars
were daily dredged off between the walls. This was continued for 10();000
tides, during which 5 per cent, of the tidal capacity was removed, or about
1,000 million cubic yards on a 30-foot tide, or 250 millions on a 15-foot tide,
which represents 7 millions annually on the 30-foot tide, or 1*8 millions on a
15-foot tide. The effect, as shown in Plan 4, tank F, Page 516, is to add
some 20 feet to the depth on a 80-foot tide, or 10 feet on a 15-foot tide.
The silting up behind the walls is the same as in tank E, and the
detriment to the navigable depth of the river is also similar.
19. Experiment V. (continued] with Tide deviating from the Simple
Harmonic in Tank E, February 23 to March 12, 1891. This was meant as a
preliminary experiment. The balance of the generator was altered to give
a rise of tide in 17 seconds and a fall in 20. The experiment was run for
about 40,000 tides, and a survey taken, which showed little or no effect.
On carefully examining the tide curves it was found that they showed very
little inequality in the rise and fall. On attempting to increase this by
further altering the balance, it was found that this could not be done. To
continue this part of the investigation it would have been necessary to
introduce complex gearing. Time did not suffice for this, and the study was
not carried further.
20. Experiments with Tides varying from Spring to Neap, Tank E, V.,
VL, VII., VIII., Tank A, XIII. Figs. 11, 12, 13, 15, pp. 514..., March 20
to August 1891.— The gearing for tank E having been modified so as to cause
a rise in the generator, varying to over an interval of 29 tides, the variation
being harmonic and adjustable, so as to admit of any relation between the
maximum and minimum rise.
These were adjusted so that the mean rise was the same as the rise in
59] ON THE ACTION OF WAVES AND CURRENTS. 499
Experiment V., the spring and neap rises being in the ratio 3 to 2. A drain
with an adjustable orifice was put in the bottom of the tank-to-drain off
nearly all the fresh water, and the scummer adjusted so as to draw off the
excess of land water at low spring tide level ; this being adjusted by trial
until, when running, the mean tide level was the same as before.
Experiment V. was then restarted, without the sand having been dis-
turbed, to afford a preliminary trial of the apparatus, the period being that
of Experiment V., 36 seconds. This was continued 18,000 tides, till the
apparatus was completely in hand ; then the sand was relaid for Experiment
VI., Fig. 11, Page 514, in which the conditions were the same as V., except
the tide. The mean rise in the generator was the same in VI. as in V., and
the ratio of the spring and neaps 3 to 2. This brought the rise in the
generator at spring tides in VI. greater than that in Experiment IV., in the
ratio of 11 to 1. The action on the sand was much more rapid than in
Experiment V. with the uniform tide, being nearly as quick as in IV. The
sand reaching the top of the river in 13,000 tides, as against 10,000 in IV.
and 25,000 in V., and the bottom of the river being swept as clean in 17,000
tides in VI., as in 14,000 in IV. In other respects the action in VI. very
closely resembled that in IV. The rate of action was a little slower, but
the action itself seemed rather stronger, as corresponding to a higher tide.
Surveys were taken at 20,000 and 34,000 tides. The experiment was then
stopped, in order to make the conditions comparable with those of Experi-
ment V.; it being quite clear that the action of spring and neap tides,
having a mean rise equal to that of a uniform tide, was not only much
more rapid, but led to a different state of final equilibrium.
Experiment VII., Plan 1, Page 515. In this the tide was adjusted until
the rise of the generator at spring tide was the same as that for the uniform
tide in V., the other conditions being all the same.
The character of the action now became identical with what it had been
in Experiment V., but the rate was decidedly slower. Thus the sand moving
up the river reaches :
Section 19 after 13,000 in V. and 39,000 in VI.
27 after 20,000 „ „ 51,000 in VI.
The survey taken after
18,000 tides in Experiment V., Tank E, and
51,000 „ „ VII., „
are almost identical, the latter being a little the forwardest.
It thus appears that spring and neap tides, having a ratio 3 to 2, produce
the same result as two-fifths the same number of tides all springs.
32—2
500 ON THE ACTION OF WAVES AND CURRENTS. [59
So far neither of these estuaries had reached the condition of final equi-
librium, but the similarity that the Plans 1, Experiments V. and VII. present,
seemed sufficient assurance that this would be the same.
It was intended to repeat Experiment V., tank A, as soon as the tank
had been re-formed to its rectangular shape ; in the meantime groins were
introduced in tank E similar to those which had been used in Experiment
VI. F, and Experiment VII. E was continued, to ascertain how far similar
effects would be produced by varying and uniform tides in estuaries with
similar but boldly irregular outlines.
Experiment VII. E, Plan 2, Page 517 was continued with groins to
123,000 tides. Similar groins had affected the condition of the sand in
the estuary and river in Experiment VI., tank F, so that further comparison
between Experiments VII. and V. cannot be made.
Experiment XIII., Tank A, rectangular without land-water, spring and
neap tides, Plan 3, Page 512, from July 10 to August 10, 1891. — In this
experiment the rates of spring and neap tides were 3 to 2, and the rise of
tide at spring tides was 0*176, the same as in Experiment V., tank A. The
tank was reduced to its original rectangular form (Report I.), namely, 4 feet
broad, and 12 feet from the generators to the top. The sand was laid as
in Experiment V., tank A, at a depth of 2 in. from Section 18 to the top of
the tank, and the mean tide was adjusted in Experiment V., tank A. The
period was 50 seconds, as in tank A. Thus the conditions of Experiment
XIII. and V., tank A, were precisely the same, with the exception that
while the tides in Experiment V. were all springs, those in Experiment
XIII. varied from springs to neap; the object of Experiment XIII. being
to compare the rate of action and final condition of equilibrium with varying
tides, with the very definite results, as to the slopes of the sand, obtained
in the rectangular tanks, and recorded in Report I., B. A. Report, 1889 (see
page 380).
These results are shown in the plans on page 510. The period in
Experiment XIII., tank A, being shorter than in V. The actual slope is
greater, but the slopes reduced to a 30-foot tide agree.
21. Experiments on Estuaries not Symmetrical. Experiment VI., in
Tank F, with large groins, Plans 1 and 4, Pages 517, 518, from April 8 to
June 16, 1891. — This experiment was started under conditions in all respects
similar to those in Experiment V., tank F, with the exception of a vertical
groin extending from the right bank to the middle of the estuary, with
an inclination of 45° towards the generator, and rising from the bottom of
the tank above high water. This groin, which appears in the charts to
represent an artificial structure, is, in fact, out of all proportion to anything
59] ON THE ACTION OF WAVES AND CURRENTS. 501
of that kind which has yet been attempted. As reduced to a 30-foot tide,
it is 11 miles long, 100 feet high up to H.W.L., and half a mile broad. Thus
it corresponds rather to such a natural feature as Spurn Head, at the mouth
of the Humber, than to a breakwater such as that at Harwich.
In starting the experiment, the end of the sand at Section 26 was 20
miles above the point of the groin at Section 36. The groin had deep
water on both sides of it, so that its only effect was to deflect the flood on
to the left bank of the estuary.
This effect was very decided, the strength of the flood on the right
carrying the sand up the estuary in spite of the effect of the ebb to bring
it down. But this in itself was not so much ; it was the large eddy caused
by the groin which produced the greatest effect. The water entering on
the left of the estuary crossed over to the right, and returned along the
right bank. In other words, during flood the right side of the estuary
for 30 miles from the generator was in back water. This back water also
gave the ebb a start down the right bank which rendered the ebb stronger
on this side.
The sand came down rapidly on the right side, and besides was carried
H\< r from the left to the right, and formed a bank along the right middle
of the estuary, reaching the generator after a very few tides. Round this
bank the water circulated, carrying the sand with it up on the left and down
on the right, the bank growing all the time. The ripple round this bank
was very striking, arranged with the ripple heads all down on the right side
and up on the left. After about 3,000 tides the sand began to pass from the
point of this bar in a fine stream across the open channel, dividing this point
from the point of the groin, and commenced the formation of a bank in
the generator corresponding to that in the tank. This bank had to be
removed from the generator, and after 6,000 tides 4 Ibs. of sand were so
removed. In Experiment V. the first sand removed from the generator was
after 120,000 tides had been run.
The sand also went more rapidly up the river in Experiment VI. than in
Experiment V. But this was accounted for by dredging in the river having
begun much earlier, after 20,000 tides as against 30,000.
In all 8 Ibs. of sand were removed from the river in Experiment VI.,
against 10 Ibs. in V., or about 0'004 of the tidal capacity in VI. against 0 08
in V. In both cases the dredging stopped when the sand began to shift up
the river after 70,000 tides.
At 100,000 tides a condition of final equilibrium had been arrived at.
The sand in the river was just the same as in V., Plan 3, Experiments V.
502 ON THE ACTION OF WAVES AND CURRENTS. [59
and VI. in tank F. There is deep water in VI. up to Section 21, 30 miles
from the generator, the levels of the sand being much the same from this
point up as in V.
A similar groin was then introduced at Section 16, extending from the
left bank to the middle of the estuary. This groin was 4| miles long and
100 feet high to H.W.L., and 50,000 more tides were run, the river all the
time slightly improving. Thus having brought deep water up to Section 14,
or about 44 miles from the generator, a groin extending from the right bank
to mid-channel at Section 8, about 2'5 miles long and 70 feet high, and
another from the left bank to mid-channel at Section 5, 2 miles long and
70 feet high, were put in.
The first effect of these groins was to raise the sand slightly in the mouth
of the river; but this improved again, and after 50,000 more tides there was
deep water extending from the mouth of the river to the generator, and the
river was better than in Experiment V. with the training walls, though not
quite so good as before these were put in.
In the meantime the banks had risen in the estuary below the groins,
extending down from nearly H.W.L. to the point of the next groin, where
there was a pass with water nearly to the bottom of the tank.
The sand carried down into the generator during the experiment
amounted to 69 Ibs., or 57 per cent, of the tidal capacity. In Experiment V.
24 Ibs. were removed in like manner, or 20 per cent, of the tidal capacity.
37 per cent, of the tidal capacity on a 30-foot tide would represent a mean
increase of depth over the entire estuary of 1 1 feet ; and as the increase was
by no means over the whole estuary, the increase in the channels and lower
estuary was much more than this, and although by this time the sand in the
estuary had for the most part become quite yellow, sand was still being
carried down into the generator.
In the meantime, as already stated, groins similar to those in Experi-
ment VI. in tank F, had been introduced into experiment VII. in tank E,
after 64,000 tides had been run with spring and neap tides. 60,000 more
tides, which would be equivalent to about 27,000 spring tides, were run,
the effect being that, notwithstanding the difference in the initial condi-
tions, the state of the lower estuary was closely approximating to the
state of VI. in F after 36,000 tides (Plan 2, Experiment VII, tank E;
VI., tank F).
In the upper estuary in VII., tank E, the distribution of the sand is
precisely similar to that in VI., tank F, but there is rather more of it,
which is explained partly by the fact of the difference in the equivalent
59] ON THE ACTION OF WAVES AND CURRENTS. 503
tides run, 30,000 in E as against 50,000 in F, after the upper groins were
put in, and partly by the much greater amount of sand still leftinlhe lower
estuary in tank E. Had it been possible to run 250,000 more spring and
neap tides in VII, tank E, there is every reason to suppose that the final
condition would have been precisely similar to that obtained in Experiment
VI. in tank F.
TABLE I. GENERAL CONDITIONS
Per-
Horizontal scale
Shape of the
Estuary
cent-
nge of
Land
Water
Experi
ment
Tank
Plan
Plan
Oil
Page
Shortest
period
in
seconds
Vertica
scale
Inches
1 in. to a
mile
50 _;
0-5
III
E
1
507
46-16
14,901 4-25
240
miles 1
V
n
F
1
507
30-53
25,844 2-45
315
'
11
IV
E
1
514
33-47
20,550
3-01
240
11
11
F
1
22-20
38,256 1-65
365
n
11
E
2
508
33-20
22,090
2-78
273
11
11
F
2
508
22-03
38,788
1 -f>3
370
11
V
E
1
515
35-6
19,558
3-24
246
11
„
F
1
509
23-68
36,310
1-74
375
«
£
11
»
E
2
509
35-6
19,972
3-172
256
"3
11
11
F
2
23-32
36,890
1-718
375
H
11
11
E
3
510
35-60
20,833
3-03
280
C
o
^3
11
11
F
3
510
23-32
38,955
1-63
416
**
g
Train- I
11
11
E
4
516
35-60
20,691
3-06
275
B
ing |
1
o
Walls (
11
11
F
4
516
23-32
39,700
1-60
435
W
1
Quick (
rise
11
11
E
5
35-60
21,285
2-97
291
I
f
11
11
11
6
35-60
19,095
3-318
234
op
11
VI
11
1
35-78
18,230
3-475
215
Spring
and
11
5)
11
2
514
36-26
20,000
3-168
252
Neap
Tides
11
VII
•i
1
515
35-10
19,756
3-207
244
1 "
»
11
2
517
35-10
20,890
3-033
273
,
1
11
»
11
4
518
—
—
—
—
Unsym-
11
VI
F
1
517
23-40
39,564
1-605
434
metrical
11
11
»
2
23-40
38,465
1-647
411
11
11
n
3
23-40
39,854
1-589
411
1
\ 11
11
11
4
518
23-40
39,280
1-613
428
, fe ( Spring and
•§ •= 1 Neap Tides
o-o
XIII
A
3
512
48-00
12,473
5-08
182
p§ g> I Uniform
« I Tides
11
V
"
1
511
50-00
11,758
5-45
170
AND RESULTS OF EXPERIMENTS.
Else of
tide in
feet
Vertical
exaggera-
tion on
a 30-feet
tide e
Criterion of
similarity
Height
of initial
sand in
feet
Height
of mean
tide in
feet
Excess
of mean
tide over
initial
sand in
feet d
Number
of tides
from the
start
Remarks
C'=h3e
C' =
(h + 2d)»e
0-125
62-00
0-121
—
0-333
0-322
—
47,183
Normal.
0-095
81-84
0-070
0-070
0-187
0-187
—
66,369
River blocked.
0-125
85-63
0-167
—
0-333
0-310
—
18,530
Eiver cleaned.
0-082
104-57
0-057
—
0-187
0-182
0-005
21,135
River blocking.
0-110
80-98
0-108
—
0-333
0-308
—
37,755
River cleaned.
0-081
104-73
0-056
—
0-187
0-179
0-008
38,719
(River nearly
\ blocked.
0-122
79-53
0-144
—
0-333
0-336
—
17,923
Slow.
0-080
96-82
0-049
—
0-187
0-203
0-016
19,416
Quicker.
0-117
77-88
0-124
—
0-333
0-321
—
37,359
River cleaned.
0-080
98-32
0-050
0-165
0-187
0-203
0-016
37,181
^Blocking —
( Dredged.
0-107
74-48
0-091
—
0-333
0-320
—
65,404
River clear.
0-072
93-49
0-035
—
0-187
0-207
0-020
95,558
River clear.
0-109
0-069
75-18
91 -32
0-097
0-030
:
0-333
0-187
0-306
0-204
0-017
167,186
255,200
[Similar.
0-103
73-08
0-080
—
0-333
0-335
—
208,264
Failure.
0-128
81-47
0-170
—
0-333
0-328
—
226,930
Preliminary.
0-139
84-46
0-2268
—
0-333
0-325
—
20,822
Quick.
0-119
79-33
0-1336
—
0-333
0-317
—
34,394
River clear.
0-123
81-00
0-1507
-
0-333
0-333
—
51,591
Normal.
0-110
76-60
0-1017
—
0-333
0-332
—
101,790
—
—
—
—
—
—
—
—
122,989
—
0-069
91-01
0-0299
—
0-187
0-192
0-005
18,972
—
0-073
93-60
0-0360
—
0-187
0-193
0-006
36,511
—
0-068
93-33
0-0284
—
0-187
0-193
0-006
99,558
—
0-070
91-66
0-0314
—
0-187
0-192
0-005
196,651
__
0-165
0-176
IJS-.-i 1
69-16
0-3084
0-3769
—
0-250
0-250
—
—
51,240
16,282
f Similar.
506
s s
£
507
I !*
, '
508
: :
r is
509
510
ii
*£
I
511
¥
MJy? /turyr yyiay jnoyutv
512
513
0. R. II.
33
514
-Ult
il
515
>
33—2
516
i#R
517
=1
d <
jfiSja
If
518
•s
60.
ON TWO HARMONIC ANALYZERS.
[From the Fourth Volume of the Fourth Series of " Memoirs and Proceedings
of the Manchester Literary and Philosophical Society." Session 1890 — 91.]
(deceived April 2nd, 1891.)
THE object of these instruments is to afford a ready means of ascertaining
the periods of free vibration of structures or members of structures. If any
portion of a material structure (i.e., an elastic structure) is disturbed from
its normal position of equilibrium and suddenly released, the structure is
thrown into a complex state of vibration, which gradually subsides. While
the vibration lasts each point in the structure goes through movements
which may be very complex, but which are, nevertheless, compounded of
simple periodic or harmonic movements, each simple movement taking place
in a definite direction, as well as having a definite period.
The art of measuring and recording the complex movements at a point
of the earth during an earthquake has long been a study, and the seismometer
of Professor Ewing has been applied to record the movements of points of
various structures when subjected to disturbances. The principle of these
seismometers consists in attaching a weight to the point of the structure to
be examined, by attachments of such slight elasticity, that the disturbances
communicated to the weight are insensibly small, and the weight remains
sensibly steady amid the surrounding vibrations, and forms a steady observa-
tory from which the vibrations may be measured. This measurement is
effected by causing pencils vibrating with the structure to describe lines on
cards attached to the steady weight, or vice versa, the cards being fixed, or
having a time movement. In this way the complex motions of the points
are beautifully recorded, as in Prof. Ewing's experiments on the Tay Bridge,
and Prof. J. Milne's numerous experiments in railway carriages, &c.
Such curves represent the complex movements of the point of the
structure examined ; and any analysis of the motion into its simple periodic
components remains to be accomplished by mathematical reduction — or by
520 ON TWO HARMONIC ANALYZERS. [60
such instrumental synthesis as that which may be effected in Sir William
Thomson's " Harmonic Analyzer."
The Harmonic Analyzers about to be described differ essentially from
the seismometer in that they do not measure or record the actual motions of
the structure, while they single out and exaggerate any component periodic
motion according to its period and direction, which are defined in the instru-
ments. The principle of these Harmonic Analyzers is that of the accumu-
lation of motion which takes place, when a weight is subject to a periodic
disturbance which coincides in period and direction with that of free vibra-
tion of which the weight is susceptible.
If a small weight w be elastically attached to a much heavier weight so
that it requires a definite force (El) to disturb the weight (w) through a
distance I, the large weight remaining at rest ; then, if released after any
disturbance, the small weight w will vibrate in the direction of disturbance,
and with a constant period of:
A /
V
— ^r seconds,
i.e. in the period of free vibration of the small weight.
If the small weight be at rest and the large weight be subject to a
periodic disturbance having a period 1/w; then, if this period is larger
than the period of free vibration of the small weight, i.e., if
- is smaller than 2-n- A / -
n V gE'
the small weight will follow essentially the movements of the larger weight
as if rigidly attached, while if the period of motion of. the larger weight is
smaller than that of the period of free vibration of the small weight, the
small weight will remain virtually at rest. But when the period of motion
of the large weight coincides with the period of free vibration of the small
weight, the small weight will take and accumulate the disturbance, oscillat-
ing with increasing amplitude until it reaches such an extent that the
energy dissipated is equal to that received from the disturbance. If the
elasticity of the connections be fairly perfect, the amplitude of the small
weight will be very considerable, although the disturbing motion is otherwise
insensible.
If the small weight (w) has only one degree of freedom, i.e., if the
elasticity of the connections is not equal in all directions, there will be three
axes of elasticity, and if the elasticities along two of these directions are
much greater than the third this is the direction of freedom ; then, when the
period of free vibration along the third axis, i.e., in the direction of freedom,
coincides with the period of disturbance, the small weight will only take up
60]
ON TWO HARMONIC ANALYZERS.
521
the disturbance when this has a component in the direction of freedom ;
that is, if the direction of the disturbance is at right angles to the direction
of freedom, there will be no vibration. So that in this way the direction of
the disturbance may be ascertained, or vice versa.
Similar results follow if, instead of the disturbance coming through the
elastic supports, the body be subject to a synchronous periodic force. If the
period of the force were not synchronous with any of the three periods of
free vibration corresponding respectively to the three axes of elasticity, the
resulting vibration would, as before, merely correspond with the time effect
of the force, but on coincidences with any one of these, unless the direction
of the disturbance were at right angles to that of the axis of elasticity, the
body would accumulate the disturbance.
It thus appears that, if a structure is in a state of vibration, the periods
of free vibration and their directions may be ascertained by an Harmonic
Analyzer consisting of a small weight with elastic attachments, so adjustable
that the period of free vibration of the weight can be varied to any required
extent, and the direction of such free vibration turned through all requisite
angles.
This may be accomplished in many ways. That which I have so far
adopted with satisfactory success has been very simple.
2l6il5"ll6'
Fig. 1.
It consists, as shown in Fig. 1, essentially of a base formed of a bar of
hard wood, one-and-a-half inches square, and two feet long, a cross notch
being cut in one end to enable this end to be held against any point of the
structure with less chance of slipping. About four inches from the notched
end, right across the axis of this bar, is a hole, in which is fitted, with
moderate tightness, a piece of straight steel wire, one-eighth of an inch in
diameter, and 18 inches long. On one end of the wire is a ball of lead,
522 ON TWO HARMONIC ANALYZERS. [60
about 2 oz., through the centre of which is a small hole at right angles to
the wire, in which is fixed a small graphite pencil. On the other end of the
wire is a carrier, to afford handhold for the purpose of adjusting the wire in
the hole.
When the carrier is pushed right up to the wood, the ball, if disturbed,
will vibrate in any direction perpendicular to the wire so as to make about
200 oscillations a minute, which is slower than any period it is required to
measure. As the carrier is pulled back, and the wire between the base and
the ball shortened, the rate of vibration increases, until, when the wire is
only 1^ inches long, the ball, when disturbed, gives out an audible note of
about 2,000 vibrations a minute.
The instrument is used by holding in one hand the longer end of the
wood, and pressing the notched end hard against the point of the structure
of which the motion is to be analyzed, the carrier having previously been
pushed up to the wood, then, with the free hand, the carrier is pulled
steadily back, the ball being carefully watched. As by the shortening of the
wire between the base and the ball the free period of vibration of the ball is
diminished, and comes near to any period amongst the vibrations in the
structure, the ball is seen to take up the vibration in beats with intervals of
rest; and a very little more careful adjustment is sufficient to bring the
period into coincidence, when the ball continues vibrating with the structure,
having the appearance in Fig. 2
The period of the Analyzer having been thus adjusted to that of one of
the periods of free vibration of the structure, the period is ascertained either
by adjusting the Analyzer so that the pencil in the ball may oscillate in con-
tact with the paper on a chronograph, or by measuring the distance of the
ball from the wood on a scale, previously adjusted by aid of the chronograph
to give the number of vibrations per minute.
Extreme accuracy of determining the periods has not so far been an im-
portant consideration. The readings on the chronograph were only taken to
about 10°/0. But that the Analyzer is susceptible of much greater accuracy
is shown by the fact that several different adjustments to the same period in
the structure brought the wire into exactly the same position.
60] ON TWO HARMONIC ANALYZERS. 523
Its power of analyzing complex vibrations is so far unqualified. It was
invented for the purpose of determining the period of a particular vibration —
in a very stiff iron structure subject to the periodic disturbance of the belts
from two engines running at high speed, and the centrifugal action of such
want of balance as there might be in heavy pulleys, three feet in diameter,
and running at 500 revolutions per minute. The vibration was very slight —
nothing more than a slight tremor could be felt with the hand. The
periodic disturbances were about 500 per minute, and these came out clearly,
but small, in the Analyzer when adjusted to these periods — but the periods
of free vibration of one of the members, 720 per minute, caused an am-
plitude of half an inch in the ball, and that of another, 1,270, was easily
identified.
The instrument already described can clearly only be used on a structure
while it is so disturbed as to set its members vibrating. Such disturbances
can generally be set up by a shock of some sort, but when it is necessary to
cause artificial disturbance, it is better to adopt a periodic disturbance of
such varying period as will come gradually into coincidence with the periods
of free vibration, bringing these vibrations out separately, when they will be
readily identified with the Analyzer, if not otherwise perceptible.
For this purpose, in 1887, I adopted the following method: — A small
cast-iron pulley, 6 inches in diameter, very much out of balance, was
mounted on a small frame that could be clipped on to any part of the
structure, and a cord passed over this pulley on to a larger wheel, which was
turned by hand. In this way the unbalanced wheel was driven at a
gradually increasing rate until steady vibrations in the structure were
observed, then these coincided with the period of the unbalanced wheel, and
this was ascertained to be about 1,200 by counting the revolutions of this
hand-wheel. At this speed the disturbing force resulting from the un-
balanced weight, 2 Ibs. on a radius of 2 inches, would be 40 Ibs. The
structure thus under examination was an iron standard, very stiff. A
theodolite was adjusted, with the cross wires on a mark on the top of the
standard, which, when the period of the small unbalanced wheel coincided
with that of free vibration, was seen to move as much as one-twentieth of an
inch. Chains were then attached to the top of the standard, and by means
of blocks, a horizontal force of a ton was thrown on to the top of the
standard, when it did not yield more than two-hundredths of an inch. So
that the deviation caused by the periodic force of 40 Ibs., in such coincidence
with the period of free vibration as could be attained with the hand-wheel,
was three times as great as that which resulted from a direct statical force of
one ton.
61.
STUDY OF FLUID MOTION BY MEANS OF COLOURED BANDS.
[From the " Proceedings of the Royal Institution of Great Britain."]
(Read June 2, 1893.)
IN his charming story of The Purloined Letter, Edgar Allan Poe tells
how all the efforts and artifices of the Paris police to obtain possession of a
certain letter, known to be in a particular room, were completely baffled for
months by the simple plan of leaving the letter in an unsealed envelope in
a letter-rack, and so destroying all curiosity as to its contents ; and how the
letter was at last found there by a young man who was not a professional
member of the force. Closely analogous to this is the story I have to set
before you to-night — how certain mysteries of fluid motion, which have
resisted all attempts to penetrate them are at last explained by the
simplest means and in the most obvious manner.
This indeed is no new story in science. The method adopted by the
minister, D, to secrete his letter, appears to be the favourite of Nature in
keeping her secrets, and the history of science teems with instances in
which keys, after being long sought amongst the grander phenomena, have
been found at last not hidden with care, but scattered about, almost openly,
in the most commonplace incidents of every-day life which have excited no
curiosity.
This was the case in physical astronomy — to which I shall return after
having reminded you that the motion of matter in the universe naturally
divides itself into three classes.
1. The motion of bodies as a whole — as a grand illustration of which
we have the heavenly bodies, or more humble, but not less effective, the
motion of a pendulum or a falling body.
61] STUDY OF FLUID MOTION BY MEANS OF COLOURED BANDS. 525
2. The relative motion of the different parts of the same fluid or elastic
body — for the illustration of which we may go to the grand phenomena pre-
sented by the tide, the whirlwind, or the transmission of sound, but which
is equally well illustrated by the oscillatory motion of the wave, as shown
by the motion of its surface, and by the motion of this jelly, which, although
the most homely illustration, affords by far the best illustration of the pro-
perties of an elastic solid.
•S. The inter-motions of a number of bodies amongst each other — to
which class belong the motions of the molecules of matter resulting from
heat, as the motions of the molecules of a gas, in illustration of which I
may mention the motions of individuals in a crowd, and illustrate by the
motion of the grains in this bottle when it is shaken, during which the
white grains at the top gradually mingle with the black ones at the bottom
—which interdiffusion takes an important part in the method of coloured
bands.
Now of these three classes of motion, that of the individual body is
incomparably the simplest. Yet, as presented in the phenomena of the
heavens, which have ever excited the greatest curiosity of mankind, it
defied the attempts of all philosophers for thousands of years, until Galileo
discdvered the laws of motion of mundane matter. It was not until he had
done this, and applied these laws to the heavenly bodies, that their motions
received a rational explanation. Then Newton, taking up Galileo's parable
and completing it, found that its strict application to the heavenly bodies
revealed the law of gravitation, and developed the theory of dynamics.
Next to the motions of the heavenly bodies, the wave, the whirlwinds,
and the motions of clouds, had excited the philosophical curiosity of man-
kind from the earliest time. Both Galileo and Newton, as well as their
followers, attempted to explain these by the laws of motion, but although
the results so obtained have been of the utmost importance in the develop-
ment of the theory of dynamics, it was not till this century that any
considerable advance was made in the application of this theory to the
explanation of fluid phenomena, and although during the last fifty years
splendid work has been done, work which, in respect of the mental effort
involved, or the scientific importance of the results, goes beyond that which
resulted in the discovery of Neptune, yet the circumstances of fluid motion
are so obscure and complex, that the theory has yet been interpreted only
in the simplest cases.
To illustrate the difference between the interpretation of the theory of
the heavenly bodies and that of fluid motion, I would call your attention
to the fact that solid bodies, on the behaviour of which the theory of the
motion of the planets is founded, move as one piece, so that their motion
526 STUDY OF FLUID MOTION BY MEANS OF COLOURED BANDS. [61
is exactly represented by the motion of their surfaces ; that they are not
subject to any internal disorder which may affect their general motion. So
surely is this the case, that even those who have never heard of dynamics can
predict with certainty how any ordinary body will behave under any ordinary
circumstances, and so much so that any departure is a matter of surprise.
Thus I have here a cube of wood, to one side of which a string is attached.
Now hold it on one side, and holding the string you naturally suppose that
when I let go it will turn down so as to hang with the string vertical ; it
does not do so, that is a matter of surprise ; I place it on the other side
and it still remains as I place it. If I swing it as a pendulum it does not
behave like one.
Would Galileo have discovered the laws of motion had his pendulum
behaved like this ? Why is its motion peculiar ? There is internal motion.
Of what sort ? Well, I think my illustration may carry more weight
if I do not tell you ; you can all, I have no doubt, form a good idea. It
is not fluid motion or I should feel bound to explain it. You have here
an ordinary looking object which behaves in an extraordinary manner, which
is yet very decided and clear, to judge by the motion of its surface, and
from the manner of the motion I wish you to judge of the cause of the
observed motion*.
This is the problem presented by fluids, in which there may be internal
motion which has to be taken into account before the motion of the surface
can be explained. You can see no more of what the motion is within a
homogeneous fluid, however opaque or clear, than you can see what is going
on within the box. Thus, without colour bands the only visual clue to what
is going on within the fluids is the motion of their bounding surfaces. Nor
is this all ; in most cases the surfaces which bound the fluid are immovable.
In the case of the wave on water the motion of the surface shows that
there is motion, but because the surface shows no wave it does not do to
infer that the fluid is at rest.
The only surfaces of the air within this room are the surfaces of the
floor, walls, and objects within it. By moving the objects we move the air,
but how far the air is at rest you cannot tell unless it is something familiar
to you.
Now I will ask you to look at these balloons. They are familiar objects
enough, and yet they are most sensitive anemometers, more sensitive than
anything else in the room ; but even they do not show any motion ; each
of them forms an internal bounding surface of the air. I send an aerial
* In this experiment a cubical box of wood, apparently a solid block, contained a heavy
spinning top.
61] STl'DY OF FLUID MOTION BY MEANS OF COLOURED BANDS. 5*27
messenger to them, and a small but energetic motion is seen by which it
acknowledges the message, and the same message travels through- the rest,
as if a ghost touched them. It is a wave that moves them. You do not
feel it, and. but for the surfaces of the air formed by the balloons, would
have no notion of its existence*.
In this tank of beautifully clear distilled water, I project a heavy ball in
from the end, and it shows the existence of the water by stopping almost
dead within two feet. The fact that it is stopped by the water, being
familiar, does not raise the question, Why does it stop? — a question to
which, even at the present day, a complete answer is not forthcoming. The
question is, however, suggested, and forcibly suggested, when it appears that
with no greater or other evidence of its existence, I can project a disturbance
through the water which will drive this small disc the whole length of the
tank.
1 have now shown instances of fluid motion of which the manner is in
no way evident without colour bands, and were revealed by colour bands, as
I showed in this room sixteen years ago. At that time I was occupied in
setting before you the manners of motion revealed, and I could only inci-
dentally notice the means by which this revelation was accomplished.
Amongst the ordinary phenomena of motion there are many which
render evident the internal motion of fluids. Small objects suspended in
the fluid are important, and that their importance has long been recognised
is shown by the proverb — straws show which way the wind blows. Bubbles
in water, smoke and clouds, afford the most striking phenomena, and it is
doubtless these that have furnished philosophers with such clues as they
have had. But the indications furnished by these phenomena are imperfect,
and, what is more important, they only occur casually, and in general only
under circumstances of such extreme complexity that any deduction as to
the elementary motions involved is impossible. They afford indication of
commotion, and perhaps of the general direction in which the commotion
is tending, but this is about all.
For example, the different types of clouds ; these have always been noticed
and are all named. And it is certain that each type of clouds is an indication
of a particular type of motion in the air; but no deductions as to what
definite manner of motion is indicated by each type of cloud have ever been
published.
Before this can be done it is necessary to reverse the problem, and find
to what particular type of cloud a particular manner of motion would give
* By means of a large box, having a hinged door on one side, and a circular aperture on the
side opposite, invisible vortex rings of air were projected towards the balloons.
528 STUDY OF FLUID MOTION BY MEANS OF COLOURED BANDS. [61
rise. Now a cloud, as we see it, does not directly indicate the internal
motion of which it is the result. As we look at clouds, it is not in general
their motion that we notice, but their figure. It is hard to see that this
figure changes while we are watching a cloud, though such a change is
continually going on, but is apparently very slow on account of the great
distance of the cloud and its great size. However, types of clouds are
determined by their figure, not by their motion. Now what their figure
shows is not motion, but is the history or result of the motion of particular
strata of the air in and through surrounding strata. Hence, to interpret
the figures of the clouds we must study the changes in shape of fluid
masses, surrounded by fluid, which result from particular motions.
The ideal in the method of colour bands is to render streaks or lines in
definite position in the fluid visible, without in any way otherwise interfering
with these properties as part of the homogeneous fluid. If we could by a
wish create coloured lines in the water, these would be ideal colour bands.
We cannot do this, nor can we exactly paint lines in the air or water.
I take this ladle full of highly coloured water, lower it slowly into the
surface of the surrounding water till that within is level with that without ;
then turn the ladle carefully round the coloured water ; the mass of coloured
water will remain where placed.
I distribute the colour slowly. It does not mix with the clear water, and
although the lines are irregular they stand out very beautifully. Their
edges are sharp here. But in this large sphere, which was coloured before
the lecture, although the coloured lines have generally kept their places,
they have, as it were, swollen out and become merged in the surrounding
water in consequence of molecular motion. The sphere shows, however, one
of the rarest phenomena in Nature — the internal state in almost absolute
internal rest. The forms resemble nothing so much as stratus clouds, as
seen on a summer day, though the continuity of the colour bands is more
marked. A mass of coloured water once introduced is never broken. The
discontinuity of clouds is thus seen to be due to other causes than mere
motion.
Now, having called your attention to the rarity of water at rest, I will
call your attention to what is apt to be a very striking phenomenon, namely,
that when water is contained, like this, in a spherical vessel of which you
cannot alter the shape, it is impossible by moving the vessel suddenly to set
up relative motion in the interior of the water. I may swing this vessel
about and turn it, but the colour band in the middle remains as it was,
and when I stop shows the water to be at rest.
This is not so if the water has a free surface, or if the fluid is of unequal
density. Then a motion of the vessel sets up waves, and the colour band
61] STUDY OF FLUID MOTION BY MEANS OF COLOURED BANDS. 529
shows at once the beautifully lawful character of the internal motion. The
colour bands move backwards and forwards, showing how the water is dis-
torted like a jelly, and as the wave dies out the colour bands remain as they
were to begin with.
This illustrates one of the two classes of internal motion of water or fluid.
Wherever fluid is not in contact with surfaces over which it has to glide,
or surfaces which fold on themselves, the internal motions are of this purely
wave character. The colour bands, however much they may be distorted,
cannot be relatively displaced, twisted, or curled up, and in this case motion
in water once set up continues almost without resistance. That wave motion,
in water with a free surface, is one of the most difficult things to stop, is
directly connected with the difficulty of setting still water in motion ; in
either case the influence must come through the surfaces. Thus it is that
waves once set up will traverse thousands of miles, establishing communica-
tion between the shores of Europe and America. Wave motion in water is
subject to enormously less resistance than any other form of material motion.
In wave motion, if the colour bands are across the wave they show the
motion of the water ; nevertheless, their chief indication is of the change of
shape while the fluid is in motion.
This is illustrated in this long bottle, with the coloured water less heavy
than the clear water. If I lay it down in order to establish equilibrium, the
blue water has to leave the upper end of the bottle and spread itself over the
clear water, while the clear water runs under the coloured. This sets up
wave motion, which continues after the bottle has come to rest. But as the
colour bands are parallel with the direction of motion of the waves, the
motion only becomes evident in thickening and bending of the colour bands.
The waves are entirely between the two fluids, there being no motion in
the outer surfaces of the bottle, which is everywhere glass. They are owing
to the slight differences in the density of the fluids, as is indicated by the
extreme slowness of the motion. Of such kind are the waves in the air,
that cause the clouds which make the mackerel sky, the vapour in the tops
of the waves being condensed and evaporated again as it descends, showing
the results of the motion.
The distortional motions, such as alone occur in simple wave motion,
or where the surfaces of the fluid do not fold in on themselves, or wind in,
are the same as occur in any homogeneous continuous material which com-
pletely fills the space between the surfaces.
If plastic material is homogeneous in colour, it shows nothing as to the
internal motion; but if I take a lump built of plates, blue and white, say a
square, then I can change the surfaces to any shape without folding or
o. R. n. 34
530 STUDY OF FLUID MOTION BY MEANS OF COLOURED BANDS. [61
turning the lump, and the coloured bands which extend throughout the lump
show the internal changes. Now the first point to illustrate is that, however
I change its shape, if I bring it back to the original shape the colour bands
will all come back to their original positions, and there is no limit to the
extent of the change that may thus be effected. I may roll this out to any
length, or draw it out, and the diminution in thickness of the colour bands
shows the extent of the distortion. This is the first and simplest class
of motion to which fluids are susceptible. By this motion alone elements
of the fluid may be, and are, drawn out to an indefinitely fine line, or spread
out in an indefinitely thin sheet, but they will remain of the same general
figure.
By reversing the process they change back again to the original form.
No colour band can ever be broken, even if the outer surface be punched in
till the punch head comes down on the table ; still all the colour bands are
continuous under the punch, and there is no folding or lapping of the colour
bands unless the external surface is folded.
The general idea of mixture is so familiar to us that the vast generaliza-
tion to which these ideas afford the key, remains unnoticed. That continued
mixing results in uniformity, and that uniformity is only to be obtained by
mixing, will be generally acknowledged, but how deeply and universally this
enters into all the arts can but rarely have been apprehended. Does it ever
occur to any one that the beautiful uniformity of our textile fabrics has only
been obtained by the development of processes of mixing the fibres ? Or,
again, the uniformity in our construction of metals ; has it ever occurred to
any one that the inventions of Arkwright and Cort were but the application
of the long-known processes by which mixing is effected in culinary opera-
tions ? Arkwright applied the draw-rollers to uniformly extend the length
of the cotton sliver at the expense of the thickness ; Cort applied the rolling-
mill to extend the length of the iron bloom at the expense of its breadth ;
but who invented the rolling-pin by which the pastry-cook extends the
length at the expense of the thickness of the dough for the pie-crust ?
In all these processes the object, too, is the same throughout — to obtain
some particular shape, but chiefly to obtain a uniform texture. To obtain
this nicety of texture it is necessary to mix up the material, and to accom-
plish this it is necessary to attenuate the material, so that the different parts
may be brought together.
The readiness with which the fluids are mixed and uniformity obtained
is a by- word; but it is only when we come to see the colour bands that we
realize that the process by which this is attained is essentially the same as
that so laboriously discovered for the arts — as depending first on the atten-
uation of each element of the fluid — as I have illustrated by distortion.
61] STUDY OF FLUID MOTION BY MEANS OF COLOURED BANDS. 531
In fluids, no less than in cooking, spinning, and rolling — this attenuation
is only the first step in the process of mixing — all involve the second process,
that of folding, piling, or wrapping, by which the attenuated layers are
brought together. This does not occur in the pure wave motion of water,
and constitutes the second of the two classes of motion. If a wave on water
is driven beyond a certain height it leaps or breaks, folding in its surface.
Or, if I but move a solid surface through the water it introduces tangential
motion, which enables the fluid to wind its elements round an axis. In these
ways, and only in these ways, we are released from the restriction of not
turning or lapping. And in our illustration, we may fold up our dough,
or lap it — roll it out again and lap it again ; cut up our iron bar, pile it,
and roll it out again, or bring as many as we please of the attenuated fibres
of cotton together to be further drawn. It may be thought that this
attenuation and wrapping will never make perfect admixture, for, however
thin, each element will preserve its characteristic, the coloured layers will be
there, however often I double and roll out the dough. This is true. But in
the case of some fluids, and only in the case of some fluids, the physical
process of diffusion completes the admixture. These colour bands have
remained in this water, swelling but still distinct ; this shows the slowness of
diffusion. Yet such is the facility with which the fluid will go through the
process of attenuating its elements and enfolding them, that by simply
stirring with a spoon these colour bands can be drawn and folded so fine
that the diffusion will be instantaneous, and the fluid become uniformly
tinted. All internal fluid motion other than simple distortion, as in wave
motion, is a process of mixing, and it is thus from the arts that we get
the clue to the elementary forms and processes of fluid motion.
When I put the spoon in and mixed the fluid you could not see what
went on — it was too quick. To make this clear, it is necessary that the
motion should be very slow. The motion should also be in planes at right
angles to the direction in which you are looking. Such is the instability of
fluid that to accomplish this at first appeared to be difficult. At last,
however, as the result of much thought, I found a simple process which
I will now show you, in what I think is a novel experiment, and you will see,
what I think has never been seen before by any one but Mr Foster and
myself, namely, the complete process of the formation of a cylindrical vortex
sheet resulting from the motion of a solid surface. To make it visible to all
I am obliged to limit the colour band to one section of the sheet, otherwise
only those immediately in front would be able to see between the con-
volutions of the spiral. But you will understand that what is seen is a
section, a similar state of motion extending right across the tank. From the
surface you see the plane vane extending half-way down right across the
tank; this is attached to a float.
34—2
532 STUDY OF FLUID MOTION BY MEANS OF COLOURED BANDS. [61
Out of the tube I now institute a colour band on the right of the vane.
There is no motion in the water, and the colour descends slowly from the
tube. I now give a small impulse to the float to move it to the right, and
at once the spiral form is seen from the tube. Similar spirals would be
formed all across the tank if there were colours. The float has moved out of
the way, leaving the revolving spiral with its centre stationary, showing that
the horizontal axis of the spiral is half-way between the bottom and surface
of the tank, in which the water is now simply revolving round this axis.
This is the vortex in its simplest and rarest form (for a vortex cannot
exist with its ends exposed). Like an army it must have its flanks protected ;
hence a straight vortex can only exist where it has two surfaces to cover its
flanks, and parallel vertical surfaces are not common in nature. The vortex
can bend, and, as with a horse-shoe axis, can rest both its flanks on the
same surface, as this piece of clay, or with a ring axis, which is its commonest
form, as in the smoke ring. In both these cases the vortex will be in motion
through the fluid, and less easy to observe.
These vortices have no motion beyond the rotation because they are
half-way down the tank. If the vane were shorter they would follow the
vane ; if it were longer they would leave it.
In the same way, if instead of one vortex there were two vortices,
with their axis parallel, extending right across, the one above another, they
would move together along the tank.
I replace the float by another which has a vane suspended from it,
so that the water can pass both above and below the vane extending right
across the middle portion of the tank. In this case I institute two colour
bands, one to pass over the top, the other underneath, the vane, which colour
bands will render visible a section of each vortex just as in the last case.
I now set the float in motion and the two vortices turn towards each other
in opposite directions. They are formed by the water moving over the
surface of the vane, downwards to get under it, upwards to get over it, so
that the rotation in the upper vortex is opposite to that in the lower.
All this is just the same as before, but that instead of these vortices standing
still as before they follow at a definite distance from the vane, which con-
tinues its motion along the tank without resistance.
Now this experiment shows, in the simplest form, the modus operandi by
which internal waves can exist in fluid without any motion in the external
boundary. Not only is this plate moving flatwise through the water, but it
is followed by all the water, coloured arid uncoloured, enclosed in these
cylindrical vortices. Now, although there is no absolute surface visible, yet
there is a definite surface which encloses these moving vortices, and separates
them from the water which moves out of their way. This surface will be
61] STUDY OF FLUID MOTION BY MEANS OF COLOURED BANDS. 533
rendered visible in another experiment I shall show you. Thus the water
which has only wave motion is bounded by a definite surface, the motion of
which corresponds to the wave; but inside this closed surface there is also
water, so that we cannot see the surface, and this water inside is moving
round and round, but so that its motion at the bounding surface is every-
where the same as that of the outside water.
The two masses of water do not mix. That outside moves over the
bounding surface, out of the way of and past the vortices, while the vortices
move round arid round inside the surface in such a way that they are
moving in exactly the same manner at the surface as the wave surface
outside.
This is the key to the internal motion of water. You cannot have a pure
wave motion inside a mass of fluid with its boundaries at rest, but you have
a compound motion, a wave motion outside, and a vortex within, which
fulfils the condition that there shall be no sliding of the fluid over fluid at
the boundary.
A means, which I hope may make the essential conditions of this motion
clearer, occurred to me while preparing this lecture, and to this I will now
;isk your attention. I have here a number of layers of cotton-wool (wadding).
Now I can force any body along between these layers of wadding. They
yield, as by a wave, and let it go through ; but the wadding must slide over
the surface of the body so moving through it. And this it must not do if it
illustrate the conditions of fluid motion. Now there is one way, and only
one way, in which material can be got through between the sheets of wadding
without slipping. It must roll through ; but this is not enough, because if it
rolls on the under surface it will be slipping on the upper. But if we have
two rollers, one on the top of the other, between the sheets, then the lower
roller rolls on the bottom sheet, the upper roller rolls against the upper sheet,
so that there is no slipping between the rollers or the wadding, and, equally
important, there is no slipping between the rollers, as they roll on each other.
I have only to place a sheet of canvas between the rollers and draw it
through ; both the flannel rollers roll on the canvas and on the wadding,
which they pass through without slipping, causing the wadding to move
in a wave outside them, and affording a complete parallel of the vortex
motion.
I will now show by colour bands some of the more striking phenomena
of internal motion, as presented by Nature's favourite form of vortex, the
vortex ring, which may be described as two horse-shoe vortices with their ends
founded on each other.
To show the surface separating the water moving with the vortex, from
534 STUDY OF FLUID MOTION BY MEANS OF COLOURED BANDS. [61
that which gives way outside, I discharge from this orifice a mass of coloured
water, which has a vortex ring in it formed by the surface as already
described. You see the beautifully defined mass moving on slowly through
the fluid, with the proper vortex ring motion, but very slow. It will not go
far before a change takes place, owing to the diffusion of the vortex motion
across the bounding surface ; then the coloured surface will be wound into
the ring which will appear. The mass approaches the disc in front. It
cannot pass, but will come up and carry the disc forward; but the disc,
although it does not destroy the ring, disturbs the motion.
If I send a more energetic ring it will explain the phenomenon I showed
you at the beginning of this lecture ; it carries the disc forward as if struck
with a hammer. This blow is not simply the weight of the coloured ring, but
of the whole moving mass and the wave outside. The ring cannot pass the
disc without destruction, with the attendant wave.
Not only can a ring follow a disc, but as with the plane vane so with the
disc, if we start a disc we must start a ring behind it.
I will now fulfil my promise to reveal the silent messenger I sent to those
balloons. The messenger appears in the form of a large smoke ring, which is
a vortex ring in air rendered visible by smoke instead of colour. The
origination of these rings has been carefully set so that the balloons are
beyond the surface which separates the moving mass of water from the
wave, so that they are subject to the wave motion only. If they are within
this surface they will disturb the direction of the ring, if they do not break
it up.
These are, if I may say so, the phenomenal instances of internal motion
of fluids. Phenomenal in their simplicity, they are of intense interest, like
the pendulum, as furnishing the clue to the more complex. It is by the light
we gather from their study that we can hope to interpret the parallel of the
vortex wrapped up in the wave, as applied to the wind of heaven, and the
grand phenomenon of the clouds, as well as those things which directly
concern us, such as the resistance of ships.
62.
ON THE DYNAMICAL THEORY OF INCOMPRESSIBLE VIS-
COUS FLUIDS AND THE DETERMINATION OF THE
CRITERION.
[Fro7n the " Philosophical Transactions of the Royal Society," 1895.]
(Read May 24, 1894.)
SECTION I.
Introduction.
1. THE equations of motion of viscous fluid (obtained by grafting on
certain terms to the abstract equations of the Eulerian form, so as to adapt
these equations to the case of fluids subject to stresses depending in some
hypothetical manner on the rates of distortion, which equations Navier*
seems to have first introduced in 1822, and which were much studied by
Cauchyt and PoissonJ) were finally shown by St Venant§ and Sir Gabriel
Stokes||, in 184=5, to involve no other assumption than that the stresses,
other than that of pressure uniform in all directions, are linear functions of
the rates of distortion, with a coefficient depending on the physical state of
the fluid.
By obtaining a singular solution of these equations as applied to the
case of pendulums in steady periodic motion, Sir G. StokesH was able to
compare the theoretical results with the numerous experiments that had
* Mem. de I'Academie, vol. vi. p. 389.
t Mem, des Sacmitx Klrnngers, vol. I. p. 40.
J Mem. de VAcademie, vol. x. p. 345.
§ B.A. Report, 1840.
|| Cambridge Phil. Tnin*., 1845.
IT Ibid., vol. ix. 1857.
536 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62
been recorded, with the result that the theoretical calculations agreed so
closely with the experimental determinations as seemingly to prove the
truth of the assumption involved. This was also the result of comparing
the flow of water through uniform tubes with the flow calculated from a
singular solution of the equations, so long as the tubes were small and the
velocities slow. On the other hand, these results, both theoretical and
practical, were directly at variance with common experience as to the
resistance encountered by larger bodies moving with higher velocities
through water, or by water moving with greater velocities through larger
tubes. This discrepancy Sir G. Stokes considered as probably resulting
from eddies, which rendered the actual motion other than that to which
the singular solution referred, and not as disproving the assumption.
In 1850, after Joule's discovery of the Mechanical Equivalent of Heat,
Stokes showed, by transforming the equations of motion — with arbitrary
stresses — so as to obtain the equations of ("Vis-viva") energy, that this
equation contained a definite function, which represented the difference
between the work done on the fluid by the stresses and the rate of increase
of the energy, per unit of volume, which function, he concluded, must,
according to Joule, represent the Vis- viva converted into heat.
This conclusion was obtained from the equations irrespective of any
particular relation between the stresses and the rates of distortion. Sir G.
Stokes, however, translated the function into an expression in terms of the
rates of distortion, which expression has since been named by Lord Rayleigh
the Dissipation- Function.
2. In 1883 I succeeded in proving, by means of experiments with colour
bands — the results of which were communicated to the Society* — that when
water is caused by pressure to flow through a uniform smooth pipe, the motion
of the water is direct, i.e., parallel to the sides of the pipe, or sinuous, i.e.,
crossing and re-crossing the pipe, according as Um, the mean velocity of the
water, as measured by dividing Q, the discharge, by A, the area of the
section of the pipe, is below or above a certain value given by
where D is the diameter of the pipe, p the density of the water, and K a
numerical constant, the value of which according to my experiments, and, as
I was able to show, to all the experiments by Poiseuille and Darcy, is for
pipes of circular section between
1900 and 2000,
* Phil. Trains., 1883, Part III. p. 935. (See this vol. p. 51.)
62] AND THE DETERMINATION OF THE CRITERION. 537
or, in other words, steady direct motion in round tubes is stable or unstable
according as
> 1900 or < 2000,
the number K being thus a criterion of the possible maintenance of sinuous
or eddying motion.
3. The experiments also showed that K was equally a criterion of the
law of the resistance to be overcome — which changes from a resistance
proportional to the velocity, and in exact accordance with the theoretical
results obtained from the singular solution of the equation, when direct
motion changes to sinuous, i.e., when
DUm
p— - = K.
P
4. In the same paper I pointed out that the existence of this sudden
change in the law of motion of fluids between solid surfaces when
P
proved the dependence of the manner of motion of the fluid on a relation
between the product of the dimensions of the pipe multiplied by the velocity
of the fluid, and the product of the molecular dimensions multiplied by the
molecular velocities which determine the value of
for the fluid, also that the equations of motion for viscous fluid contained
evidence of this relation.
These experimental results completely removed the discrepancy previously
noticed, showing that, whatever may be the cause, in those cases in which
the experimental results do not accord with those obtained by the singular
solution of the equations, the actual motions of the water are different.
But in this there is only a partial explanation, for there remains the
mechanical or physical significance of the existence of the criterion to be
explained.
5. [My object in this paper is to show that the theoretical existence of
an inferior limit to the criterion follows from the equations of motion as
a consequence : —
(1) Of a more rigorous examination and definition of the geometrical
basis on which the analytical method of distinguishing between molar-
motions and heat-motions in the kinetic theory of matter is founded ; and
538 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62
(2) Of the application of the same method of analysis, thus definitely
founded, to distinguish between mean-molar-motions and relative-molar-
motions, where, as in the case of steady-mean-flow along a pipe, the more
rigorous definition of the geometrical basis shows the method to be strictly
applicable, and in other cases where it is approximately applicable.
The geometrical relation of the motions respectively indicated by the
terms mean-molar-, or MEAN-MEAN-MOTION, and relative-molar-, or RELATIVE-
MEAN- MOTION, being essentially the same as the relation of the respective
motions indicated by the terms molar-, or MEAN-MOTION, and relative-, or
HEAT-MOTION, as used in the theory of gases.
I also show that the limit to the criterion obtained by this method of
analysis, and by integrating the equations of motion in space, appears as a
geometrical limit to the possible simultaneous distribution of certain quantities
in space, and in no wise depends on the physical significance of these quan-
tities. Yet the physical significance of these quantities, as defined in the
equations, becomes so clearly exposed as to indicate that further study of
the equations would elucidate the properties of matter and mechanical
principles involved, and so be the means of explaining what has hitherto
been obscure in the connection between thermodynamics and the principles
of mechanics.
The geometrical basis of the method of analysis used in the kinetic
theory of gases has hitherto consisted : —
(1) Of the geometrical principle that the motion of any point of a
mechanical system may, at any instant, be abstracted into the mean-motion
of the whole system at that instant, and the motion of the point relative to
the mean-motion ; and
(2) Of the assumption that the component, in any particular direction,
of the velocity of a molecule, may be abstracted into a mean-component-
velocity (say u) which is the mean-component-velocity of all the molecules
in the immediate neighbourhood, and a relative-velocity (say £), which is
the difference between u and the component- velocity of the molecule*;
u and £ being so related that, M being the mass of the molecule, the
integrals of (M%), and (Mug), &c., over all the molecules in the immediate
neighbourhood are zero, and 2 [M (u, + £)L>] = 2 [M(u? + £2)]t-
The geometrical principle (1) has only been used to distinguish between
the energy of the mean-motion of the molecule, and the energy of its internal
motions taken relatively to its mean-motion ; and so to eliminate the internal
motions from all further geometrical considerations which rest on the as-
sumption (2).
* "Dynamical Theory of Gases," Phil. Trans., 1866, p. 67.
t Phil. Trans., 1866, p. 71.
62] AND THE DETERMINATION OF THE CRITERION. 539
That this assumption (2) is purely geometrical, becomes at once obvious,
when it is noticed that, the argument relates solely to the distribution in
space of certain quantities at a particular instant of time. And it appears
that the questions as to whether the assumed distinctions are possible under
any distributions, and, if so, under what distribution, are proper subjects for
geometrical solution.
On putting aside the apparent obviousness of the assumption (2), and
considering definitely what it implies, the necessity for further definition at
once appears.
The mean-component-velocity (u) of all the molecules in the immediate
neighbourhood of a point, say P, can only be the mean-component-velocity
of all the molecules in some space (S) enclosing P. u is then the mean-
component-velocity of the mechanical system enclosed in S, and, for this
system, is the mean-velocity at every point within S, and, multiplied by the
entire mass within S, is the whole component momentum of the system.
But according to the assumption (2), u with its derivatives are to be con-
tinuous functions of the position of P, which functions may vary from point
to point even within S', so that u is not taken to represent the mean-
component-velocity of the system within S, but the mean-velocity at the point
P. Although there seems to have been no specific statement to that effect,
it is presumable that the space S has been assumed to be so" taken that P
is the centre of gravity of the system within S. The relative, positions of P
and S being so defined, the shape and size of the space S requires to be
further defined, so that u, &c., may vary continuously with the position of
P, which is a condition that can always be satisfied if the size and shape of
S may vary continuously with the position of P.
Having thus defined the relation of P to $ and the shape and size of the
latter, expressions may be obtained for the conditions of distribution of u, for
which S (^/£) taken over S will be zero, i.e., for which the condition of mean-
momentum shall be satisfied.
Taking Slt u1} &c., as relating to a point Pl and S, u, &c., as relating to P,
another point, of which the component distances from P] are x, y, z; Pl is
the C.G. of Si, and by however much or little S may overlap Si, S has its
centre of gravity at x, y, z, and is so chosen that u, &c., may be continuous
functions of x,y,z\ u may, therefore, differ from Ui even if P is within $,.
Let u be taken for every molecule of the system Si. Then according to
assumption (2), 2 (Mu) over Si must represent the component of momentum
of the system within Sl} that is, in order to satisfy the condition of mean-
momentum, the mean-value of the variable quantity u over the system St
must be equal to HI the mean-component-velocity of the system Si, and this
is a condition which, in consequence of the geometrical definition already
540 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62
mentioned, can only be satisfied under certain distributions of u. For since
u is a continuous function of x, y, z, M(ii-u^) may be expressed as a
function of the derivatives of u at Pl multiplied by corresponding powers
and products of #, y, z, and again by M ; and by equating the integral of this
function over the space Si to zero, a definite expression is obtained, in terms
of the limits imposed on x, y, z, by the already-defined space Si for the
geometrical condition as to the distribution of u under which the condition of
mean-momentum can be satisfied.
From this definite expression it appears, as has been obvious all through
the argument, that the condition is satisfied if u is constant. It also appears
that there are certain other well-defined systems of distribution for which
the condition is strictly satisfied, and that for all other distributions of u the
condition of mean-momentum can only be approximately satisfied to a degree
for which definite expressions appear.
Having obtained the expression for the condition of distribution of u, so
as to satisfy the condition of mean-momentum, by means of the expression
for M(u-u'), &c., expressions are obtained for the conditions as to the
distribution of £, &c., in order that the integrals over the space SL of the
products Jf(ttff), &c. may be zero when 2 [M (u.— Wi)] = 0, and the con-
ditions of mean-energy satisfied as well as those of mean-momentum. It
then appears that in some particular cases of distribution of u, under which
the condition of mean-momentum is strictly satisfied, certain conditions as
to the distribution of £, &c., must be satisfied in order that the energies of
mean- and relative-motion may be distinct. These conditions as to the
distribution of £, &c., are, however, obviously satisfied in the case of heat-
motion, and do not present themselves otherwise in this paper.
From the definite geometrical basis thus obtained, and the definite
expressions which follow for the condition of distribution of u, &c., under
which the method of analysis is strictly applicable, it appears that this
method may be rendered generally applicable to any system of motion by a
slight adaptation of the meaning of the symbols, and that it does not
necessitate the elimination of the internal motion of the molecules, as has
been the custom in the theory of gases.
Taking u, v, w to represent the motions (continuous or discontinuous) of
the matter passing a point, and p to represent the density at the point, and
putting u, &c., for the mean-motion (instead of u as above), and u', &c., for
the relative-motion (instead of £ as before), the geometrical conditions as to
the distribution of u, &c., to satisfy the conditions of mean-momentum and
mean-energy are, substituting p for M, of precisely the same form as before,
and as thus expressed, the theorem is applicable to any mechanical system
however abstract.
62] AND THE DETERMINATION OF THE CRITERION. 541
(1) In order to obtain the conditions of distribution of molar-motion,
under which the condition of mean-momentum will be satis_fied, so that
the energy of molar-motion may be separated from that of the heat-
motion, u, &c., and p are taken as referring to the actual motion and density
at a point in a molecule, and Sl is taken of such dimensions as may corre-
spond to the scale, or periods in space, of the molecular distances, then the
conditions of distribution of u, under which the condition of mean-momentum
is satisfied, become the conditions as to the distribution of molar-motion,
under which it is possible to distinguish between the energies of rnolar-
motions and heat-motions.
(2) And, when the conditions in (1) are satisfied to a sufficient degree of
approximation by taking u to represent the molar-motion (u in (1)), and the
dimensions of the space S to correspond with the period in space or scale of
any possible periodic or eddying motion, the conditions as to the distribution
of u, &c. (the components of mean-mean-motion), which satisfy the condition
of mean-momentum, show the conditions of mean-molar-motion, under
which it is possible to separate the energy of mean-molar-motion from the
energy of relative-molar- (or relative-mean-) motion.
Having thus placed the analytical method used in the kinetic theory on
a definite geometrical basis, and adapted so as to render it applicable to all
systems of motion, by applying it k> the dynamical theory of viscous fluid,
I have been able to show :— Feb. 18, 1895.]
(a) That the adoption of the conclusion arrived at by Sir Gabriel Stokes,
that the dissipation function Represents the rate at which heat is pro-
duced, adds a definition to the meaning of u, v, w— the components of mean
or fluid velocity — which was previously wanting.
(6) That as the result of this definition the equations are true, and are
only true, as applied to fluid in which the mean-motions of the matter,
excluding the heat-motions, are steady.
(c) That the evidence of the possible existence of such steady mean-
motions, while at the same time the conversion of the energy of these mean-
motions into heat is going on, proves the existence of some discriminative
cause, by which the periods in space and time of the mean- motion are
prevented from approximating in magnitude to the corresponding periods
of the heat-motions, and also proves the existence of some general action by
which the energy of mean-motion is continually transformed into the energy
of heat-motion, without passing through any intermediate stage.
(d) That as applied to fluid in unsteady mean-motion (excluding the
heat-motions), however steady the mean integral flow may be, the equations
542 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62
are approximately true in a degree which increases with the ratios of the
magnitudes of the periods, in time and space, of the mean-motion, to the
magnitude of the corresponding periods of the heat-motions.
(e) That if the discriminative cause and the action of transformation are
the result of general properties of matter, and not of properties which affect
only the ultimate motions, there must exist evidence of similar actions as
between the mean-mean-motion, in directions of mean-flow, and the periodic
mean-motions taken relative to the mean-mean-motion but excluding heat-
motions. And that such evidence must be of a general and important kind,
such as the unexplained laws of the resistance of fluid motions, the law
of the universal dissipation of energy, and the second law of thermo-
dynamics.
(/) That the generality of the effects of the properties on which the
action of transformation depends, is proved by the fact that resistance, other
than proportional to the velocity, is caused by the relative (eddying) mean-
motion.
(g) That the existence of the discriminative cause is directly proved by
the existence of the criterion, the dependence of which on circumstances
which limit the magnitudes of the periods of relative-mean-motion, as com-
pared with the heat-motion, also proves the generality of the effects of the
properties on which it depends.
(k) That the proof of the generality of the effects of the properties
on which the discriminative cause, and the action of transformation depend,
shows that — if in the equations of motion the mean-mean-motion is dis-
tinguished from the relative-mean-motion in the same way as the mean-
motion is distinguished from the heat-motions — (1) the equations must
contain expressions for the transformation of the energy of mean-mean-
motion to energy of relative-mean-motion ; and (2) that the equations, when
integrated over a complete system, must show that the possibility of relative-
mean-motion depends on the ratio of the possible magnitudes of the periods
of relative-mean-motion, as compared with the corresponding magnitude of
the periods of the heat-motions.
(i) That when the equations are transformed so as to distinguish
between the mean-mean-motions, of infinite periods, and the relative-mean-
motions of finite periods, there result two distinct systems of equations, one
system for mean-mean-motion, as affected by relative-mean-motion and heat-
motion, the other system for relative-mean-motion as affected by mean-mean-
motion and heat-motions.
(j) That the equation of energy of mean-mean-motion, as obtained from
the first system, shows that the rate of increase of energy is diminished by
62] AND THE DETERMINATION OF THE CRITERION. 543
conversion into heat, and by transformation of energy of mean-mean-motion
in consequence of the relative-mean-motion, which transformation is ex-
pressed by a function identical in form with that which expresses the
conversion into heat ; and that the equation of energy of relative-mean-
motion, obtained from the second system, shows that this energy is in-
creased only by transformation of energy from mean-mean-motion expressed
by the same function, and diminished only by the conversion of energy of
relative-mean-motion into heat.
(k) That the difference of the two rates (1) transformation of energy of
mean -mean-motion into energy of relative-rneari-motion as expressed by the
transformation function, (2) the conversion of energy of relative-mean-motion
into heat, as expressed by the function expressing dissipation of the energy
of relative-mean-motion, affords a discriminating equation as to the conditions
under which relative-mean-motion can be maintained.
(I) That this discriminating equation is independent of the energy of
relative-mean-motion, and expresses a relation between variations of mean-
rnean-motion of the first order, the space periods of relative-mean-motion,
and fji/p, such that any circumstances which determine the maximum periods
of the relative-mean-motion, determine the conditions of mean-mean-motion
under which relative-mean-motion will be maintained, that is, determine the
criterion.
(m) That as applied to water in steady mean-flow between parallel
plane surfaces, the boundary conditions, and the equation of continuity,
impose limits to the maximum space periods of relative-mean-motion, such
that the discriminating equation affords definite proof that when an in-
definitely small sinuous or relative disturbance exists, it must fade away if
is less than a certain number, which depends on the shape of the section of
the boundaries, and is constant as long as there is geometrical similarity.
While for greater values of this function, in so far as the discriminating
equation shows, the energy of sinuous motion may increase until it reaches to
a definite limit, and rules the resistance.
(??) That besides thus affording a mechanical explanation of the existence
of the criterion K, the discriminating equation shows the purely geometrical
circumstances on which the value of K depends, and although these circum-
stances must satisfy geometrical conditions required for steady mean-motion
other than those imposed by the conservations of mean-energy and momentum,
the theory admits of the determination of an inferior limit to the value of K
under any definite boundary conditions, which, as determined for the par-
ticular case, is
517.
544 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62
This is below the experimental value for round pipes, and is about half what
might be expected to be the experimental value for a flat pipe, which leaves
a margin to meet the other kinematical conditions for steady mean-mean-
motion.
(o) That the discriminating equation also affords a definite expression
for the resistance, which proves that, with smooth fixed boundaries, the con-
ditions of dynamical similarity under any geometrical similar circumstances
depend only on the value of
where b is one of the lateral dimensions of the pipe ; and that the expression
for this resistance is complex, but shows that above the critical velocity the
relative-mean-motion is limited, and that the resistances increase as a power
of the velocity higher than the first.
SECTION II.
The Mean-motion and Heat-motions as distinguished by Periods.— Mean-
mean-motion and Relative-mean-motion. — Discriminative Cause and
Action of Transformation.— Two Systems of Equations.— A Discrimi-
nating Equation.
6. Taking the general equations of motion for incompressible fluid,
subject to no external forces to be expressed by
du f d . d
~di=~\fa (PXX + puu) + dy
dv ( d , d
dw _ ( d d d
~dt~ ~ \dx (PXZ + pwu>+ dy (JV« + Pwv"> + fa (P
...... ax
with the equation of continuity
0=du/dx+dv/dy + dw/dz ....................... '....(2),
where pm, &c., are arbitrary expressions for the component forces per unit of
area, resulting from the stresses, acting on the negative faces of planes
perpendicular to the direction indicated by the first suffix, in the direction
indicated by the second suffix.
62]
AND THE DETERMINATION OF THE CRITERION.
545
Then multiplying these equations respectively by u, v, w, integrating by
parts, adding and putting
2E for p(u*+v* + w2)
and transposing, the rate of increase of kinetic energy per unit of volume is
given by
d , d . d
T~ ('Upxx) + -j- (uPvx) + j~ '
ax ay dz
d_
•t \vfJXIl) I ~J (.vPw) '
dx dy
d , d , d
(d d d d\ T d
U + * 'Si + ' dy + '" Tz) E = ~ 1 + dec
du
du
du1
dv dv dv
~7 "~ Pw J r Pzv ~j~
dx yy dy y dz
dw dw . dw
.(3).
The left member of this equation expresses the rate of increase in the
kinetic energy of the fluid per unit of volume at a point moving with the
fluid.
The first term on the right expresses the rate at which work is being
done by the surrounding fluid per unit of volume at a point.
The second term on the right therefore, by the law of conservation of
energy, expresses the difference between the rate of increase of kinetic
energy and the rate at which work is being done by the stresses. This
difference has, so far as I am aware, in the absence of other forces, or any
changes of potential energy, been equated to the rate at which heat is being
converted into energy of motion, Sir Gabriel Stokes having first indicated
this* as resulting from the law of conservation of energy then just established
by Joule.
7. This conclusion, that the second term on the right of (3) expresses
the rate at which heat is being converted, as it is" usually accepted, may be
correct enough, but there is a consequence of adopting this conclusion which
enters largely into the method of reasoning in this paper, but which, so far as
I know, has not previously received any definite notice.
* Cambridge Phil. Trans., vol. ix. p. 57.
o. B. n.
35
546 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62
The Component Velocities in the Equations of Viscous Fluids.
In no case, that I am aware of, has any very strict definition of u, v, w,
as they occur in the equations of motion, been attempted. They are usually
defined as the velocities of a particle at a point (x, y, z) of the fluid, which
may mean that they are the actual component-velocities of the point in the
matter passing at the instant, or that they are the mean- velocities of all the
matter in some space enclosing the point, or which passes the point in
an interval of time. If the first view is taken, then the right-hand member
of the equation represents the rate of increase of kinetic energy, per unit of
volume, in the matter at the point ; and the integral of this expression over
any finite space S, moving with the fluid, represents the total rate of increase
of kinetic energy, including heat-motion, within that space ; hence the differ-
ence between the rate at which work is done on the surface of S, and the rate
at which kinetic energy is increasing can, by the law of conservation of energy,
only represent the rate at which that part of the heat which does not consist
in kinetic energy of matter is being produced, whence it follows : —
(a) That the adoption of the conclusion that the second term in equation
(3) expresses the rate at which heat is being converted, defines u, v, w, as not
representing the component-velocities of points in the passing matter.
Further, if it is understood that u, v, w, represent the mean velocities of
the matter in some space, enclosing x, y, z, the point considered, or the
mean-velocities at a point taken over a certain interval of time, so that
2 (pu), 2 (pv), 2 (pw) may express the components of momentum, and
22 (pv) — ;?/2 (pw), &c., &c., may express the components of moments of
momentum, of the matter over which the mean is taken ; there still remains
the question as to what spaces and what intervals of time.
(b) Hence the conclusion that the second term expresses the rate of conver-
sion of heat, defines the spaces and intervals of time over which the mean-
component-velocities must be taken, so that E may include all the energy of
mean-motion, and exclude that of heat-motions.
Equations Approximate only except in Three Particular Gases.
8. According to the reasoning of the last article, if the second term on
the right of equation (3) expresses the rate at which heat is being converted
into energy of mean-motion, either pu, pv, pw express the mean components
of momentum of the matter, taken at any instant over a space S0 enclosing
the point x, y, z, to which u, v, w refer, so that this point is the centre of
gravity of the matter within S0 and such that p represents the mean density
of the matter within this space; or pu, pv, pw represent the mean components
of momentum taken at x, y, z over an interval of time T, such that p is
62] AND THE DETERMINATION OF THE CRITERION. 547
the mean density over the time r, and if t marks the instant to which n, v, w
refer, and t' any other instant, 2 [(t — t') p], in which p is the actual density,
taken over the interval T is zero. The equations, however, require, that so
obtained, p, u, v, w shall be continuous functions of space and time, and
it can be shown that this involves certain conditions between the distribution
of the mean-motion and the dimensions of S0 and T.
Mean- and Relative-motions of Matter.
Whatever the motions of matter within a fixed space S may be at any
instant, if the component- velocities at a point are expressed by u, v, w, the
mean-component-velocities taken over S will be expressed by
(4).
If then u, v, w are taken at each instant as the velocities of x, y, z, the
instantaneous centre of gravity of the matter within S, the component
momentum at the centre of gravity may be put
pu = pu + pu ................................. (5 ),
where u' is the motion of the matter, relative to axes moving with the mean
velocity, at the centre of gravity of the matter within S. Since a space S of
definite size and shape may be taken about any point x, y, z in an indefinitely
larger space, so that x, y, z is the centre of gravity of the matter within S,
the motion in the larger space may be divided into two distinct systems of
motion, of which u, v, w represent a mean-motion at each point and u', v', w'
a motion at the same point relative to the mean-motion at the point.
If, however, u, v, w are to represent the real mean-motion, it is necessary
that 2 (pu'}, 2 (pv), 2 (pwr) summed over the space 8, taken about any point,
shall be severally zero ; and in order that this may be so, certain conditions
must be fulfilled.
For taking x, y, z, for 0, the centre of gravity of the matter within S, and
x ', i/, z' for any other point within S, and putting a, b, c for the dimensions
of S in directions x, y, z, measured from the point x, y, z; since u, v, w
are continuous functions of x, y, z by shifting S so that the centre of gravity
of the matter within it is at x ', y', z', the value of u for this point is given by
+ &c ................... (6),
where all the differential coefficients on the left refer to the point x, y, z; and
in the same way for v and w.
Subtracting the value of il thus obtained for the point x' , y', z', from that
35—2
548 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62
of u at the same point, the difference is. the value of u at this point, whence
summing these differences over the space 8 about G at x, y, z, since by
definition when summed over the space 8 about G
2[p(tt-ue)] = 0 and 2[p(V-#)] = 0 (7),
> (8 A).
s (P,,-) . - iS o (. - .')'] +
That is
it \ G I \M U \ n
1+5- + &c.
rV<? 2 \dz2Je
In the same way if 2 be taken over the interval of time T including t ;
and for the instant t
2 ,
u = ~, ' , and pu = pu + pu ;
then since for any other instant t'
where 2 [p (t - 1')] = 0, and 2 [p (ut - u)] = 0.
It appears that
(8B).
From equations (8 A) and (8B), and similar equations for S(pv') and
2 (pw'), it appears that if
2 (pu) = S (pw) = 2 (pw') = 0,
where the summation extends both over the space 8 and the interval r, all
the terms on the right of equations (8A) and (8B) must be respectively and
continuously zero, or, what is the same thing, all the differential coefficients
of u, v, w with respect to x, y, z and t of the first order must be respectively
constant.
This condition will be satisfied if the mean-motion is steady, or uniformly
varying with the time, and is everywhere in the same direction, being
subject to no variations in the direction of motion ; for suppose the direction
of motion to be that of x, then since the periodic motion passes through a
complete period within the distance 2a, 2(pw') will be zero within the
space
2or . dy . dz,
62] AND THE DETERMINATION OF THE CRITERION. 549
however small dy.dz may be, and since the only variations of the mean-
motion are in directions y and z, in which b and c may be taken- zero, and
</>/ dt is everywhere constant, the conditions are perfectly satisfied.
The conditions are also satisfied if the mean-motion is that of uniform
expansion or contraction, or is that of a rigid body.
These three cases, in which it may be noticed that variations of mean-
motion are everywhere uniform in the direction of motion, and subject to
steady variations in respect of time, are the only cases in which the condi-
tions (8 A), (8fi), can be perfectly satisfied.
The conditions will, however, be approximately satisfied, when the
variations of n, v, w of the first order are approximately constant over the
space S.
In such case the right-hand members of equations (8 A), (8fi), are
neglected, and it appears that the closeness of the approximations will be
measured by the relative magnitude of such terms as
d?u d*u .,, du du
adtf> &C" Td? M comPared Wlth dx> dt' &C'
Since frequent reference must be made to these relative values, and, as
in periodic motion, the relative values of such terms are measured by the
period (in space or time) as compared with a, b, c and r, which are, in a
sense, the periods of u', v', w', I shall use the term period in this sense, taking
note of the fact that when the mean-motion is constant in the direction of
motion, or varies uniformly in respect of time, it is not periodic, i.e., its
periods are infinite.
9. It is thus seen that the closeness of the approximation with which
the motion of any system can be expressed as a varying mean-motion
together with a relative-motion, which, when integrated over a space of
which the dimensions are a, b, c, has no momentum, increases as the magni-
tude of the periods of w, v, w in comparison with the periods of u', v', w', and
is measured by the ratio of the relative orders of magnitudes to which these
periods belong.
Heat- 1 notions in Matter are Approximately Relative to tlie Mean- motions.
The general experience that heat in no way affects the momentum of
matter, shows that the heat-motions are relative to the mean-motions of
matter taken over spaces of sensible size. But, as heat is by no means the
only state of relative-motion of matter, if the heat-motions are relative to
all mean-motions of matter, whatsoever their periods may be, it follows —
that there must be some discriminative cause which prevents the existence
of relative-motions of matte]' other than heat, except mean-motions with
550 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62
periods in time and space of greatly higher orders of magnitude than
the corresponding periods of the heat-motions — otherwise, by equations
(8A), (8B), heat-motions could not be to a high degree of approximation
relative to all other motions, and we could not have to a high degree of
approximation,
du du du \
dv dv dv
d~x*PyyJy+p*yTz
dw dw dw
where the expression on the right stands for the rate at which heat is con-
verted into energy of mean-motion.
Transformation of Energy of Relative-mean-motion to Energy of Heat-
motion.
10. The recognition of the existence of a discriminative cause, which
prevents the existence of relative-mean-motions with periods of the same
order of magnitude as heat-motions, proves the existence of another general
action by which the energy of relative-mean-motion, of which the periods
are of another and higher order of magnitude than those of the heat- motions,
is transformed to energy of heat- motion.
For if relative-mean-motions cannot exist with periods approximating to
those of heat, the conversion of energy of mean-motion into energy of heat,
proved by Joule, cannot proceed by the gradual degradation of the periods
of mean-motion until these periods coincide with those of heat, but must, in
its final stages, at all events, be the result of some action which causes the
energy of relative-mean-motion to be transformed into the energy of heat-
motions, without intermediate existence in states of relative-motion, with
intermediate and gradually diminishing periods.
That such change of energy of mean-motion to energy of heat may be
properly called transformation, becomes apparent when it is remembered
that neither mean-motion nor relative-motion has any separate existence,
but are only abstract quantities, determined by the particular process of
abstraction, and so changes in the actual-motion may, by the process of
abstraction, cause transformation of the abstract energy of the one abstract-
motion, to abstract energy of the other abstract-motion.
All such transformation must depend on the changes in the actual-motions,
and so must depend on mechanical principles and the properties of matter,
and hence the direct passage of energy of relative-mean-motion to energy of
62] AND THE DETERMINATION OF THE CRITERION. 551
heat-motions is evidence of a general cause of the condition of actual-
motion which results in transformation — which may be called ~ihe cause of
transformation.
The Discriminative Cause, and the Cause of Transformation.
1 1 . The only known characteristic of heat-motions, besides that of being
relative to the mean-motion, already mentioned, is that the motions of
matter which result from heat are an ultimate form of motion which does
not alter so long as the mean-motion is uniform over the space, and so long
as no change of state occurs in the matter. In respect of this characteristic,
heat-motions are, so far as we know, unique, and it would appear that heat-
motions are distinguished from the mean-motions by some ultimate properties
of matter.
It does not, however, follow that the cause of transformation, or even the
discriminative cause, are determined by these properties. Whether this is
so or not can only be ascertained by experience. If either or both these
causes depend solely on properties of matter which only affect the heat-
motions, then no similar effect would result as between the variations of
mean-mean-motion and relative-mean-motion, whatever might be the
difference in magnitude of their respective periods. Whereas, if these
causes depend on properties of matter which affect all modes of motion,
distinctions in periods must exist between mean-mean-motion and relative-
mean-motion, and transformation of energy take place from one to the other,
as between the mean-motion and the heat-motions.
The mean-mean-motion cannot, however, under any circumstances stand
to the relative-mean-motion in bhe same relation as the mean-motion stands
to the heat-motions, because the heat-motions cannot be absent, and in
addition to any transformation from mean-mean-motion to relative-mean-
motion, there are transformations both from mean- and relative-mean- motion
to heat-motions, which transformation may have important effects on both the
transformation of energy from mean- to relative- mean-motion, and on the
discriminative cause of distinction in their periods.
In spite of the confusing effect of the ever present heat-motions, it would,
however, seem that evidence as to the character of the properties on which
the cause of transformation and the discriminative cause depend, should be
forthcoming as the result of observing the mean- and relative-mean-motions
of matter.
12. To prove by experimental evidence that the effects of these
properties of matter are confined to the heat- motions, would be to prove a
negative ; but if these properties are in any degree common to all modes of
matter, then at first sight it must seem in the highest degree improbable
that the effects of these causes on the mean- and relative-mean-motions
552 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62
would be obscure, and only to be observed by delicate tests. For properties
which can cause distinctions between the mean- and heat-motions of matter
so fundamental and general, that from the time these motions were first
recognized the distinction has been accepted as part of the order of nature,
and has been so familiar to us that its cause has excited no curiosity, must,
if they have any effect at all, cause effects which are general and important
on the mean-motions of matter. It would thus seem that evidence of the
general effects of such properties should be sought in those laws and
phenomena known to us as the result of experience, but of which no rational
explanation has hitherto been found ; such as the law that the resistance
of fluids moving between solid surfaces and of solids moving through fluids,
in such a manner that the general-motion is not periodic, is as the square of
the velocities, the evidence covered by the law of the universal tendency of
all energy to dissipation, and the second law of thermodynamics.
13. In considering the first of the instances mentioned, it will be seen
that the evidence it affords as to the general effect of the- properties, on
which depends transformation of energy from mean- to relative-motion, is
very direct. For, since my experiments with colour bands have shown that
when the resistance of fluids, in steady mean flow, varies with a power of
the velocity higher than the first, the fluid is always in a state of sinuous
motion, it appears that the prevalence of such resistance is evidence of the
existence of a general action, by which energy of mean-mean-motion, with
infinite periods, is directly transformed to the energy of relative-mean-
motion, with finite periods, represented by the eddying motion, which
renders the general mean-motion sinuous, by which transformation the state
of eddying- motion is maintained, notwithstanding the continual transforma-
tion of its energy into heat-motions.
We have thus direct evidence that properties of matter which determine
the cause of transformation, produce general and important effects which
are not confined to the heat-motions.
In the same way, the experimental demonstration I was able to obtain,
that relative-mean-motion in the form of eddies of finite periods, both as
shown by colour bands and as shown by the law of resistances, cannot be
maintained except under circumstances depending on the conditions which
determine the superior limits to the velocity of the mean-mean-motion, of
infinite periods, and the periods of the relative-mean-motion, as defined in
the criterion
DUm/f* = K (10),
is not only a direct experimental proof of the existence of a discriminative
cause which prevents the maintenance of periodic mean-motion except with
periods greatly in excess of the periods of the heat-motions, but also indicates
that the discriminative cause depends on properties of matter which affect
the mean-motions as well as the heat-motions.
62] AND THE DETERMINATION OF THE CRITERION. 553
Expressions for the Rate of Transformation and the Discriminative Cause.
14. It has already been shown (Art. 8) that the equations of motion
approximate to a true expression of the relations between the mean-motions
and stresses, when the ratio of the periods of mean-motions to the periods of
the heat-motions approximates to infinity. Hence it follows that these
equations must of necessity include whatever mechanical or kinematical
principles are involved in the transformation of energy of mean-mean-
motion to energy of relative-mean-motion. It has also been shown that
the properties of matter, on which depends the transformation of energy of
varying mean-motion to relative-motion, are common to the relative-mean-
motion as well as to the heat-motion. Hence, if the equations of motion are
applied to a condition in which the mean-motion consists of two components,
the one component being a mean-mean-motion, as obtained by integrating
the mean-motion over spaces Si taken about the point x, y, z, as centre of
gravity, and the other component being a relative-mean-motion, of which the
mean components of momentum taken over the space ^ everywhere vanish,
it follows : —
(1) That the resulting equations of motion must contain an expression for
the rate of transformation from energy of mean -mean-motion to energy of
relative-mean-motion, as well as the expressions for the transformation of the
respective energies of mean- and relative-mean-motion to energy of heat-
motion.
(2) That, when integrated over a complete system these equations must
shod,' that the possibility of the maintenance of the energy of relative-mean-
motion depends, 'whatsoever may be the conditions, on the possible order of
magnitudes of the periods of the relative-mean-motion, as compared with the
periods of the heat-motions.
The Equations of Mean- and Relative-mean-motion.
15. These last conclusions, besides bringing the general results of the
previous argument to the test point, suggest the manner of adaptation of the
equations of motion, by which the test may be applied.
Put u = u+u', v = v
where u = ~ , ^,&c., &c .......................... (12),
*(P)
the summation extending over the space £>j of which the centre of gravity is
at the point a, y, z. Then since u, v, w are continuous functions of x, y, z,
554 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62
therefore u, v, w, and u, v, w', are continuous functions of x, y, z. And as p
is assumed constant, the equations of continuity for the two systems of
motion are :
.(13);
du dv dw _ _. du dv' dw' _
~T~ T ~r~ H j — " and -= — | — j- -f- — = = (}
dx dy dz dx dy dz
also both systems of motions must satisfy the boundary conditions, whatever
they may be.
Further putting pxx, &c., for the mean values of the stresses taken over
the space Sl and
P'XX=PXX~PXX (14),
and defining Sl to be such that the space variations of u, v, w are approximately
constant over this space, we have, putting u'u', &c., for the mean values of the
squares and products of the components of relative-mean-motion, for the
equations of mean-mean-motion,
du ( d . _
P ~jl = ~ l T ( Pxx
r at [dx ^
pu'v')
, d < - -7-»v1
+ ~T ( Pzx + PUW + OUW)\
dz ' )
&c. = &c.
&c. = &c.
.(15),
which equations are approximately true at every point in the same sense as
that in which the equations (1) of mean-motion are true.
Subtracting these equations of mean-mean-motion from the equations of
mean-motion, we have
du!
d
-T- [p'xx f P (UU + U'll) + p (it'll! - u'u')}
Ty {p'yx + P (UV> + u'^ + P (u'v' ~ ^)1
(P'zx + P (UW' + U'w) + p (tl'w' - vfw'
>&c.,
which are the equations of momentum of rclative-mean-motion at each
point.
Again, multiplying the equations of mean-mean-motion by u, v, w
respectively, adding and putting 'IE =p(u~+ v- + vjfi), we obtain
62]
AND THE DETERMINATION OF THE CRITERION.
555
dt
*L
dx
d
-^,
dy dz
pxx + pu'u')]
-J- tM (P'JX + PU'V')] + j- LM (.Pzs + ptt'«0]
—, ,^+d_r-,-
dz
du
_ du
PyxTy
dv dv
. div dw
^ du
dv
dz
. dw
—r-, du —r, du -7— / du
u u j- + u v -,- 4- u w -=-
dx dy dz
— , dv — dv —, dv
+ vu -j- + v v -j- + vw -j-
ax ay dz
—T-, dw —r-, dw —f—,dw
+w u -r- +w v -j- +w w T-
dx dy dz
...(17),
which is the approximate equation of energy of mean-mean-motion in the
same sense as the equation (3) of energy of. mean-motion is approximate.
In a similar manner multiplying the equations (16) for the momentum of
relative-mean-motion respectively by u, v', w', and - adding, the result would
be the equation for energy of relative-mean-motion at a point, but this would
include terms of which the mean values taken over the space $, are zero, and,
since all corresponding terms in the energy of heat are excluded, by sum-
mation over the space S0 in the expression for the rate at which mean-motion
is transformed into heat, there is no reason to include them for the space St ;
so that, omitting all such terms and putting
2#' = /3(^ + 72+w70 .............................. (18),
we obtain
( - , -
\-J* + UJ-+V-J- + W^-
\dt dx dy dz
- d\ ~
+
JLK(
d ,
j \P \P vx •
ax
^[^(P« +
, du' , du'
p**fa+p»*^7
d , ,
+ j- Cw
dy1
c^
dytv
d
+
d.
d
*'(?'.
+ pv'v')] + -j- [v' (p'zy + pv'w')]
+ [w' (p'yz + pw'v')] + [w' (p'u + pw'w')]
dy
, dv
dx +Py»dj,
dv'
du'\
dv>
dx
dy
dw'
f>u'u'fx+pU'V' cfy
-7—. dv -r-, dv
+ pvu d;K+pVv ^
—r-, dw —f-, dw —. u, u
4- pw u -j- + pw v -j— + pww -j-
i dx ay dz
puw j—
—j dv
, da-
556 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62
where only the mean values, over the space S1} of the expressions in the
right member are taken into account.
This is the equation for the mean rate, over the space 8lt of change in
the energy of relative-mean-motion per unit of volume.
It may be noticed that the rate of change in the energy of mean-mean-
motion, together with the mean rate of change in the energy of relative-
mean-motion, must be the total mean-rate of change in the energy of
mean-motion, and that by adding the equations (17) and (19) the result
is the -same as is obtained from the equation (3) of energy of mean-motion
by omitting all terms which have no mean value as summed over the
space $j.
The Expressions for Transformation of Energy from Mean- mean- motion to
Relative-mean-motion.
16. When equations (17) and (19) are added together, the only expres-
sions that do not appear in the equation of mean-energy of mean-motion are
the last terms on the right of each of the equations, which are identical in
form and opposite in sign.
These terms, which thus represent no change in the total energy of
mean-motion, can only represent a transformation from energy of mean-
mean- motion to energy of relative-mean-motion. And as they are the only
expressions which do not form part of the general expression for the rate
of change of the mean energy of mean- motion, they represent the total
exchange of energy between the mean-mean-motion and the relative-mean-
motion.
It is also seen that the action, of which these terms express the effect,
is purely kinematical, depending simply on the instantaneous characters of
the mean- and relative-mean-motion, whatever may be the properties of
the matter involved, or the mechanical actions which have taken part in
determining these characters. The terms, therefore, express the entire
result of transformation from energy of mean-mean-motion to energy of
relative-mean-motion, and of nothing but the transformation. Their exist-
ence thus completely verifies the first of the general conclusions in Art. 14.
The term last but one in the right member of the equation (17) for
energy of mean-mean-motiori, expresses the rate of transformation of energy
of heat-motions to that of energy of mean-mean-motion, and is entirely
independent of the relative-mean-motion.
In the same way, the term, last but one on the right of the equation (19)
for energy of relative- mean-motion, expresses the rate of transformation from
energy of heat-motions to energy of relative-mean-motion, and is quite in-
dependent of the niean-niean-motion.
62]
AND THE DETERMINATION OF THE CRITERION.
557
17. In both equations (17) and (19) the first terms on the right express
the rates at which the respective energies of mean- and relative-mean-motion
are increasing on account of work done by the stresses on the mean- and
relative-motions respectively, and by the additions of momentum caused by
convections of relative-mean-motion by relative-mean-motion to the mean-
and relative-mean-motions respectively.
It may also be noticed that while the first term on the right, in the
equation (19) of energy of relative-mean-motion, is independent of mean-
mean-motion, the corresponding term in equation (17) for mean-mean-motion
is not independent of relative-mean-motion.
A Discriminating Equation.
18. In integrating the equations over a space moving with the mean-
mean-motion of the fluid, the first terms on the right may be expressed as
surface integrals, which integrals respectively express the rates at which
work is being done on, and energy is being received across the surface, by
the mean-mean-motion, and by the relative-mean-motion.
If the space over which the integration extends includes the whole
system, or such part that the total energy conveyed across the surface by
the relative-mean-motion is zero, then the rate of change in the total
energy of relative-mean-motion within the space, is the difference of the
integral, over the space, of the rate of increase of this energy by trans-
formation from energy of mean-mean-motion, less the integral rate at
which energy of relative-mean-motion is being converted into heat, or,
integrating equation (19),
- d d _ d\ =,,, , j
+ « j- + » -j- 4 WT- 1 E dxdydz
dx dy dz)
-I
' ' au _T7T> du , ^77^7, du
dz
,— dv -,— dv -7-7 dv
+ pv'u' -.- + pvv + pvw
dx dy dz
a
7—7 dw — — dw
T + PW V -j—
ix ay
, du' , du' , du'
dv , dv' , dv'
, dw' , dw' , dw'
,dw
dz I
dxdydz (20).
558 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62
This equation expresses the fundamental relations : —
(1) That the only integral effect of the mean-mean-motion on the relative-
mean-motion is the integral of the rate of transformation from energy of
mean-mean-motion to energy of relative-mean-motion.
(2) That, unless relative energy is altered by actions across the surface
within which the integration extends, the integral energy of relative-mean-
motion will be increasing, or diminishing, according as the integral rate of
transformation from mean-mean-motion to relative-mean-motion is greater,
or less than, the rate of conversion of the energy of relative-mean-motion into
heat.
19. For p'xx, &c., are substituted their values as determined according
to the theory of viscosity, the approximate truth of which has been verified,
as already explained.
Putting
du' dv dw'\ du g
+ + / , &'., &c.
we have, substituting in the last term of equation (20), as the expression for
the rate of conversion of energy of relative-mean-motion into heat,
[f[d/ IT\J j i /Y/T /<&*' M du/\
- If I -j- (pH) dxdydz = ||U_+_ + _)
JJJ at JJJ [_ \dac dy dz )
du' dv dw'\2
,M *V /*/ AA. ,M Af-^-l ......
\dy dz] \dz dx ) \dx dy J jj
in which JJL is a function of temperature only ; or since p is here considered
as constant,
du'\'
whence substituting for the last term in equation (20) we have, if the energy
of relative-mean-motion is maintained, neither increasing nor diminishing,
7-7 du —f-, du -j—: du
\UU -j- + U V j- + UW' -1-
dx dy dz
-P
4- v'u - - + v V 1- v'w' -,-\ dxdydz
dx dy dz j
—r-jdw , ,dw —r-,dw\
+ wu ,— +wv -j- + ww -j-
dx dy dz !
62] AND THE DETERMINATION OF THE CRITERION. 559
dxdydz = Q ...(24),
dw dv'\2 du
d/uf\z
dx dy 1
which is a discriminating equation as to the conditions under which relative-
mean-motion can be sustained.
20. Since this equation is homogeneous in respect to the component
velocities of the relative-mean-motion, it at once appears that it is independent
of the energy of relative-mean-motion divided by the p. So that if fijp is
constant, the condition it expresses depends only on the relation between
variations of the mean-mean-motion and the directional, or angular, distri-
bution of the relative-mean-motion, and on the squares and products of
the space periods of the relative-mean-motion.
And since the second term expressing the rate of conversion of heat
into energy of relative-mean-motion is always negative, it is seen at once
that, whatsoever may be the distribution and angular distribution of the
relative-mean-motion and the variations of the mean-mean-motion, this
equation must give an inferior limit for the rates of variation of the
components of mean-mean-motion, in terms of the limits to the periods
of relative-mean-motion, and p/p, within which the maintenance of relative-
mean-motion is impossible. And that, so long as the limits to the periods
of relative-mean-motion are not infinite, this inferior limit to the rates of
variation of the mean- mean -motion will be greater than zero.
Thus the second conclusion of Art. 14, and the whole of the previous
argument is verified, and the properties of matter which prevent the main-
tenance of mean-motion, with periods of the same order of magnitude as
those of the heat-motion, are shown to be amongst those properties of
matter which are included in the equations of motion of which the truth
has been verified by experience.
The Cause of Transformation.
21. The transformation function, which appears in the equations of
mean-energy of mean- and relative-mean-motion, does not indicate the cause y
of transformation, but only expresses a kinematical principle as to the effect
of the variations of mean-mean-motion, and the distribution of relative-
mean-motion. In order to determine the properties of matter and the
mechanical principles on which the effect of the variations of the mean-
mean-motion on the distribution and angular distribution of relative-mean-
motion depends, it is necessary to go back to the equations (16) of relative-
5GO THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62
momentum at a point; and even then the cause is only to be found by
considering the effects of the actions which these equations express in detail.
The determination of this cause, though it in no way affects the proofs of the
existence of the criterion as deduced from the equations, may be the means
of explaining what has been hitherto obscure in the connection between
thermodynamics and the principles of mechanics. That such may be the
case, is suggested by the recognition of the separate equations of mean- and
relative-mean-motion of matter.
The Equation of Energy of Relative-mean-motion and the Equation of
Thermodynamics.
22. On consideration, it will at once be seen that there is more than an
accidental correspondence between the equations of energy of mean- and
relative-mean-motion respectively, and the respective equations of energy of
mean-motion and of heat in thermodynamics.
If instead of including only the effects of the heat-motion on the mean-
momentum, as expressed by pxx, &c., the effects of relative-mean-motion are
also included by putting pxx for pxx + pu'u' , &c., and pyz for pyz + pw'v , &c.,
in equations (15) and (17), the equations (15) of mean-mean-motion become
identical in form with the equations (1) of mean-motion, and the equation
(17) of energy of mean-mean-motion becomes identical in form with the
equation (3) of energy of mean-motion.
These equations, obtained from (15) and (17), being equally true with
equations (1) and (3), the mean-mean-motion in the former being taken
over the space $1 instead of 80 as in the latter, then, instead of equation (9),
we should have for the value of the last term —
du „ d(pH) -7— , rfw 0
ft-SE + *"•• = - ti+>"lu £+&c (2o)'
in which the right member expresses the rate at which heat is converted
into energy of mean-mean-motion, together with the rate at which energy
of relative-mean-motion is transformed into energy of mean-mean-motion ;
while equation (19) shows whence the transformed energy is derived.
The similarity of the parts taken by the transformation of mean-mean-
motion into relative-mean-motion, and the conversion of mean-motion into
heat, indicates that these parts are identical in form ; or that the conversion
of mean-motion into heat is the result of transformation, and is expressible
by a transformation function similar in form to that for relative-mean-motion,
but in which the components of relative-motion are the components of the
heat-motions, and the density is the actual density at each point. Whence
it would appear that the general equations, of which equations (19) and (16)
are respectively the adaptations to the special condition of uniform density,
62] AND THE DETERMINATION OF THE CRITERION. 561
must, by indicating the properties of matter involved, afford mechanical
explanations of the law of universal dissipation of energy and of the second
law of thermodynamics.
The proof of the existence of a criterion, as obtained from the equations,
is quite independent of the properties and mechanical principles on which
the effect of the variations of mean-mean-motion on the distribution of
relative-mean-motion depends. And as the study of these properties and
principles requires the inclusion of conditions which are not included in the
equations of mean-motion of incompressible fluid, it does not come within
the purpose of this paper. It is therefore reserved for separate investigation
by a more general method.
The Criterion of Steady Mean-motion.
23. As already pointed out, it appears from the discriminating equation
that the possibility of the maintenance of a state of relative-mean-motion
depends on p/p, the variation of mean-mean-motion, and the periods of the
relative-mean-motion.
Thus, if the mean-mean-motion is in direction # only, and varies in
direction y only, if u', v', w' are periodic in directions x, y, z, a being the
largest period in space, so that their integrals over a distance a in direction
x are zero, and if the co-efficients of all the periodic factors are a, then
putting
and taking the integrals, over the space a3, of the 18 squares and products in
the last term on the left of the discriminating equation (24) to be
/O— 2
-18M — ) a'a3,
\ a /
the integral of the first term over the same space cannot be greater than
Then, by the discriminating equation, if the mean-energy of relative-mean-
motion is to be maintained,
pC? is greater than 700 . -% ,
CL
P-V(S' = 700 . ...(26)
is a condition under which relative-mean-motion cannot be maintained in a
o. K. ii. 36
562 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62
fluid, of which the mean-mean-motion is constant in the direction of mean-
mean-motion, and subject to a uniform variation at right angles to the
direction of mean-mean-motion. It is not the actual limit, to obtain which
it would be necessary to determine the actual forms of the periodic function
for u, v', w', which would satisfy the equations of motion (15), (16), as well
as the equation of continuity (13), and to do this the functions would be of
the form
^ r A \ ( • 2?r '
2i A. cos \r \nt-\ x
1 V a
where r has the values 1,2, 3, &c. It may be shown, however, that the
retention of the terms in the periodic series in which r is greater than unity
would increase the numerical value of the limit.
24. It thus appears that the existence of the condition (26) within
which no relative-mean-motion, completely periodic in the distance a, can be
maintained, is a proof of the existence, for the same variation of mean-mean-
motion, of an actual limit of which the numerical value is between 700 and
infinity.
In viscous fluids, experience shows that the further kinematical con-
ditions imposed by the equations of motion do not prevent such relative-
mean-motion. Hence for such fluids equation (26) proves that the actual
limit, which discriminates between the possibility and impossibility of
relative-mean-motion completely periodic in a space a, is greater than 700.
Putting equation (26) in the form
/(pV-TOoA.
V \dy/ pa?
it at once appears that this condition does not furnish a criterion as to the
possibility of the maintenance of relative-mean-motion, irrespective of its
periods, for a certain condition of variation of mean-mean-motiori. For by
taking a2 large enough, such relative-mean-motion would be rendered
possible whatever might be the variation of the mean-mean-motion.
The existence of a criterion is thus seen to depend on the existence of
certain restrictions to the value of the periods of relative-mean-motion — on
the existence of conditions which impose superior limits on the values of a.
Such limits to the maximum values of a may arise from various causes.
If dujdy is periodic, the period would impose such a limit, but the only
restrictions which it is my purpose to consider in this paper, are those which
arise from the solid surfaces between which the fluid flows. These restric-
tions are of two kinds— restrictions to the motions normal to the surfaces,
62] AND THE DETERMINATION OF THE CRITERION. 563
and restrictions tangential to the surfaces — the former are easily defined, the
latter depend for their definition on the evidence to be obtained -from experi-
ments such as those of Poiseuille, and I shall proceed to show that these
restrictions impose a limit to the value of a, which is proportional to D, the
dimension between the surfaces. In which case, if
/Y*?Y - E
V\dy) ''=D'
equation (26) affords a proof of the existence of a criterion
of the conditions of mean- mean-motion under which relative or sinuous-
motion can continuously exist in the case of a viscous fluid between two
continuous surfaces perpendicular to the direction y, one of which is main-
tained at rest, and the other in uniform tangential-motion in the direction x
with velocity U.
SECTION III.
The Criterion of the Conditions under which Relative-mean-motion cannot be
maintained in the case of Incompressible Fluid in Uniform Symmetrical
Mean-flow between Parallel Solid Surfaces. — Expression for the Resist-
ance.
25. The only conditions, under which definite experimental evidence as
to the value of the criterion has as yet been obtained, are those of steady
flow through a straight round tube of uniform bore ; and for this reason
it would seem desirable to choose for theoretical application the case of a
round tube. But inasmuch as the application of the theory is only carried
to the point of affording a proof of the existence of an inferior limit to the
value of the criterion, which shall be greater than a certain quantity deter-
mined by the density and viscosity of the fluid and the conditions of flow,
and as the necessary expressions for the round tube are much more complex
than those for parallel plane surfaces, the conditions here considered are
those defined by such surfaces.
Case I. Conditions.
26. The fluid is of constant density p and viscosity p, and is caused to
flow, by a uniform variation of pressure dp/dx, in direction x between parallel
surfaces, given by
y=—b0, y = b0 (28),
the surfaces being of indefinite extent in directions z and x.
36—2
564 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62
The Boundary Conditions.
(1) There can be no motion normal to the solid surfaces, therefore
v = 0 when y=±b0 ............................. (29).
(2) That there shall be no tangential motion at the surface, therefore
u = w=Q when y = ± b0 ......................... (30);
whence by equation (21), putting u for u', pyx = — fsdu/dy.
By the equation of continuity du/da; + dv/dy + dw/dz = 0, therefore at
the boundaries we have the further conditions, that when y = ± b0,
du/dx = dv/dy = dwfdz = 0 ........................ (31).
Singular Solution.
27. If the mean-motion is everywhere in direction x, then, by the
equation of continuity, it is constant in this direction, and as shown (Art. 8)
the periods of mean-motion are infinite, and the equations (1), (3), and (9)
are strictly true. Hence if
v = w = u' = v' = w' = 0 ......................... (32),
we have conditions under which a singular solution of the equations, applied
to this case, is possible whatsoever may be the value of b0, dpjdx, p and /*.
Substituting for pxx, pyz, &c., in equations (1) from equations (21), and
substituting u for u', &c., these become
This equation does not admit of solution from a state of rest*; but
assuming a condition of steady motion such that du/dt is everywhere zero,
and dp/dx constant, the solution of
* In a paper on the "Equations of Motion and the Boundary Conditions of Viscous Fluid,"
read before Section A at the meeting of the B. A., 1883, I pointed out the significance of this
disability to be integrated, as indicating the necessity of the retention of terms of higher orders
;o complete the equations, and advanced certain confirmatory evidence as deduced from the
theory of gases. The paper was not published, as I hoped to be able to obtain evidence of a
more definite character, such as that which is now adduced in Articles 7 and 8 of this paper,
which shows that the equations are incomplete, except for steady motion, and that to render then.
mtegrable from rest the terms of higher orders must be retained, and thus confirms the argument
I advanced, and completely explains the anomaly. (See Paper 46, page 132.)
62] AND THE DETERMINATION OF THE CRITERION. 565
•(34).
fj, /d*u d*u\ _ 1 dp _
p \dy~ dz*J p dx
if u = du/dz = 0 when y = ± b0,
_ 1 dp y2 - 602
IS U — -j— x
IJL dx 2
This is a possible condition of steady motion, in which the periods of u,
according to Art. 8, are infinite ; so that the equations for mean-motion as
affected by heat-motion, by Art. 8, are exact, whatever may be the values of
u, b0, p, p,, and dpjdx.
The last of equations (34) is thus seen to be a singular solution of the
equations (15) for steady mean-flow, or steady mean-mean-motion, when
u, v', w', p, &c., have severally the values zero, and so the equations (16) of
relative-mean-motion are identically satisfied.
In order to distinguish the singular values of w, I put
rb
u = U, I udy = Zb0Um ;
j- (35).
dp ZHTT TT 3TT &02-;
whence j=~lhm' ^ = n^m — IT
According to the equations, such a singular solution is always possible where
the conditions can be realized, but the manner in which this solution of the
equation (1) of mean-motion is obtained affords no indication as to whether
or not it is the only solution — as to whether or not the conditions can be
realized. This can only be ascertained either by comparing the results as
given by such solutions with the results obtained by experiment, or by
observing the manner of motion of the fluid, as in my experiments with
colour bands.
The fact that these conditions are realized, under certain circumstances,
has afforded the only means of verifying the truth of the assumptions as to
the boundary conditions, that there shall be no slipping, and as to /* being
independent of the variations of mean-motion.
Verification of the Assumptions in the Equation of Viscous Fluid.
28. As applied to the conditions of Poiseuille's experiments and similar
experiments made since, the results obtained from the theory are found to
agree throughout the entire range so long as u', v, w' are zero, showing that
if there were any slipping it must have been less than the thousandth part
of the mean-flow, although the tangential force at the boundary was 0'2 gr.
566 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62
per square centimetre, or over 6 Ibs. per square foot, the mean flow
376 millims. (1*23 feet) per second, and
du/dr = 215,000,
the diameter of this tube being 0'014 millim., the length T25 millims., and
the head 30 inches of mercury.
Considering that the skin resistance of a steamer going at 25 knots is not
6 Ibs. per square foot, it appears that the assumptions, as to the boundary
conditions and the constancy of p, have been verified under more exigent
circumstances, both as regards tangential resistance and rate of variation
of tangential stress, than occur in anything but exceptional cases.
Evidence that other Solutions are possible.
29. The fact that steady mean-motion is almost confined to capillary
tubes, and that in larger tubes, except when the motion is almost insensibly
slow, the mean-motion is sinuous and full of eddies, is abundant evidence
of the possibility, under certain conditions, of solutions other than the singular
solutions.
In such solutions u', v, w have values, which are maintained, not as a
system of steady periodic motion, but such as has a steady effect on the mean-
flow through the tube ; and equations (1) are only approximately true.
The Application of the Equations of the Mean- and Relative-
mean-motion.
30. Since the components of mean-mean-motion in directions y and z
are zero, and the mean flow is steady,
v = 0, w = Q, duldt=Qt du/dx=0 ............. (36),
and as the mean values of functions of u', v, w' are constant in the direction
of flow,
......
dx dx dx
By equations (21) and (37) the equations (15) of mean-motion become
du dp /d*u d»u\ (d ,-r-,. d ,-r-,.}\
-- -
dw dp
~
62] AND THE DETERMINATION OF THE CRITERION.
The equation of energy of mean-mean-motion (17) becomes
d (E) dp (d fdu\ d I dii\]
\ • = - u JL + p 4 j- [u -j- ) + -j- (u -j- )}•
dt ax [dy \ dy] dz \ dz)}
d r -r* . d ,- -T-A) (/du\* . /du^
507
....(39).
— du -7-7 du
+ p luv -j- + uw -j-
{ dy dz
Similarly the equation of mean-energy of relative-mean-motion (19)
becomes
dE' d
dt ~ dy
-—[
(iz[
'v} + W<
-^
+ pUw) + V ( p'zy + pv'l<j')+ W ( pzz +
-r-
dz
(du' dw'V /do' du'\*~\
\dz dx) \dx dz ) }
du — — ,du
.(40).
Integrating in directions y and z between the boundaries and taking note
of the boundary conditions by which M, u, v', w vanish at the boundaries
together with the integrals, in direction z, of
(ill
(L
the integral equation of energy of mean-mean-motion becomes
[CdE, , ff[~dP^_ ttd*\* . (faVl
j.- dydz =- ruj+Atij~+T~r
JJ dt JJ L dx \\dyj \dz) j
(-r~, du —r-.du]~] , 7
— p\uv -r+uw -j- H dydz .............. (41).
I dy dz}\
The integral equation of energy of relative-mean-motion becomes
\dy dz)
dx
568 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62
If the mean-mean-motion is steady it appears from equation (41) that
the work done on the mean-mean-motion u, per unit of length of the tube,
by the constant variation of pressure, is in part transformed into energy of
relative-mean-motion at a rate expressed by the transformation function :
ff f-r/du , -r-,dw\ , ,
— II p( uv -•=- + uw -j- dydz,
JJr\ dy dy)
and in part transformed into heat at the rate :
du\* , (du\*~\ , ,
j- 1 +-7- dydz.
dyj \dzj J y
While the equation (42) for the integral energy of relative-mean-motion
shows that the only energy received by the relative-mean-motion is that
transformed from mean-mean-motion, and the only energy lost by relative-
mean-motion is that converted into heat by the relative-mean-motion at the
rate expressed by the last term.
And hence if the integral of E' is maintained constant, the rate of
transformation from energy of mean-mean-motion must be equal to the
rate at which energy of relative-mean-motion is converted into heat, and
the discriminating equation becomes
ff
U
J.
f—du —,du\ , , f[[n(fdu'\* fdtf\* fdu/\*}
plu'v -J- + u'w -j- dydz = -/i2-M-r- +-7- + ( T~ J f
V dy dz) ^n\_ \\doc) \dyj\dz)}
dw' dv'\2 /du' dw'y fdv' du'V'] 7
-j-+-3r +(-r: + -T-J +(TL+TL \dyd
dy dz) \dz dxj \dx dy ) J
The Conditions to be Satisfied by u and u, v, w.
31. If the mean-mean-motion is steady u must satisfy : —
(1) The boundary conditions
w = 0 when y=±bQ ........................... (44);
(2) The equation of continuity
du/dx = 0 ................................. (45) ;
(3) The first of the equations of motion (38)
dp (d*u d*u\ (d .-r,. t d ,
» + -U+U
62] AND THE DETERMINATION OF THE CRITERION. 569
or putting w = U + u — U,
and -f- = /AT-T as in *he singular solution,
dx ay*
equation (46) becomes
(4) The integral of (47) over the section of which the left member is
zero, and
the mean value of fjudu/dy = fidUjdy when // = + b0 ...... (48).
From the condition (3) it follows that if u is to be symmetrical with
respect to the boundary surfaces, the relative-meau-motion must extend
throughout the tube, so that
/•GO I- J J _ -1
I . - (fo1) + -5- (u'w) \dz is a function of y2 ......... (49).
J _ 30 \_ay az J
And as this condition is necessary, in order that the equations (38) of mean-
mean-motion and the equations (16) of relative-mean-motion may be satisfied
for steady mean-motion, it is assumed as one of the conditions for which the
criterion is sought.
The components of relative-mean-motion must satisfy the periodic
conditions as expressed in equations (12), which become, putting 2c for
the limit in direction z,
fa fa fa \
(1) u'dx=\ v'dx=\ w'dx=0\
Jo JQ Jo
/&„ re
u'dydz = 0
-b0J -c
(2) The equation of continuity
du'/dtc + dv'/dy + dw'/dz = 0.
(3) The boundary conditions which with the equation of continuity give
u' = v' = w' = du'/dx = dv'/dy = dw'Jdz = 0 when y=±bv ...... (51).
(4) The condition imposed by symmetrical mean-motion
dz= 2c-/(2/2) ............ (52)-
These conditions (1 to 4) must be satisfied, if the effect on u is to be
symmetrical however arbitrarily u', v, w' may be superimposed on the mean-
motion which results from a singular solution.
.(50).
570 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62
(5) If the mean-motion is to remain steady u', v', w' must also satisfy
the kinematical conditions obtained by eliminating p from the equations of
mean -mean-motion (38) and those obtained by eliminating p' from the
equations of relative-mean-motion (16).
Conditions (1 to 4) determine an inferior Limit to the Criterion.
32. The determination of the kinematic conditions (5) is, however,
practically impossible ; but if they are satisfied, u', v', w must satisfy the
more general conditions imposed by the discriminating equation. From
which it appears that when u', v', w' are such as satisfy the conditions
(1 to 4), however small their values relative to u may be, if they be such
that the rate of conversion of energy of relative-mean-motion into heat
is greater than the rate of transformation of energy of mean-mean-motion
into relative-mean-motion, the energy of relative-mean-motion must be
diminishing. Whence, when u, v', w' are taken such periodic functions of
a, y, z, as under conditions (1 to 4) render the value of the transformation
function relative to the value of the conversion function a maximum, if this
ratio is less than unity, the maintenance of any relative-mean-motion is im-
possible. And whatever further restrictions might be imposed by the
kinematical conditions, the existence of an inferior limit to the criterion is
proved.
Expressions for the Components of possible Relative-mean-motion.
33. To satisfy the first three of the equations (50) the expressions for
u', v', w, must be continuous periodic functions of x, with a maximum periodic
distance a, such as satisfy the conditions of continuity.
Putting
I = 2-TT/a ; and n for any number from 1 to oo ,
, Voo (fdan dyn\ /d/3n d$n\ . , , .} \
and u = S0 4 i-j- + -/- cos (nix) + -£— + ~ sin (nlx)\
[\dy dz J \dy dz I ')
v' = 2^ {nlan sin (nix) — nlftn cos (nix)}
w' = 2o [nlyn sin (nix) - nlSn cos (nix)}
u', v, w' satisfy the equation of continuity. And, if
a = /3 = <y = 8 = da/dy = d/3/dy = dyfdz = d8/dz = 0 when y=±b0)
\ (54),
and a/3, ay, aS are all functions of y2 only,
it would seem that the expressions are the most general possible for the
components of relative-mean-motion.
62] AND THE DETERMINATION OF THE CRITERION. 571
Cylindrical-relative-motion.
34. If the relative-mean-motion, like the mean-mean-motion, is re-
stricted to motion parallel to the plane of xy,
y = & = w' = Q, everywhere (55),
and the equations (53) express the most general forms for u, v' in case
of such cylindrical disturbance.
Such a restriction is perfectly arbitrary, and having regard to the kine-
matical restrictions, over and above those contained in the discriminating
equation, would entirely change the character of the problem. But as no
account of these extra kinematical restrictions is taken in determining the
limit to the criterion, and as it appears from trial that the value found for
this limit is essentially the same, whether the relative-mean-motion is
general or cylindrical, I only give here the considerably simpler analyses for
the cylindrical motion.
The functions of Transformation of Energy and Conversion to Heat for
Cylindrical Motion.
35. Putting -j-(pH') for the rate at which energy of relative-mean-
motion is converted to heat per unit of volume, expressed in the right-hand
member of the discriminating equation (43),
J/j ^
'\* fdv'\z} (du\ (dv\ , etdu'dv'~\ , , ,
+ - r + [-*- + j- +2 j- -j- dxdydz ...... (56).
J \dy)) \dy ) \dx) dy dx\
= u
dx
Then substituting for the values of u', v', w from equations (53), and
integrating in direction x over Sir/I, and omitting terms the integral of
which, in direction y, vanishes by the boundary conditions,
' + (f
In a similar manner, substituting for u', v', integrating, and omitting
terms which vanish on integration, the rate of transformation of energy
572 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62
from mean-mean-motion, as expressed by the left member in the discrimi-
nating equation (43), becomes
[[ -r,du , , * f/U f ,/ dfin 0 dan\ du~] . . , .
j)pu>v Ty dydz = * j j 2 [rf ^ -^ - & ^J ^ J dy <fc. . .(58).
And, since by Art. 31, conditions (3) equation (47),
"as^-^-'iJ^ ......................... (59)'
integrating and remembering the boundary conditions,
pj-(u-U) = pu'v', fi(u-U) = pl u'v'dy ......... (60).
y * ~ ^o
And since at the boundary u — U is zero,
(MV)dy«0 ........................... (61).
Whence, putting ?7+ u— U for w in the right member of equation (58),
substituting for u — U from (60), integrating by parts, and remembering that
=-3-, which is constant ................. (62),
df 602
(fj /D , /ft \ ^
«n^-/3n?n}\ ..................... (63),
ay ay / )
we have for the transformation function :
If u', v are indefinitely small, the last term, which is of the fourth degree,
may be neglected.
Substituting in the discriminating equation (43) this may be put in
the form
o i'b" j[yv\lf»d(t» d&n\} J
3 dy 2,\ nl 0n r" - an -~ [ dy
J -ba J -i,0 ( \ dy dyj)
...... (65).
62] AND THE DETERMINATION OF THE CRITERION. 573
Limits to the Periods.
36. As functions of y, the variations of «.„, ftn are subject to the restric-
tions imposed by the boundary conditions, and in consequence their periodic
distances are subject to superior limits determined by 260, the distance
between the fixed surfaces.
In direction x, however, there is no such direct connection between the
value of b0 and the limits to the periodic distance, as expressed by Sir/ril.
Such limits necessarily exist, and are related to the limits of otn and ftn in
consequence of the kinematical conditions necessary to satisfy the equations
of motion for steady mean-mean-motion ; these relations, however, cannot be
exactly determined without obtaining a general solution of the equations.
But from the form of the discriminating equation (43) it appears that no
such exact determination is necessary in order to prove the inferior limit to
the criterion.
The boundaries impose the same limits on an, ftn whatever may be the
value of nl; so that if the values of an, ftn be determined so that the value of
m .
- is a minimum
for every value of nl, the value of rl, which renders this minimum a mini-
mum-minimum may then be determined, and so a limit found to which the
value of the complete expression approaches, as the series in both numerator
and denominator become more convergent for values of nl differing in both
directions from rl.
Putting I, a, ft for rl, o^, /3r respectively, and putting for the limiting
value to be found for the criterion
(66)
_ da dft
where a and ft are such functions of y that Kl is a minimum whatever the
value of I, and I is so determined as to render /T, a minimum-minimum.
Having regard to the boundary conditions, &c., and omitting all possible
terms which increase the numerator without affecting the denominator, the
most general form appears to be
574 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62
a = 20" [a>ag+i sin (2s+l)p],\
' (68)'
where p = 7n//2&0 )
To satisfy the boundary conditions
s = 2r, when s is even, s = 2r + 1, when s is odd.
t = 2r + 1, when t is odd, £ = 2 (r + 1), when t is even.
Since a = 0, when p = ± ^TT,
ioo / \ A
.(69).
and since dfi/dy = 0, when p = ± %TT,
20" {- (4r + 2) 64r+2 + 4 (r + 1) 64r+4} = 0 j
From the form of Kl it is clear that every term in the series for a and /3
increases the value of K1 and to an extent depending on the value of r. K1
will therefore be a minimum, when
a = «! sin p + as sin
ft = b2 sin 2p + 64 sin
which satisfy the boundary conditions if
(70),
.(71).
Therefore we have, as the values of a and /8, which render Kl a minimum
for any value of I
a.1 a-i = sin p + sin 3p, /3/62 = sin 2p + ^ si
And
0 a 0
~r = cos « + 3 cos %>, -^ -y- = 2 cos 2w + 2 cos 4w
Trttj ay 7rb2 ay
26
/arf/8 /3da\ 1 f
-T2- ~ ^j— )= - {- 3 sm » - 3 sin 3p + sm 5» + sm lp\
\ a dt/ / 4 l
...(72)
and integrating twice
j - ba J -ba\ dy dy / TT
T> .Li.' f T C 7
rutting —j- L for I,
the denominator of ^ KI, equation (67), becomes
- l-325Xo162.
/TOX
(73).
62] AND THE DETERMINATION OF THE CRITERION. 575
In a similar manner the numerator is found to be
£- Y {L< (2a,2 + 1-25622) + 2# (lO^2 + 8&22) + 82^ + 80622},
\ZOo/
and as the coefficients of a^ and 6S are nearly equal in the numerator, no
sensible error will be introduced by putting
2 = - o,,
3 Z< + 2 x 5'53£2 + 50
2> 0-408Z
which is a minimum if
£ = 1-62 ................................. (75)
and ^ = 517 ................................. (76).
Hence, for a flat tube of unlimited breadth, the criterion
p2b0Um/fji is greater than 517 ..................... (77).
37. This value must be less than that of the criterion for similar
circumstances. How much less it is impossible to determine theoretically
without effecting a general solution of the equations ; and, as far as I am
aware, no experiments have been made in a flat tube. Nor can the experi-
mental value 1900, which I obtained for the round tube, be taken as
indicative of the value for a flat tube, except that, both theoretically
and practically, the critical value of Um is found to vary inversely as the
hydraulic mean depth, which would indicate that, as the hydraulic mean
depth in a flat tube is double that for a round tube, the criterion would
be half the value, in which case the limit found for Kl would be about
This is sufficient to show that the absolute theoretical limit found /< = ~^r ~
is of the same order of magnitude as the experimental value ; so that the
latter verifies the theory, which, in its turn, affords an explanation of the
observed facts.
The State of Steady Mean-motion above the Critical Value.
38. In order to arrive at the limit for the criterion it has been necessary
to consider the smallest values of u't v, w', and the terms in the discriminating
equation of the fourth degree have been neglected. This, however, is only
necessary for the limit, and, preserving these higher terms, the discriminating
equation affords an expression for the resistance in the case of steady mean-
mean-motion.
576 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62
The complete value of the function of transformation as given in equation
(64) is
Whence putting U + u — U, for u in the left member of equation (77), and
integrating by parts, remembering the conditions, this member becomes
^npf dy r pu'v'dy + £-f° (u'vjdy (78),
PO J -60 J -&o /* ' -&o
in which the first term corresponds with the first term in the right member
of equation (64), which was all that was retained for the criterion, and the
second term corresponds with the second term in equation (64), which was
neglected.
Since by equation (35)
we have, substituting in the discriminating equation (43), either
( [dldt(pH')dy ? [*• (Mv
2 bjdp 2V U ^ +^l-bn(UV)d
»V.«6 3 ' A p _,
ay uv ay
J-b0 yJ-ba
dzu dp
»3f-£-° <80>-
Therefore, as long as - p — -f-
3r p* dx
is of constant value, there is dynamical similarity under geometrically similar
circumstances.
The equation (79) shows that,
when —~p~-f- is greater than K,
3 r p2 dx
uv' must be finite, and such that the last term in the numerator limits the
rate of transformation, and thus prevents further increase of u'v'.
62] AND THE DETERMINATION OF THE CRITERION. 577
The last term in the numerator of equation (79) is of the order and
degree
p2Z4a4//i- as compared with Z4a2,
1 t?
the order and degree of - -r (pH'} the first term in the numerator.
It is thus easy to see how the limit comes in. It is also seen from
equation (79) that, above the critical value, the law of resistance is very
complex and difficult of interpretation, except in so far as showing that
the resistance varies as a power of the velocity higher than the first.
o. R. u.
37
63.
EXPERIMENTS SHOWING THE BOILING OF WATER IN AN
OPEN TUBE AT ORDINARY TEMPERATURES.
(Exhibited before Section A, Brit. Assoc., 1894, at Oxford.)
AMONG the many phenomena, the secrets of which have been preserved
by the deadening influence of familiarity on curiosity, there is perhaps none
more remarkable than that of the ' singing of the kettle on the hob,' which
has many times been the subject of sentiment and verse but not, it would
seem, hitherto a subject of physical study which like the study of the rain-
bow might afford evidence as to the conditions under which we exist.
That the cheering evidence of the readiness of the social gathering is not
the only evidence to be obtained from the song of the kettle will in the first
place be demonstrated in these experiments. Thus, having analyzed by
experiment the physical causes of this sound and its variations, the purpose
of the experiments is to demonstrate the relation which exists between
sounds in the kettle and sounds produced by the motion of water, or any
liquid, under certain common conditions. And, in the third place, to
demonstrate the general fact that liquids flowing between fixed boundaries
emit no sound as long as they continuously occupy the space between the
boundaries, and thence to demonstrate that when such sound occurs it is
evidence of the boiling of the water.
If we place a kettle on the top of a fire, the first evidence of action is that
of a somewhat feeble and intermittent hissing sound which at first increases
and becomes continuous and then again subsides as the temperature in-
creases.
This is followed by a much more definite and harsher sound which
comes on suddenly, somewhat increases in volume, then suddenly softens
63] EXPERIMENTS SHOWING THE BOILING OF WATER, ETC. 579
and is immediately followed by the exit of steam showing that the water is
boiling.
If a glass flask is substituted for the opaque kettle the causes of the
sound and its variations become apparent.
The water in the flask is under the pressure of the atmosphere at its
upper surface, which pressure is increased at points below the surface by
the water above ; so that the boiling-point at the bottom of the kettle is
somewhat above that higher up.
The water receives its heat from the fire below by conduction through
the metal, or glass, and the water between the bottom and the point con-
sidered.
The conduction through water is very slow ; so that the water in imme-
diate contact with the hot surface at the bottom becomes much hotter than
the water immediately above. Water expands with heat. Hence this hot layer
on the bottom is in unstable equilibrium, and vertical convection currents are
set up which carry the hot water from the bottom into the colder water
above. Owing however to the eddying motion which is a consequence of
the resistance offered to the ascending currents by the water above, these
currents do not follow a straight course but, somewhat rapidly, interweave,
as thin sheets, with the surrounding water ; so that the heat is soon diffused
through the flask, leaving very little variation of temperature except close to
the bottom of the flask or kettle. These convection currents are most
vigorous soon after the kettle is put on the fire, when there is the greatest
difference of temperature between the water on the bottom and the water
above. In this condition however there is no sound, since the vigour of the
currents, owing to the greater density of the water above, carry away the
water, heated on the bottom, before it has reached a sufficient temperature.
Then as the water above acquires heat through the agency of these currents,
these currents diminish in vigour but still continue.
When a certain temperature, about 174° F., at the upper surface is
reached (which depends on the amount of air occluded in the water) bubbles
begin to collect on the surface at the bottom of the flask and then to rise in
increasing numbers. These bubbles do not vanish but rise to the surface,
increasing in size as they ascend. They are a consequence of the tension of
the occluded air added to that of the vapour.
When a bubble first appears there is a sharp but slight click and these
clicks, as they become numerous, constitute the preliminary hiss, which nearly
subsides before the temperature reaches 200° F.
At about 10° below the boiling-point the harsh hiss comes on suddenly
37—2
580 EXPERIMENTS SHOWING THE BOILING OF WATER [63
and, in the glass flask, it may be observed that, simultaneous with this sound,
there appear again bubbles on the bottom of the flask, which bubbles grow
on the bottom gradually until they leave the surface, and start to rise, when
unlike the previous bubbles of air they suddenly collapse with a sharp click,
which being rapidly repeated causes the harsh hiss. The reason of the
collapse of these bubbles is that they are bubbles of steam at the tem-
perature of the boiling-point at the bottom of the flask, formed between the
surface of the glass, or metal, and held down by capillary action until they
are large enough to break away and ascend, when their ascension brings the
steam into contact with the colder water above, when, being free from air,
their collapse is sudden and sonorous.
As the temperature still further increases and the difference of tem-
perature between the water at the bottom and that which is above diminishes,
the bubbles rise higher and higher before condensing but still collapse
suddenly, until the bubbles rise to the surface, when the water boils and
the sharp sound subsides as suddenly as it came on.
This analysis of the sound phenomenon of the kettle, which owing to our
familiarity with it, has hitherto attracted but little notice, throws very
definite light on a fact of the greatest importance to physics, which it
would appear has met with partial recognition only.
The question as to whether the motion of continuous liquid between
solid boundaries with which it is everywhere in contact can produce sound,
as a consequence of the motion, has not I believe hitherto received any
definite answer.
The general association of sound with running water has doubtless
obscured the subject, although for the most part where it occurs the source
of such sound may be easily traced to the variation of the positions of the
surface of the water, and particularly where the surface is discontinuous
or intermittent.
But, apart from such sources of sound, it is a matter of familiar obser-
vation that the flow of water through pipes under great pressure, as when, in
the water supply of a town, the water is brought from below the surface of
a reservoir on a continuous slope into houses or mains several hundred feet
below the reservoir, and is generally attended with a hissing noise ; and of this
I believe no explanation has hitherto been given. Nor have I ever heard
anyone suggest that there is any connection between the singing of the
kettle and the hiss which almost invariably attends the opening of a tap in
a pipe under considerable pressure as in a town's service. Yet when
observed the hiss of the pipe closely resembles the harshest sound of the
kettle.
63] IN AN OPEN TUBE AT ORDINARY TEMPERATURES. 581
It is now some years since I was led, as the result of hydrodynamical
analysis applied to a fluid having the physical properties of water, to the
conclusion that both these sources of sound have the same origin.
In hydrodynamics it is customary to consider the physical properties of
the fluid as consisting of incompressibility and perfect fluidity only, no
account being taken of internal cohesion or of adhesion to solid surfaces, as
between water and glass, and still less of any vapour tension in spaces not
occupied by the water.
With these limited properties the hydrodynamical problem only admits
of solution when the circumstances are such that the pressure is every-
where positive, so that there could be no possibility of disruption of the
fluid.
The case however is entirely changed when we recognise that the water
has cohesion, depending on its freedom from occluded air as well as viscosity,
and that where the water is discontinuous the spaces are filled with vapour
at a tension corresponding to the temperature.
It has long been known, as shown by Bernoulli, that when water flows
along a contracting channel which it completely occupies, the pressure falls
approximately according to the law that the sum of the intensity of pressure p
and the product of the density of mass multiplied by the half of the vis viva
is constant, or
p + pv* = a constant.
Thus if water flows from below the surface of a reservoir, of unlimited
dimensions, through a conical tube, the small end of which is in connection
with the receiver of an air-pump from which all air has been removed,
the small end of the pipe being at the level of the surface of the reservoir,
supposing that there is no vapour tension, and the pressure of the atmosphere
1470 Ibs. on the square inch, the water enters the receiver with a velocity of
46 "5 feet per second.
If however the temperature of the water is 59° F. the vapour tension is
0'241 Ibs. per sq. inch ; so that on entering the receiver the water would boil,
and if the water and vapour were continually removed the experiment might
be continued indefinitely — the water enters in the receiver at 59° F. and
boiling, so as to maintain the vapour tension something less than 0'241 Ibs.
per square inch.
In this case we have a continuous stream of water boiling at the ordinary
temperature 59° F. But this cannot be said to be boiling in an open tube.
And it is important to notice that although the water at the neck entering
the receiver would be boiling, the temperature of the receiver would be
582 EXPERIMENTS SHOWING THE BOILING OF WATER [63
maintained at, or about, 59°, so that there would be no condensation of
bubbles in the stream of water, and hence the only hissing sound would
be that resulting from the disruption of the water as in the preliminary hiss
in the kettle when the air bubbles are coming off.
If instead of withdrawing the water and vapour by means of the air-pump
we can by taking off the receiver and connecting the small end of the
conical tube, so far contracting in the direction of flow, with a similar
conical tube the other way about, i.e., expanding in the direction of flow and
discharging into a reservoir at a lower level than that of the supply, make
such arrangements that the momentum of the water entering the diverging
pipe at the minimum section would be sufficient to sweep out the water and
bubbles of vapour and air which had been formed in the contracting tube,
and secure in the expanding pipe a law of pressure and velocity somewhat
similar to that of the contraction :
p + pip = a constant.
Then as the bubbles of air and vapour in the stream would be carried with
great velocity from the low pressure at the neck, where they formed, into the
higher pressure in the wider portion of the expanding tube ; so that the
pressure being greater than the vapour tension, condensation would ensue
and the bubbles would collapse, producing the hiss of the kettle before
boiling, and in this case we should have water boiling in an open tube.
Although certain conditions are necessary a simple experiment shows that
these may be realized.
Take a glass tube, say, half-an-inch internal diameter and six inches
long, and draw it down in the middle so as to form a restriction with easy
gradual curves so that the inside diameter in the middle is something less
than the tenth of an inch, leaving the parallel ends of the tube something like
2£ inches each. And then connect one of these parallel ends by flexible hose
to a water main which is controlled by a tap. Then, on first opening the tap,
the water entering from the main at A will fill the tube as far as the
restriction, and pass through the restriction, but it will not, in the first
instance, of necessity fill the tube on the far side of the restriction. If the
water is turned on very slowly and the open end of the tube is inclined
upwards, then the water will accumulate and fill the tube, displacing the air.
But if the water is turned on sharply so that when it reaches the neck it
has a velocity of 40 or 50 feet a second, the water after passing the minimum
section will preserve its velocity and shoot out as a jet from a squirt,
not touching the sides of the glass, while if the open end of the tube be held
downwards the water, whatever the velocity, will, after passing the restriction,
run out of the tube without filling it.
63]
IN AN OPEN TUBE AT ORDINARY TEMPERATURES.
583
In neither of these cases is there any hiss or sound except such as is
caused by the free jet passing through the air.
But on holding the open end of the tube upwards and quietly filling both
limbs of the tube by opening the tap very quietly, as in case (1), and then
turning on more water, the water will not shoot out in a jet but will come
out like any other stream — as it might do if there were no restriction.
At first, while the velocity through the neck is below 50 feet per second,
there is no sound, but as soon as a velocity of 54 feet per second is attained,
or a little more, a distinct sharp hiss is heard — exactly resembling that of the
kettle or the hiss of the water through a tap.
So far however this is no proof that the hiss is the result of the boiling
or disruption of the water. But the hiss is not the only evidence afforded
by the experiment. If the glass tube, through which the water is flowing at
velocities below that at which the sound comes on, be carefully examined
against a black ground to see whether there are any imperfections in the
glass in the region of the neck, such as minute bubbles, and the positions of
any such carefully located ; and if then, after increasing the flow so that the
584 EXPERIMENTS SHOWING THE BOILING OF WATER [63
hiss just begins, it be carefully examined again, a small white speck will be
observable somewhere in the region of the minimum section a little before
where the water enters the neck. This is always observable unless obscured
by imperfection in the glass. And a crucial test is afforded by varying the
tap so as to bring the hiss on and off; when it will be seen that the appear-
ance and disappearance of the spot and the starting and stopping of the hiss
are simultaneous.
The white spot against a dark ground indicates reflection of light by a
frost-like surface such as would be afforded by bubbles coming on and going
off rapidly, and is thus a crucial proof of disruption in the water or between
the water and the glass.
The sound, when it first comes on, is generally loud enough to be heard
distinctly over a lecture room, and any increase in the flow augments the
sound as well as the size of the spot. See Fig. 4, page 583.
During the experiment the water is quietly flowing out of the tube
running with a full bore but with rather an uneven surface, which indicates
some internal disturbance. If however the parallel parts of the tube leading
to the open end be examined both when the hiss is on and off another
phenomenon will be observed, which again furnishes evidence of the effects
of boiling.
When the hiss is on, the water in the tube will be somewhat opaque —
rather foggy — which fog disappears after the hiss is stopped.
This fog is caused by the separation of the air occluded in the water,
and corresponds exactly to the separation of the air, as when the tem-
perature of the water in the kettle is above 174° F. In the case of the tube
the bubbles of air, which separate out, are very much smaller than those
in the kettle on account of the greater violence of the action.
If however instead of holding the tube with the mouth inclined upwards
the tube be immersed in a beaker of water, the water coming out of the
tube into the beaker will present the appearance of clouds of white smoke
from a chimney. On close examination it is seen that the whiteness is due
to minute bubbles, while the cloud-like appearance in the beakers is owing
to the fact that the air in these bubbles is being somewhat rapidly again
occluded by the water in the beaker, while the motion of the water in the
beaker, disturbed by the flow from the tube, wafts the foggy water as it
leaves the tube in directions which are continually changing until the
reocclusion terminates their existence, leaving those parts of the water
farthest from the mouth of the tube practically clear.
In showing this experiment it is not my object to enter into the hydro-
dynamical and physical considerations, on which the explanation of increase
63]
IN AN OPEN TUBE AT ORDINARY TEMPERATURES.
585
of pressure as the water flows along the diverging tube depends. And
I will conclude by pointing out that these considerations are entirely distinct
from those on which the fall of pressure in the water proceeding along the
converging channel depends.
In the latter eddies or tumultuous motion the water has no function
other than that of diminishing the rate at which the pressure falls, while in
the former the rate of increase in the pressure depends entirely on this
sinuous eddying or tumultuous motion.
It has been proved definitely that water moving between solid boundaries
has two manners of motion depending on whether the value of the quantity
expressed by
is greater or less than a certain numerical constant K.
In a parallel pipe water in tumultuous motion entering with a velocity
V such that
- is greater than 1400,
will, as it flows in the pipe at a steady rate, convert all the eddying motion
into heat.
While if water enters a pipe without tumultuous motion, such motion
will be generated if
- r\ir
is greater than 1900.
These limits have been for some time established as the limits of the
criterion K for straight smooth pipes, and having thus found the limiting
values of the purely numerical physical constant, it still remains to find the
form of the function corresponding to £- under other boundary conditions
586 EXPERIMENTS SHOWING THE BOILING OF WATER [63
such as parallel pipes with sections other than round and smooth and for
converging or diverging boundaries. Such determinations present analytical
difficulties which have not been altogether overcome. But it has been
possible to obtain from analysis evidences that in converging pipes the critical
velocity increases very rapidly with the rate of convergence and, on the
other hand, that the critical velocity in diverging pipes diminishes very
rapidly as the divergence increases.
When the mean velocity of the water taken over the section of the pipe,
whether parallel, converging or diverging, is greater than the critical velocity,
there is a steady fall of pressure all along the channel and no rise of pressure
in any part, as long as the flow is horizontal.
But as soon as the rate of mean flow exceeds the critical velocity the
motion becomes tumultuous — the water moving in all directions across the
channel as well as along the channel ; so that the continual mixing up of
the water which has high forward velocity with that which has less, effected
by the lateral motion, ensures a nearly uniform velocity of mean flow across
the channel between the boundaries, except at the actual boundaries.
The eddies or tumultuous motion represent an irreversible loss of head or
vertical energy in the outflowing stream, but this loss is definitely controlled
by the laws of momentum, and were it not for the resistance at the boundaries
this law would admit of analytical expression.
Thus, taking u for the velocity of flow, /(tan 6} as expressing the diverg-
ence of the boundaries, # as the direction of flow, A for the area of the section,
Ac the area at the neck,
PC + pue* - /(tan 0) pu* . jl - ^J j = gpH.
Such law is only approximately fulfilled on account of our want of
definite knowledge of the resistance at the boundaries. But it is com-
paratively easy to experimentally determine the value of the function /(tan 0}
for some particular arrangement, and it is found that the same law holds
for all geometrical similar arrangements however different the dimensions
may be, provided that the velocity is inversely proportional to the linear
dimensions.
It also appears that if the divergence, as expressed by tan 0, is small,
owing to the greater length the water has to traverse in the diverging
channel to attain equal total divergences, the loss of head owing to the
resistance at the boundaries exceeds the resistance where tan 6 is greater ;
so that there is a particular value of tan 6 for which the loss of head is a
minimum. And it is found by experiment that when tan 6 is such that the
loss is a minimum the loss of head is about 0'4. Taking this to be the total
loss of head in the whole arrangement, it follows as a direct consequence
63] IN AN OPEN TUBE AT ORDINARY TEMPERATURES. 587
that with the pressure of the atmosphere 1470 Ibs. per square inch, and the
temperature 59° F. giving a vapour tension 0'241 per square -inch, the
minimum pressure necessary to reduce the pressure at the neck to the
vapour tension would be
1470 -0-241
0-6
+ 0-241 = 24-34,
or subtracting the pressure of the atmosphere the excess of pressure in the
reservoir over and above that of the atmosphere is 9'64 Ibs. per square inch.
In this case, supposing there were very little air occluded in the water
there would be no boiling or rupture in the water, but with the usual amount
of air the ruptures would occur under a somewhat less difference of head,
such rupture corresponding to the preliminary discharge of air in the kettle.
If the head is increased the point of rupture takes place earlier, that is,
at a point before the neck is reached, and the supply of air being strictly
limited the pressure will fall until the water boils, sending forth the hissing,
or it may be screaming, sound resulting from the sudden condensation of the
vapour entering the higher pressure after passing the neck and producing
the further evidence of disruption already pointed out. And thus demon-
strating that the only sound due to the flow of water between solid
boundaries results from the boiling or disruption of the water, whether
the actual source of the sound is the disruption or the subsequent condensa-
tion of the vapour in the vacuum produced.
64.
ON THE BEHAVIOUR OF THE SURFACE OF SEPARATION
OF TWO LIQUIDS OF DIFFERENT DENSITIES.
[From the Ninth Vol. of the Fourth Series of the " Memoirs and Proceed-
ings of the Manchester Literary and Philosophical Society." Session
1894—95.]
(Read March 19, 1895.)
THE paradox first noticed by Benjamin Franklin which was brought
before the Society by Dr Schuster at the last meeting, namely, that when
a glass vessel containing water and oil, so that the oil floats on the top of the
water, forming two surfaces, one the upper surface of the water and lower
surface of the oil, the other the surface between the oil and the air, is moved
with a periodic motion, the surface separating the two fluids is much more
sensitive and much more disturbed than the upper surface, is very striking,
even when the motion of the vessel is somewhat casual — such as may be
imparted by the hand. And the paradox becomes even more pronounced
when the vessel is, by suspension or otherwise, subject to regular harmonic
motion in one plane, and compared with a vessel similar in all respects and
similarly situated, except that it contains one fluid only. For while the
upper surface of the oil appears to follow the motion of the vessel, remaining
very nearly perpendicular to the line of suspension, as it would if the whole
mass were a solid, the free surface of the water in the vessel without oil
has a decidedly greater amplitude than that of the line of suspension, though
the oscillations are exactly in the same phase and the amplitude is still
small. On the other hand the surface separating the oil and water has an
oscillatory motion about the line of suspension, much greater in magnitude
than that of the surface of the water in the vessel without oil, and in exactly
the same or the opposite phase. Another very striking fact is that all the
surfaces appear to remain plane surfaces when the motion is within certain
64] ON THE BEHAVIOUR OF THE SURFACE OF SEPARATION, ETC. 589
considerable limits. These motions, however, do depend on the relations
between the length of the pendulum, the size of the vessel, and the depths
of the fluids, the phase of the separating surface changing from the same
to the phase opposite to that of the line of suspension if the pendulum
is shortened. The solution of the problem presented by this paradox,
although not altogether confirmed or fully worked out, appears to be in-
dicated by the fact that the oscillations of all the surfaces are steady, and
in the same or opposite phase with the line of suspension, that is, in the
same or opposite phase with the disturbance, together with the fact that
the free surfaces of the fluids remain nearly plane. For if any material
system is, when disturbed, capable of oscillating in a particular period (its
natural period), and such oscillation is subject to a viscous resistance, then if
subject to a very gradually increasing disturbance, having a period longer
than the natural period, the system will oscillate in the period of disturbance
always in the same phase as the disturbing force. But if the disturbance
has a period shorter than the natural period, the system will oscillate in the
same period as the force, but in the opposite phase. Now, in the vessel with
oil and water three systems of oscillation, or wave motions, are possible. If
the vessel were completely full, so that there were no free surface, and if
there were no oil, no oscillation would be possible except (1) the pendulous
motion. If half full of oil and filled up with water, then, if disturbed and
left, a wave motion (2) in its natural period would be set up in the surface
between the oil and water. In the same way (3) if the vessel were half
full of water without oil. But in the latter case (3) the natural period would
be two or three times less than (2) between the oil and water. Now,
when the vessel contains oil and water, disturbances (2) and (3) will both
be set up, and might continue, till destroyed by viscosity, in their natural
periods if these were the same, but the periods being different, the oscilla-
tions in the period (3) would cause periodic disturbance in (2), and the
natural period of (3) being much shorter than that of (2), the oscillation
so maintained in (2) would be in opposite phase to (3), but, owing to
viscosity, such maintenance would be of short duration. If, however, the
natural period of the pendulous motion (1) of the vessel were in magnitude
between the periods (3) and (2), smaller than (2) and greater than (3), then it
would maintain an oscillation in the same period as the pendulous motion
in (3) and also in (2), that in (3) having the same phase as the pendulum,
that in (2) having the opposite phase. So far this explanation is only
partial, as it is assumed that there will be a disturbance in (2) in the
same phase as in (3). That this must be the case, however, becomes
evident when it is considered that the motion of the water cannot be that
of a solid, but must be irrotational, and that the disturbance arises from
the non-spherical form of the surfaces of the fluids. If the surface of the
vessel were flexible, the motion of the fluids would be essentially that of a
590
ON THE BEHAVIOUR OF THE SURFACE OF SEPARATION, ETC.
[64
particular portion of the water in a long wave adjacent to the surface as
shown in the figure. In this, the plain lines indicate lines in the water at
rest, which take the position of the dotted lines when the wave surface has
the position of the thick dotted line. The black circle indicates the surface
of the spherical vessel ; and the dotted curve shows the shape this surface
would become if it were subject to the same distortion as the water. In
fact, the vessel is rigid, and the surface of the water must conform to it,
which requires further internal distortional motion of the water. It is seen
there is an excess of water at the top on the higher side and a deficiency on
the lower, to supply which the upper surface must be still further tipped,
while there is a deficiency on the higher side below and an excess on the
lower side, to remedy which the lower surface must tip in the opposite
direction. This is exactly what is seen with the oil and water, and is there
though it cannot be seen in the water, although not to so great an extent
because there is no possibility of an internal wave as between the oil and
water.
65.
ON METHODS OF DETERMINING THE DRYNESS OF
SATURATED STEAM AND THE CONDITION OF
STEAM GAS.
[From Volume 41, Part I. of " Memoirs and Proceedings of the Manchester
Literary and Philosophical Society." Session 1896-97.]
(Read November 3, 1896.)
WHEN, after all air has been expelled from a vessel partially filled with
water and kept at rest at a constant temperature, equilibrium is established,
the vapour is said to be dry saturated steam.
It is easy to show that under these circumstances the pressure of the
steam is a definite function of the temperature. But it has been found very
difficult to show, by direct means, that the density of the steam is also an
invariable function of the temperature, although many experiments, from
the time of Watt, have indicated that this is the case; those of Fairbairn
and Tate being the least open to criticism.
That the density of dry saturated steam is a constant function of the
temperature has, however, been completely established indirectly by the
experiments of M. Regnault on the total heat of evaporation, although
these experiments do not directly furnish a measure of the density. These
experiments consisted in maintaining a vessel containing a definite quantity
of water in steady constant condition as to temperature and pressure and
quantity of water, by the steady admission of water at any constant tem-
perature, and the withdrawal of the vapour in an upward direction, with a
slow motion so as to preclude the convection of water out of the vessel by the
steam, the steam so withdrawn being condensed in a calorimeter back again
592 ON METHODS OF DETERMINING THE DRYNESS OF [65
to water at any constant temperature. The results proving that the total
amount of heat given up by the steam for each temperature in the boiler
is consistently proportional to the weight of steam condensed.
It thus appears that the density of saturated steam at constant tem-
perature must be constant, and that gravity alone is sufficient to free the
saturated steam from any water that may have been entangled with it by the
action of boiling, provided the rate of flow over the surfaces is not sufficient
to carry along with the steam any water there may be on the surfaces. It
was only after the utmost care in securing these conditions that Regnault
succeeded in obtaining consistent results — which results have since been
confirmed by many researches, including that of Messrs Harker and Hartog
read before the Society last year.
It is to be noticed that the whole theory of the properties of steam, as at
present accepted, and all the steam tables, are founded on these experiments
of Regnault's on the total heat of evaporation, so that if any other definition
is given of dry saturated steam, than that of the vapour of water which
results from boiling the water under constant pressure after it is drained of
entangled water by gravitation, these properties and tables will not apply.
Wet Steam.
For the most part the precautions taken by Regnault are precisely those
under which steam is produced in practice. That is to say, in practice the
conditions in the boiler are maintained, as far as practicable, steady, and
the steam is withdrawn in a vertical direction from the steam space over the
water, where it is drained by gravitation. Owing, however, to exigencies
as to space and weight, a great deal more steam is often generated in
proportion to the space than was the case in the experiments. Also the
velocity of the steam after entering the steam pipes is, in practice, often so
great that, even where these are ascending, any water that may have been
drawn in with the steam, or produced by condensation owing to the radiation
of heat from the pipes, is swept along with the steam ; and where, as in cases
like the locomotive, the engine is under the boiler, so that the pipes are
descending, this must be so. Under such conditions the steam as it enters
the engine will be accompanied by some water, and is then variously called
"wet steam," "nearly dry steam," or "super-saturated" steam, though the
last name is apparently intended to imply that, notwithstanding Regnault's
experiments, the density of steam after drainage is not necessarily a definite
function of the temperature or pressure.
Whatever may be the cause of the water entering the engine with the
steam, its presence in unknown quantity prevents Regnault's formula for the
65] SATURATED STEAM AND THE CONDITION OF STEAM GAS. 593
total heat of evaporation from being used to form a correct estimate of the
quantity of heat received by the engine. For the only measures q£the steam
supplied to the engine are obtained from the measures of the teed- water
supplied to the boiler, or the water discharged from a surface condenser, so
that, if an unknown quantity of water enters with the steam, estimates so
formed must be in excess.
This is a matter of very serious consideration in all attempts to obtain a
comparison of the actual performance of an engine in work done, as com-
pared with the theoretical performance under ideal conditions. And, as the
modern practice of steam engineering is largely guided by the results of such
attempts, methods of assuring dry steam or, failing that, of in some way
measuring the percentage of water passing with the steam into the engine,
have attracted a great deal of attention.
For purely experimental purposes, it is always possible to supply the
engine with dry steam, even where the boiler is at a distance, by passing the
steam through a sufficiently large vessel close to the engine, so that the water
may be disentangled by gravitation before the steam enters the engine.
These are called water-separators. In some cases such separators form part
of the engine, but, although their employment is becoming more common,
it is only in comparatively few cases that this is practicable.
In other cases, that is, in the great majority of cases, the desire to obtain
some experimental evidence of the percentage of water in the steam as it
enters the engine, has led to the use of methods of testing samples of the
steam drawn continuously from the steam pipe close to the engine.
Sampling the Steam.
In such methods, the question of getting a fair sample of the steam as it
enters the engine is quite distinct from that of testing the sample so
obtained The water in the pipe, although moving in the direction of the
steam, will not be uniformly distributed throughout the steam, and will, to a
great extent, merely drift along the surface of the pipe and mostly on the
lower surface, so that unless a sample taken from the lowest part of the pipe
is found to be dry, in which case the steam is dry, such methods afford
but little evidence as to the percentage of water entering the engine with
the steam.
Testing the Samples.
For absolute dryness such samples may, where the pressure in the steam
pipe is steady, be tested by allowing the sample to flow quietly through
a separator, so as to drain out the water, the weight of which is then
o. K. ii. 38
594 ON METHODS OF DETERMINING THE DRYNESS OF [65
observed. But any attempt to estimate the percentage of water in the
sample involves the subsequent condensation and weighing of the steam
in the sample, as well as the drained water, which are difficult and com-
plicated operations. Besides this, the pressure in the steam pipe near the
engine is generally subject to considerable periodic alterations, owing to the
intermittent and periodic demand for steam in the engine, which introduces
complications of unknown extent.
Wire-drawing Calorimeters.
With a view to obtaining a test for the samples of steam, which should be
independent of the separator, the so-called Wire-drawing Calorimeter has
been introduced. In this, the sample of steam, whether it has been first
drained or not, is received quietly in a vessel at the same pressure as the
steam pipe, where it is at steady known pressure ; from this it is allowed to
escape continuously through a small orifice into a second larger vessel, main-
tained at greatly lower pressure than the first. In this its temperature and
pressure are measured, the steam then passing on into a condenser or into
the atmosphere.
The quantity of water present is then estimated from the observed pressures
in the two vessels, and the difference between the observed temperature in the
second vessel and the temperature of saturation at that pressure, as taken
from Regnault's tables.
Such calculations are at once seen to be based on Regnault's deter-
mination of the relations between the pressure and temperature of saturated
steam, together with the heat relations, whatever they may be, between
saturated steam and superheated steam. And, as the second relation does
not appear to be known except as a very rough approximation, the results so
obtained must be doubtful.
Results.
The results obtained with these calorimeters have apparently revealed
the presence of anything up to 5 per cent, more water in the samples than
revealed by the simple separator, and this even when the steam has been
drained in the separator before passing into the calorimeter.
This apparent experimental evidence of previously unsuspected water
carried by steam has necessarily excited great interest, and is naturally
welcomed, as it apparently brings the engines by so much nearer per-
fection.
On second thoughts, however, a very serious consideration will present
65] SATURATED STEAM AND THE CONDITION OF STEAM GAS. 595
itself, namely, that if the drained steam from a separator contains latent
water, the drained steam from the separator on which Regnauit "made his
experiments must also have contained similar latent water, and hence the
theoretical volumes of steam, which are based solely on these experiments,
must be subject to identically the same corrections as the observed results,
so that the discovery, if true, would thus leave the percentage of theoretical
performance unchanged, while it would upset the truth of Regnault's results
as to the properties of steam — and, moreover, upset all other deductions
from these properties, including the deductions involved in these estima-
tions.
That such is the case cannot be admitted until after the fullest con-
sideration and verification of the experiments, and of the method of
reduction by which the novel results have been obtained.
These experiments are many, and the methods of reducing the results
have not been very fully, although widely, published, but in all that I have
seen the results have been deduced by means of the properties of steam as
determined by Regnault's experiments, by a formula which is based on a
misunderstanding of the meaning of " the specific heat, at constant pressure,
for steam when in the gaseous state," as determined by Regnauit. And that
this must have been the case with the other results would seem to follow
from the fact that this formula, when based on the correct meaning, affords
no definite result at all under the circumstances of the experiments.
It has thus seemed to me important not only to call attention to the error
in reduction by which certain of these results have been obtained, but also
to indicate, and if possible to verify, a method by which experiments could
be made, so that Regnault's determination of the specific heat of steam gas
could be correctly used to ascertain whether or not such latent water does
exist in drained steam — that is, to ascertain whether Regnault's experiments
on the specific heat of steam gas are consistent with his experiments on
the latent heat of steam.
In the present paper the purpose is limited to pointing out the theory of
the reductions, and to giving indications of the method of experimenting,
the general character of the apparatus, and the precautions necessary.
The Theory of the Reductions.
By the law of conservation of energy, when a steady stream of matter
flows through a chamber with fixed walls, so that the condition within the
chamber is steady, the energy of the matter which enters (potential and
actual) is equal to the energy which leaves in the same time, and hence is
equal to the energy of the matter which leaves, together with such energy as
38—2
596
ON METHODS OF DETERMINING THE DRYNESS OF
[65
may escape into the walls of the chamber. Thus, if a stream of fluid flows
in a horizontal direction through a fixed passage and if
at A,
Pi = pressure,
TI = temperature,
Vi= volume per Ib. of fluid,
HI — P]' Vi = mechanical equivalent of heat per Ib. of fluid,
%! = velocity of fluid
P2' = pressure, \
TJ = temperature,
F2' = volume per Ib. of fluid, \ at B,
HZ — P/F2' = mechanical equivalent of heat per Ib. of fluid,
u2 = velocity of fluid
and Hj = ihe mechanical equivalent of heat received through the surface
per Ib. of fluid passing through ;
then
Also, if the fluid at A consists, per Ib., of
51 Ib. of steam and (1 — Si) Ib. of water,
and at B consists of
52 Ib. of steam and (1 — $2) Ib. of water,
and if ^ and h2 are put for the mechanical equivalents of heat per Ib. of water
respectively at the temperatures, T^ and T2, of saturated steam at pressures of
PI and P2' respectively, then T-f = Tlt where P/ and Tl are pressure and
temperature corresponding to the initial state of saturated steam at A, and
T2 may be taken to correspond to the temperature of saturated steam at
pressure Pa'. And if, further, H^ equals the equivalent of the total heat
of evaporation at pressure P/ per Ib., then
H^S^H.-h^ + h, (2).
And if, similarly, Hz and A2 correspond to the temperature of saturated steam
at pressure P2', then
T9) (3),
65] SATURATED STEAM AND THE CONDITION OF STEAM GAS. 59*7
where K is the mean specific heat of steam at constant pressure between
the temperatures TJ the actual temperature at B, and T.2 the temperature of
saturated steam at the actual pressure (P2') at B. It being noticed that, if
1 - S2 is greater than nothing, T.2' = T2, so that the last term in (3) vanishes.
While, if (1 — $2) is zero, this last term expresses the heat, whatever it may
be, requisite to raise steam, at constant pressure PI, from the temperature of
saturation T2 to the observed temperature T2'.
Substituting from equations (2) and (3) in equation (1), this becomes
If then w1( u2, and Hj are small enough to be neglected, since the values of
Hlt hlt HZ, h2, T.2 are obtainable from Regnault's tables, when P/, P2' or f\
are observed, all the remaining quantities may be known except S1} S2, and
K. And either, if S.2 is not equal to unity, (TV — T2) = 0, and
S^-AO + AI = &(#•-*.) + *• .................. (5),
or, if (1-S2) = 0,
TJ .................. (6).
Equation (5) gives Sl in terms of S.2 when T2' — T2, but, since S2 is un-
known, this is of no use ; while, if T2 is greater than T2, equation (6) gives
Si in terms of K which is a function of T2 and TV, which has not been
determined.
If it were possible to determine the exact value of TV at which Sz— 1 = 0,
then
But, here again, this is practically impossible, since the only indication
that S.2 — 1 = 0 is that T2 is greater than T2 as given by Regnault's tables
for steam at P.,', and, for any such excess as can be observed, the value of
K(T2 — T.2) is considerable, since, at the point of saturation, K is apparently
infinite, so that neither of these determinations is practical.
With a view to getting over these difficulties, the course that has
apparently been adopted is to obtain a condition such that the temperature
(TV) after wire-drawing is from 10° to 20° F. higher than the saturation
temperature (T2), and then to assume that K is equivalent to the specific
heat at constant pressure of steam gas as determined by Regnault, or that
K = 772x048,
an assumption which constitutes the error in reduction to which I have
referred.
598 ON METHODS OF DETERMINING THE DRYNESS OF [65
The possibility of obtaining an accurate estimate.
This depends on obtaining a certain condition in the experiment, and
reducing by a formula proved by Rankine (Trans. Roy. Soc. Edinb., 1849,
1855).
Rankine's formula is that the total heat to convert water from a liquid
state at any particular temperature, say 32°, to steam gas at any temperature
(T2'), the operation being completed under constant pressure, is expressed by
TT I
@i + a ^Y ~ 32°,
Gl being a quantity depending only on the initial state, and a being the
specific heat at constant pressure of the steam gas, determined by Regnault
to be
0-48.
Taking the initial state to be at 32°, Rankine obtained, as the most probable
value,
G! = 1092.
It is to be noticed, however, that although this value 0'48, as obtained
by Regnault, has been universally accepted, the experiments by which he
obtained it were independent of the method by which he determined the
total heat of evaporation of saturated steam, and that, as Regnault observes*,
the smallness of the scale, as compared with that by which the total heats
were determined, rendered it necessarily less accurate, as regards the measure-
ment of the total quantities of heat observed, although the extreme care with
which the numerous experiments in the four cases were made, seems to
assure their relative accuracy. The experiments consisted in determining
the total heat necessary to raise water from 32° F. or 0° C. to temperatures
of about 120°C. and 220° C. under the pressure of the atmosphere, then
taking the differences as being the heat necessary to raise water from 120° C.
to 220° C. It thus involves the assumption that steam at 20° C. (or 36° F.)
above the boiling point, is in the condition of steam gas. This is probably
very near the truth. Had, however, the experiments been as absolutely
accurate as those for the total heat of saturated steam, they would have
afforded the means of comparing the two methods of Regnault by Rankine's
thermodynamical formulae. As it is, such a comparison can be made. Thus,
substituting the total heats as obtained in the experiments for specific heat
iu Rankine's formula, the constant d is found to be not 1092, as given by
Rankine, but between 1076'4 and 1053'7, with a mean of about 1055.
* Him. Acad. Sci., Vol. xxvi. pp. 170, 909.
65] SATURATED STEAM AND THE CONDITION OF STEAM GAS. 599
Taking this value, the heat necessary to raise water from 32° to 248° F. at
constant pressure of 14'7 Ibs. per square inch is
1055 + 0-48 (216) =1158-68.
To raise water from 32° to saturated steam at 212° requires by Regnault's
formula for total heat of saturated steam
10917 + '305 (180) = 1146-6.
Hence, to raise saturated steam from 212° to 248° at constant pressure would
require 12*08 T.U., which, divided by the difference of temperature, gives for
the mean specific heat of steam from saturation at 212° to 248° F. at constant
pressure
12'08 --335
"36" 3i>'
which shows that the specific heat, at constant pressure, of steam rises with
the temperature. And this, although in accordance with the results obtained
by Regnault for other vapours, presents great thermodynamical difficulties ;
since many experiments have shown that the steam, on being heated from
saturation to 36° F. above, expands three or four times as much as it would
if it were gas. It is to be noticed that an error of 3 per cent, in estimating
the total quantity of steam, which in these experiments would only mean an
error of
0-0004
in the actual weighings, would account for the differences in the values of (7,
as determined by Rankine, and as estimated from Regnault's experiment on
specific heat, while such an error on the determination of the specific heat
would fall within the limits of experimental accuracy. It thus seems probable
that Rankine's determinations of the constants in his formula are approxi-
mately right.
In order to make use of this formula in the reduction of the experiments
under consideration, all that is necessary is to bring about, by means of wire-
< Ira wing, the condition that T.,' shall be sufficiently larger than T.2 to insure
that the final condition approximates to that of steam gas. That this differ-
ence must be more than 20° F. has been shown, but it would appear that
with this difference the error is not great.
To use the formula,
(1092 + 0-48 (ZV-21,)} 772
is substituted for the right member of equation (6).
?/ ^ // "
Hj . - being small, therefore
} 2<7 2^7
&(#!-/»!) + /*, = 772 {1092 + 0-48 (Tj - T°}} (7),
which only requires the experimental determination of Tl and T2' to give the
value of Slt provided that the final condition is that of steam gas.
600 ON METHODS OF DETERMINING THE DRYNESS, ETC. [65
The means of assuring the condition of Steam Gas.
Perhaps the most important fact to which attention is herein directed is
that, although, as already stated, the limiting relations of temperature and
pressure of steam gas are not known with any degree of precision, the wire-
drawing experiments are capable of affording simple and direct evidence of
the existence of such a final state. As the pressure of steam is reduced by
wire-drawing, which is gradually increased, at first, owing to the great
expansion, the temperature falls considerably, but, as the wire-drawing in-
creases, by the diminution of pressure in the receiving vessel the fall of
temperature gradually diminishes, until the gaseous state is produced, when
the temperature T2' will be unaffected by still greater wire-drawing.
So that to insure a gaseous state, all that is necessary is to gradually
diminish the pressure in the receiving vessel, maintaining that in the first
vessel, until the temperature T£ in the receiving vessel becomes constant.
The only doubt is whether this point can be practically reached, and this
can only be determined by experiments.
. The remarkable circumstance in the flow of gases, of which I published
the explanation in a paper read before this Society in 1885, that when steam
or gas flows through a restricted channel from one vessel into another, in
which the pressure is less than half that of the first, the quantity which passes
is independent of the pressure on the receiving side, must have an important
place in simplifying the apparatus required for such experiment.
Thus, with boiler pressure on one side of an orifice, opening into a vessel
from which its escape is allowed by an adjustable valve, the whole experiment
can be regulated by this valve, the quantity flowing through remaining con-
stant for all pressures after the half is reached.
The only precautions necessary for accuracy, are those to secure approxi-
mately small velocities at the points where the temperature is measured, and
those to render small the loss of temperature in the steam by radiation.
And, although these must complicate the appliances, they appear to be
practical. I may also notice that, should such experiments be accomplished,
they will afford the means of verifying or correcting Rankine's value for Cl}
which he has only given as a probable approximate value.
I hope these experiments may shortly be made, as Mr J. H. Grindley,
B. Sc., Fellow of Victoria University, has undertaken the research in the
Whitworth Engineering Laboratory, Owens College.
66.
BAKERIAN LECTURE.— ON THE MECHANICAL EQUIVALENT
OF HEAT.
[From the "Philosophical Transactions of the Royal Society of London," 1897.]
(Read May 20, 1897.)
PART I.
By Professor OSBORNE REYNOLDS, F.R.S., and W. H. MOORBY, M.Sc., late
Fellow of Victoria University and 1851 Exhibition Scholar.
ON THE METHOD, APPLIANCES AND LIMITS OF ERROR IN THE DIRECT
DETERMINATION OF THE WORK EXPENDED IN RAISING THE TEM-
PERATURE OF ICE-COLD WATER TO THAT OF WATER BOILING UNDER
A PRESSURE OF 29'899 INCHES OF ICE-COLD MERCURY IN MANCHESTER.
—BY OSBORNE REYNOLDS.
The Standard of Temperature for the Mechanical Equivalent.
1. THE determination by Joule, in 1849, of the expenditure of mechanical
effect (772-69 Ibs. falling 1 foot) necessary to raise the temperature of 1 Ib.
of water, weighed in vacuo, 1° Falir. between the temperatures of 50° and
60° Fahr. (at Manchester), together with the second, in 1878, 772'55 ft.-lbs.,
to raise the temperature of 1 Ib. (weighed in vacuo) from 60° to 61° Fahr.,
at the latitude of Greenwich, established once for all the existence of a
physically constant ratio between the work expended in producing heat
and the heat produced; while the extreme simplicity of his methods, his
marvellous skill as an experimenter, and the complete system of checks he
adopted, have led to the universal acceptance of the numbers he obtained
602 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
as being within the limits he himself assigned (I foot), of the true ratio of
work expended in his experiments in producing heat and the heat produced
as measured on the scale of the thermometer on which he spent so much
time and care.
The acceptance of J = 772, as the mechanical equivalent of heat, amounts
to the acceptance of the scale between 50 and 60 on Joule's thermometer b
as the standard of temperature over this range.
Joule's thermometers are now in the custody of the Manchester Literary
and Philosophical Society (having been confided to its care by Mr A. Joule);
so that this material standard is available. But the standard of temperature
actually established by Joule is universally available wherever the British
standard of length is available, together with pure water and the necessary
means and skill of expending a definite quantity of work in raising the
temperature of water between 50° and 60° Fahr., since in this way the scale
on any thermometer may be compared with that on Joule's.
The difficulty of access to Joule's thermometer, and the inherent difficulty
of making an accurate determination of the equivalent, have limited the
number of such comparisons.
The most serious attempts have been made with the very desirable object
of determining the mechanical equivalent of a thermal unit, measured on
the scale of pressures of gas at constant volumes, first recognised by Joule
as the nearest approximation to absolute temperature.
The results of these comparisons have been various, all having apparently
shown that Joule's standard degree of temperature is less than the one-
hundred-and-eightieth part between freezing and boiling points on the scale
of pressure of gas at constant volume, the differences being from 01 to I'O
per cent. Joule himself contemplated comparing his thermometer with the
scale of air pressures, but did not do so. So that only indirect comparisons
have been possible.
Him, who was the first to follow Joule, in one of his researches introduced
a method of measuring the work done which afforded much greater facility
for applying the work to the water than the falling weights used by Joule
in his first determination, and this was adopted by Joule in his second
determination. But notwithstanding the greater facilities enjoyed by sub-
sequent observers, owing to the progress of physical appliances, the inherent
difficulties remained. The losses from radiation and conduction could only
be minimised by restricting the range of temperature, and this insured
thermometric difficulties, particularly with the air thermometer, which, it
seems, does not admit of very close reading. This, together with certain
criticisms, of which some of the methods employed admit, appear to have
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 603
left it still an open question what exact rise in the temperature in the scale
of air pressures corresponds to the 772 ft.-lbs.
2. The research, to the method and appliances for which this paper
relates, has been the result of the occurrence of circumstances which offered
an opportunity, such as might not again occur, of obtaining the measure, in
mechanical units, of the heat in water between the two physically fixed
points of temperature to which all thermometrical measurements are referred,
and of thus placing the heat as defined in mechanical units, on the same
footing as the unit of heat as defined by temperature, without the inter-
vention of scales, the intervals of which depend on the relative expansions
of different materials such as mercury and glass.
It has been, so far as I am concerned, undertaken with considerable
hesitation, on account of the responsibility even in attempting such a deter-
mination, and the harm to science that might follow from further confusion
owing to error in what, in spite of opportunities, must be the extremely
difficult task of making such complex determinations within less than the
thousandth part. These considerations, together with my inability to find
the large amount of time necessary for making the observations, prevented
any attempt until July, 1894. At that time Mr W. H. Moorby offered to
devote his time to the research, and so relieve me of all responsibility except
that which attached to the method and the appliances ; and having, from
experience, the highest opinion of Mr Moorby 's qualifications for carrying
out the very arduous research, there seemed to be no further excuse for
delay, particularly as after seeing the appliances in the laboratory both Lord
Kelvin and Dr Schuster expressed strongly their opinion as to the value of
the research.
The Opportunity for the Research.
3. This consisted in the inclusion in the original equipment of the
laboratory, in 1888, of the following appliances : —
(1) A set of special vertical triple-expansion steam-engines, with separate
boiler, closed stoke-hole, and forced blast ; these engines being specially
arranged to give ready access to the shafts, 3 feet above the floor, and being
capable of running at any speeds up to 400 revolutions per minute, and
working up to 100H.-P. (Plate 1).
(2) Three special hydraulic brake dynamometers, on separate shafts,
between and in line with the engine shafts, with faced couplings, so that
one brake shaft could be coupled with the shaft of each engine to work its
own shaft ; or the brakes on the high-pressure and intermediate engines
could be removed, and their shafts coupled by means of intermediate shafts,
604 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
so that all three engines worked on the brake connected with the low-
pressure engine. These brakes, which are shown (Plate 1), are separately
capable of absorbing any power up to a maximum of 30 horse-power at
100 revolutions, and increasing as the cube of the speed ; so that a single
brake is capable of absorbing the whole power of the engine at any speed
above 100 revolutions a minute.
The whole of the work is absorbed by the agitation of the water contained
in the brake, while the heat so generated is discharged by a stream of water
through the brake, with no other functions than of affording the means of
regulating, independently, the temperature of the brake and the quantity
of water in the brake. The moment of resistance of the brake at any speed
is a definite function of the quantity of water in the brake. And as, except
for this moment, the unloaded brake is balanced on the shaft, the load being
suspended from a lever on the brake at 4 feet from the axis of the shaft, if
the moment of resistance of the brake exceeds the moment of the load, the
lever rises, and vice versa. By making the lever actuate the valve which
regulates the discharge from the brake, and thus regulate the effluent
stream, the quantity of water in the brake is continually regulated to that
which is just sufficient to suspend the load with the lever horizontal, and a
constant moment of resistance is maintained whatever may be the speed of
the engines.
(3) Manchester town's water, of a purity expressed by not more than
3 grams of salts in a gallon, brought into the laboratory in a 4-inch main
at town's pressure (50 to 100 feet head), and distributed either direct from
the main or at constant pressure from a service tank 10 feet above the floor
of the laboratory.
(4) Two tanks, each capable of holding 60 tons of water, one in the
tower, 116 feet above the floor, the other 15 feet below the floor, connected
by 4-inch rising and falling mains, each 500 feet long, passing in a chase
under the floor. The rising main is in communication with a special
quadruple centrifugal pump, 2 feet above the floor, capable of raising a
ton a minute from the lower to the upper tank. (Shown in Plate 5.) Also
a set of mercury balances, showing continually the levels of water in the
two tanks, and the pressures in the rising, falling, and town's mains. (Shown
in Plate 2.)
(5) A special quadruple vortex turbine, supplied from the falling main
and discharging into the lower tank, capable of exerting 1 H.-P., and available
for steady speeds at all parts of the laboratory. (Shown in Plate 5.)
(6) A supply of power to the laboratory by an engine and boiler, quite
distinct from the experimental engine, and distributed by convenient shafting
which is always running. (Shown in Plate 1.)
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 605
The Measurement of the Work.
4. Of the appliances mentioned, the brake on the low-pressure engine
is the centre of interest, as it was by this that the work was measured, as
well as converted into heat.
The existence of the appliances was largely due to the interest in educa-
tional work taken by Mr William Mather, who, together with the other
members of the firm of Mather arid Platt, not only placed at my disposal
the facilities of their works, but, inspired the enthusiasm which alone
rendered the execution of such novel and special work possible.
The development of the brake dynamometer, from its introduction by
Prony, has an interesting and important history, but into this it is not
necessary to enter. The purpose of these dynamometers is to afford con-
tinuous frictional resistance, adapted to the power exerted by the prime
mover in causing a shaft to revolve, and of a kind that is definitely measure-
able. To fulfil the first of these conditions, the mean moment of resistance
of the brake must just balance the mean moment of effort of the engine,
and the means of escape of heat from the brake must be sufficient to allow
all the heat generated to depart, without accumulating to an extent which
may interfere with the action of the appliances. In the first brakes the
resistance was obtained by the friction of blocks or straps pressed against
a cylindrical wheel on the shaft, and, small powers being used, radiation
and air-currents round the brake were found sufficient to carry off the heat,
but, when larger powers were used, these sources of escape failed to keep
the temperatures down to practical limits, which necessitated the application
of currents of water to carry off the heat.
The measurement of the work was invariably accomplished by attaching
the brake blocks, or straps, to a lever, or arm, so that the whole brake would
be free to revolve with the brake-wheel, except for the moment of the weight
of the parts which, adjusted to the power of the engine, was kept in balance
by the adjustment of the pressure of the blocks on the wheel. Then, since
the work done is equal to the product of the mean moment of resistance,
over the angle turned through, multiplied by the angle, if the resistance is
constant over time, the moment of the brake, multiplied by the whole angle,
measured the work done.
It is however to be noticed that the assumption, that the time-mean of
the moment on the brake is the same as would be the angle-mean of this
moment, might involve an error of any extent, provided the resistance and
the angular velocity varied in conjunction. And as steam engines invariably
exert an effort, varying within the period of the revolution, while the friction
and the pressure causing it are apt to respond to any variations of speed, it
606 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
is probable that there has been some error from this cause in all such
measurements, although not previously noticed.
Hirn appears to have been the first to recognise that in a steady condition
the resistance of fluid between the brake-wheel and the brake would answer
instead of the solid friction, so that the mean time moment of effort exerted
in turning a paddle in a case, containing water, with bafflers, would be
strictly measured by the mean time moment of the case. And although
subject to the same error from periodic motion as the friction brake, the
facility this fluid brake offered for cooling arid regulating led to its simul-
taneous adoption and development by several inventors, for measuring power
— the late William Froude, for the purpose of measuring the work of large
engines, inventing that arrangement of paddle vanes and bafflers which gives
the highest resistance, regulating the resistance by thin sluices between
the vanes and bafflers, and always working with the case full of water.
The brake under consideration differs from that of Mr Froude in only
one fundamental particular — the provision by which a constant pressure in
the interior of the brake is secured by the admission of the atmosphere to
that part of the brake where the dynamical effect of the water is to cause the
lowest pressure — this admits of working the brakes with any quantity of
water from nothing to full, and thus allows of the regulation of the re-
sistance, by regulating the quantity of water in the brakes, without sluices.
The description of this brake has already been published, together with
that of the engines*, but it will be convenient to give a short description.
This brake consists primarily of (1) a brake wheel, 18 inches in diameter,
fixed on the 4-inch brake shaft by set pins, so that it revolves with the shaft
(Figs. 2 and 3), and (2) a brake (or brake case) which encloses the wheel, the
shaft passing through bushed openings in the case which it fits closely, so as
to prevent undue leakage of water while leaving shaft and brake-wheel free
to turn in the case, except for the slight friction of the shaft (Figs. 1, 2
and 3).
The outline of the axial section of the brake- wheel is that of a right
cylinder, 4 inches thick. The cylinder is hollow — in fact, made of two discs
which fit together, forming an internal boss for attachment to the shaft, and
also meet together at the periphery, forming a closed annular box, except for
apertures to be further described (Fig. 3). In each of the outer disc faces of
the wheel are 24 pockets, carefully formed, 4£ inches radial, and 1£ inches
deep measured axially, but so inclined that the narrow partitions or vanes
(^ inch) are nearly semicircular discs inclined at 45° to the axis ; the vane
on one face being perpendicular to the vane on the opposite face (Fig. 2).
* " Triple Expansion Engines," by Professor Osborne Reynolds, Minutes of Proceedings, Inst.
C. E., vol. 99, 1889, p. 18. (See Paper 56, page 336.)
66]
ON THE MECHANICAL EQUIVALENT OF HEAT.
607
The internal disc faces of the brake case, as far as the pockets are con-
cerned, are the exact counterparts of the disc faces of the wheel,- except that
there are 25 pockets, so that the partitions in the case are in the same planes
as the partitions meeting them in the wheel, there being ^ inch clearance
between the two faces.
Fig. 1.
The pairs of opposite pockets, when they come together, form nearly
closed chambers, with their sections, parallel to the vanes, circular. In such
spaces vortices in a plane inclined at 45° to the axis of the shaft may exist, in
which case the centrifugal pressure on the outside of each vortex will urge
the case and the wheel in opposite directions inclined at 45° to the direction
of motion of the wheel, which will give a tangential stress over the disc
faces of the wheel of 1/V2 of the sum of these vortex pressures. The
existence and maintenance of these vortices is insured by the radial
centrifugal force of the water in the pockets in the wheels owing to its
motion.
This is the late Mr W. Froude's arrangement. But an essential feature
608
ON THE MECHANICAL EQUIVALENT OF HEAT.
[66
of the brake is the provision which insures the pressure of the atmosphere
at the centre of the vortices, even when the pockets are only partially
filled.
Fig. 2.
The vortex pressure is greatest at the outsides of the vortices, which
occurs all over the annular surfaces of the pockets, but the actual pressure
on these surfaces is not determined solely by the vortex motion unless the
state of pressure at the centre of the vortices is fixed, for the vortex motion
only determines the difference between these pressures. To insure the
constant pressure, and at the same time to allow of the pockets being
only partially full — that is, to allow of hollow vortices with air cores at
atmospheric pressure, it is necessary that there should be free access of
air to the centres of the vortices, and as this access cannot be obtained
through the water, which completely surrounds these centres, it is obtained
by passages (^ inch diameter) within the metal of the guides, which lead to a
common passage opening to the air on the top of the case (Figs. 2 and 3).
To supply the brake with water there are similar passages in the vanes of
the wheel leading from the box cavity, which again receives water through
ports which open opposite an annular recess in one of the disc faces of the
66]
ON THE MECHANICAL EQUIVALENT OF HEAT.
609
case into which the supply of water is led, by means of a flexible indiarubber
pipe, from the supply regulating valve.
Fig. 3.
Fig. 4.
The water on which work has been done leaves the vortex pockets by the
clearance between the disc surfaces of the wheel and case, and enters the
annular chamber between the outer periphery of the wheel and the cylindrical
portion of the case, which is always full of water when the wheel is running,
whence its escape is controlled by a valve in the bottom of the case, from
which it passes to waste.
By means of linkage connected with a fixed support and the brake case,
an automatic adjustment of the inlet and outlet valves, according to the
position of the lever, is secured without affecting the mean moment on
the brake case. And this also affords means of adjusting the position of
the lever. To admit of adjustment for wear, the shaft is coned over that
portion which passes through the bushes, the bushes being similarly coned,
and screwed into short sleeves on the casing, so that by unscrewing them the
wear can be followed up and leakage prevented.
The brake levers for carrying the load and balance weight, are such as to
allow the load to be suspended from a groove parallel to the shaft, at 4 feet
from the shaft, by a carrier with a knife edge, the carrier and the weights
each being adjusted to 25 Ibs. (shown figs. 1 and 4). In addition to this
load, a weight is suspended from a knife edge on the lever nearer the shaft,
this weight being the piston of a dash-pot in which it hangs freely, except
39
O. B. II.
610 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
for the viscous resistance of the oil. This weight being adjusted to exert a
moment of 100 ft.-lbs., and again a travelling weight of 48 Ibs., is carried on
the lever and worked by a screw with £ inch pitch, so that one turn changes
the moment by 2 ft.-lbs., while a scale on the lever shows the position.
A shorter lever on the opposite side of the case carries a weight of 74'6 Ibs.,
which is adjusted to balance the lever and sliding weight when the load is
removed.
The Accuracy of the Brake.
5. The principle of these hydraulic dynamometers is that when moment
of momentum is introduced into a fixed space without altering the moment
of momentum within that space, the rate at which moment of momentum
leaves the space must equal the rate at which it enters. The brake- wheel
imparts moment of momentum to the water within the case, and the friction
of the shaft imparts moment of momentum to the case. The water in the
case, when its moment of momentum is steady, imparts moment of momentum
to the case as fast as it receives it, and the time mean of the moment of the
load is equal to the time mean of the moment of the effort of the shaft.
This is not affected by water entering and leaving the case at equal rates,
provided it enters and leaves radially.
The condition of steadiness is, however, essential, in order that the
moment of effort shall be at each instant equal to the moment of resistance
on the case ; any change in the moment of momentum of the water in the
case being the result of the difference of the moment of effort on the shaft
and that of resistance on the case.
The Time-Mean of the Moment of Effort.
6. When, however, the shaft is run over an interval of time, the mean
moment of resistance on the case, less the difference of the moments of
momentum of the water, at the end and beginning of the interval, divided
by the time, is the time-mean moment of effort on the shaft.
The possible limit of this error may be estimated when the maximum
moment of momentum of the water is known as well as the minimum
moment of resistance, and the minimum interval of time.
Thus taking the limits to be 30 Ibs. of water, with radius of gyration
0'66 foot, at 300 revolutions a minute (< 14), the interval of running 3600
seconds, the moment of the load 400 ft.-lbs., the limit of the time-mean of
change of moment of momentum of the water is 14/3600, and this divided
by the mean moment of resistance gives as the limits of relative error,
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 611
+ O'OOOOl. This is supposing the whole of the water to be absent at the
beginning or end of the trial, while the actual difference never amounts to
more than 2 or 3 Ibs., so that the limits do not exceed O'OOOOOl, which is
neglected.
The Angle-Mean of the Moment of Effort on the Shaft.
7. As already pointed out in Art. 4, when both the angular velocity of,
and the moment of effort on, the shaft are subject to fluctuations of speed,
the time-mean of the moment of effort may differ from the angle-mean. This
applies to all brakes, but in hydraulic brakes, in which the resistance is
proportional to the square of the speed, although lagging by an unknown
interval, it becomes possible to estimate the possible limits of this error
when the limits of fluctuation of speed are known.
Taking <u the angular velocity of the shaft and &>0 the time-mean of the
angular velocity, 2a2&>0 the extreme differences of speed, and assuming the
variation to be harmonic,
= a>02 l
(
<u = <»0{l-f a2 cos ?? (tf-Tj)} ........................ (1),
l + ^ + 2a2 cos n (t - 1\) + £a4 cos 2n (t - T)\ ...... (2).
2 J
Then to a second approximation, neglecting a6, if T^ is the interval of
lagging in the resistance, and M the moment of resistance at the time t,
M=M0{1 + 2aacos n (t - 2\ - rl\} + ^a4cos 2w (t-T>- T2)} ...(3),
where M0 is the time-mean of the moment of resistance. Also the rate at
which work is done with uniform velocity, is J/o>0, of which the mean is
M0(i)0, and is the rate of work as measured by the mean moment on the case,
multiplied by the mean-angular velocity.
To a second approximation the rate of work with varying speed is
Mo> = M0a>0 { 1+ 2a2 cos n (t - rl\ - Ta) + £a4 cos 2n (t - T, - rl\)}
(H-OaCOSW^-^)} ...... (4),
and from this it appears that the mean rate of work is
6>0JUQ (1 + a* cos nl'a),
which shows that the relative error in taking this as Jt/0<«>o is +tt4coswjT2.
Thus the error arising from fluctuations in speed of 2a2o) is within the limits
± a4, when the resistance varies as the square of the speed, as in the hydraulic
brakes.
Where, as in the brake under consideration, there is an automatic adjust-
ment, by which the quantity of water in the brakes is adjusted to the speed,
39—2
612 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
so as to maintain the resistance constant, there will be no error caused by
such gradual variations of speed as result from changes in the boiler pressure,
since the automatic adjustment can keep pace with them. But it takes time
for the water to get in and out, and any variations, so rapid that, owing to
the inertia of the brake case with its load, their effect has been reversed
before the case has moved sufficiently to affect the water in the brake, will
produce errors.
Such cyclic variations of speed attend all motions derived from recipro-
cating engines, and it is only these, and not the secular variations, that
produce errors.
The Variations in the Speed of Rotation of the Steam- Engine.
8. The cyclic variations all go through one or two complete periods in
the time of revolution of the engine, and are approximately simple harmonic
functions of the time.
They arise from three distinct causes : —
(1) The varying energy of motion of the reciprocating parts ;
(2) The varying moment of the effort of the steam pressures on the
cranks ;
(3) The effect of gravitation on the unbalanced parts in the engine.
In the case of a simple vertical engine, unbalanced and working with
moderate expansion, these variations of speed may be severally estimated
when /, the moment of inertia of the revolving parts, r the half-stroke of the
reciprocating parts, and W the weight of these parts are known, together
with N the number of revolutions per minute, and U the work done per
stroke.
For, considering the variations as existing separately, we may assume
that the angular motion would be steady but for the particular effect, thus :
(1) The moment of effort on the crank being constant, and the resistance
constant, and equal to the effort, the energy of motion of all the parts is
constant.
Putting ft>= 277-^/60, and i = r*W/g,
£/<os + £ r«2 sin2 nt = C,
where C is constant, t is the time since the axis of the crank-pin has crossed
the axis of the cylinder and n is <w0, the mean value of a* or
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 613
Whence neglecting i as compared with I, the extreme variation of o> is
approximately
whence
O^-ij.
(2) In the same way, considering the effect of the crank effort alone,
with a moderate expansion, the energy that has to be absorbed and given out
by the revolving parts is about one-fourth part of the work per stroke, and
£/«2 - £ £7 cos 2n (t-T)= C,
where nT, say — is the angle of the crank at which <o2 is a minimum.
The extreme fluctuations in velocity are
U m° U
<o = o)0 ll 4- ft ,— 2 cos 2 (nt -
( Ol Wo
(3) The effect of the weight of the reciprocating parts acting alone,
causes a fluctuation on the revolving parts of 2rW; thus approximately
and
Wr
o> = <u0 1 +
r \
— - cos nt ,
too )
giving an extreme fluctuation on the angular velocity of
Wr
agx-ijr^*.
The equation of velocity is thus approximately expressed by
[I i U Wr ~\
1 + j 7 cos 2«^ + R j — -2 cos 2 (nt — £TT) 4- j •— -t cos nt .
In the low-pressure engine used in these experiments, the values of the
several quantities are, the units being linear feet, lb., seconds,
7=126, 1 = 2-47, r = 0-025, TF=200, rTT=125, tf=
li -00049 J7!-
~ IJ' -
614 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
whence, substituting
CD = o)0 f 1 + 0-0049 cos 2a>0£ + -^-t cos 2 f &>0< - - J + ^ cos w^J ,
from this the approximate joint error can be found. But it is sufficient here
to show that the individual errors are negligible.
The first gives an error in the mean moment
< ± 0-000024 (Mafi.
The second and third are inversely proportional to N\ If N is 800,
which is the lowest value,
the second error is between
< + 0-0000025 (Ma,4).
The third
< + 0-0000001 (Ma*}.
These are all negligible quantities, and, as the corresponding effects in
the high-pressure and intermediate engines, owing to the cranks being set
at angles of 60°, would only be to compensate those of the low-pressure
engine, the greatest error would not exceed w<yoot
9. Besides the errors resulting from the terminal differences in the
moment of momentum of the water and the fluctuations of speed in the
engine, error in the measurement of the work may arise from imperfect
balance of the brake, from the frictional resistance of the automatic gear,
from unequal resistance in rising and falling of the piston of the dash-pot,
and from the end oscillation of the brake.
The Error of Balance of the Brake.
Although, when the shaft is running, the brake levers are perfectly free
between the stops, yielding to the slightest force even when carrying a load
of 400 pounds in addition to the weight of the brake-case of over 300 pounds,
yet, when the shaft is standing, it requires a moment of some 40 ft.-lbs. to
move the lever in either direction, so that the balance can only be obtained
as the difference of these moments, and this can only be obtained to about
1 foot pound. But, it is to be noticed that as long as the distribution of
weights is unaltered and the lever is in the same position, any error of
balance, whatever might be its cause, would be the same for all trials, no
matter what might be the difference in the suspended load ; so that, in taking
the difference of the trials, the error would be eliminated, and, to insure
this, the automatic adjustment was so arranged that, by a screw adjustment,
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 615
the lever could be raised or lowered without affecting the automatic adjust-
ment of the valves (see fig. 4, p. 609). Also an index was arranged adjacent
to the end of the lever, to which it might be always adjusted (shown in
Plates 2 and 3).
The Error of Balance resulting from Friction of the Automatic Gear.
This had been a matter of serious consideration in designing the brakes,
for, although it was obviously possible to so balance the parts of such gear
that there should be no pressure, arising from the weight of this gear,
against the fixed support, it was not obvious that the friction of these valves
and their gear would not allow of a steady resistance to motion being
maintained, that is, would not allow the brake to lean against the fixed
support within the limits of friction. However, after careful consideration
of various contrivances, I came to the conclusion that, if the gearing between
the support and the valve were inelastic, the joints being an easy fit, the
tremor of the shaft and the brake, when running, might be depended
upon to release any frictional resistance in this gear; so that, after any
change, the gear would rapidly return to equilibrium. This proved to be
the case, even to an unexpected extent, as was shown by the freedom of all
the pins.
It was subsequently found by experiment that, even when the valves were
so tight that it required a moment of 30 ft.-lbs. on the brake to move the
automatic gear alone, with the shaft standing, in either direction, when the
shaft was running any tendency to lean upon the support in either direction
was the result of imperfect balance in the gear; and that, by adjusting this
balance to an extent which would not cause a moment on the brake of
O'Ol ft.-lb., the tendency of the brake to lean either in one direction or the
other might be reversed, showing that, with a load of 600 ft.-lbs., the relative
limits of error are < + 0'000016, and in the difference of the trials would be
zero.
The Work done in the Brake by End Play in the Shaft.
The clearance in the brake-case would allow of nearly ^-inch end play
along the shaft ; and when the brake is running, owing to the slight end
play of the engine-shaft, there is at times a slight back wards-and- forwards
movement, in the period of the engine, of the brake-case on the shaft, but
not more than the 64th of an inch at the greatest. This end play, when it
existed at 300 revolutions and 1200 ft.-lbs. load, could always be prevented
by an end pressure on the case of < 50 Ibs. Hence the limit of work done
616 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
on the brake is < 2 x 50/12 x 64 = 0'13 ft.-lb., which, compared with the
work in one revolution with a load of 1200 ft.-lbs., is
0-13/1200 x 27r = 0-000017.
This would be the limit if the error is proportional to the load, while if
constant, the error on the difference of two trials would be zero ; so that the
greatest relative error is less than
+ 0-000017.
The Error from the Dash-Pot.
Since the piston is suspended freely in the oil-cylinder, and the resistance
of the oil is viscous and expressed by fj,vs/at where /JL is the coefficient of
viscosity, v the velocity of the piston, .9 the area of surface, and a the
distance between the surfaces, the total resistance is thus [*.s/a multiplied by
the total displacement (which never exceeds O'l ft.) divided by the time
(3600 seconds). This is infinitesimal. Besides which, with 1200 or 600 ft.-lbs.
load at 300 revolutions, the lever remains perceptibly steady, there being no
vertical vibration perceptible to the finger on the lever. Hence; as long as
there are no oscillations, the limit of error from the dash-pot, if any, is im-
perceptibly small.
The only circumstances under which the lever oscillates is when the
water flowing through is less than about 4 Ibs. a minute ; then a slow oscilla-
tion appears, the lever moving some half-inch, which causes the automatic
gear to lean on the fixed support, and may cause a small error.
The Development of the Thermal Measurements.
The appliances were originally designed, in 1887, solely for the purpose
of the study of the action of steam in the engines, and certain problems in
hydraulics and dynamometry, without any intention of their being used for
the purpose of measuring the heat equivalent of the work absorbed, but
rather the other way.
It was, of course, obvious that, as the primary purpose of the brakes was
to afford accurate measurement of the work spent in heating water, it was
only necessary to measure the change of temperature of the water between
entering and leaving the brake, as well as its quantity, to obtain an approxi-
mate estimate of the heat equivalent of the work done. But the recognition
of the extreme difficulty of obtaining any first-hand assurance as to the
accuracy of scales of thermometers, and the fear of creating erroneous
impressions as to the value of the equivalent, made me reluctant to allow
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 617
such determinations. For this reason, as well as to avoid complicating the
brake, in the first instance I made no provision for the introduction of ther-
mometers, as may be seen in Plate 1.
But, after the engines and brake had been in use for two years, and had
been found to possess attributes in steadiness of running, delicacy of adjust-
ment and balance, beyond what I had dared to expect, and particularly in
being able to work with an almost absolutely steady supply of water
between steady temperatures, and the same temperatures for different
powers, arising either from differences of speed, or differences of load,
I realized that by working with the same thermometers on the same parts of
their scales, and with the same loads and temperatures at different speeds,
since the relative error of balance would be the same, if the surrounding
temperatures were the same, the difference of two trials would afford the
means of determining the loss of heat by radiation, and, this being determined,
the difference of two trials made at the same temperatures as the previous
trials, and both at the same speeds, but with different loads, would afford
data for determining the error of balance without introducing the value of
the equivalent or the use of the scales of the thermometers, except to identify
equal temperatures.
I then yielded to the very general wish on the part of those who worked
in the laboratory, and added such provision to the brake on the low-pressure
engine as would admit of the measurement of the heat carried away by the
effluent water, but only for the purpose of verifying the accuracy of balance
as determined by mechanical means.
The Thermal Verification of the Balance of the Brakes.
10. The desirability of such independent determination of the balance
arose in the first instance from the circumstances already described (Art. 9),
viz., that the statical balance could only be determined to 1 ft.-lb., while the
absence of effect from the friction of the automatic gear, &c., was only
arrived at by somewhat complicated considerations.
The supply of water to the brake came from the service tank, 10 feet
above the floor, and 7 feet above the shaft, the tank being supplied direct
from the town main, and regulated by a ball-cock. The pipe from the tank
passes beneath the concrete floor to a point conveniently close to the brake,
whence a branch, in which is a hand-cock, rises vertically to a height of
4 feet above the floor, at which height is the automatic inlet valve, and from
this the pipe is bent over, so that its mouth is directly over the inlet opening
into the brake, with which the pipe is connected by a flexible indiarubber
tube.
618
ON THE MECHANICAL EQUIVALENT OF HEAT.
[66
Fig. 5.
The first provision made for measuring the temperature of the entering
water was an opening in the bend of the pipe over the
inlet valve, with a vertical f-inch brass tube soldered in,
about 4 inches long. This admitted of an indiarubber
cork, through the centre of which a thermometer was
passed into the pipe, as shown in Fig. 5. This was after-
ward replaced by a glass thermometer chamber, as shown
in Plate 3.
To measure the temperature of the water leaving the
brake it was necessary, by means of a pipe fixed to the
mouth of the outlet valve, to bring the effluent water
above the balancing lever of the brake, and to one side
of it. This pipe was arranged so as to admit the intro-
duction of a vertical thermometer into the ascending pipe,
much in the same way as the other. In the first instance,
the- extension passage and the thermometer were all rigidly attached to the
brake, and moved with ifc, which entailed a re-balance of the brake. Sub-
sequently another arrangement was made. The thermometers used were
divided to one-fifth of a degree Fahrenheit ; they were both immersed
in the flowing water to within a few degrees of the top of the mercury.
They were compared at equal temperature, but otherwise subjected to no
tests for accuracy of scale.
In making the experiments the link connecting the inlet valve with the
automatic gear was removed and the valve was set open, the supply being
adjusted by the hand-cock below. The head on the inlet being constant,
when the cock was set the flow was practically steady. Tlie quantity of
water in the brake then depended on the outlet valve, which, with the
exception of a little trouble at starting and stopping, soon overcome, kept
the brake lever steady.
To catch the water after leaving the outflow thermometer, the extension
pipe turned horizontally over the lever and then turned downwards into a
basin, the lip of which was above the mouth of the pipe, and from the basin
flowed in a short trough, from which it was caught in buckets. In these
it was taken to the scales and carefully weighed. This was a primitive
arrangement, and required several assistants, but was found capable of
considerable accuracy up to about 40 Ibs. a minute.
In making these experiments the engines were kept running at nearly
constant speed by keeping constant pressure in the boiler. The speed being
indicated on the speed gauge as well as recorded on the counter.
The water entering the brake, coming, as it did, from the town's main,
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 619
was at nearly constant temperature between 40° and 50° Fahr., according to
the time of the year, and varying less than a degree throughout several
trials.
The rise of temperature was adjusted by the quantity of water admitted,
according to the work, so that the final temperatures as well as the initial
were as nearly as possible the same in the different trials.
This rise was such as admitted of the temperature of the brake being the
same as that of the laboratory, which could always be adjusted to about
70° Fahr., so that the rise was from 25 to 30 degrees. This, with 40 Ibs. a
minute, required from 25 to 30 H.-P.
Before commencing the actual trial everything was adjusted, and the
engines running with steady load and steady speed until the thermometer
showed the heat to be steady at the desired temperature, then, at a signal,
the counter was put in and the water caught, each of the thermometers,
and one giving the temperature of the laboratory, being then read at
minute intervals over 15 or 30 minutes, when, on a signal, the counter was
removed and also the last bucket.
The results of these tests were very consistent, within about 0'3 per cent,
which was within the limits of accuracy then aimed at.
Trials with equal loads and different speeds showed that the loss by
radiation was very small, while those at the same speed with different loads
showed the balance was within the limits determined by mechanical tests.
In these trials the only correction was that for the lubricating water
which escaped from the brake bushes. This was caught at each bearing,
and the temperature taken so that the heat might be added, this being
seldom more than 3 per cent. It may also be noticed that in these trials
the heat lost or gained by conduction to or from the shaft was included in
the radiation. As the brake is on an overhanging shaft which extends no
farther than the outer bush of the brake case (Plate 1), the only conduction
is on the side at which the shaft is continuous, where the brake bush is only
some 4 inches from the brass of the shaft bearing. As the temperature of
the brake on this side, which is opposite to that at which the cold water
enters, was kept by the lubricating water at the temperature at which the
water left the brake, and this was at the temperature of the laboratory, there
would be no cause of conduction unless the friction of the shaft in its bearing
caused its temperature to rise above that of the laboratory. When the
lubrication was good this was small, although on one or two occasions it
made itself felt.
620 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
The Idea of Raising the Temperature from 32 to 212.
11. These tests became an annual exercise in the laboratory, and a very
instructive exercise. But, as the subject — the value of the equivalent — was
attracting much attention, the desire to obtain measures of it from these
trials, by those engaged in them, resulted in Mr T. E. Stanton, M.Sc., then
Senior Demonstrator, effecting, for his own satisfaction, a comparison of the
scales of the thermometers used in the experiments with a thermometer
used in the Physical Laboratory, which had been compared with the air
thermometer, and introduced these corrections into the results of the trials,
which so gave values very close to what might be expected. I could not see
however that determinations made with thermometers so corrected could
have any intrinsic value, but, as the matter was exciting great interest in
the laboratory, I carefully considered the conditions which would be necessary
in order to render the great facilities, which this brake was thus seen to
afford, available for an independent determination.
The institution of an air thermometer was carefully considered and
rejected. But it occurred to me that it might be possible to avoid the
introduction of scales of the thermometers, just as before, and yet obtain
the result. If it could be so arranged that the water should enter the
brake at the temperature of melting ice and leave it at the temperature
of water boiling under the standard pressure, all that would be required
of the thermometers would be the identification of these temperatures. At
first the difficulties appeared to be very formidable. But on trying, by
gradually restricting the supply of water to the brake when it was absorbing
some 60 H.-P., and finding that it ran quite steadily with its automatic
adjustment till the temperature of the effluent water was within 3° or 4° of
212° Fahr., I further considered the matter and formed preliminary designs
for what seemed the most essential appliances to meet the altered circum-
stances.
These involved —
(1) An artificial atmosphere, or a means of maintaining a steady
air pressure in the air passages of the brake of something like
one-third of an atmosphere above that of the atmosphere.
(2) A circulating pump and water cooler, by which the entering
water (some 30 Ibs. a minute) could be forced through the
cooler and into the brake, at a temperature of 32°, having
been cooled by ice from the temperature of the town main.
(3) A condenser by which the effluent water leaving the brake at
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 621
212° Fahr. might be cooled down to atmospheric temperature
before being discharged into the atmosphere and weighed.
(4) Such alteration in the manner of supporting the brake on the
shaft as would prevent excess of leakage from the bushes in
consequence of the greater pressure of the air in the brake,
since not only would the leaks be increased, but when the rise
of temperature of the water was increased to 180°, the quantity
for any power would be diminished to one-sixth part of what
it would be for 30°, so that any leakage would have six times
the relative importance.
(5) Some means which would afford assurance of the elimination of
the radiation and conduction, as, with a rise of 140° Fahr.
above that of the laboratory, these would probably amount to
two or three per cent, of the total heat.
(6) Scales for greater facility and accuracy in weighing the water,
with a switch actuated by the counter.
(7) A pressure gauge or barometer, by which the standard pressure
for the boiling point might be readily determined at 3° or
4° Fahr. above and below the boiling point, so as to admit
of the ready and frequent correction of the thermometers used
for identifying the temperature of the effluent water.
(8) Some means of determining the terminal differences of tem-
perature and quantity of water in the brake, which would be
relatively six times larger with a rise of 180° than with 30°.
The Special Appliances and Preliminaries of the Research.
12. Having convinced myself by preliminary designs, not only of the
practicability of the appliances, but also of the possibility of their inclusion
in this already much occupied space adjacent to the brake, there still
remained much to be done in the way of experimental investigation to
obtain data from which the requisite proportions of these appliances could
be determined, and these preliminary investigations were not commenced
till the summer of 1894, when Mr Moorby undertook to devote himself to
the research.
Weighing Machine and Tank.
13. The first step consisted in obtaining a somewhat special table
weighing machine (Plates 2 to 4), having two rider weights on independent
622
ON THE MECHANICAL EQUIVALENT OF HEAT.
[66
scales, one divided to 100 Ibs. from 0 to 2200, the other to 1 Ib. from 0 to
100. Also a galvanized iron tank, 5' x 2' 9" x 2' 9", capable of holding above
one ton of water, with a 4-inch screw valve at the bottom, opening inwards
by a handle above the top of the tank, the top of the tank being covered
with carefully fitted, but separate, ^-inch pine boards, previously steeped
in melted paraffin-wax, to prevent adhesion or absorption of water. This
machine and tank, which is a large affair, was placed in the only position
available, opposite the end of the shaft and behind the standing pipes for
supplying the condensing water to the engine, thus leaving the passage
between these pipes and the end of the shaft open, an important matter,
as this passage was the only place from which the observations on the
brakes could be made. This entailed the carrying the outflow from the
brake over the passage, about 6 feet 6 inches from the floor.
Design of the Outflow.
14. This extension of the pipe further entailed the necessity of making
this pipe a fixture, and connecting it with the outlet below the automatic
cock by a bent wire-bound flexible indiarubber pipe, so as to prevent any
moment on the brake. (See Fig. 6.)
Fig. 6.
The Thermometer Chambers.
15. A glass chamber for the outflow thermometer was introduced as
shown (Fig. 6), and another for the inlet, somewhat similar. These were
arranged so that the bulbs of the thermometers were down in the full
current while the scale was in the glass tube, through which a portion of
the water was allowed to flow, that from the inlet thermometer being con-
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 623
ducted away to waste, while that from the outlet was conducted back again
into the outflow pipe. In this way, not only the bulbs of the thermometers,
but the entire thermometers were immersed in the flowing water.
The Two-way Switch.
16. A switch, as shown in Plate 3, was also constructed for diverting
suddenly the stream of effluent water from waste to the tank, or vice versa,
without exposing the stream for more than an inch, and without any splash-
ing or uncertainty.
Experience in Making Observations.
17. When these arrangements were completed, and whilst the other
appliances were progressing, Mr Moorby commenced a series of experiments
similar to those which had been previously made, using the water from the
tank at the temperature of the town's water, and raising it to temperatures
which were successively increased. This was with a view of testing the
improved facilities, and also of gaining experience and facility in making
and recording the observations.
The engines and brakes were occupied two or three times a week in
the ordinary work of the laboratory, so that there were only one or two
days a week available for these experiments, and every opportunity was
valuable.
The Design of the Condenser.
18. At the same time he made experiments to determine the necessary
length of pipe in order that the water flowing along it at the rate of 20 Ibs.
a minute would be cooled from 212° to 70°, when the pipe was jacketed by
a stream of town's water at 50° Fahr. ; by the result of which experiments
the condenser in which the effluent water is cooled to 75° was designed
(Plates 2 to 5).
Design of the Ice-Cooler.
19. To cool the water to 32°, or as near as practicable, I had, on account
of the danger of some ice being carried through with the water if the ice were
once put into the water, decided to pass the water through a long coil of
ordinary water piping, immersed in water, towards the top of a tank with ice
under the coil, and from experiments made by Mr Moorby, I decided on the
coil and arrangements shown. The coil consists of |-inch composition pipe,
200 feet long, the tank being 2 feet 6 inches wide and deep and 4 feet long, the
624
ON THE MECHANICAL EQUIVALENT OF HEAT.
[66
coil being placed near the surface of the water on a shelf, with a wire netted
space at the end for the introduction of the ice, which is pushed down
under the shelf, and with a paddle which is kept in continual motion by
Fig. 7.
a cord from the line shaft, thus securing a rapid circulation of the water.
The tank is constructed of 1-inch pine saturated with paraffin wax, in
preference to a metal tank.
In this design account had to be taken of the requisite head of water
necessary to force some 20 Ibs. a minute through the coil. It was estimated
that this would require some 30 Ibs. on the square inch, which, together
with the 5 Ibs. excess of pressure in the brake above the atmosphere, and a
margin of some 25 Ibs. in order to secure steadiness of flow, made a total of
60 Ibs. on the square inch, or 122 feet of head.
The Circulating Pump.
20. It was essential that this head should be approximately steady, and
under control during the trials, also that the water should be drawn as
directly as possible from the town's mains, in order to secure both the low
temperature and great purity of this water. This precluded the direct use
of the water from the large tank in the tower, which would otherwise have
just afforded this head. It also precluded the use of such head as there
might be in the town's mains, as this was insufficient and continually varying,
so that some special means of imparting the steady head to the water after
drawing it from the mains was necessary. This involved pumping the
water through the ice-cooler and brake. It might be done by pumping it
from the service tank in the laboratory into an accumulator under a constant
load, or by passing the water through a centrifugal pump, running at a steady
speed, on its way to the brake.
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 625
The facilities in the laboratory decided this question. There already
existed the quadruple vortex turbine, with four three-inch wheels in series,
worked from the water in the tower, which would work steadily up to 1 h.-p.,
in a position which would be convenient for driving a centrifugal pump in
the in-circuit of the pipe leading to the brake ; I also had a quintuple
centrifugal pump with five 1^-inch wheels in series which was adapted to
the purpose. It was decided, therefore, to lead the water from the surface
tank, 9 feet above the floor, into the quintuple pump, driven by the turbine
under a constant and controllable head, so that the head would be raised to
the required amount. Then, to lead the water through the cooling coils to
a pressure gauge close to the brake, and thence through a regulating valve
into the passage, with the thermometer, leading into the brake. (See Plates 4
and 5.)
The Outlet from the Condenser.
21. In order to prevent the formation of steam, owing to the presence
of air in the water, before it had passed the outlet thermometer, it was
necessary to maintain a certain pressure in the effluent water as it passed
the bulb of this thermometer. At first it was thought that a head of
5 or 6 feet would suffice. In order to secure this, the level of the con-
(1. user being some 3 feet above the bulb, the pipe leading from the condenser
was carried up vertically about 3 feet higher, then turned over and led down
again to an orifice immediately over the switch, while from the top of the
bend a vertical branch extended upwards about 3 feet, with its mouth open,
to the air. This was subsequently raised. (See Plate 2.)
Preliminary Experiments at 212° under Pressure.
22. The preliminary investigations and the construction of the appliances
so far described were not completed till May, 1895. It then became possible
to make some experiments as to the working of the brake under pressure and
at high temperature, so as to obtain guidance as to the artificial atmosphere
and means of controlling the leakages at the bearings. From these experi-
ments two things came out clearly. It was found that all that was necessary
for an artificial atmosphere was to connect the outlet of the air passage on
the top of the brake by means of a flexible indiarubber pipe capable of
bearing the pressure to a vessel of very moderate capacity.
The Artificial Atmosphere.
23. A tin can, holding about 3 gallons, with the bottom and top coned
upwards, and strong enough to stand the full pressure of 60 pounds, was
o. R. ii. 40
626
ON THE MECHANICAL EQUIVALENT OF HEAT.
[66
adopted. The air connection with the can was at the top, at which there
were also two side openings, one with a cock, to admit of air being pumped
into the can, and the other with a fine screw stop for allowing a slow and
definite escape of air. An opening at the bottom, with a cock for drawing off
water, was also provided. For forcing the air in, a syringe for inflating
bicycle tyres was used in the first instance and proved ample ; in fact, when
once the pressure was raised, the small amount of air released from the
water was more than sufficient to maintain the pressure, so that it was
continually allowed to escape.
The Stuffing-box and Cap to prevent Leakage.
24. The thing that was revealed by the experiments at high tempera-
tures, was that the leakage of water at the coned bushes of the brake was
so much increased by the pressure within the brake, that even when these
bushes were adjusted to run, as close as was practicable, on the cones of the
shaft, this leakage was very considerable, so that some other method of
controlling this escape became necessary.
This matter threatened to present great difficulties. It was apparently
66]
ON THE MECHANICAL EQUIVALENT OF HEAT.
627
impossible to close in the bushes with stuffing-boxes and stop the leakage
altogether, as that would prevent the lubrication of the shaft, :m<l. apart
from this, would cause the temperature on the shaft side of the brake to
rise to the temperature of the brake, 212° Fahr., which would cause a large
escape of heat along the shaft. Besides this, the adaptation of stuffing-
boxes to the existing brake presented such difficulties that it almost seemed
as though it would be necessary to have a new brake, which, besides the
delay, would entail an addition of some £200 to the expenses, which were
otherwise very considerable.
Fig. 9.
To avoid this I determined to try a stuffing-box on the shaft side,
constructed in halves to be bolted together on the shaft, and then sweated
into one, this stuffing-box to screw on to the exposed screw of the bush, and
make a joint against the lock ring ; then to open a passage through the box
inside the packing-ring, with a tap to control the escape of water, and at
the other end to screw a cap on to the bush, entirely inclosing the end of the
shaft, with an aperture and a tap to regulate the water, also a small stuffing-
box in the cap, to allow of a spindle for connecting the shaft with the
counter.
These entailed very difficult and exceptional work, but were beautifully-
executed by Mr Foster, in the laboratory (Fig. 9).
However, the result was very doubtful, as the water flowing from the
brake through the aperture in the stuffing-box not only raised the tempera-
ture of the shaft, but was itself of uncertain temperature.
40—2
628 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
It was in July, 1895, that this experience was obtained, and for a time
the success of the research seemed doubtful. During the vacation, however,
an idea occurred to me which at once promised to do away with the whole
difficulty.
The Cooling and Lubricating of the Bushes.
25. This idea consisted of what seemed to be a practicable plan of
forcing a relatively small, but sufficient portion of the ice-cold water into
the brake through each of the bearings, the quantities being strictly under
control.
That this plan should not have presented itself as soon as the addition of
the stuffing-box and the cap were contemplated, becomes intelligible when
it is remembered that the main object in the invention of this brake had
been to secure a constant pressure in the air space within the vortices, so
that by admitting the water through passages in the vanes directly into this
air space a constant resistance, whether that of the atmosphere, or artificial
atmosphere, on the entering water would be secured, and that the possibility
of maintaining an even flow through the brake, so essential to any success
in the research, depended entirely on the realization of this constant resist-
ance. Except the inlet passage, the interior of the wheel, and the air space
in the vortices, all the spaces in the brake and brake-case are under the full
vortex pressure, excepting where, as in the bush on the closed side of the
brake, and that between the solid disc faces on the inlet side, the pressure
is relaxed by the escape of the water. This vortex pressure depends on the
load on the brake, and may be anything up to 25 pounds on the square inch
greater than that in the air cores. It thus seemed like starting de novo to
interfere with this arrangement ; and it was only when one came to realize
that the possibility of preventing all leakage by the introduction of the
stuffing-box and the cap had rendered it possible, by controlled subsidiary
supplies under pressure, to reverse the flow of the lubricating water, and
so to do away with leakage, and not only to secure lubrication, but also
to cool the bushes, and then only after considering the amounts of water
required, and the provision in the way of pumping appliances, separate
supplies of water and thermometers, &c., that the altered facilities afforded
by the circulating pump came to be recognized.
The By-channels and Regulator admitting Cooled Water to the Bushes.
26. Since the main supply must enter, as before, at the same pressure
as the air within the vortices, while, in order to reverse the flow through the
bushes, that entering the cap must enter at a little, but only a little, above
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 629
that of the air within, while that entering on the brake side of the packing-
ring in the stuffing-box must enter at any pressure up to 20 Ibs., according
to the load, above that of the air within, it was clear that there must be
three supplies of water at different pressures under separate control ; and it
was equally clear that these supplies must all be at the same temperature.
Fortunately, the arrangements already made for the new supply afforded
ready means of securing these conditions, as, in order to insure steadiness in
the supply through the regulating valve, it had been provided, in arranging
the pump, that there should be an excess of 20 Ibs. on the square inch above
that necessary to force the maximum water through the coil and to overcome
the air pressure in the brake ; also, as the regulating cock was only an inch
or two from the thermometer chamber, the water would be subject to little
heating by radiation after leaving the cock, while the effect of radiation to
the by channels would be of secondary importance, as it is eliminated with
the rest of the radiation in the difference of the trials.
It thus became possible, by leading cooled water through two short by-
branches, with separate regulators, from the supply pipe, before passing the
main regulator respectively into the aperture through the stuffing-box on
the inside of packing-ring, and into the cap on the inlet end, to secure
controlled inflows of ice-cold water between each of the bushes and the
shaft, and so to adjust the temperature of the bearing and insure lubrication
of the shaft (Fig. 9).
In order to render such inflows steady and constant, it was desirable that
the pressures before passing the regulator should be kept at a considerable
and constant quantity above the vortex pressure in the brakes.
From the first preliminary trials made with the branches it appeared
that the turbine and pump were capable of supplying sufficient pressure
for this, so that the only additions necessary were the branches. These
were made of ^-inch brass pipe from the main pipe from the cooler as far
as the branch regulators, and thence continued by £-inch indiarubber vacuum
tube f inch outside wrapped with tape. The branch regulators have cocks,
with provision for fine adjustment, so that the very small quantities which
passed might be definitely regulated to great nicety (Plate 3). With these
it was found practicable to maintain the temperature of the bushes from
anything a few degrees above 32 to any required temperature.
It is to be noticed that the work done by pressure over and above the
pressure pa in the inlet thermometer chamber, is that due to the difference
between the pressure in the main pipe before passing the regulators and pa,
through whichever passage the water enters. And since in that water which
passes into the thermometer chamber through the main regulator this work
630 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
has been converted into heat, and is measured as entering heat by the inlet
thermometer, the assumption that the water through the branches enters
at the pressure pa, and the temperature given by the inlet thermometer,
involves no other error than that resulting from radiation, which is constant
for all trials, and is eliminated in the difference.
The Regulation of the Temperature of the Bushes.
27. In the preliminary trials this temperature was only ascertained by
touch, and regulated so as to be as nearly as possible that of the laboratory,
the branch cocks being set with a definite opening, and the excess of pressure
maintained as nearly as possible constant, a plan which was found to give
consistent results. But it also appeared that in order to maintain the same
temperature in the stuffing-box for the large and small trials with the same
pressure in the main pipe, it was necessary to open the branch cock wider in
the large trials. This was to be expected from the greater vortex pressure
in the large trials. And as owing to the greater resistance of the cooler
in the large trials there was difficulty in maintaining a great excess of
pressure over the vortex pressure, it was decided to run both large and
small trials with the same setting of the cock, and the same head in the
cooling pipe, keeping a record until some means was obtained of estimating
the comparative slopes of temperature in the shaft in the large and small
trials.
The Measurement of the Comparative Slopes of Temperature in the Shaft.
28. The desirability of some more definite knowledge of the slope of
temperature in the shaft between the brass of the nearest shaft bearing and
the stuffing-box was strongly felt, but it was not at first apparent how this
might be done, the shaft being 4 inches in diameter, and the gap between
the end of the stuffing-box and the brass of the bearing being only 3 inches.
However, as it became more evident, with the branch cocks set at a
constant opening and the same pressures in the supply pipe, that the
temperatures in the stuffing-box were greater in the large than in the
small trials, and that a small difference in the adjustment of the branch
cock to the stuffing-box affected the apparent loss of heat to the extent
of some 01 or 0"2 per cent, of the total heat, I determined to try and
obtain some definite evidence of the relative slopes of temperature in the
two trials, by measuring the relative temperatures of the brass and the
stuffing-box as far as was practicable. For this purpose, I had thick brass
tubes, radiating outwards, sweated on to the end of the stuffing-box, to
hold thermometers. Two such tubes were necessary on account of the
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 631
screwing-up of the box, which had to be done whenever it began to leak;
and although this was riot done during a trial, one tube would sometimes
face downwards, which was inconvenient. In a similar manner two tubes
were attached, one to the top and one to the bottom brass of the bearing,
holes being bored into the brass and the tubes screwed in. These tubes
are shown in Fig. 9.
In this way, with a thermometer in one of the tubes on the stuffing-box
and one in each of the tubes on the bearing, although the thermometers
might not give the actual temperatures of anything in particular, still the
steadiness of the conditions of the brake warranted the conclusion that the
differences in the readings of the thermometers would serve to identify
similar conditions as to slope of temperature, and this turned out to be
the case.
These thermometers threw a Hood of light on to conditions which had
before been hardly perceptible. Thus, after reading the thermometer during
three large trials and three small trials, with the cocks set as before without
having been displaced, and with the same pressures, it was found that the
mean of the three large trials indicated 13° Fahr. greater slope from the
stuffing-box to the brass than that indicated by the mean of the three small
trials.
The Constants and Limits of Error of Conduction.
29. It thence became possible in the subsequent trials, by adjusting the
cocks, to bring about a mean condition in which the mean slope in the large
trials was the same as that in the small, and by comparing the mean results
of those trials in which the difference of slope had been in one direction
with the mean of those in which it had been in the opposite, to obtain a
constant expressing the quantity of heat lost for each degree of the recorded
slope.
These thermometers, read to 1° Fahr. 7 times during the trial of each
sort, would give a limit of error of the f of a degree, which, taking 12 thermal
units per hour as the loss per degree, would give as limit of relative error
on 100,000 thermal units of, on one trial,
0-00002,
and these being casual, when taken over 40 trials would be less than a
millionth.
The Hand- Brake for Regulating the Speed of the Engines.
30. Although it had been found possible to maintain the speed of the
engine constant within 2 or 3 per cent, when the engines were working
632 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
with a considerable margin of pressure in the boiler, by maintaining the
pressure in the boiler constant, the care and attention required on the part
of Mr J. Hall, who had charge of the engine, became excessive when the
engines were indicating over 80 H.-P., particularly as he could not be
attending to the fire and lubrication, and at the same time watching the
speed indicator. To meet this difficulty, as there is no known automatic
governor which will regulate an engine working against a resistance which
is independent of the speed, without fluctuations, I arranged a hand-brake
on the rope pulley, 3 feet in diameter, on the brake shaft, to be applied by
one of the assistants in the laboratory during the trial. The amount of
power to be absorbed by this being less than 2 H.-P. at the most, a |-inch
cotton rope, with one end fast, passed round in one of the grooves of the
pulley, the other end being attached to a spring balance, the position of
which could be regulated with a screw, would answer the purpose (shown in
Plate 3).
In this way, as the natural variations of speed of the engines are very
slow, Mr Matthews was able, after a little experience, to keep the speed to
within something like one revolution, or 0'3 per cent.
The Corrections for the Terminal Heat of the Brake.
31. As the temperature of the effluent water could be continually
regulated by regulating the supply of water to the brake, whatever might
be the speed, the chief importance of keeping the speed regular arose from
the errors (1) caused by small differences of temperature in the brake
together with the water it contained at the commencement and end of the
trial, and (2) by small differences in the weight of water in the brake at
the commencement and end of the trial.
Such errors belong to the class of casual errors to be eliminated in
the mean of a number of trials. Still, it seemed desirable to have some
assurance that such elimination was effected, and, in order to obtain this,
I proposed that the actual quantity of water in the brake for each of the
loads used in the experiments should be determined experimentally at
several speeds covering the range of variations likely to occur, and so to
obtain a curve for each load, showing the water at each particular speed ;
this to be done by running the brake as in the trials, steadily, at a particular
speed, the water passing as in the trial. Then, suddenly, by forcing down
the lever, to close the automatic outlet valve, and, at the same time shutting
the inlet valve and stopping the engines, and thus trapping the working
charge of water in the brake. The water could then be drawn out and
weighed.
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 633
Putting B for the capacity for heat of the metal of the brake, w for the
weight of water, and T for the temperature observed on the effluent thermo-
meter, the total heat in the brake is expressed by
and, if Wi, T? refer to the weight of water and temperature at starting, and
Wf, T/ to the corresponding quantities at the end of the trial, the correction
which has to be subtracted from the heat observed is expressed by
The Method of Conducting the Trials — Elimination of Radiation.
32. The entire system of working was designed to secure the most
perfect elimination of radiation possible. Thus, it was arranged in the
first place that the trials be made in pairs, one heavy trial and one light
trial, made under circumstances as nearly similar as possible, except in
respect of load and water. The loads in the first instance being 1200 and
600 foot-pounds, and the quantities of water such that the final temperature
should be as nearly as possible 212° Fahr., and, after the preliminary trials,
300 revolutions per minute was adopted as the speed for all the trials,
60 minutes as the time of running. The inlet and outlet thermometers to
be read after the first minute, and every two minutes ; also the temperature
of the laboratory as shown by a thermometer in a carefully-chosen place.
This temperature to be maintained as nearly constant as possible. The
setting of the regulators during each trial to be recorded ; also the pressure
of the artificial atmosphere, and that in the supply pipe after passing the
coil ; and, subsequently, the reading of the thermometers in the stuffing-box
and bearings taken every five minutes, and the speed gauge every two
minutes. The observations and incidents being recorded by the rules in
surveying, in ink, in a book, and distinct from any reductions. The initial
and final reading on the scales and counter being included, as were also the
initial and final readings of the inlet and outlet thermometers and speed
gauge for the purpose of determining the terminal differences of the heat
in the brakes.
As it was impossible to make trials simultaneously, and so secure similar
conditions in the laboratory, it was at first arranged that the trials should be
made in groups, including four pairs of trials.
The regular work in the laboratory monopolised the engines and brakes
on all days in term time, except Mondays and Thursdays, so that the trials
were confined to two days in the week. There was a certain likelihood of
the state of temperature of the walls and objects in the laboratory being
634 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
systematically different on the Mondays, after the laboratory had been
without steam all Sunday, from what it would be on the Thursday, after
the steam had been on for three days. And besides this, there would be
a systematic difference in the temperature of all the objects during the
first trial in the day, although the brake had been running for an hour
before, from that which would hold in the following trials. In the first
instance, therefore, it was arranged that a heavy and a light trial should
be made on the same day, and a light and a heavy trial on the next available
day, under as nearly similar circumstances as possible, except for the inversion
of the order. Then again, a light and a heavy trial on the next day, followed
by a heavy and a light on the following, so as to break the order and secure
the same arrangement, in days of the week as well as in hours of the day,
for the four light trials as for the four heavy trials.
As the results of any group of four pairs of trials would furnish a
tolerably close approximation to the loss of heat by radiation, assuming
this to be proportional to the observed mean difference of temperature
between the laboratory and the brake, it was easy to obtain an approximate
constant, R, for radiation for each degree of difference of temperature, and
so to introduce a correction, R(T2— Ta), in each trial for the radiation
resulting from the observed mean difference of temperature of laboratory
and brake, T2-Ta.
These corrections would serve two purposes — first, affording a better
comparison of the results of the separate trials for future guidance, and
secondly, by recording the mean difference of temperature, would show
how far the mean differences of temperature in the large trials had differed
from those in the small trials, and thus how far the radiation had been
eliminated.
Lagging the Brakes.
33. In order to obtain still more definite assurance as to the elimination,
it was arranged that after consistent results had been obtained in several
groups of four pairs of trials, as above, with the brake naked, the brake
should be covered with non-conducting material, in the best way practicable,
so as greatly to reduce the radiation, at the same time leaving it definite,
and then similar trials should be run.
If the coefficient of radiation could in this way be reduced to one-fourth
that of the naked brake, such error as there might be remaining in the
mean results with the naked brake would be reduced to one-fourth with the
lagged brake.
In this, however, there was danger of introducing errors of other kinds.
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 635
The non-conducting material would absorb heat slowly and take a long
time to arrive at a state of equilibrium, and during the interval the rate
of loss of heat from the brake would be irregular. The total error that
could result from this cause would be the product of the specific heat of
the material used multiplied by the weight, and again by the 75°, or the
half of whatever was the difference in temperature of the brake and the
air. This decided the choice of the material to include cotton-wool. Two
pounds of this would, if not too tightly pressed, cover the brake about
If inches thick, and the total heat it would absorb would be less than
0'4 Ib. of water raised from 32° to 212° Fahr., and would then be only
0'0008 of the heat generated by 30 H.-P. in an hour, while it would reduce
the radiation to about |. But as the cotton-wool would gradually collapse
if subjected to any elastic pressure, it was decided only to use this to such
thickness as it could be protected by light cotton strings extending in axial
planes round the brake, and to prevent absorption of moisture by the cotton-
wool, to cover it with thick anti-rheumatic flannel about 1 inch to 1^ inches
in thickness, as shown in Plate 5, which would raise the capacity for heat
of the entire lagging to about ^fa that of the heat generated in the small
trials, and as the brake was kept at steady temperature for about one hour
or more before the trial commenced, the actual differences would not exceed
some one ten-thousandth part.
The Conduction by the Levers.
This lagging only extended over the body of the brake covering all the
brass-work, leaving the levers and balance weights on the levers bare.
These levers being in metallic contact with the brass of the brake assumed
at these points the temperature of the brake, and would conduct the heat
along to the balance weights till it was lost by radiation. As the temperatures
were constant in all the trials this loss of heat would merely form part of
the radiation and be eliminated as the rest ; but, owing to the masses of the
balance weights and the length of the levers, it must take a long time for
the balance weights and the further parts of the levers to arrive at a steady
temperature, a fact which would account for a greater loss of heat in the
first trial made in the day.
In order to obtain assurance that this delay produced no error it was
arranged that after the completion of the series of trials with the brakes
lagged, corresponding to that with the naked brakes, that the balance
weights should be removed, and only the load at 4 feet from the brake
left, and a third series of trials made.
636 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
Starting and Stopping the Trials.
34. Having adopted an hour as the length of each trial, and 300 revo-
lutions as the normal speed, the engines having been running for an hour
previously, while the water entering the brake was being adjusted, and
afterwards, so as to ensure the temperature, not only of the brake, but of
the surrounding objects, having become approximately steady at the time
of starting the trial, all that was necessary was that the counter should be
pushed into the gear, and at the same time the water-switch pushed over,
and the reverse operation at the end of the trial. These operations, simple
Fig. 10.
as they were, entailed errors, which arose partly from the impossibility of
instantaneous engagement of the counter simultaneously with the switching
of the water. In order to diminish these as far as possible, the spindle
of a counter, on which was the worm which drove the worm wheel, was
wrapped with a spiral spring of steel wire, which gripped the spindle so
tight that it would not slip, the end of the wire being bent, so as to form
a clutch standing off the shaft half-an-inch, the end of the wire being
pointed, the shaft of the counter projecting a little beyond the wire. Facing
the end of this shaft, and in line with it, was a socket in the end of the
engine shaft, which was brought down to three-quarters of an inch diameter
and carried two round pins, a sixteenth of an inch diameter, standing out
radially, the engagement being effected by pushing the counter forward till
the wire crank engaged on one of the pins. (Owing to the wire being
pointed and the pins rounded, the chance of the wire striking plumb on
to the pin and so preventing engagement was reduced to a minimum.)
This engagement was the result of a great deal of experience, and
answered perfectly, but it involved the mean chance of a quarter of a
revolution of the engine-shaft after the wire had passed the pin before
the actual engagement was effected, whereas on coming off the disengage-
ment was instantaneous, the counter stopping by the friction of the worm
before the momentum had carried it through any appreciable angle.
This would leave a mean error of the work done during one-fourth of
a revolution on each trial, whence, the number of revolutions during the
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 637
trial being 18,000, the relative mean correction would be one seventy-two
thousandth part, or (V000013. As, however, when the two operations were
executed by different observers on a signal, the personal equations might
amount to more than this, although it involved a difficult piece of linkage,
an automatic connection was effected, as shown in Plate 3, the pushing of
the counter into engagement shifting the switch, so that in making the
trials no error was introduced.
The Leakage of Water.
35. As the loss of any of the water, which had entered the brake before
it was weighed, would constitute a corresponding error in the results, the
perfect tightness, not only of all the fixed joints, but of the casting and
the pipes, was a matter of first consideration and of continual care. This
was one of the reasons why the lagging was delayed till after consistent
results had been obtained ; for, as long as the brake and pipes were naked,
such leakage could not fail to be observed on close inspection, and before
lagging it was arranged to test the brake and pipes to an excess of pressure,
so as to insure perfect soundness. Besides the fixed joints there were only
two working joints, in addition to the openings into the switch and again
into the tank.
(1) The working joints were: The stuffing-box on the main shaft and
the stuffing-box on the automatic cock on the outlet from the brake.
Any leakage from these was open to observation both before and after
lagging, as they were in no way covered ; and arrangements were made so
that such leakage could be separately conducted by pipes and caught in
bottles. With care such leakage could be reduced to insignificant quantities.
The absolute loss of heat resulting from a leakage of WSB Ibs. of water
from the stuffing-box on the shaft was equal to the product of the difference
of temperature of the stuffing-box TSB°, and inlet (Tf) multiplied by WSB,
and in the few trials in which this became a sensible quantity it was to be
added as a correction.
The Loss of Heat by the Leakage of Water from the Automatic Cock.
36. This was the product of (wc), the weight of water which escaped,
multiplied by the total rise of temperature. Since the water passing the
cock was on its way to the high temperature thermometer, where any such
water was caught it was put into the tank, and so required no correction.
638 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
This leakage was very small, at most 2 oz. in a trial, but as there must be
some evaporation as the water escaped through the hot gland, which, though
small, might be of some importance on account of the latent heat of
evaporation, it was desirable in some way to enclose this stuffing-box in
an indiarubber bag closing on the spindle, so that the vapour could not
escape, and this was eventually accomplished very effectively and neatly
by Mr Foster, in a way which did not interfere at all with the free action
of the cock (Art. 14, Part II.).
The result of this, besides preventing any subsequent loss of water, in
this way, was to show that any error that had previously existed from
evaporation was inappreciable.
The Loss of Water at the Switch.
37. Apart from evaporation, which would result from the exposure to
the air, and in passing the air gap into the switch, there was no loss, as
the water descended almost tangentially on to the surface of the tube on
the switch which received it, the switch itself being a vertical knife-edge
extension of this surface, which passed through the vertically descending
water at starting and stopping ; and further, to prevent any minute drops
of water going astray from the bursting of an occasional bubble in passing,
a sheet brass hood was placed round the descending pipe directly the trial
started.
The outside of the weighing tank is completely exposed to observation,
and is perfectly tight. The valve in the bottom, being a 4-inch leather-
faced screw-valve on a brass seat, is also tight, but for satisfaction it was
arranged to place a clean tin dish under the valve before starting a trial,
and only to remove it after the water was weighed, so that there should
be absolutely no loss of water from any of these causes.
That there must be some loss of water by evaporation to the air as long
as the temperature of the water, after leaving the condenser, was above that
of the dew-point of the surrounding air, was certain. By using sufficient
cooling water it would be possible to bring the temperature down to that
of the dew-point; but it was found that this could not be done under all
circumstances without a larger condenser, for which room was wanting, and,
as long as the water lost by evaporation was the same in both trials, all
error would be eliminated in the difference of the large and small trials.
After careful consideration, it was arranged that the condensing water
should be adjusted so that the water in all trials entered the tank at a
temperature as nearly as possible 85°; it being probable, as the surface
exposed to the air was nearly the same in the large and small trials, if the
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 639
differences in temperature between the air and the water were the same, the
evaporation would be the same, or would at least differ by~ a constant
amount. In order to test this, it was further arranged that, after the trials
were finished, the centrifugal pump should be temporarily re-arranged so that
it could be used to draw water out of the tank and force it round through
the condenser and switch, and so back again into the tank at rates cor-
responding to those of the large and small trials, and at the same temperature
(85°), the water in the tank being at this temperature, the arrangement of
the pump being such that, when stopped, all the water in the pipes would
run back again into the tank. This would practically insure the same loss
of water by evaporation during one hour's pumping as during one hour's
trials, and any difference (we) thus established between the large and small
trials would then be treated as a standing correction on the difference of
the heavy and light trials. This relative correction, taking W as the mean
difference of water in the heavy and light trials, would be
W.
W
The Standards of Measurement.
38. In these experiments the expressions obtained for the work done
in heating the water and the heat generated are, respectively,
Wi and
where R, W, T°, S are respectively length, weight, temperature, and capacity
for heat.
Since these expressions both represent the same absolute quantity of
energy, the difference in the numerical values of these expressions results
only from the difference in the units in the two expressions. These units
may be considered as the unit of work and the unit of heat respectively,
as it is the inverse ratio of these units, measured in absolute quantities of
energy, that is expressed by the ratio obtained from
But, as there are no actual standards either of work or heat with which
quantities of work and heat can be respectively compared by a simple
measurement, such comparisons can only be accomplished by the comparison
of the several factors involved in each of these expressions with the several
absolute standards which exist for such factors.
These standards are the standards of mass, length, and force, on the one
hand, and of mass, quality of matter, and temperature, on the other.
640 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
Thus, work being defined as the mean product of force multiplied by the
distance, and the standard of force being the force of gravitation on the unit
of mass wherever it occurs, the work is represented by W . h, where W
expresses the number of units of mass, and h the number of units of length
through which it has been raised. Taking (M) and (L) as expressing these
units, the unit of work is expressed as (ML).
Again, the unit of heat is defined to be one nth part of that quantity
which is required to raise one unit of mass (M) of a standard substance (pure
water) from one definite state of temperature to another definite state. And
calling this interval 6, the unit of temperature is defined to be 0/n. And,
taking S to express the ratio of the number of units of heat required to raise
Wu units of mass of matter from T° to T° compared with Wu (Ts° - T°), the
/ B\
heat expressed bv SWU (T2 — TJ is in units (M -} .
\ n/
So that, from the physical equivalence of the absolute energy expressed
in the respective forms, it appears that the unit of heat as defined by
/ 0\
[Jf-| is equivalent to
V n)
ZtrNRW
T — T\ umts °* WOI>k as defined by (ML),
J-% — -L i)
or that the heat required to raise one unit of mass of pure water through the
definite interval of temperature 0 is equivalent to
U W (ML).
SW (T ^
This is the definition of the mechanical equivalent of heat in Manchester,
adopted by Joule, if n = 1, and 6 is 1° Fahr. between 50 and 60, as deter-
mined on his thermometer. But, since the absolute kinetic value of the unit
of force as here defined varies with the latitude and height of the place,
while that of the unit of heat is constant, this mechanical equivalent varies
from place to place with 1/g, where g is the expression, in kinetic units, for
the unit of force (M).
Thus, expressing the work in kinetic units, the unit of heat, as already
defined, is equivalent to
2-rrNRW _
m
where the dimensions of C are
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 04-1
Whence, since g has dimension (LT~-},
2-n-NRW tf
where the dimensions of C/g are (LnO~l).
The object in this research being to replace the standard of temperature,
as defined by the scale on a particular thermometer, by the standard obtained
from the states physically defined by melting ice and by water boiling under
a standard pressure, 6 is here defined to express this interval, and 8 is, in
accordance with the definition already given, used to express the ratio which
the heat required to raise unit mass over any interval, per degree of rise,
bears to that required to raise pure water over the interval 0, per degree
of rise.
The Standards Involved.
39. It appears from the dimensions of Cfg, as obtained in the last
article, that the only general standards to which reference need be made
are those of length and temperature.
It is, however, to be noticed that the determination of the work and
the heat involve the determination of separate masses, and that the units
only disappear on the condition that they are equal.
The Measurement of Mass.
40. Since it was not necessary to refer the mass to a general standard,
the weights used were only referred to a Board of Trade standard for
convenience.
Thirteen of the 25 Ib. weights used for loading the brake were adjusted
to the Board of Trade weight, then carefully balanced against each other, till,
balanced in groups of four in any arrangement, there was less than 0 01 Ib.
difference. Four of these weights were then taken as the standard.
The compound lever machine, which had two scales on the same lever,
one notched to each 100 Ibs. for the position of the large rider, the other with
a flat scale for every 1 Ib. for the position of the small rider, was taken
to pieces and the knife edges re-ground and re-set (by Mr Foster) till con-
sistent results were obtained to the one-hundredth of 1 Ib. Another rider was
made to work on the same scale as the small rider, being adjusted to one-
hundredth of the weight, so as to road O'Ol Ib.
o. R. n. 41
642 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
The scales were then carefully surveyed by the standard 100 Ib. weight,
the original small rider being adjusted till the difference between its extreme
positions on the scale balanced the standard to < O'Ol Ib., and the cor-
rections for each V-notch into which the feather on the large rider fitted
ascertained by balancing the standard to a like degree of accuracy.
The dead load on the scales, including the empty tank, came to 340 Ibs.,
about, and between this and 2200 Ibs. the scales would weigh any quantity
with the lever swinging to O'Ol Ib.
The weights to which the scales had been adjusted Avere then exclusively
used on the brake. Thus the brake was balanced by the same weights as
were used as the standard in weighing the water, with a sensitiveness which
gave the error less than one forty-thousandth part of the weight of water in
the smallest trials, while the casual error, which would not exceed this in
a single weighing, would be eliminated in the mean of a large number of
weighings. Thus the relative limits of error in weighing would not exceed
•000025.
The Correction for the Weight of the Atmosphere.
41. The balances being made in air, it is necessary to add the weight of
air displaced in each case.
As the relative weights only are concerned, if Da is the weight of a unit
volume of air, Dw that of water, and D{ that of cast-iron, the weights in air
of unit masses are : —
1 — Da/Dw .................. for water,
1 — Da/Di .................. for cast-iron.
The load on the brake is therefore subject to the correction expressed by
the factor (l—Da/Di), while that of the water balanced against cast-iron
weights, has the correction factor
and the relative correction for the actual weight of water, as against the load
on the brake in air, is
1 ( 1 — yr ) or approximately 1 + ~ ,
\ "w/ -L/w
for the temperature 67° Fahr., Da = 0'0752, Dw = 624.
Hence, the relative correction factor for the equivalent is
(1 -0-001205).
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 643
The Correction for g in Latitude of Greenwich and 45°.
42. Since the latitude of Manchester is 53° 29', Greenwich 51° 29', the
value of g being (Memoires sur le Pendule, Soc. Francaise de Physique)
#450(1 - 0-00259 cos 2X) = g^0(\ + 0-0007558) at Manchester,
= #450 (1 + 0-0005814) at Greenwich,
whence the correction factor is (1 + 0'0001746) at Greenwich,
and for 45° (1 +0-0007558).
The Specific Heat of the Water.
The standard capacity for heat being that of distilled water, the obvious
course would have been to have used distilled water in the trials, had this
been practicable ; but as it was apparent from the first that the quantity of
water which would have to pass through the brakes during the trials would
amount to some 20,000 gallons, or, say, 100 tons, all of which would have to
be brought down to a temperature of 32° Fahr. ; and that to do this,
using distilled water, whether or not the water was used over again, the
necessary appliances for producing, storing and cooling the water, were
impracticable in the laboratory, the last 40° must be removed with ice, and
this would require some 25 or 30 tons of ice. While using the town's water
direct from the main, the average temperature, from February to June, would
not exceed 45°, so that only 12° or 13° would have to be removed by ice,
which would require from 7 to 10 tons, with no appliances except the relatively
small appliance for cooling.
The only practical course, therefore, was to use the town's water. And
had it not been for the known purity of this, the research would never have
been undertaken.
As affording definite assurance of the consistent purity of this water, as
delivered in the college, Professor Dixon kindly undertook to furnish the
mean results of the analyses which he makes periodically for the Manchester
Corporation, of the water drawn from the supply in the college. These show
that the impurities are almost negligible, and taking 0'2 as the specific heat
of the salts, the relative correction is 0"8.s, where s is the relative weight of
the salts.
41—2
644 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
The Effect of Air in the Water.
43. Even distilled water contains air unless special precautions are
taken for its removal ; so that any effect such air may have on the capacity
for heat as measured would not have been avoided by using distilled
water.
The direct effect of the same 0'00323 per cent, of air which water
exposed to the atmosphere usually contains at normal temperatures, is so
small as to be altogether negligible, and it would seem to be an open
question whether the standard condition of water, as regards the capacity
for heat, does not involve the inclusion of this air. But the indirect effect
of such air on the heat necessary to raise water from normal temperatures to
near the boiling-point, is by no means negligible.
It does not appear that any definite study has hitherto been made of this
effect ; but it is a matter of common observation that as water reaches
a temperature some 40° Fahr. below the boiling-point, bubbles appear on the
sides and bottom of the vessel, which gradually increase in size and rise to
the surface, increasing rapidly in size as they rise. The bubbles are usually
referred to as bubbles of gas or air. But, a moment's consideration will show
that, although the air or gas is the immediate cause of the premature
formation and subsequent expansion of the bubble, it is none the less certain
that the space occupied by the bubble is filled with saturated steam at the
temperature of the water, the function of the air being merely that of
balancing the excess of pressure of the surrounding water over the pressure
of the saturated steam.
It thus appears that every bubble so formed represents a quantity of
heat, which is the latent heat of the volume of the saturated steam in
the bubble, over and above the heat of the weight of water in this steam.
Thus, if bubbles of air exist in water at a temperature of 212° Fahr., the
weight of air per Ib. of water being a, and p the pressure of the water in
inches of mercury, then, since the pressure of the air is p - 30, and the
volume of 1 Ib. of air at 212° Fahr. under 30 inches of mercury is
16 '9 cubic feet, the volume of air per Ib. of water is
T, 16-9 x 30
:T^3o-xa'
or, if p = 40, V= 50-7 x a.
This is the volume, in cubic feet, of saturated steam at 212°; whence,
since the latent heat per cubic foot is 36'6 at 212°, the excess of heat will be
per Ib. of water
Fx :16-6 = 1855 x a,
66] ON THE MECHANICAL EQUIVALENT OF HEAT. G45
and this, divided by 180°, gives a relative error
10-31 x a.
If a = 0-0000323, the error is
0-00033, or 0'033 per cent.
The water, after being exposed to the atmosphere in the service reservoir,
where it discharges any excess of air, enters the brake cold with this normal
air, there it is heated by work, under the pressure of the artificial atmosphere
at pressure p, to maintain which it parts with some of the air, which, in
passing out into the flexible pipe, carries out saturated steam, which is
condensed by radiation from the pipe. The water, with the remainder of
the air, is then carried by the centrifugal pressure into the outer chamber
in the brake case, under a pressure of about 50 inches of mercury. It then
passes the automatic cock, into the flexible pipe, at 41 inches pressure,
thence rising to the thermometer bulbs at 40 inches. In passing the
automatic cock with a difference of pressure of 9 inches, the pressure will
be further reduced, probably 9 inches below that in the pipe, so that any air
that might have been retained would come out at that point, and expand
further as it approached the thermometer bulb.
In the first instance, it was thought that a pressure of 5 feet of water
would prevent the formation of bubbles, and the air gap in the pipe leading
from the condenser was placed at this height above the thermometer. But
many, and sometimes large, bubbles of air were observed passing up the
thermometer chamber ; and Mr Moorby observed that he could detect the
passage of a large bubble by a fall in the thermometer before the bubble
appeared in the glass chamber.
To prevent this, the air-gap was raised till it was 12 feet above the
thermometer bulb ; so that the error is limited to three ten-thousandths. Even
so, as it is much larger than any of the errors of constant sign, it was
important to try, by assimilating the conditions under which the water leaves
the brake, to obtain experimental evidence which would narrow the limits.
It may appear at first sight as though these losses from the air in the
water would, like the radiation, be eliminated in the difference of the large
and small trials, but this is not so, since the quantity of heat so lost is
pmportional to the amount of water used, or it may be greater in the heavy
trials.
The Standard of Length.
44. The measures of length that the research involves are —
(1) The horizontal distance of the centres of gravity of the adjustable
loads on the brake from the axis of the shaft.
046 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
(2) The vertical heights of the barometer at which the boiling-points of
the water were determined.
In order to secure a definite reference of these to the British standard,
recourse was had to two carefully-preserved and independent measures
derived from this standard.
(1) A set of gauges by Sir Joseph Whitworth and Co., consisting of
three steel bars, 9, 6 and 3 inches respectively, with parallel plane ends f inch
in diameter, adapted to a 20,000th of an inch measuring machine, which
constitute the standards used in the engineering laboratory.
(2) A brass bar by Elliott and Co., 39 inches long, and graduated in
inches, used as the standard in the physical department in Owens College.
From the Whitworth gauges, two steel bars, f inch in diameter and 9 inches
long, with parallel plane ends, were made by Mr Foster, and compared with
the 9-inch Whitworth bar by the measuring machine.
With these and the Whitworth gauges, placed end to end, an outside
gauge consisting of two surfaced angle-plates on a surfaced cast-iron bed
was set out, and then a steel bar f inch in diameter with plane ends fitted
to these. Careful comparison showed that this bar did not differ from the
sum of the lengths of the gauges by Tofiny Pai>ts of an inch. This length
was then carefully laid off by the surfaced angle plates on the surface plate,
and was so compared with the scale of the Elliott brass bar, account being
taken of the temperature, and found to agree within less than 10^00 of an
inch.
The 30-inch bar so obtained was then taken as the standard both for the
levers of the brake and the barometer, to be carefully preserved.
Lengths of the Levers.
45. The V-groove, in which the knife-edge of the carrier, by which the
load on the brake was suspended, rested, was originally made at a distance
of four feet from the axis of the shaft at ordinary temperatures, and as,
whatever the error might be when the brakes were hot, it would be the
same for all the trials, since the temperatures were the same, it was decided
to take this as the length of the levers in estimating the loads during
the progress of the research, and to treat whatever error there might be
as a standing correction on the final results. Such correction to be obtained
by laying off four feet, less the radius of the shaft, from the carefully squared
end of a steel plate 3 inches broad and T3^ inch thick, then placing this, flat,
in a vertical plane perpendicular to the shaft, with its edge horizontal, as
near as practicable to the knife-edge groove with the squared end touching
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 647
the shaft. Then by means of a theodolite, set so that its line of collimation
was in a vertical plane parallel to the axis of the shaft, and intersecting the
vertical line on the plate, to observe the distance of the groove from the
line on the plate, while the brake was running under the same conditions
of temperature, and load as in the trials ; but with the carrier temporarily
displaced further along the shaft, so as to leave the bottom of the V-groove
visible through the theodolite, and in this way to obtain the actual distance
of the groove from the axis of the shaft, as affected by the expansion of the
brake, and any displacement of the bearing on the shaft which might result
from the running.
By using a scale divided to the one-hundredth of an inch, and taking
several readings, this could be determined to a thousandth of an inch, so
that the limits of accuracy would be
± 0-00002.
The Standard of Temperature.
46. As the most general standard is the difference between the two
physically fixed points of temperature, corresponding to the temperature
of ice melting under the pressure of the atmosphere, and that of water
boiling under a pressure corresponding to 760 millims. of ice-cold mercury
in the latitude of 45°, taking account of the variation of g, the standard in
Manchester is the interval between melting ice and water boiling under a
pressure of 760 x 1 '0001 7 21 millim. of ice-cold mercury, which corresponds
to 29'899 inches. And this interval divided by 180 is one degree Fahr.
According to Regnault's tables, a divergence of one thousandth of an
inch from the boiling point would correspond to an error of 0'0017° Fahr.,
and this would be less than the one-hundred-thousandth part of 180°.
In order to obtain this degree of accuracy in comparing the pressure of
the vapour of pure water, in which thermometers could be placed, with the
height of mercury over a range of two or three degrees above, and two or
three below the point, at almost any time, irrespective of what might be
tin' actual pressure of the atmosphere, it was necessary that the barometer,
or pressure gauge, while in free communication with the vapour chamber,
should be shut off from the atmosphere, and at the same time so far
removed, that the temperature of the mercury should not be affected by
the heat from the gas or boiling water. And, further, although in direct
communication with the vapour, this must be such that no moisture could
reach the mercury ; and, such as involved no current in the passages which
might affect the relative pressures, as would result by the interposition of
a condensing vessel.
648 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
It was also necessary that the arrangements for reading the vertical
distances between the upper and lower surfaces of the mercury should not
only give absolute differences of height, but also that they should afford
ready means of at any time determining the presence of vapour or gas,
other than that of mercury, in the upper limb of the barometer.
The Barometer.
47. To meet these requirements, the barometer shown (Plate 8) was
designed. The vessel which holds the mercury consists of a bottle-shaped
casting of iron, 3 inches in diameter. Through a stuffing-box in the neck
of this, the stem of the barometer tube passes. To admit of reading the
level of the surface of the mercury in the bottle, two parallel plate-glass
windows are arranged, f inch diameter, having their axis f inch from the
axis of the bottle. These are sunk into the casting so as to leave the outer
cylindrical surface of the bottle clear, the joints between the glass and the
cast-iron being faced and made tight with a trace of beeswax, the other
openings into the bottle being one for the admission and abstraction of
mercury, fitted with a screwed valve, and one for the admission of air, with
a mouthpiece for the attachment of a tube from the vapour chamber.
The glass stem of the barometer is drawn down into a neck towards the
lower end, and this is bent through 180° so as to bring the mouth upwards,
and thus admit of its introduction into vthe mercury in the bottle without
letting in air. This bend has to be passed through the stuffing-box, then
the tube is secured by screwing the gland on to the beeswax stopping. A
brass guard tube is then screwed into the neck, to support the glass tube,
to a height of 24 inches from the mercury in the vessel.
For reading the height of the lower limb, a cylindrical brass curtain,
with a conical contraction on the top, the aperture in which is threaded
internally at twenty threads to an inch to correspond to the screw on the
outside of the neck of the bottle, is screwed on to this neck, the lip or
bottom of the curtain being truly turned so that, when screwed down to
the level of the mercury, it cuts off the light through the windows from
a white sheet behind.
To the top of the brass casting, which forms the curtain, a brass cylin-
drical tube is rigidly attached coaxial with the curtain which fits over the
brass guard round the barometer tube, this extends to a height of 26 inches
from the lower lip, the internal diameter for the last inch being a little
smaller and internally screwed at twenty threads to an inch. Into this is
screwed a brass tube, externally screwed throughout its length, about
6 inches long, with parallel opposite slots J inch wide extending to within
66] ON THK MKCHANICAL EQUIVALENT OF HEAT. 649
an inch at either end, to form windows through which to see the light
over the upper limb of the mercury. And on to the upper portion of this
tube there is screwed a long cap, capable of screwing down to the bottom
of the slot. The lower lip of this cap forms the curtain which cuts off the
light when the lip is level with the upper limb of the mercury.
By this arrangement the variation of the distance between the lips of
the lower and the upper curtains depends only on the change in their
relative angular positions. For, since the slotted tube has a uniform thread,
it can be turned, screwing into the lower curtain and out of the upper, both
of which remain unmoved. Thus the position of the windows may be
fixed, while the curtains are moved. So that for reading the distances it
is only necessary to measure the relative angle.
This angle is measured by dividing the circumference of the cap just
above the lip into five equal divisions, from 0 to 5, and these again into ten,
then a turn through one of the smaller divisions means an alteration in the
distance of one-fiftieth of one-twentieth of an inch. As this angle is
measured relatively to the lower curtain, a vertical brass scale, divided to
tenths and twentieths of an inch, is fixed externally to the top of the
extension of the lower curtain, extending vertically just outside the gradu-
ated limb of the upper curtain, and thus serves for reading the angular
distance of the index mark on the limb of the upper curtain, on any
particular thread, and the number of threads from the index on the scale.
The Adjustment of the Indices on the Barometer.
48. The lower curtain, together with the slotted tube and cap, is un-
screwed from the neck of the cast-iron bottle and lifted off over the tube.
Then the 30-inch standard bar is set on end upright on a surface plate, and
the lower curtain, &c., are lowered over the bar until the lower lip of the
curtain rests on the surface plate, and the top of the bar is 30 inches from
this lip. The cap is then screwed down until light is seen over the top of
the bar through the slot just cut off. Then a vertical line drawn on the
cap just above the lip, at the edge of the scale, is the index on the cap,
and a horizontal line, drawn on the scale level with the lip of the cap, is
the index point on the scale. And, when these two lines are brought into
this position, the distance between the lips will equal the length of the bar.
In order to check this the curtain is raised, and two thin pieces of
chemical paper are placed on the surface plate, one on each side of the bar,
so as to leave a space between the paper and the bar. Then the curtain is
replaced so that it rests on the paper, and light can be seen through the
interval between the paper and the bar. Then light should be seen to an
650 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
equal extent over the bar, and by screwing down the cap till the light
disappears, the thickness of the paper will be measured by the angle turned
through.
The construction of this barometer, the first of its kind, was undertaken
by Mr Foster, who has produced a very beautiful instrument by which direct
reading can be taken to the ten-thousandth of an inch. The mercury having
been re-evaporated for the purpose, in an apparatus belonging to Dr Schuster,
by his assistant, Mr S. Stanton.
This barometer could be used as a pressure gauge for pressure up to
34 inches and down to 26 inches, and by connecting the mouthpiece with
a receiver in connection with a mercury or water syphon gauge, with the
other limb open to the atmosphere, the differences of reading of the
barometer for different pressures in the receiver can be readily compared
with the corresponding differences in the syphon gauge, and by such
comparisons, taken at intervals till the mercury reaches the closing in of
the tube, a test is obtained as to the absence of anything but mercury
vapour above the mercury.
When the barometer is in connection with the vapour chamber in
which the thermometer is immersed, the passage of moisture back into
the barometer is prevented by connecting the tube by a branch with an
air receiver, in which the pressure is maintained higher than that in the
vapour chamber ; the branch pipe communicating with the chamber through
a piece of quarter-inch glass pipe, 3 inches long, plugged as tightly as
possible throughout its length with cotton-wool, through which the air
has to pass from the receiver into the vapour chamber. In this way, an
indefinitely slow current of clean dry air can be maintained into the
passage from the vapour chamber to the valve which controls the exit
of the steam into the atmosphere, so that the air does not enter the vapour
chamber in which the thermometers are, but directly passes out with the
overflow steam.
There is necessarily some resistance to the air passing along the pipe to
the vapour chamber, but this could easily be tested by removing the pipe
from the vapour chamber, and leaving it. open to the atmosphere, so that
the barometer would adjust itself to that of the atmosphere, plus the
pressure due to the resistance of the current in the pipe ; then, stopping
the current by closing the branch pipe, and reading again, the difference
would give the pressure due to the current. With the plug as described
this was so small as to be negligible, even when the pressure in the
receiver was two atmospheres. As during the testing of the thermometers
the pressure in the vapour chamber was generally greater than that of the
66] ON THE MECHANICAL KQUIVALENT OF HEAT. 651
atmosphere, in order to maintain this steady, a governor on the gas burner
was necessary, as well as an accurately adjustable exit valve.
With these appliances the scale of the high temperature thermometer
could be tested at intervals, over a sufficient interval on each side of the
boiling point (212° Fahr.), the corrections for surface tension, temperature,
and gravitation being applied to within the thousandth of an inch of
mercury.
This gives the limits of error + O'OOOOl.
Correction of the Low Temperature Thermometer.
49. The correction on the thermometer for 32° would be at any time
obtained in the usual way by immersing the thermometer vertically in a
bath of soft snow, but as there was no ready means, as with the scale
about 212", of testing the scale at 32°, while this would be used for one
or two degrees, this correction could only be made by comparison with a
thermometer already corrected with the air thermometer, which comparison
Dr Schuster allowed to be made in the physical department.
Corrections of the Thermometers for Pressure.
50. The pressures in the thermometer chambers of the brake being both
some 10 or 15 inches of mercury above that of the atmosphere, it would be
necessary to determine the corrections on each of the thermometers under
the pressures and temperatures at which they had to work.
Thus, if elt e.2 are the corrections per unit of pressure in the initial and
final thermometers, the correction for the heat is (e^p^ — e.2p.^).
The Range of Temperature over which the Specific Heat would be Measured.
51. The temperature of the effluent water from the brake can be
regulated either up or down to any required extent, and although there
would necessarily be some divergence from the boiling-point, with care
and experience it would be possible to bring the mean result in a number
of trials within a close approximation of 212° Fahr.
On the other hand, there has been no means provided of regulating the
temperature of the water entering the brake. This is determined by the
rate at which the water passes through the iced coil and the temperature
at which it entered, as determined by the temperature in the town's mains,
which varies from 38° in the winter to 55° in the summer. Thus the
temperature in the light trials would be from half to a degree above 32°,
and that of the heavy trials from a degree to two degrees.
652 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
In calculating the heat of each trial, the actual difference with the
correction for the thermometers is taken, but if, as is shown by previous
investigations by Regnault and others, the specific heat at and near 32° is
less than the mean specific heat between 32° and 212° by something like
0*5 per cent., there would be errors in taking the results so obtained as the
mean specific heat between 32° and 212°.
Owing to the extreme difficulty of determining the specific heat over a
very short range of temperature to such high degrees of accuracy as '01 per
cent., the experimental evidence as to the exact value of the specific heat
within a few degrees of 32° is but vaguely surmised from the general fall
of the specific heat with the temperature.
The law of the thermal capacity of water between 0° C. and t°, as deduced
by Regnault from his experiments, is avowedly vague as to the lower tem-
peratures. It shows no singular point at the maximum density, as would be
expected ; and Rankin deduced another law from these experiments, making
the minimum specific heat coincide with the point of maximum density.
Also other experimenters have obtained higher specific heats near 32° than
are given by Regnault's formula. It would seem probable, therefore, that
the difference between the specific heat at 32° and the mean between 32°
and 212°, as given by Regnault's formula, is too large.
In that case, the correction obtained by this formula in order to reduce
the specific heat between the observed temperature in the trials to that
between the standard points, would probably be too large, and thus afford
an outside limit of error.
Thus, putting s for the mean specific heat between 32° and 212°,
s(l+X) for the specific heat between T,° and 212°, when T± is small
compared with 180°, and, by Regnault, taking s(l — O005) for the specific
heat at T^, then the total heat from T,° to 212° is
s (1 + X ) (212 - 2V) = s (180 - (T° - 32) (1 - O'OOo)}
= s (212 - 2V) (l - ^ - *vo x 0-005) ,
or, neglecting (Tl — 32)2,
T ° — S9
X = 0-005 * = 0-000028 (T,0 - 32).
loU
Thus, taking the mean capacity of water between the temperatures of
32° and 212° as the standard capacity, the mean specific heat between TI
and 212° would be
1 + X = 1 + 0-000028 (2V - 32) ;
and, if 2\° is the mean initial temperature of the water of any number of
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 653
trials, 1 + X is the mean specific heat of the water in all the trials. The
mean specific heat of the difference of two trials would be 1 + X ; this
appears as follows : —
Suppose 1 4- TI to be the mean specific heat for a set of heavy trials, and
ir, the mean weight of water, and (1 + X2) to be mean specific heat of a
corresponding set of light trials, and W2 the mean weight of water, 2\°,
T2° being respectively the initial temperatures of W1 and TF2, the difference
of the total heats would be
(1 + Z,) (212 - T,°) W, - (1 + X9) (212 - T2°) Ws,
and the mean specific heat would be approximately
(212 - TV) F, -(212 - TV) W3+lSO(XlW1-XtW9)
(212 - 2\) Wi - (212 - T2) W,
ISO(X1W1-X,W.2)
and, as in the heavy and light trials TFj =• 2 W» approximately, the mean
specific heat by Regnault's formula would be
1 + 2X, - X2 = 1 + 0-000028 [2 (Z\ - 32) -(Tz - 32)].
This result is obtained by merely summing the trials, but counting the
water in the light trials as negative,
JT-
2 (IT)
The Gradual Rising of the Indices of the Thermometer.
52. Where, as is generally the case, the indices of the thermometers are
gradually rising, if they are used between the intervals at which they are
corrected, the last observed correction being applied, there will be an error
which will be negative, and of magnitude equal to the rate of rise during
the interval multiplied by the interval. Thus, if the trials are uniformly
distributed between the intervals of correction, the correction would be
0'5«, where a is the observed rise in the interval, hence the relative cor-
rection on the equivalent, taking a^ arid «», as the mean rises between the
intervals of correction of the initial and final thermometers, would be
0-5
The Work done by Gravity on the Water.
53. The difference of pressure on the bulbs of the initial and final
thermometers which are at the same level, expressed in feet of water, is
654 ON THE MECHANICAL EQUIVALENT OF HEAT. [G6
the work done by gravity per Ib. of water. If pl and p., express these
pressures in inches of mercury, the work done by gravity is
which gives as the relative correction for the equivalent, approximately,
+ 0-000008 S [ W ( P! - pj]/ 2(W).
The Work absorbed in Wearing the Metal of the Bushes and tihaft.
[54. During the six years the brake had been in use, before the trial
commenced, the shaft and bushes were occasionally lubricated with oil,
chiefly to prevent oxidation of the shaft when standing, and, up to the
commencement of the trials, there was hardly any appreciable sign of wear.
After the closing of the bushes by the stuffing-box and cap, when the use
of oil was purposely discontinued, there was no means of observing the
wear of the metal as long as the brake worked satisfactorily, as it did
during all the trials. But when, after the completion of the trials, the
stuffing-box and cap were removed, in order to return to the original
manner of working, the excess of leaking through the bushes showed that
there had been considerable wear.
At that time it did not occur to me that the proportion of this wear,
which took place during the actual running of the trials, would represent
a certain amount of work absorbed in disintegrating the metal, or a certain
amount of heat developed by the oxidation of the metal, and no attempt
was then made to form a definite estimate of the amount of metal which
had disappeared. As, however, the worn metal was replaced by a coating
of white metal, the thickness of this (less than ^nd of an inch) and the
extent of surface (less than 124 square inches) subsequently showed that it
could not be more than 1 Ib.
This was after it occurred to me that however small might be the effect
of this wear, since it was definitely observed to have taken place during the
twelve months when the bushes were closed for the purpose of the trials,
it was desirable, in order to complete the research, that some outside
estimate should be obtained of the limits to its possible effect, whether
from disintegration or from oxidation.
In as far as the loss of metal was due to the abrasion of the clean metal
surfaces, it would be proportional to the number of revolutions, while in as
far as it was owing to the oxidation of the metal surfaces, left bright after
each run, it would be probably proportional to the number of runs.
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 655
The number of revolutions with the bushes closed, counting ordinary
work as well as the trials, is found from the records to be less than
300 x 60 x 360,
and the number of runs to be 80, the mean time being 4'5 hours. The
revolutions during any one of the accepted trials were 300 x 60. And the
trials were made in threes, so that the coefficient for oxidation would be -%fa.
Hence, the metal worn by abrasion in a single trial would be less
than s^yth of 1 Ib. = 0'0028 lb., and the metal oxidised in one trial less than
siiyth = 0'004 lb. So far the estimate is fairly definite, but, for its completion,
it is necessary to arrive at some conclusion as to the work absorbed in dis-
integrating the metal, and of the heat developed by its oxidation.
There does not seem to be any reason why there should be more oxidation
of the bright surfaces in a light trial than in a heavy trial, so that there
would have been no error from this cause in their difference.
As regards the abrasion and the oxidation of the abraded metal, there
would be a difference, as the weight on the shaft in a heavy trial is T23 of
the weight in a light trial. Thus the differences of abrasion would have
been
0-0006 lb.
The work necessary to produce a state of disintegration, such as exists in
the vapour of the metal, would be the total heat of vaporization, less the
kinetic energy and work [fcv/(T— 32) + PV~\, and, although the heat of
vaporization of the metal is not known, it would seem that it cannot greatly
exceed, when subject to the deductions mentioned, the heat of vaporization
of ice subjected to like deductions (1,000,000 ft.-lbs.).
Assuming this, since the difference in the work of two trials is about
70,000,000 ft.-lbs., the correction would be
- 0-00001,
which, considering that the disintegration would be very imperfect, may be
taken as an outside limit, while the effect may have been even reversed by
the oxidation of the degraded metal. — Nov. 9, 1897.]
Accidents.
55. In contemplating such an extensive and complex research, the result
of which depends on the mean of a number of experiments, it was impossible
to overlook the question as to how such accidents, as would probably occur,
should be dealt with.
656 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
It was clear that, whatever the rule might be, it must be definite and
rigorously applied.
Two other things were also clear, that, as in surveying, accidents might
occur, say in reading the counter or the scales, which would only be apparent
from the reduction of the results after the trial was finished. Also, that in
these experiments there would be no such rigorous check on the results as in
surveying ; so that, without danger of sorting the results, anomalous results,
the cause of which was not noted during the trial, could only be rejected
when the results themselves contained evidence of the cause of the anomaly,
say an abnormal difference between the mean speeds by the counter and the
speed gauge.
It was therefore, from the first, decided to reject all trials in which there
was definite evidence either during the trial or in the results, of uncertainty
to which no definite limits could be assigned, in any one of the measurements,
without regard for the apparent consistency of the results, and in the same
way to retain all other trials.
56. The following table contains a summary of all those circumstances
on which the accuracy of the result of the investigation depends, together
with references to the several Articles in which they have been discussed. In
line with each circumstance is placed the formula for the relative correction
in the equivalent, necessary in consequence of the observed deviation from
the conditions of equality between the heavy and light trials. In the same
line with each circumstance are also given, to the millionth part, the limits
of relative error as deduced in the corresponding Articles.
ON THE MECHANICAL EQUIVALENT OF HEAT.
657
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658 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
PART II.
ON AN EXPERIMENTAL DETERMINATION OF THE MECHANICAL EQUIVALENT
OF THE MEAN SPECIFIC HEAT OF WATER BETWEEN 32° AND 212° FAHR.,
MADE IN THE WHITWORTH ENGINEERING LABORATORY, OWENS COLLEGE,
ON PROFESSOR OSBORNE REYNOLDS' METHOD. — BY WILLIAM HENRY
MOORBY, M.Sc.
1. In view of the frequent and extremely careful and accurate deter-
minations of the value of the mechanical equivalent of heat which have been
made of late years by different experimenters using different methods the
present series of experiments may on first thoughts seem superfluous. There
did, however, seem to be sufficient disagreement between the results pre-
viously published — more particularly between values of the equivalent, as
derived from the direct methods described by Joule, Rowland, and Miculescu,
and the indirect electrical methods of Griffiths, and Gannon, and Schuster, to
warrant a new investigation into the value of this important constant, if the
proposed new method of working should carry with it advantages not available
in previous investigations. I was accordingly very glad to fall in with the
wishes of Professor Reynolds that I should undertake a research bearing on
this point on lines which he suggested to me in July, 1894.
2. In Part I., par. 3, a full description is given of the apparatus whose
existence in the Whitworth Engineering Laboratory led up directly to the
institution of this research into the value of the mechanical equivalent of
heat.
The advantages which the proposed method offered were briefly : —
(1) The possibility of obtaining a result which in no way depended
for its accuracy on the value of the scale divisions of the ther-
mometers used in the measurements of temperature (Part I.,
par. 11).
This was done by supplying a stream of water to the brake at a
temperature of 32° Fahr., and there raising its temperature to
212° Fahr. before admitting it to the discharge pipe where its
temperature was again taken.
(2) A means of eliminating from the result all losses of heat due
to radiation and conduction from the calorimeter employed
(Part I., par. 32). The manner in which this elimination was
accomplished is indicated below.
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 659
Let U and u represent the quantities of work done in two trials which
differed only in the moment of resistance offered by the brake — the number
of revolutions of the engine shaft and the duration of the trials being the
s;ime in each case.
Also let //' and h' be the apparent quantities of heat generated in the
brake in these trials. These quantities will be less than the true equivalents
of the works U and u by quantities which represent the losses of heat from
the brake by conduction, radiation, &c. These losses were made as nearly as
possible equal by keeping the temperatures of the brake and its supports and
surroundings at the same levels in the two trials.
Then the quantity of work (U - u) should be exactly equivalent to the
quantity of heat (H' — h'), and by dividing the h'rst of these by the second,
a value of the constant required is obtained.
The power available for the purposes of the investigation enabled me to
deal with quantities approaching the following values in trials of one hour's
duration : —
Revolutions, 18,000.
Total work done, 135,000,000 ft.-lbs.
Total weight of water raised 180°Fahr. = 960 Ibs.
Total apparent heat generated = 170,000 B.T.U.
In quantities so large as these some of the small errors inevitable to all
physical experiments became quite or nearly negligible.
Preliminary Apparatus and Trials.
3. It will, perhaps, be sufficient to indicate the general arrangement of
the apparatus as first set up. This is illustrated in the annexed sketch.
The water was supplied from the mains through the iron stand-pipe, A, and
the regulating cork, B. Before it entered the brake its temperature was
measured by means of the thermometer, C, inserted through a cork in the
stand-pipe, the part of the stem on which readings were taken being exposed
to the atmosphere. After being discharged from the brake, D, the water
entered a flexible rubber pipe, E, bent through an angle of 90°, which con-
nected a horizontal nipple at the bottom of the brake with a vertical one
forming the lower end of a fixed line of copper piping, F. The temperature
of discharge of the water was indicated by the thermometer, G, which was
enclosed in a glass tube opening through a stuffing-box into the discharge
pipe, the whole length of the stem being therefore kept at the temperature
of discharge. On leaving the copper discharge pipe the water was directed
42—2
660
ON THE MECHANICAL EQUIVALENT OF HEAT.
[66
at will by the two-way tipping switch, K, either to the left to waste or to
the right into the tank, L, standing on the platform of the weighing
machine, M.
Fig. 1. Preliminary Apparatus. Course of water shown by arrows.
A series of trials were made with this apparatus, the water being raised
through varying intervals of temperature between 35° Fahr. and 100° Fahr.
For obvious reasons the results were not satisfactory, and are therefore not
published. Experience was gained, however, which helped very materially
in the design of the final apparatus.
Common thermometers were used, and calibration errors on the com-
paratively small range of temperature through which the water was raised
were of sufficient importance to vitiate all results. Again, the exposure of
the stem of the thermometer, 0, was a weak spot in the apparatus. I was
much troubled also with leakage of water from the two bushed bearings of
the brake.
In so far as could be judged, the bent rubber pipe, E, was found to be a
satisfactory connection between the brake and the copper discharge pipe, and
this has been retained in the subsequent apparatus.
DETAILS OF THE CONSTITUENT PARTS OF THE FINAL APPARATUS.
Artificial Atmosphere. — (Part I., par. 23.)
4. To prevent loss of water by evaporation at the centres of the vortices
formed in the brake, the ports in the vanes of the outer casing were connected
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 661
through a flexible rubber tube some 4 feet long, with an artificial atmosphere
formed in a tin receiver, the pressure in which was maintained by means of
a cycle tyre inflator at about 9 inches of mercury, as measured on a U -gauge.
The shape of this vessel is made clear in the sketch (Part I., Fig. 8). The
ends were made conical for greater strength. The receiver was also provided
with an air valve, with which to relieve the pressure when too high, and
a cock, with which water accidentally lodging inside could be drained
away.
The Ice Cooler.— (Part I., par. 19.)
5. Some preliminary experiments indicated that a length of about
200 feet of f-inch diameter lead piping would, when immersed in a mixture
of ice and water, be sufficient to cool a stream of some 16 Ibs. of water per
minute very nearly to 32° Fahr.
The ice cooler was accordingly made as follows: A wooden box,
4' 0" x 2' 3" x 2' 0", and lined inside with waxed cloth, was fitted with a
horizontal wooden shelf about 2 feet 6 inches long, and on this was laid a
flat oval coil of j|-inch composition piping nearly 200 feet in length, the
left-hand end of the coil and shelf stopping short at a distance of 1 foot
from the end of the box, the right-hand end of the coil reaching the end of
the box, but the shelf stopping some 6 inches short of that point. The coil
was about o inches diameter, vertically, and over it were placed the wooden
guide plates shown (Part I., Fig. 7). An 8-inch diameter paddle, having 6
wooden floats, was placed about the middle of the box, at a height just
sufficient to ensure the lower edges of the floats clearing the coil of pipe
below it. A galvanized iron wire netting, extending from the shelf upwards
to the top, separated the well at the left-hand end of the box from the com-
partment to the right containing the coil and paddle.
When working, the well and space beneath the shelf contained broken
ice, well rammed in ; while the level of the water was automatically kept at
about 3 inches above the top of the coil. The paddle, driven by a cord from
the line shafting in the engine-room, revolved in the direction shown by the
arrow, and caused a circulation of water up through the ice in the well,
and then horizontally through the coil and back to the ice under the
shelf.
Circulating Pump. — (Part I., par. 20.)
6. In order to supply sufficient water to the brake against the resistance
offered by the 200 feet of pipe in the cooler and the augmented pressure in
the brake itself, it was necessary to use a circulating pump. This was a
small Mat her- Reynolds centrifugal pump with four 1^-inch wheels, driven
662 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
by a turbine available for this purpose in the engine-room. This pump was
capable of supplying 16 Ibs. of water per minute, against a pressure of 25 Ibs.
per square inch at the supply valve.
Some difficulty was encountered in the summer of 1896 with this com-
bination, because the excessive demand for condensing water for the engine
hardly left sufficient flow in the falling hydraulic main to work the turbine
at the requisite speed to maintain the above pressure.
On the whole, however, the combination was exceedingly efficient, and
with a graduated supply valve afforded a very delicate means of regulating
the flow of water into the brake.
Water-tight Joints between the Brake and the Engine Shaft.
7. In Part I., par. 24-29, the necessity of obtaining control over the
leakage of water at the bearings of the brake, and the methods by which
this was accomplished, are fully discussed. The bearing on the up-shaft end
of the brake was provided with a stuffing-box, while the shaft end was covered
with a cap. The annexed sketches show the general design of the stuffing-
box and cap : —
A — The engine crank shaft.
B — The outer skin of the brake.
C — Conical brass bushes screwed into the outer skin of the brake.
D — Lock nuts on these bushes.
E, F, and G — Stuffing-box, ring and cover.
K — Set screws fastening stuffing-box to the lock nut.
L — Cap covering the end of the shaft.
M — Small spindle driven by a pin on the end of the engine shaft,
passing through a stuffing-box on the cap, and required to drive
the revolution counter.
The cap completely stopped all leakage from the bearing to which it was
fixed, and, when the stuffing-box had worked for a short time, only a few
drops of water escaped from the up-shaft bearing.
The brass bush bearings needed lubricating, and this was accomplished
by supplying a small stream of water to each bearing through the pipes N
and P, each provided with a regulating cock. This water carne from the
supply pipe between the ice cooler and the regulating valve controlling the
main supply to the brake. It was consequently under considerable pressure
and at a temperature very little over 32° Fahr. The water thus supplied
66]
ON THE MECHANICAL EQUIVALENT OF HEAT.
663
had, of course, to enter the brake, and the amount supplied afforded a very
convenient means of controlling the temperatures of the bearings.
Fig. 2. Joints between brake and shaft.
At a distance of 2f inches from the cap of the stuffing-box was the
end of one of the main bearings, R, carried on the cast-iron pedestal, S.
It was important that I should have some control over the loss of
heat by conduction along this length of shaft. Accordingly, two pieces
of brass pipe were soldered on to the cap of the stuffing-box, while two
others were screwed, the one in the upper and the other into the lower
brass forming the main bearing. Thermometers were placed inside the tube
affixed to the stuffing-box cap, which happened to be uppermost at the
time, and into the two pipes screwed into the main bearing. It was then
assumed that the loss of heat along the shaft would vary with the difference
of temperature between the stuffing-box cap and the bearing. In order
that the losses of heat occurring in this way in any two trials should be
identical, it was sufficient under the above assumption that this difference
of temperature should be the same in both trials, and the temperature of
the stuffing-box was regulated to this end by means of the amount of cold
water passing into it.
Considerable difference of temperature was observed between the upper
and lower brasses of the bearing, and as it seemed probable that the
lower one approximated the more closely to the temperature of the shaft,
that thermometer was the one used in determining the loss of heat by
conduction.
In the later trials I endeavoured to keep the temperatures of the
664
ON THE MECHANICAL EQUIVALENT OF HEAT.
[66
stuffing-box and the bearing at the same level, thus entirely eliminating
this cause of loss from the experiments.
Water Jackets for the Low and High Temperature Thermometers. —
(Part I., par. 15.)
8. It was evident that the temperatures of the water would be much
more easily and accurately taken if the whole stem of each thermometer
was kept at one temperature. To this end each of the principal thermo-
meters was completely jacketed with a stream of the water whose tempera-
ture was required.
Main supply
regulating valve
from
ice cooler
to condenser
and tank
Fig. 3. Cold water thermometer jacket.
{from brake
I
Fig. 4. Hot water thermo-
meter jacket.
The arrangements adopted for this purpose are illustrated in the annexed
sketches. (Figs. 3 and 4.)
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 665
After leaving the main regulating valve the cold supply water entered
a vertical brass T, shown at A. The main volume of the water flowed on
to the brake through the horizontal arm of this T. At its upper end
the T carried a small stuffing-box, B, into which was fixed a vertical ^-inch
diameter glass tube, C. This tube was closed at its upper end by means
of a rubber stopper, held in place by the brass cap, D, screwed on to the
upper end of a f-inch slotted copper pipe surrounding the glass tube. The
stopper and cap were both penetrated by a short length of |-inch diameter
brass tube, which carried a gas-cock at its upper end. The thermometer
was hung by a piece of string from the lower end of the £-inch pipe — the
graduated part of the stem being all clearly visible through the glass walls
of the chamber while the bulb was well in the main stream of water flowing
through the brass T.
A small stream of water was allowed to run to waste through the small
gas-cock at the top, thus ensuring the whole of the stern of the thermometer
being kept at the proper temperature.
The hot water discharged by the brake flowed from the bent rubber
tube, previously mentioned, into the lower end of the vertical 1-iuch diameter
copper pipe, A. This pipe carried a brass cross, It, at its upper end, while
fitted to the top of the cross was the stuffing-box, C, in which was fixed
a piece of f-inch diameter glass tubing, D, forming the thermometer chamber.
The upper end of this chamber was closed by a rubber stopper penetrated,
as before, by a piece of |-inch diameter brass pipe, connected by a piece of
rubber tubing to the main discharge pipe above.
The left arm of the cross carried an upward-turning elbow, and that
again a f-inch diameter copper pipe, up which most of the water flowed.
The thermometers, two of which were used, were hung to the lower end
of the Jj-inch pipe in the rubber stopper, so that the bulbs were immersed
in the whole stream of water flowing up the ]-inch copper pipe from the
brake. One of these thermometers was only used as a finder to indicate the
temperature of the water as it rose after first starting the engine, and no
record of its readings was kept.
The Con denser.— (Part I., par. 18.)
9. In order that there should not be a large loss of water before weighing,
by evaporation from the tank into which it flowed from the brake, it was
necessary to cool the stream to a temperature approaching that of the
atmosphere.
For this purpose a condenser was constructed after the ordinary chemical
666 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
pattern. It consisted of a length of 21 feet of |-inch diameter pipe inserted
in an equal length of li-inch diameter iron pipe.
Stuffing-boxes were used to form the joints between the two pipes. The
hot water from the brake flowed through the inner tube, while a supply
of condensing water flowed in the opposite direction through the annular
space between the two pipes. By means of this condenser the water entering
the tank was always cooled at least to 100° Fahr., and to lower temperatures
in the earlier experiments when the water available in the mains was con-
siderably colder.
The Rising Pipe. — (Part L, par. 21.)
10. The thermometer indicating the discharge temperature often gave
readings more or less above 212° Fahr.
To provide against any fall in temperature at the thermometer bulb,
which might occur by reason of the formation of bubbles of steam in the
water, it was found desirable to keep some pressure on the water at that
part of its course.
Accordingly, instead of discharging the water directly from the con-
denser into the tank, it was conducted up a vertical pipe, which was open
at the top through a T to the atmosphere. The water then drained down
another pipe provided with a nozzle at its lower end, opening into the
two-way switch, to be described later. By this means a head of ITS feet
of water was maintained at the thermometer bulb, and at a temperature
of 220° Fahr. I had not much trouble with bubbles of vapour.
The Two-way Tipping Switch.— (Part L, par. 16.)
11. This was constructed to provide a means of rapidly diverting the
water at will, either to waste or into the tank. It consisted, as shown in the
sketch, of two curved copper pipes of rectangular section, meeting at
their upper ends at an angle of about 30°. Their common side was pro-
duced for about £ inch, and formed "into a knife-edge, separating the two
orifices.
These pipes were rigidly connected to a wooden link which worked about
a horizontal axis, distant 25 inches below the knife-edge. Wooden stops
were provided to limit the swing of the switch to rather less than 2 inches.
One arm of the switch worked in a funnel forming the top of a pipe leading
to waste, while the other worked through a hole in the cover of the tank.
The whole arrangement was fixed so that when in the central position the
66]
ON THE MECHANICAL EQUIVALENT OF HEAT.
667
knife-edge was J inch vertically below the nnxzle at the end of the dis-
charge pipe.
from Brake
Fig. 5. Tipping Switch.
This switch worked exceedingly well, diverting the stream of water
almost instantaneously, without making any perceptible splash.
In the later trials this switch was connected by a chain of links with the
revolution counter, so that when the latter was pushed into gear with the
engine shaft the switch simultaneously directed the water into the tank, and
vice verm.
Weighing Machine and Tank. — (Part I., par. 13.)
12. To facilitate the weighing, the stream of water was led during each
experiment into ;i galvanized iron tank which stood on the platform of a
writhing machine. The tank was 4 feet long by 2 feet 9 inches deep, by
2 feet t) inches wide. During the experiments it was kept covered by a lid
of thin boards, steeped in paraffin wax. These boards were always weighed
with the tank, so that any water they might absorb was accounted for.
A 2^ inch valve in the tank bottom was used for discharging the water after
weighing.
668 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
The weighing machine was graduated up to 2200 Ibs., and was supplied
with three rider weights.
No. 1, the largest, was provided with a knife-edge which fitted into
grooves cut in the lever of the machine, each division representing 100 Ibs.
No. 2 worked on another scale on the lever, each division representing
1 lb., and graduated up to 100 Ibs.
No. 3 was made by Mr Foster, in the laboratory, and indicated O'Ol lb.
per division of the second scale. The lever was 32^ inches long, and readings
were taken only when the middle of the swing of a pointer fixed to the end
of the lever coincided with a line marked on a brass plate alongside it.
It was quite easy in each individual weighing to set the machine to
O'Ol lb., but owing, no doubt, to shifting of the platform, levers, &c., I do not
think the readings taken were reliable beyond the -J^th of a lb.
This machine was not at first quite as sensitive as was necessary to attain
the high degree of accuracy required for the purposes of the research. On
examination this was found to be due to the slightly imperfect adjustment of
the knife-edges attached to the graduated lever. The fault was rectified by
Mr Foster, and since then the performance of the machine has been highly
satisfactory.
The Rubber Pipe Connections to the Brake.
13. On account of the very considerable pressure to which all the fittings
of the brake were subjected, it was found necessary to bind with tape the
rubber pipes supplying the water to ensure them against bursting.
The extra stiffness thus given to these pipes did not much affect the free
working of the brake, since none of them had a leverage of more than
4 inches from the centre of the shaft.
The case was, however, different with the bent rubber connection between
the brake and the discharge pipe, since in this case the leverage is about
1 foot 6 inches. This pipe was eventually inserted in a cage consisting of a
spiral of copper wire, 1^ inches in diameter, through the coils of which were
threaded two longitudinal wires to prevent elongation of the cage and rubber
tube. By this arrangement the flexibility of the rubber tube was almost
unimpaired.
66]
ON THE MECHANICAL EQUIVALENT OF HEAT.
669
The Device for Catching the Leakage at the Bottom Regulating Cock.—
(Part I., par. 36.)
14. It was found impossible to prevent leakage taking place, generally
to a small extent, from the automatic cock controlling the amount of water
in the brake. It was, therefore, necessary to provide some means of catching
this water, and it was very important that no impediment should be placed
in the way of the free working of the cock spindle.
Fromfbrake
To discharge pipe
Fig. 6.
A tight joint was made between the valve seating, B, and the bracket, C,
which carried the overhanging end of the valve, A. All the leakage, there-
fore, occurred along the valve spindle at cc. The method adopted to catch it
was to solder a brass ring on to the bracket at D, and fit a ring of cork of the
same diameter tightly on to the spindle at E. A piece of thin rubber tubing,
F, was bound tightly to the ring, D, and the cork, E.
This tube caught all the leakage, which then drained down the smaller
tube (shown in the sketch) into a bottle standing on the floor.
To prevent evaporation, the end of this small tube contained a short
length of glass tube, the capillarity of which always kept the end closed by a
bead of water.
070
ON THE MECHANICAL EQUIVALENT OF HEAT.
[66
General Arrangement of the Final Apparatus.
15. The general arrangement of the apparatus, as finally set up, is shown
in the drawings at the end of the paper and in the annexed diagram. The
course of the water was as follows : —
Fig. 7. Final apparatus.
It was drawn from the mains by the circulating pump, A, and forced
through the ice cooler, B, to the main regulating valve, C. Between the ice
cooler and this valve there was a Bourdon pressure gauge and a branch-pipe,
D, supplying water to the bearings of the brake. Entering the vertical stand-
pipe, E, the water flowed round the bulb of the initial temperature thermo-
meter, a small stream being diverted to waste through the jacket. The
straight flexible rubber pipe, F, then led the stream to the brake, 0, from
which the water flowed through the automatic valve, H, and the bent rubber
pipe, K, to the vertical stand-pipe, L, carrying the thermometer for measuring
the temperature of discharge. Then passing through the condenser, M, and
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 671
the rising pipe, N, the two-way switch, P, directed the water either to waste
or into the tank, R, standing on the platform of the weighing machine, S. At
T is shown the tin vessel forming the artificial atmosphere. A small Bourdon
gauge was fitted on to the top of the brake because the mercury gauge, indi-
cating the pressure in the air-vessel, was not visible to the observer when
taking readings of the thermometers, and it was important that this pressure
should be kept constant.
The Hand Brake and Speed Indicator. — (Part I., par. 30.)
16. In addition to the separate parts of the apparatus already mentioned
there was a hand brake by which a moment of about 50 ft.-lbs. could be
gradually applied to the engine shaft, and by this means a delicate adjust-
ment of the speed of revolution was obtained.
To make this speed evident a small speed gauge was driven by a gut
band from the engine shaft. It consisted of a paddle rotating about a vertical
spindle in a cylindrical case. The case contained coloured water, and the
pressure generated forced a column of the water up a glass tube, to a height
which varied with the speed of revolution.
In Part I. Professor Reynolds has referred in one or two instances to the
excellent manner in which various parts of the apparatus were constructed by
Mr Foster, to whom my thanks are also due for the valuable assistance he
often rendered at critical moments in the research, and further for the advice
and help he was always willing to give in the construction of apparatus for
which I was mainly responsible.
The Method of conducting the Experiments finally adopted — using the
Completed Apparatus.
17. During the progress of the experiments I had at my disposal the
services of two men and a boy. Of the men, the first, Mr J. Hall, was fully
engaged in attending generally to the needs of the engine and boiler, and had
besides to maintain the boiler pressure at a point which ensured the steady
running of the engine. I am bound to state that very much of the success
met with must, bo attributed to the very admirable manner in which Mr Hall's
part of the work was performed.
The duties of the second assistant Mr J. W. Matthews consisted in
regulating the engine speed by means of the hand brake, more particularly
at the commencement and ond of each trial, and also in keeping a constant
pressure of 9 inches of mercury in the artificial atmosphere.
672 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
The boy's time was occupied in breaking up the ice and feeding it as
required into the ice cooler.
In the last series of experiments three similar trials of 62 minutes
duration each were made per day, and the engine having been once started
was not stopped till the three trials were completed. Consequently what I
say below as to the starting of the engine does not refer to every trial, for
after emptying the tank at the close of any one all the necessary adjustments
were ready made for the next.
I. The pump and engine were started simultaneously, the brake being
therefore supplied with a stream of cold water through the ice cooler. The
brake then automatically adjusted the weight of contained water till the load
floated clear of the engine floor. The speed was then adjusted till the speed
indicator gave the required reading, viz., in all recorded trials 300 revolutions
per minute.
II. Since all the work done was expended on the stream of water passing
through the brake, its final temperature rose more or less quickly, and by
adjusting the regulating valve on the supply pipe the temperature of dis-
charge finally remained steady at 212°Fahr. nearly. In the meantime the
supply of water to the stuffing-box was regulated till the temperature of
the cover was at the required level.
These adjustments took from a quarter to half an hour, and when made,
the engine was allowed to run for some half-hour longer to ensure a steady
condition being attained.
The water supply to the condenser had also been regulated till the stream
of water issuing from the rising pipe and flowing to waste had the requisite
temperature.
III. Readings were then taken of —
(a) The revolution counter.
(6) The weight of the empty tank and its cover.
IV. When a steady condition was reached, the revolution counter at a
given signal was pushed into gear with the small spindle previously mentioned,
making connection through the cap with the engine shaft, and simultane-
ously the two-way tipping switch, which had hitherto been directing all the
water to waste, was pulled over and diverted the whole stream into the tank.
In the later trials all leakage that did sometimes take place from the stuffing-
box, and a slight leakage that always occurred at the automatic cock below
the brake, were collected in two bottles kept for that purpose. These were
6(j] ON THE MECHANICAL EQUIVALENT OF HEAT. 673
put under the drain pipes in each case as soon as possible after the
signal.
The speed of the engine as indicated by the gauge was read when the
signal was given, and as soon as possible afterwards a reading was taken of
the temperature in the discharge pipe.
V. At intervals of two minutes thirty observations were then taken of
the temperatures of supply and discharge of the water to and from the brake,
and also at each of these intervals a note was made of the reading of the
speed gauge.
At intervals of four minutes fifteen observations were made of a thermo-
meter registering the temperature of the room. Also at intervals of eight
minutes readings were taken of the two thermometers in the stuffing-box
and on the main bearing.
VI. When sixty-two minutes had elapsed the counter was freed from the
shaft, at the same time the water being again diverted to waste.
The drain pipes from the stuffing-box and cock were removed from their
respective bottles.
Readings were taken of the speed indicator and of the temperature of
discharge.
VII. Fresh observations were made of —
(a) The reading of the revolution counter.
(b) The weight of the tank and water received during the trial, to
which had been added the water caught from the regulating
cock.
A record was also made of—
(c) The weight of water which had been caught from the stuffing-
box.
18. These observations were afterwards reduced as follows : —
Let Tl = mean temperature of water supplied to the brake.
T., = „ „ discharged by the brake.
Wi = weight of tank and contents before the trial.
Wt = „ „ after the trial.
w = weight of water caught from the stuffing-box.
t = rise of reading of the thermometer in the discharge pipe during
the trial,
o. R. ii. 43
674 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
Ts = mean temperature of the stuffing-box cover.
TB = „ „ lower brass of the main bearing.
TA= „ „ air.
N-i = reading of revolution counter before the trial.
JV2 = tt „ after the trial.
M = moment in ft.-lbs. carried by the brake.
Therefore we have for the total heat generated
The determination of the quantity X and of the constants C and R,
representing the losses by conduction and radiation, will be dealt with later
(pars. 30, 43 and 45).
Also the total work done
where m = error in balance of the brake. This error will be dealt with sub-
sequently (par. 29).
If the capitals H and U refer to trials with a large turning moment on
the brake, and the small letters h and u refer to trials with a small turning
moment, then for our value of the mean specific heat of water in mechanical
units we have
U-u
K =
H-h'
This quantity K is not strictly the same as the mechanical equivalent
of heat, of which other determinations have been made, since we are here
dealing with the mean specific heat of water between freezing and boiling-
points.
For this reason it has been decided not to use the usual symbol J, at any
rate at this stage of the research.
19. As an illustration of the method of tabulating and reducing the
observations, I append all that were taken in trials 69 and 72 made on the
7th and 8th July, 1896, respectively.
It will be seen that all the observations of temperature, together with
the readings of the speed indicator, which were made during the actual
progress of each trial, are given on pages 679 and 681 respectively.
With the exception of the two readings of the speed indicator taken at
the moments of starting and finishing each trial, and shown in brackets at
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 675
the top and bottom of column No. 8, I was personally responsible for all
observations recorded. These two observations were made by the assistant
in charge of the hand brake and artificial atmosphere.
In the tables of temperature and speed observations
Col. 1 gives the times at which observations became due, the whole
period of 62 minutes being divided into 31 two-minute
intervals.
Col. 2 gives the temperatures of supply of the water to the brake.
Col. 3 „ „ discharge of the water from the brake.
Col. 4 „ „ the air in the engine room.
Col. .") „ „ the stuffing-box cover.
Col. G „ „ the lower brass of the main bearing.
Col. 7 „ fall of temperature between the stuffing-box and
bearing, being the difference of Cols. 5 and 6.
Col. 8 gives the readings of the speed indicator.
Observations of the revolution counter and of the weight of the tank
before and after each trial, are given on pages 678 and 680 respectively.
As I had to take all the observations myself, it was, of course, impossible
to make them simultaneously at the times indicated in Col. 1. " They were,
however, always taken in the same order, as follows.
When the time for the next ensuing series of observations had arrived
as given by a watch lying on the table at my side, I immediately read the
temperatures of supply and discharge and the speed gauge in the order
named, and after reading the three I entered them in the note-book. This
generally took about a quarter of a minute. If then a reading of the
atmospheric temperature was due, it was next taken and entered. After
that the temperatures of the stuffing-box cap and of the bearing were
noted in their turn, the whole series of observations being made in 1 or
l£ minutes.
The interval which then elapsed before the next series of observations
became due was often fully occupied in making adjustments of the regulat-
ing valve controlling the main water supply to the brake ; of the cock
regulating the supply to the stuffing-box ; and of the speed of the turbine
driving the pump, small alterations at all these points being frequently
necessary.
At the head and foot of Cols. 3 and 8 will be seen observations in
brackets. These observations were taken at the moments of starting and
43- 2
676 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
ending the trials, and were required in the calculation of a terminal correction
to be referred to later.
At the close of each trial a mean of the observations occurring in Cols.
2, 3, 4, 5 and 7 was made, the two observations in brackets in Col. 3 being
omitted in calculating these means.
On pages 678 and 680 additive corrections to the weights and to the
mean temperatures of supply and discharge are given. These will be referred
to later.
It will be noticed that in neither of the trials chosen was there any leakage
of water from the stuffing-box.
The observations are given again in the partially reduced form which has
been adopted for the final tabulation of the results on p. 682.
Cols. 1 to 8 should be self-explanatory.
Col. 9 gives the first approximation to the heat generated, obtained
by multiplying the weight of water by its mean rise in tem-
perature.
Col. 11 gives the difference of the temperature of the stuffing-box
(supposed to be a measure of that of the water leaking from
it), and the temperature of supply.
Col. 12 gives the loss of heat due to this leakage, and represents the
product of Cols. 10 and 11.
Col. 13 gives the rise of temperature of the brake during the trial,
and is assumed to be equal to the difference of the two
temperatures given in brackets in the table of temperature
observations (Col. 3).
Col. 14 gives the terminal correction to the heat required on account
of the increase of heat in the brake itself during the trial.
Col. 15 gives the difference between the mean temperature of the
stuffing-box and of the shaft bearing. As already explained
the loss of heat by conduction has been assumed proportional
to this difference, and a determination of its amount will be
given later. At present it is sufficient to say that a loss of
12 thermal units occurred per trial per unit fall of tempera-
ture along the shaft.
Col. 10 gives, therefore, the product of this difference x 12, which
represents the total loss by conduction.
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 677
Col. 17. The difference of temperature between the brake and the
surrounding air was taken as being equal to the difference
of the mean discharge temperature of the water and that
of the air. The determination of the constant representing
the loss of heat per unit difference of temperature is given
later, and consequently,
Col. 18 gives the product of this constant x the difference of tem-
perature in Col. 17.
Col. 19 gives the sum of the heat in Col. 9 added to all the corrections
afterwards given.
A further Table (p. 682) gives the work done, and the corrected values
of the heat generated in these two trials, and the differences between them.
The value of K in the last column is then found by dividing the difference
of work in Col. 4 by the difference of heat in Col. 6.
A slight inaccuracy has been pointed out to me by Professor Reynolds
in the method of finding the mean temperatures of supply to and discharge
from the brake. It was originally intended that the trials should be of
exactly one hour's duration, and that the first series of readings should be
taken one minute after the start. It was found impossible to do this, on
account of the number of points requiring attention in the first few minutes,
and consequently I made all trials 62 minutes long, and took the first
reading two minutes after starting. The mean used has not therefore been
obtained strictly in accordance with the middle breadth rule. Any error
introduced would be of the occasional type, and should be eliminated in the
mean of a number of trials.
678 ON THE MECHANICAL EQUIVALENT OF HEAT. [60
July 7, 1896.
Trial No. 69 (A).
Moment on the brake ....... 600 ft.-lbs.
Trial began at 11.17 A.M., and ended at 12.19 P.M.
Reading of revolution counter after trial .... 92,948
„ „ „ before trial .... 75,400
Number of revolutions during trial ..... 17,548
Weight of tank and water after trial .... 81T94— '5 Ib.
„ „ „ before trial . . . 342*16 + '4 „
Weight of water discharged by brake during trial,
including leakage from bottom cock . . . 468'88 Ibs.
Mean temperature of water in the discharge pipe . 212'007° F. + '04
„ „ „ supply pipe . 33'595° —'52
Mean rise of temperature of the water . . . 178'972° F.
Weight of water caught from stuffing-box . . . . = 0 Ib.
Temperature of water entering the tank = 100° F.
06]
ON THE MECHANICAL EQUIVALENT OF HEAT.
679
1
2
3
4
5
6
7
8
Timee
Temperatures
Fall of
tempera-
ture
between
stuffing-
box and
bearing
Readings
of speed-
gauge
(revolu-
tions per
minute)
Water-
supplied
to
brake
Water
discharged
from
brake
Air
Stuffing-
box
cover
Lower
brass of
bearing
Began 11.17
19
0
33-57
(212)
211-9
74-4
0
0
0
(302)
300
21
33-5
212-0
...
...
302
23
33-r> 7
212-3
75-7
107
107
302
25
38-68
211-3
...
303
27
33-58
211-5
76-0
...
...
...
302
29
33-58
212-2
• • *
304
31
33-57
211-1
76-4
109
110
-1
302
33
33-6
211-0
• . •
299
35
33-6
211-0
76-5
299
37
33-6
214-9
...
• • •
303
39
33-6
213-7
77-5
109
Ill
-2
301
41
33-62
213-3
301
43
33-6
213-2
76-8
299
45
33-59
212-2
301
47
33-64
211-5
77-0
lib
111
-i
301
49
33-62
211-8
303
51
33-64
212-0
78-1
« »•
»••
304
63
33-59
212-3
...
• .*
299
55
33-59
212-1
76-5
lib
111
-i
301
57
33-58
212-2
...
...
301
59
33-6
211-8
77-8
• ••
• • •
301
12.01
33-62
211-9
...
. ..
302
3
33-61
212-0
78-3
115
113
2
301
5
33-62
211-5
. ••
300
7
33-6
212-0
79-0
. ..
. . .
300
9
33 -:.7
211-6
• • •
• . .
300
11
33-59
211-6
76-8
112
113
-1
297
13
33-57
211-5
...
...
...
300
15
33-6
211-3
77-1
...
...
301
17
33-66
211-5
...
...
301
Ended 19
...
(212)
...
...
...
(302)
Means
33-595
212-007
76-9
110-3
...
-•57
...
680 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
July 8, 1896.
Trial No. 72 (A).
Moment on the brake 1200 ft.-lbs.
Trial began 11.11 A.M., and ended 12.13 P.M.
Reading of revolution counter after trial .... 146,311
before trial .... 129,000
Number of revolutions during trial 17,311
Weight of tank and water after trial . . . 1283'50 - 1'31 Ibs.
„ „ „ before trial. . . 347'21 + '41b.
Weight of water discharged by brake during trial,
including leakage from bottom cock . . . 934'58 Ibs.
Mean temperature of water in the discharge pipe . 212'46° F. + '04
„ „ supply pipe . 34706° - '55
Mean rise of temperature of the water . . . 178'344°F.
Weight of water caught from stuffing-box . . . = 0 Ib.
Temperature of water entering tank . . . . = 101° F.
06]
ON THE MECHANICAL EQUIVALENT OF HEAT.
681
1
2
3
4
5
6
7
8
Times
Temperatures
Fall of
tempera-
ture
between
stuffing-
box and
bearing
Readings
of speed-
gauge
(revolu-
tions per
minute)
Water
supplied
to
brake
Water
discharged
from
brake
Air
Stuffing-
box
cover
Lower
brass of
bearing
Began 11.11
13
34-74
(212-4)
212-3
72-0
0
"
0
(300)
302
15
34-8
211-5
300
17
34-71
212-8
73-7
97
99
-2
304
19
34-7
212-9
• . •
303
21
34-69
211-7
74-0
• • •
...
299
23
34-72
212-0
• ••
302
25
34-7
212-6
73-3
161
101
303
27
34-77
212-8
...
307
29
34-78
213-5
74-4
...
302
31
34-77
214-0
300
33
34-69
213-2
747
ibi
103
-i
301
35
35-0
213-2
...
299
:57
34-6
214-0
75-6
...
303
39
34-7
214-4
...
307
41
34-76
214-0
74-7
104
103
i
302
43
34-79
212-8
...
304
45
34-66
213-0
74-8
...
301
47
34-7.-)
212-3
...
300
49
34-66
211-6
75-7
105
104
i
297
51
34 -(58
211-2
302
53
34-68
212-0
75-4
• • •
302
55
34-66
211-6
• * •
299
57
34-66
211-0
75-3
104
105
-i
297
59
34-58
211-3
• • .
• ••
302
12.01
34-6
212-3
76-0
305
3
34-59
212-9
...
...
299
5
34-67
211-8
76-0
107
106
i
301
7
34-7
211-4
• ••
302
9
34-69
211-9
75-8
...
. ..
...
301
11
34-68
211-8
...
302
Ended 13
...
(211-6)
...
...
...
(300)
Means
34-706
212-46
74-8
102-7
...
-0-14
...
a
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a^Baq jo ajn^aadiua^ jo asiy;
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a8T!5{B9j X!q jijaq jo ssoq
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i— I
('^o) ^^q
aqj ui aan'jBjadraa; jo asig
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(•sqi) xoq-8utgnijs
raoaj ^qSn^o aa^A\ jo ^q8i9^i
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(•n'x'a)
•a?j 'uoi^'Bip'BJ o; snp
S9SSOJ ss^i 'pg^BjgugS ?'89H
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130,522,170
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Determina-
tion No.
()()] ON THE MECHANICAL EQUIVALENT OF HEAT. 683
The Barometer, — (Part 1., par. 47.)
20. Before dealing with the thermometers and their corrections, it
becomes necessary to describe a combined barometer and manometer which
was constructed to measure the pressures of steam employed in the de-
termination of the boiling-points on the thermometer used to measure the
discharge temperature.
The structural details of this instrument are given in Professor Reynolds'
paper. At present it is sufficient to say that it consisted of a cast-iron,
bottle-shaped reservoir, through the neck of which the glass tube holding
the mercury column was carried in a stuffing-box, which made a perfectly
air-tight joint between the glass and the reservoir. The pressure to be
measured was introduced through a small iron pipe, which penetrated
horizontally the cast-iron wall of the reservoir, and then turned vertically
upwards till its open rnouth stood above the level of the mercury inside.
Two circular plate-glass windows in the reservoir walls provided a means
of ascertaining the level of the mercury surface. In order to measure
the height of the mercury column supported by any external pressure,
a brass sleeve was made, which fitted outside the glass tube and the upper
p.nt, of the reservoir. This sleeve consisted of a piece of |-inch diameter
brass pipe fixed into a conical brass casting, which carried a truly-turned
bevelled edge at its lower extremity. This conical casting engaged by an
internal screw of twenty threads to 1 inch with the neck of the cast-iron
reservoir. The upper part of the sleeve carried an internal thread of the
same pitch, and into this was screwed a second piece of pipe through which
two long narrow slits were cut at opposite extremities of a diameter. A third
piece of brass pipe engaged with the upper end of the piece just mentioned,
and was provided at its lower end with a truly-turned bevelled edge.
In use the bevelled edge on the conical brass casting was first adjusted
to the surface of the mercury in the reservoir, and then the upper bevelled
edge was adjusted to the surface at the top of the mercury column.
Suitable horizontal and vertical scales were provided to enable me to
measure the vertical distance between these two bevelled edges to f^^ of
an inch.
It was necessary to standardise this scale (Part I., par. 44). There is
;i Whitworth measuring machine in the laboratory, which is provided
amongst others with standard end gauges of 9 inches and 3 inches long
respectively.
Two new steel standards were made by Mr Foster as nearly as possible
of the same length as the 9-inch Whitworth, and by means of the measuring
684
ON THE MECHANICAL EQUIVALENT OF HEAT.
[66
machine I determined their exact lengths as follows, three comparisons being
made of the two new gauges with the standard. The table shows the
readings obtained.
Whitworth standard
9-inch gauge
Laboratory standard
gauge, No. 1
Laboratory standard
gauge, No. 2
Readings on di-j
vided wheel of >
machine )
O'OOll
0-00112
0-00114
0-00105
o-ooio
0-00097
0-00095
0-0009
0-00098
Mean readings ...
0-00112
0-001007
0-000943
True lengths
9 inches
9 inches -0-000113
9 inches -0*000177
These three 9-inch standards, together with the 3-inch Whitworth,
therefore gave a length when placed end to end of
30 inches - 0'00029 inch.
The next operation was to construct a single steel standard with a length
of approximately 30 inches. This bar being made, and the measuring
machine not being long enough to accommodate 30 inches, the measurements
were made between the centres of a large lathe in the laboratory. Two
centres were made with polished flat ends. The one was put in the fixed
headstock, while the second was carried by the movable sleeve of the loose
headstock which had previously been securely bolted to the lathe bed in
a convenient position. A temporary wooden trough was made to carry
our four short standards, and correctly line them between the two centres.
The reciprocating centre in the loose headstock was then gradually screwed
up till the gravity piece of the measuring machine just floated between
the end of the adjacent standard and the centre. A mark on the hand-
wheel actuating the centre was next fixed by means of a pointer. The
four standards were then removed, and the 30-inch bar substituted for
them, and the operation of bringing up the centre repeated. The circum-
ferential distance separating the pointer from the mark on the hand-wheel
was then carefully measured.
A series of five of these observations were made, and the following
readings taken, viz. : —
(1) -0-1 inch, (3) +0-09 inch, (5) + 0'03inch.
(2) -0-05 inch, (4) + 0'02 inch,
Mean = - 01)02 inch.
60]
ON THE MECHANICAL EQUIVALENT OF HEAT.
685
The hand-wheel had a diameter of 9£ inches, and was fixed to a screw
of i-inch pitch.
The 30-inch bar was therefore short of the length of the four steel
standards by 0-0000138 inch.
Its correct length was, therefore,
30 inches - 0*0003 inch.
As the barometer was only graduated to O'OOl inch, no error was intro-
duced in assuming the bar to be exactly 30 inches long.
(Part I, par. 48.) — For the purpose of transferring this standard 30 inches
to the brass sleeve forming the scale of the
barometer, a circular cast-iron surface plate
was made. This plate had two pieces cut out
of it, as shown in the sketch. The plate was
fixed with its surface level, and then the brass
sleeve was placed centrally upon it, standing
upright on its lower bevelled edge. In this
position the portion of the surface between
the two grooves cut in the plate corresponded
exactly to the surface of the mercury in the
barometer between the two windows pre-
viously mentioned. As it was probable that
in actual use the lower bevelled edge would
be slightly above the mercury surface, the
sleeve was packed up by means of some very
fine sheets of tissue paper till a line of light
could be seen under it. Four sheets were necessary to effect this; one
of these was removed, and then the standard 30-inch bar was placed inside
the brass tube, standing with one end on the surface plate. The upper
bevelled edge was then adjusted till the line of light between it and the
top of the steel standard was obscured, and the scale was made to read
30 inches in that position.
Together with Mr Foster I made this adjustment a number of times,
but after once fixing the 30-inch mark, the reading of the length of the
steel standard never varied by as much as 0'0003 inch from 30 inches.
Unfortunately, the comparison was made at a temperature of 67° Fahr.,
while the standard temperature of the Whitworth gauges was 60' Fahr.
A formula of reduction of the readings of the barometer therefore became
necessary at all temperatures.
Section A B
086 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
Taking for the coefficient of linear expansion of brass per ° Fahr. 0-000012
steel „ 0-0000066
„ „ „ „ the mercury
column of the barometer .......................................... O'OOOl.
Then at 67° Fahr. the true length of the brass barometer scale
_ 1 + 35 x 0-0000066
1 + 28 x 0-0000066
= 30-000138 inches.
To find T, the temperature at which the scale gives correct readings,
we have, if T = t + 32°,
l+<x 0-00001 2 30
1 + 35 x 0-000~01~2 ~ 30-000138 '
which gives £ = 31° and T= 63° Fahr.
The coefficient of expansion of the mercury column relative to the brass
scale is 0*000088.
Now if H = readin of barometer in inches at T° Fahr., and as before
then the corresponding corrected height of the column at a temperature of
63° Fahr.
1 + 31 x 0-000088 T]
__.
— Jl as =:
1 + t x 0-000088
1-002728
„.
T>
1 + 1 x 0-000088 T>
and if //0 = the corresponding pressure reduced to inches at the freezing-
point, then
#63 = #0 (1+0-0031).
Therefore for any required pressure H0 inches at a temperature of
32° Fahr., the corresponding reading at T° Fahr. is
_1 + 0-000088*
T002W
or, allowing for the capillarity depression in a half-inch tube, this becomes
HT = (1-00037 + 0-0000880 H0 - 0'009.
This formula has been used throughout to determine the steam pressures
required for the verification of boiling-points to be discussed later (pars. 23
and 24).
66]
ON THE MECHANICAL EQUIVALENT OF HEAT.
687
The Thermometers.
21. The thermometers used for the measurement of the temperatures
of supply and discharge of the stream of water passing through the brake
were supplied by Mr J. Casartelli of Manchester.
Their indications were read through the glass walls of their respective
chambers by eye simply, parallax being avoided by the use of a small mirror
placed behind the thermometer in each case.
Freezing-point Thermometers.
22. Two similar thermometers were obtained, one only of which was
ever used during the experiments. This was a chemical thermometer,
bearing the laboratory mark 2Q, with a ^-inch diameter stem having its
scale very plainly etched in black lines on the glass. The length was
11 \ inches over all, the bulb being l£ inches long, and then at a distance
of 2£ inches from the top of the bulb the graduations began. The scale
extended from 30° to 45° Fahr., 6| inches of the stem being occupied by
the 15° mentioned. Each degree was divided into tenths, and it was easy
to estimate to the hundredth of a degree.
The index error of this thermometer was repeatedly checked during the
whole period occupied by the research by being immersed in a mixture of
pounded ice and water.
The table appended gives the corrections and the dates on which tests
were made : —
Date
Beading
Correction
5th December, 1895
20th December, 1895
9th January, 1896
17th January, 1896
31st January, 1896
5th February, 1896
20th February, 1896
16th March, 1896 .
31-7
31-71
31-67
31-67
31-57
32-48
32-46
32-46
+0-3
+0-29
+ 0-33
+ 0-33
+ 0-43
-0-48
-0-46
-0-46
21st April, 1896
32-47
-0-47
25th June, 1896
32-47
-0-47
7th July, 1896
32-52
-0-52
Before making the test on January 31st the hot water from the brake
backed up round this thermometer, so that the sudden alteration in the
reading is accounted for to some extent.
688 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
Also up to this time part of the mercury had remained stuck in the
upper bulb, but Dr Harker, of the Physical Department, now succeeded in
bringing the separated mercury down into contact with the column below.
By permission of Dr Schuster the scale of this thermometer was com-
pared by Dr Harker on the 27th April, 1896, with a standardised thermometer
(Baudin, No. 12,771) in his possession between the points 32° and 35°Fahr.
This comparison showed that the correction of — 0 47 as obtained on
April 21st was correct between 33° and 34°, which was the part of the scale
used in most of the experiments up to that date.
At 35°, however, the correction increased to — 0'5, and consequently in
the later experiments, when the temperature of supply in the heavy trials
approached this point, a suitable correction was made to that already ob-
tained by immersion in the mixture of pounded ice and water.
Boiling-point Thermometers.
23. In the first instance two similar thermometers were made to order
to be ready for use in the discharge tube, but on one of these being broken,
two additional ones were obtained. Only one of the four was, however,
used in the research, viz., PI.
This was a chemical thermometer with a ^-inch stem, having the scale
engraved as already described. The length was 16| inches over all, the
bulb being 1^ inches long, and a blank space of 5^ inches separating the
top of the bulb from the first graduation. The scale extended from 200°
to 220° Fahr., the 20° occupying 8| inches of the stem.
During the course of an experiment the reading of this thermometer was
continually altering slightly. This fluctuation made it almost impossible to
read the temperatures to y^jth of a degree. So that only the nearest ^th
of a degree has been recorded throughout.
The English standard boiling point, viz., 212° Fahr., is defined to be the
temperature of saturated steam under a pressure which would sustain a
column of mercury 29*905 inches long at the temperature of melting ice at
the sea level in the latitude of Greenwich.
This corresponds exactly, on being corrected for the variation in the
value of gravity, to the modern definition of the boiling point on the
Centigrade scale, the pressure in this case being equivalent to a column of
mercury 7600 millims. long in latitude 45°, the other conditions being as
before.
It was consequently possible to use Regnault's steam table in the
66]
ON THE MECHANICAL EQUIVALENT OF HEAT.
689
neighbourhood of the atmospheric boiling point as a standard of comparison
for the scale of this thermometer.
In order to conduct the comparison in Manchester, a knowledge of the
relative value of gravity was necessary.
This was deduced from a formula given in ' Memoires sur le Pendule '
(Societe Franchise de Physique), which is given below,
cos 20),
i/45
where — is the ratio of the value of gravity in latitude 0 to its value in
latitude 45°.
The latitude of Manchester being 53° 29', this gives
^ = 1-000756.
#45
The altitude of the Owens College, Manchester, has no appreciable effect
on the value given by the above formula.
I give below the table of steam pressures used in the calibration of the
scale of the thermometer PI.
Pressure of steam in
Pressure of steam in inches
Temperature on
Centigrade scale
Temperature on
Fahrenheit scale
millims. of mercury
reduced to 0° C. and sea
of mercury reduced to 0° C.
and sea level in latitude of
level in lat. 45°
Manchester
99
210-2
733-305
28-849
100
212-0
760-000
29-899
101
213-8
787-590
30-984
102
215-6
816-010
32-102
24. The general arrangement of the apparatus used to check the scale
of the thermometer PI will be gathered from the annexed sketch (Fig. 8).
A is an ordinary copper boiling-point apparatus, the steam from the
boiling water passing up an inner tube in which the thermometer
to be tested is hung, and then flowing down again so as to jacket
this tube, finally escaping into the atmosphere through the cock
shown. The top of the inner tube is closed by a cork having two
holes, in one of which is fitted a half-inch brass tube for connection
with the manometer, the other carrying the thermometer.
B is a glass flask containing an artificial atmosphere, of which the
pressure is under control.
44
O. R. II.
690
ON THE MECHANICAL EQUIVALENT OF HEAT.
[66
C is the combined barometer and manometer used to measure the
pressure in A and B.
D is the tin receiver previously described, the pressure in which is
kept at about 18 inches of mercury, as measured on a U-gauge.
This receiver is in free communication through a capillary glass
tube with the tube connecting the flask B and the manometer G.
The bore of the capillary tube just mentioned is just sufficient to admit
a very small stream of air from the receiver through the flask B, and so out
into the atmosphere by way of the cock on the boiler. The object of this
stream of air was to counteract the tendency of the steam in the boiler to
diffuse down the connecting rubber tube into the flask, where condensation
would occur, and possibly some water might get into the barometer, it
having been found quite impossible to keep a steady pressure in the
apparatus whenever the steam made its way as far as the glass flask, B.
A B
Fig. 8. Apparatus for checking boiling-points.
Fig. 9.
The boiler was well lagged and protected as far as practicable from
draughts. A thermometer was hung alongside the brass scale tube of the
barometer, and its reading was assumed to be the temperature of the
66]
ON THE MECHANICAL EQUIVALENT OF HEAT.
691
barometer. Allowance having been made for this temperature, the steam
escape cock was adjusted till the pressure inside the apparatus, as measured
in the barometer, was at the required level. A reading was then taken of
the thermometer under examination. The stem was pushed as far as possible
into the boiler, the reading standing about a quarter inch above the top of
the cork. Since there was always some escape of steam which blew up the
hole in which the thermometer was inserted, it was not thought necessary
to attempt to make any correction for the exposed part of the stem.
The annexed table gives the readings taken from this thermometer when
immersed in steam of various known temperatures and the dates on which
the tests were made : —
Readings obtained from thermometer PI when immersed
in steam at temperature Correction
Date
used m
experi-
212°
213°-8
215°-6
ments
28 Nov., 1895
211-43
213-26
215-01
+ 0-57
4 Dec., 1895
211-44
213-28
215-03
+ 0-56
5 Dec., ls!t:>
211-5
213-33
215-07
+0-5
(i Dec., 1895
211-51 (rising)
211-53(falling)
...
j +0-48
12 Dec., 1895
At tempera
ture 2 10° -46 reading
was 210°-05
9 Jan., 1896
213-38 (rising)
213-40(falling)
...
'- +0-44
17 Jan., 1896
...
213-49
...
+ 0-34
23 Jan., 1896
213-49
...
+ 0-34
31 Jan., 1896
...
213-49
...
+0-34
8 Feb., 1896
211-76
213-57
215-3
+ 0-24
20 Feb., 1896
211-78
213-6
215-34
+0-22
At
21T-34 reading was
211°-1
16 Mar., 1896
211-86
213-66
215-4
+ 0-14
At
211°-07 reading was
210°-87
18 April, 1896
» • •
213-7
215-45
+0-11
15 June, 1896
211-94
213-74
215-5
+0-06
6 July, 1896
211-96
213-75
215-52
+0-04
25. In the case of each of these thermometers, viz., Q2 and Pi, the
water surrounding them was under a very considerable pressure, and it was
therefore necessary to determine the effect of pressure on the reading given
by each.
A piece of strong glass tube, Fig. 9, about 1 foot in length and g inch
inside diameter, having one end fused up, was provided with a slightly
wider mouth, in which was inserted a small branch pipe, A. This bran, h
again split up into two anus, one of which, B, was connected through a rubber
tube with an air receiver in which the pressure was indicated by a U -gauge,
while the other, C, communicated directly with the atmosphere. Each of the
44—2
692 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
branches B and C could be closed at will by means of a screw clip on the
rubber tubing.
The pressure tube having been about half filled with water, the thermo-
meter under consideration was fixed inside it by means of a cork, D.
In the case of the freezing-point thermometer, Q2, the pressure tube was
then surrounded with pounded ice. After the contained water had cooled
sufficiently for the thermometer inside to remain steady, the communication
with the atmosphere was closed, and the full pressure of the air receiver put
on the thermometer bulb by opening the clip on the tube, B. The rise in
the reading due to the known rise of pressure was then noted. A number
of these observations were made, using different additional pressure in each
case. The result obtained was that for a rise in pressure on the bulb due to
1 inch of mercury, the rise in the reading was 0'0072°.
In the case of the boiling-point thermometer, PI, the pressure tube was
immersed in the steam generated in the copper boiler previously alluded to.
Similar procedure gave in this case a mean rise of 0'0066° per inch rise of
pressure.
After applying corrections (to be dealt with later — par. 62), rectifying the
thermometric indications on this account, I think that no error of greater
magnitude than O'()l° can have existed in the calculated mean rise of tem-
perature in any trial.
On 180° this gives accuracy of 1 part in 18,000.
26. In addition to the thermometers just dealt with, three others were
used, on the readings of which depended the additive corrections to the heat
already referred to. One of these indicated the atmospheric temperature,
while two others were placed one on the stuffing-box and the other on the
shaft bearing.
On the differences of heat which were used as the divisors in the deter-
mination of the equivalent from each pair of trials, these corrections all
became extremely small quantities, and therefore it was of no importance
that small errors should exist in these thermometers. Their scales were
therefore never calibrated. Still another thermometer was used to determine
the temperature of the stream of water entering the tank. As it was only
necessary to keep this temperature in each pair of trials at the same level,
errors in this thermometer were negligible.
66]
ON THE MECHANICAL EQUIVALENT OF HEAT.
693
Weighing Machine and 25-lb. Weights used on the Brake. — (Part I., par. 40.)
27. The absolute value of the unit used in the graduation of the lever
of the weighing machine was a matter of indifference, hut it was of vital
importance that the same unit should be used for the weighing machine and
for the 25-lb. weights used on the brake.
A set of iron weights were, however, sent down to the Manchester Town
Hall, and there compared with the Board of Trade standards.
The comparison of the 25-H>. weights with our standard 25 Ibs. was one
of the first things undertaken in the course of the investigation. This was
done by first balancing the standard placed on the platform of a small
weighing machine in the laboratory by adjustment of the rider weights on
the lever of the machine. The standard was then removed, and one of the
25-lb. weights substituted, a balance being made by adding to or drilling out
some of the lead inserted in the weight.
This adjustment was accepted as perfectly satisfactory till towards the
close of the experiments, when a small difference in the value of the equiva-
lent as derived from trials in which different numbers of the weights were
used, seemed to suggest an error in the weights themselves.
Accordingly, on the 9th June, 1806, I again compared the weights with
the standard on a temporary balance, consisting of a simple lever with three
knife-edges in a straight line, with the following result : —
Weight number
True weight
1
25-00
2
25-02
3
25-03
4
25-02
5
25-01
6
24-99
7
25-02
8
25-02
9
25-03
10
25OO
11
25-04
Hanger
24-99
And a lead balance weight to be referred
to later, which weighed 13 '98 Ibs. in-
stead of 13-97 Ibs. as assumed.
On the 17th of January, 1896, a set of four of these 25-lb. weights, at
694
ON THE MECHANICAL EQUIVALENT OF HEAT.
[66
that time all supposed accurate, were used as a standard 100 Ibs., by which a
series of corrections to the 100-lb. scale of the weighing machine were
obtained. These corrections have been used throughout the investigation,
and are given below : —
Reading
300
400
500
600
700
800
900
1000
1100
1200
1300
Correction . . .
0-4
-0-12
-0-42
-0-5
-0-65
-1-12
-1-22
-1-31
-178
Rider weights Numbers 2 and 3 were at the same time made correct on
their whole range.
In June another comparison was made, and the set of four weights,
Numbers 2, 8, 9, and 10 were found to give substantially the same list of
corrections as previously obtained.
The complete set of weights were then again weighed on the weighing
machine, using the list of corrections given, together with the true value of
the standard 100 Ibs. The result was a verification of the list of their values
already given.
The maximum error that might possibly be produced by using the weights
on the brake in specially arranged groups was found to be —
In a pair of trials carrying moments of 1200 and 600 ft.-lbs. respectively,
— 0'037 per cent, or -f 0'043 per cent., and in a pair of trials run with
moments of 1200 and 400 ft.-lbs. respectively, - 0'025 per cent, or + 0'03
per cent.
The value of the equivalent obtained from a set of six trials in which the
weights had been specially arranged to eliminate the above possible error
entirely, gave a result which did not differ at all from that previously obtained,
and it may therefore be safely assumed that in the first series of trials this
error did not occur to any sensible extent.
I think that, especially with the above result in view, the loading of the
brake may be taken as absolutely accurate.
As to the limit of accuracy of the weighings in the 600 ft.-lb. trials, the
weight of water dealt with was approximately 470 Ibs. On this quantity
the maximum probable error was 0'02 Ib. in any trial. This gives greater
accuracy than 1 part in 20,000.
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 695
The Adjustments of the Brake.
(1) Length of the Lever. — (Part I., par. 45.)
28. This length was required between the centre line of the engine
shaft traversing the brake and the V-groove carried by the lever.
It had been previously observed that both the shaft and the brake shifted
a little horizontally when the engine was started, from the positions occupied
with the engine stationary. It was therefore necessary to make the com-
parison between the length of the lever and our standard 4-feet with the
engine running. Also, since the length of the lever varied with the tem-
perature of the brake, this temperature was maintained, as in all the trials,
at 212°Fahr.
Between the brake and the adjacent bearing the shaft is 4 inches diameter
within of an inch.
At a distance of 3 feet 10 inches from one of its square ends a fine line
was scribed on a steel straight edge. This straight edge was then held with
the square end aforesaid butting against the shaft, the length being horizontal
and perpendicular to the line of shafting, and the distance between the
straight edge and the lever being 10 inches. At a distance of 11 feet from
the other side of the lever a theodolite was set up and adjusted so that the
vertical plane of collimation of the instrument was parallel with the shaft
and contained the line scribed on the face of the straight edge.
A steel scale, graduated to ^ of an inch, was fixed firmly on to the lever,
and a reading of this scale was taken through the telescope without altering
the adjustments mentioned. This reading, of course, referred to the point
on the scale just 4 feet distant from the centre line of the shaft. By a slight
rotation about the vertical axis the line of collimation was then made to cut
the centre line of the groove, and then a vertical rotation enabled a second
reading of the scale to be taken.
A number of these observations were made while the brake was subjected
to moments of 1200, 600, and 400 ft.-lbs., and they all indicated that the
length of the lever in the trials made was 4' + 0'02''.
A correction to the value of the equivalent derived directly from the
trials is therefore necessary on this account. It amounts to + 0'0417 percent.
With this correction added, 1 think that the length of the lever can be
assumed accurate to ^ inch, or 1 part in 10,000 nearly.
696 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
(2) The Balance of the Brake. — (Part I., par. 9.)
29. If a pair of trials are run, the one with a heavy indicated load, Mlt
and the other with a lighter one, M^ and if m be the moment carried by the
brake on account of its initial want of balance, then the works done in the
two trials are
U2 = 271-^ (M2 + m),
where N^ and Nt are the revolutions in the two cases.
The difference of the work done
= 27r [N.M, - N,M2 + m (N, - N,)}
and the relative error involved in writing for this
which has been done in these experiments, is
£jfr^|, very nearly.
This error is 0 when Nl = Nz.
The speed of the engine was therefore always regulated to the end that
the number of revolutions in each of a pair of trials which were afterwards to
be compared together should be approximately the same. As a general rule,
this object was very nearly attained.
The maximum value of Nt - Nz was about 300, the values of A\ and N2
being approximately 18,000.
Under these circumstances, in trials carrying loads of 1200 and 600 ft.-lbs.
respectively, the above error amounts to
300 1
18000 x 600 = 36000 < )03 per cent Per ft'"lb' of error
in the balance of the brake.
The method pursued to determine the want of balance was as follows :—
The lever was freed from all extraneous loads.
The brake arid its pipe connections were then all filled with water, so as
to be in the same condition as during the progress of a trial.
The lever was then lifted till its end was in its mean position opposite a
pointer at a fixed height from the ground. A load was then gradually added
to the front side of the brake till the friction of the bearings was overcome,
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 697
and the lever fell. An observation of the moment required to cause the
motion was then made. A series of twenty of these observations w^re made
for the front and then a second series of twenty for the back of the brake, in
which case the load on the back had to lift the lever from its mean position.
On taking the difference of the means of these two series of observations,
the friction is eliminated and the resulting moment represents the error of
balance of the brake.
Since in the course of a trial the lever oscillates a little from its mean
position, the brake will, when in motion, be working against the resistance
offered by the linkage connected with the regulating cock. When at rest,
however, this resistance will not affect the load at all. In view of this fact,
two determinations of the error in balance were made, the first with the
brake working free of the linkage, by allowing the small motion to take place
in the slack of the pin-joints, the second with the brake working against the
resistance of the regulating apparatus,,^ The results obtained were
In the first case, error in balance = 45'5 ft.-lbs.
In the second case, error in balance = 4173 „
A mean of these two quantities would probably be approximately correct
viz., 43-615 ft.-lbs.
The lead balance weight previously mentioned, and weighing 13 97 Ibs.
was substituted for one of the 25-lb. weights, on the removal from the lever
of the brake of a rider weight and a balance weight whose combined moment
(par. 40) was calculated at — 4412 ft.-lbs.
The actual uncompensated error in the balance appears therefore to be
practically ^ ft.-lb. This is so small, and the balancing of the brake such a
very difficult operation to perform with any approach to accuracy, that any
error there may be has been ignored, and the balance assumed perfect in all
the calculations.
The end of the lever has always been kept at the level of the pointer
indicated before, and by this means all error due to the varying horizontal
position of the centre of gravity of the brake has been avoided.
Terminal Corrections to the Apparent Heat Generated. — (Part I., par. 31.)
30. In order that the work done in any trial should be exactly equivalent
to the heat generated in the water used, it was necessary that the total heat
contained in the brake itself should be the same at the beginning and end of
the trial.
698
ON THE MECHANICAL EQUIVALENT OF HEAT.
[66
This condition was rarely fulfilled, since ifc required that the weight of
water in the brake, together with its temperature, should be unaltered at the
close of the trial.
A determination was made of the amount of water contained by the
brake at various speeds by suddenly stopping the engine when running at
any <jiven speed, simultaneously shutting off the water supply to the brake,
and afterwards draining off and weighing the water shut in.
The results are shown in the annexed curves.
170 180 190 200 210 220 230 240 250 260 270 280 290 300
Revolutions pep minute
Fig. 10. Curves showing water contained by and water equivalents of brake and contents at
varying speeds.
The weight of brass in the brake is 368 Ibs. Taking 0'094 for its specific
heat, the water equivalent is 34'6 Ibs.
To obtain a scale of weights representing the water equivalents of the
brake at different speeds, we have to add 34 6 to the weights of water
contained at the different speeds.
This scale is given at the right of the curves just alluded to (see above).
A correction to the heat obtained is now very easily deduced.
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 699
Let wl = water equivalent of brake at commencement of trial.
M>a= „ „ end „
ti = temperature of water in discharge pipe at commencement of trial.
^2 = » >j » » end ,,
Therefore, additional heat generated in the brake = w^t2-Witlt and this
quantity is added to the heat already calculated as generated in the water.
The speed indicator, which was used in the determination of the number
of revolutions per minute required as the ordinate in the curve of water
equivalents, was not reliable to one or two revolutions, and, therefore, unless
a large difference of speed was indicated between the commencement and
end of a trial, this difference was altogether ignored, and the rise in tem-
perature was multiplied by the constant corresponding to any particular load
at 300 revolutions to obtain the terminal correction.
The speed gauge required a negative correction of 11 at 300 revolutions,
and, consequently, the curves give 57'6 and 54'6 as the water equivalent of
the brake when loaded with 1200 and 600 ft.-lbs. respectively.
By interpolation from the above values 53'6 was obtained and used as the
water equivalent in trials carrying a moment of 400 ft.-lbs.
Loss of Water by Evaporation and Leakage from the Discharge Pipe
and Tank. — (Part I., par. 37.)
31. In order to test the general efficiency of the discharge pipe as a
conveyer of the water used, it was disconnected in June, 1896, from the
brake, and the circulating pump was arranged to pump the water out of the
tank and through the discharge pipe, which emptied itself again into the
tank by means of the tipping switch.
The stream of water was regulated so as to correspond exactly with the
quantities passed in trials carrying loads of 400, 600, and 1200 ft.-lbs. In a
period of 62 minutes it was found that in each of these cases the loss
approximated very closely to a quarter of a pound of water when its tem-
perature was between 90° and 100°. Since this loss was the same in all the
trials it has not been thought necessary to make a correction rectifying the
heats on this account, for it would be completely eliminated in the differences
of heat used in the calculation of the values of K given in the tables, if the
interval of temperature through which the water was raised in the brake was
the same in corresponding light and heavy trials.
When, however, I examined the results after the final reduction had been
700 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
made, I found that the mean temperature of supply in the light trials was
0'7° lower than that in the heavy trials.
Consequently the mean difference of heat would require a slight cor-
rection, which, however, is less than — 0'0000()2 relatively to the whole.
This, being quite outside our limits of accuracy, has been ignored.
The Main Experiments.
32. In December, 1895, the apparatus, though not yet quite complete,
was in a sufficiently advanced state to make it possible to commence the
main K experiments.
The observations were taken and reduced in every experiment in sub-
stantially the same manner that I have described (paras. 17, 18, and 19).
Some of the particulars mentioned were, however, omitted in the earlier
trials, and were only recorded subsequently after their importance had come
to be recognised.
In all, 80 trials were made on which any reliance has been placed, and
these will be dealt with in different series, between any consecutive two of
which some slight alteration had been made in the apparatus, the method of
taking the observations, or of reducing the same ; all these alterations
leading up to the finally adopted methods which have been described.
33. I must first mention two sets of trials which do not appear in the
tables. They were commenced in December, 1895, and were made mainly
with the object of gaining experience in the behaviour of the apparatus, and
of determining the most favourable conditions under which the experiments
could be conducted.
The moments carried by the heavy and light trials in each set were 1200
and 600 ft.-lbs. respectively.
The speed was in the first set 230 revolutions per minute, and in the
second set 180 revolutions per minute.
With the following exceptions the apparatus and methods were the same
as described.
I. Omissions and faults in apparatus.
(1) There were no thermometers on either the stuffing-box cover
or on the main bearing, and consequently no effectual attempt
could be made to keep these parts of the shaft at the same
temperature in a pair of trials.
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 701
(2) There was no means of catching the leakage from the stuffing-box,
or from the bottom regulating cock.
(3) The rising pipe at this time only maintained a head of about
5 feet of water over the thermometer in the discharge pipe.
(4) The hand brake had not been fitted to the shaft.
II. Omissions and faults in the methods employed.
(1) No corrections were added to the heat as given by the formula
(Wi-WJx^-Tj.
(2) The heavy trials were of only half-an-hour's duration, in order
that the second reading taken of the weight of the tank should
lie on the same part of the scale of the weighing machine,
which had not up to this time been corrected, in both heavy
and light trials.
The results obtained were not very consistent, but, perhaps largely on
that account, the trials admirably fulfilled the purpose for which they were
made.
The importance of the terminal corrections were clearly indicated when
the results were considered, and consequently means were at once taken
to apply these corrections to the preliminary reduction of all subsequent
trials. These included the provision of the hand brake, by means of which
the engine speed on starting and finishing the trials could be easily con-
trolled, and the observations of the speed of the engine and the tempera-
ture of the brake which were taken at the moments of starting and ending
the trials.
Again, the terminal corrections and other incidental errors had very
unequal weights when acting on the quantities obtained in the hour light
trials and in the half-hour heavy trials — which latter quantities required
doubling before the subtraction requisite to eliminate losses of heat could be
effected.
It was therefore decided that in future all trials should be of equal
duration (viz. 02 minutes), and this necessitated the immediate careful
checking of the scale of the weighing machine, which was thereupon pro-
ceeded with. Furthermore, it was probable that many of the discrepancies
which occurred were due to thi; small quantities of water it was possible to
deal with at the low speeds hitherto used, and to remedy this defect a larger
amount of work was done and heat generated by increasing the speed in all
the recorded trials to 300 revolutions per minute. Incidentally this increase
of speed was conducive to the steadier running of the engine.
702 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
I was much troubled with bubbles of steam in the discharge pipe, and to
prevent their formation the rising pipe was lengthened till it gave a head of
11 '3 feet over the thermometer bulb.
These trials also furnished information which led to the adoption of a
pressure of 9 inches of mercury in the artificial atmosphere. It was found
that with higher pressures than this the air by some means found its way
into the discharge pipe, even with the lengthened rising pipe in position.
During the first few trials the only regulation of the water supplied to
the bearings of the brake consisted of screw clips on the rubber pipes carry-
ing the water. These were found to be very inefficient, and two cocks were
substituted, each of which carried a scale which showed the amount to
which it was open at any time.
34. Before dealing with the tables showing the final reduction of the
experiments made, it is necessary to mention a preliminary reduction of trials
Nos. 1 to 42 shown in Table A (p. 722), from which the constants used in the
determination of the losses of heat by conduction along the shaft, and also by
radiation, were deduced.
In this table the actual observations are as far as possible omitted, since
they will appear later in the completely reduced tables.
It will be seen that the table consists of three similar parts, referring
respectively to the heavy trials, the light trials, and the differences.
In each part
Col. 1 gives the number of the trial.
Col. 2 gives the work done, calculated in the ordinary way.
Col. 3 gives the heat generated, as calculated from the formula
(W2 — W ]) (T2 — T^, all corrections being omitted.
Col. 4 gives the terminal corrections, for which, as I have said, the
necessary observations were always taken.
Cols. 5 and 6 give respectively the mean differences of temperature
observed between the stuffing-box and the top and bottom
brasses of the main shaft bearing.
The quantities in brackets are not actually observed differences, but were
deduced in the manner to be hereafter explained (par. 43).
These differences are + or — according as the stuffing-box was hotter or
colder than the adjacent bearing.
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 703
Col. 7 gives the mean difference of temperature observed between the
brake and the surrounding air. These differences are, of course,
all positive.
The quantities given in the part of the table headed "differences" are in
every case the remainders which are left on subtracting the corresponding
quantities under the heading "light trials" from those appertaining to the
" heavy trials."
In the last column are given the values of X, obtained by dividing the
work occurring under the heading differences, by the heat, to which has first
been added the terminal correction.
The conditions under which each series of trials given in Table A was run
are enumerated below.
In every case the engine speed was 300 revolutions per minute, as read
on the speed-gauge.
In all heavy trials the moment was 1200 ft.-lbs., with the exception
of Series IV., in which the moment was 1244*12 ft.-lbs.
In all the light trials the load was 600 ft.-lbs.
Series I.
35. This series contains trials Nos. 1 to 11, No. 5 being omitted on
account of an accident to the revolution counter.
In all these trials the outer brass skin of the brake was exposed directly
to the atmosphere, and consequently the loss of heat by radiation was very
large.
No attempt was made to catch the small quantities of leakage occurring
at the stuffing-box and the bottom regulating cock.
The water supply to the stuffing-box was only regulated to the end that
the bearing should not become unduly hot, and no record was kept of the
temperature gradient along the shaft till trial No. 10 was reached.
In order to avoid any bias which might be given to the experiments by
always combining a trial of one type with one of another type, trials of both
of which types were always made at the same relative part of any day,
the relative order of running was changed as indicated by the dates and
times given in Table B (Part I., par. 32). This method of combining the
trials was adopted because at this time it was not as a rule possible to make
more than two trials a day successfully, for breakdowns of a more or less
serious nature were of frequent occurrence.
704 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
Referring now to the preliminary reduction shown in Table A, Series I. :
The values of K, Nos. I., III., IV., and V. are seen to be in close
agreement, notwithstanding the comparatively rough method of reduction
used.
Determination No. II., however, stands out as very distinctly higher than
the others, and the cause of this was fortunately evident.
In order to prevent the attempted rotation of the small handle shown in
the illustrations at the end of the brake lever, one revolution of which
altered the load on the brake by 1 ft.-lb., one of my assistants had tied it to
the hanger carrying the load. The string making the connection was very
tight, and the load was pulled perceptibly out of the perpendicular plane
passing through the groove on the lever.
This fault was sufficient to condemn the two trials Nos. 3 and 4, and
they do not appear in the final table on that account.
A wooden clip was subsequently added to prevent the rotation of the
handle and its attached screw.
Lagging. — (Part I., par. 33.)
36. The results given by the four accepted determinations of Series I.
were so consistent that it was decided to proceed at once with the lagging of
the brake, which, up to the present time, had been deferred on account of
want of confidence in the apparatus generally.
The lagging consisted of a layer of about 1£ inches of loose cotton wadding
with which the whole of the exterior of the body of the brake was covered,
together with the discharge pipe between the brake and the thermometer
chamber. The cotton was all tied firmly in position, and the whole was
enclosed in a covering of thick flannel.
As will be seen later, this lagging reduced the radiation by nearly
75 per cent. Its weight, about 2 Ibs., was inappreciable, and, being evenly
distributed, could not affect the balancing of the brake to any extent which
it would be possible to detect.
The lagging was, I believe, of use, more especially in that it protected
the bare metal from the strong draughts which often occurred in the engine-
room. It required very careful attention, however, to protect it against
dampness, and on this account I am not certain that better results would
not have been obtained without it.
60] ON THE MECHANICAL EQUIVALENT OF HEAT. 705
Series II.
37. With the exception of the addition of the lagging, no alteration
was made in either apparatus or method between trials 11 and 12.
Sufficient experience and confidence in the apparatus had now been
gained to enable me to make three trials per day, as a rule two being
made in the morning and one in the afternoon, a stop of about one hour
being made after the second trial. The brake was not allowed to cool down
during this interval ; the hot water contained on finishing the morning's run
being shut in.
In Table A, the value 787'4 is given as the result of the combination of
trials 12 and 14. There was -evidently something amiss with this result, and
as the combination of trials Nos. 13 and 14 gave the result 779'4, which
agrees fairly closely with those given in Series I., the explanation which at
once suggested itself was that the new lagging was damp when the day's
running began and had dried before the commencement of trial 13. On this
account trial No. 12 has been expunged from the final Table B, and takes no
further part in the investigation.
Series III.
38. As it had by this time been found possible to run three satisfactory
trials per day, the most obvious way of combining them was to make three
trials, all carrying the same load, on the first day ; while the trials required
to complete the three determinations were run on the next convenient day.
This method was pursued during the whole of the subsequent course of
the investigation.
From this series onward 1 made an attempt to keep the temperature
gradient along the shaft, between the brake and the adjacent bearing, the
s;inie in each pair of trials. In trial No. 21 I took observations for the first
time of the temperature of the lower brass in the main bearing. In these
trials also the possible importance of the small leakage of water, occurring
along the spindle of the lower regulating cock, for the first time became
apparent. The weight of water actually leaking away had not, I think, any
appreciable effect, but owing to its high temperature it was nearly all
evaporated, and, consequently, may have had a sensible effect in the lowering
of the temperature of the water discharged from the brake. No successful
means were yet devised for catching this water. So, in this series, it still
remains as a possible source of error.
o. R. ii. 45
706 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
Series IV.
39. For use in the regular engine trials the brake is provided with a
rider weighing 48 Ibs., which can be traversed along a graduated scale on the
lever by means of a leading screw. In order to maintain the balance of the
brake it carries at the back a second fixed load of 74'6 Ibs.
These two large masses of iron had hitherto been left on the brake, but it
seemed probable that they would very much affect the flow of heat away from
it between any pair of consecutive trials (Part I., par. 38), for they continued
to rise in temperature during the whole of any day on which experiments
were made, and evidently they would absorb heat more rapidly when cold in
the early part of the day than when hot later. It was therefore decided to
remove them. Their combined moment about the engine shaft was
-44.-12ft.-lbs.
No allowance was made for this alteration in the loading of the brake,
and, consequently, the moment in these trials was 1244'12 ft.-lbs., this figure
having been used in the calculations given.
In order to bring the trials under some general denomination, this series
has not been further reduced, nor combined with a corresponding set of light
trials.
With the intention of stopping the leakage at the bottom cock, I had
had some more packing placed in the gland surrounding the cock spindle.
This did, to some extent, reduce the leakage, but it also had another effect
which will be referred to under Series V.
Series V.
40. For the purpose of keeping the loads on the brake at the values
carried by trials preceding the removal of the rider and balance weights, one
of the 25-lb. hanger weights was removed, and for it were substituted some
lead sheets weighing 13'97 Ibs.
This lead weight then corresponded with the initial want of balance to a
moment of 100 ft.-lbs., made up as follows :— -
Want of balance 44'12 ft.-lbs.
Moment of lead weight 55-88 „
100
After these trials had been made, I determined, with Professor Reynolds,
by means of a spring balance, the force necessary to move the bottom cock.
Gfi] ON THE MKCHAN1CAL EQUIVALENT OF HEAT. 707
This was found to amount to a moment of 30 ft.-lbs. on the brake, and on
this account this series of trials, though appearing in the final tables, have
not been allowed any weight in the calculation of the final mean value of K.
The preliminary reduction of Table A gave what were apparently very good
values of K, but this only shows the small effect on the mean moment pro-
duced by variations in the resistance offered to the brake's motion, and this
although its period of oscillation was very long.
Series VI.
41. These trials differ from those of Series V. only in the fact that the
extra packing had been removed from the gland on the cock spindle, while a
means of catching the whole of the leakage, and at the same time preventing
its evaporation, had been provided (par. 14). The whole of the leakage
was credited with the temperature of the water in the discharge pipe, and
was weighed with the main stream of water which had been caught in
the tank.
Series VII.
42. These trials were made under similar conditions to those in Series IV.
In the two last trials, however, viz., Nos. 39 and 42, some leakage was
observed and caught from the stuffing-box.
An approximate estimation of the loss of heat due to this leakage is given
in Table B, and has been included in the heats given in Table A.
Determination of the Loss of Heat by Conduction along the Shaft.
43. In the trials enumerated in Table A, the varying values of the
temperature gradient, existing in the shaft leaving the brake, might evidently
be a cause of comparatively large losses of heat which were not eliminated
in the differences of heat, so far assumed to be equal to the corresponding
differences of work.
It therefore became important to determine, at least approximately, what
was the loss of heat by conduction along the shaft in each trial.
I have already said that the temperature of the shaft in the main bearing
was assumed to be the same as that of the lower brass, while the tem-
perature on leaving the brake was similarly taken as that of the stuffing-
box cover.
Unfortunately, before trial No. 21, I had made no record of the tem-
perature of the lower brass.
45—2
708 ON THE MECHANICAL EQUIVALENT OF HEAT. [06
It was, however, found that in trials Nos. 21 to 41 the mean temperature
of the lower brass exceeded that of the upper brass by about 7° Fahr.
Consequently, in Column 6, in the parts of Table A, where no obser-
vations had been taken, an estimation of the difference of temperature
between the stuffing-box and the lower brass was made by subtracting seven
from the difference occurring in Column 5. In this manner the differences
entered in brackets were obtained for trials Nos. 10 to 20.
It appears that we have, therefore, ten determinations, viz., V., VI., VII.,
VIII, IX., X., XL, XII., XIII, and XVIIf., in which the differences of heat
generated require a positive correction on account of the unbalanced con-
duction along the shaft, and four determinations, viz., Nos. XIV., XV.,
XVI., and XVII., in which those differences require a negative correction.
Assuming, as is very nearly the case, that the losses of heat by radiation
are eliminated in the differences of the heats, it follows that by taking
C = loss of heat per trial, by conduction along the shaft, per unit differ-
ence of temperature between the stuffing-box and lower brass,
G is given by the equation
675844869 271143956
867995 + 75-6(7 ~ 348866 - 22-56'
where the numerators represent the sums of the differences of work in the
sets enumerated above, while the first terms of the denominators represent
the sums of the differences of heat in the same sets, to which the terminal
corrections have been added. The second term in each denominator repre-
sents the correction to be applied to the differences of heat for unbalanced
conduction along the shaft.
On solving the equation we get
C = 12, very nearly.
This agrees very closely with the value C = 13'61, which may be calculated
from the dimensions of the conducting shaft, viz., 4 inches diameter and
2| inches long, and Forbes' value of the conduction coefficient for iron, viz. :
(01429 in C.G.S. unit).
Since nothing was known as to the internal thermal condition of the
shaft, the figure 12 has been used throughout as a sufficiently close approxi-
mation to the constant required.
The corrections to the heat for conduction along the shaft in each trial
were then obtained by multiplying the fall of temperature between the brake
and bearing by 12.
66]
ON THE MECHANICAL EQUIVALENT OF HKAT.
709
The sign of the correction varies, of course, with the sign of the tempera-
ture gradient along the shaft.
Determination of the Loss of Heat by Radiation.
44. Under this heading are included all losses of heat not already dealt
with under the headings " terminal corrections," " loss by conduction," and
" loss by leakage of water."
Radiation in the Unjacketed Trials. — Series I.
45. Determination No. IF., consisting of a combination of trials 3 and 4,
is omitted, for the reasons given. A constant R, representing the loss of heat
by radiation per trial per unit difference of temperature between the brake
and surrounding air is required.
In Tables B and C the corrections to the heat are given for terminal
errors and conduction along the shaft, the calculation of which has been
explained.
The quantities given in the annexed table are sums obtained by adding
together the corresponding quantities in Series I. of Tables B and C.
In trials 1, 6, and 9 the loss by conduction has been assumed the same
as in trial 10 ; while in trials 2, 7, and 8 this loss has been given the same
value as calculated for trial No. 11.
SERIES I. — Unjacketed Trials.
Work done
Heat
Terminals
Conduction
Diff. of tempera-
ture between
brake and air
Heavy trials . . .
Light trials ...
542,876,020
272,418,189
677,309
330,280
+ 19
-131
+ 116
-496
556-4
558-4
We have, therefore, the same value of K given by
K =
542,876,020
272,418,189
677,444 + 556-4 R 329,653 + 558'4 R '
and, solving for R, we get
R = 36-86,
or, using this value of R and solving for K,
tf = 777-81,
which is the mean value deduced from this series of eight unjacketed trials.
710
ON THE MECHANICAL EQUIVALENT OF HEAT.
[66
Radiation Coefficient for Jacketed Trials, Nos. 12 to 42.
46. As in Series I., we get the sums of work, heat, &c., shown in the
annexed table: —
Work done
Heat
Terminals
Conduction
Diff. of tempera-
ture between
brake and air
Heavy trials . . .
Light trials ...
1,752,718,746
874,319,846
2,236,681
1,108,013
- 64
-183
- 886
-1369
1862-6
1872-5
In this table the sums are given of the respective quantities in the
trials used in Determinations VI. to XVIII. inclusive, Series No. V. being
included, because no error was apparent in the quantities obtained ; Series
No. IV. being omitted, since the moment given could not be guaranteed
correct with any certainty.
We thus get the following equation for R : —
874,319,846 1,752,718,746
1,106,461 + 1872-5 R 2,235,731 + 1862'6 R '
which, on solution, gives
R = 9-33,
and, substituting for R,
#=777-91.
47. The loss of heat by radiation from the brake, as given in the
Tables B, C, &c., was determined by multiplying the difference of tempera-
ture between the brake and the air by the radiation constants, calculated
as just described.
The Tables B, C, and D, giving the results of trials 1 to 42 inclusive,
should now be self-explanatory.
The mean value of K given by the eight unjacketed trials I have
mentioned was 77 7 '81.
48. The best way of stating the values of K obtained throughout
seemed to be as follows: —
The sums of the differences of the works and of the corrected heats
were taken for each series of trials, and then a mean value of K for the
series was found by dividing the first of these quantities by the second.
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 711
The values of K given as the mean for each series in Table D have
been calculated in this way.
49. A mean value of K can be obtained from the jacketed trials
contained in Scries II., III., VI., and VII. (Series V. being kept out of
the determination on account of the possible error already noticed), by
finding the sums of the respective differences of work and heat given with
each of these series in Table D, and then dividing the work by the heat
so obtained.
The sum of the differences of work in Series II., III., VI., and VII.
= 676,259,560,
and the sum of the corresponding differences of heat
= 869,396;
therefore the mean value of K given by the accepted jacketed trials so
far considered is
„ 676,259,560 _
869,396
From this mean none of the values obtained from any one of the above
series differs by as much as 0'03 per cent.
Closer agreement than this could not possibly be expected, and it was
consequently decided to vary the trials somewhat, in order to determine it
any errors had been overlooked. For this purpose I made two fresh series
of six trials each, the light trials carrying a moment of 400 ft.-lbs. only, none
of the other conditions being altered in any way.
50. The full reduction of these Series (Nos. VIII. arid IX.) is shown in
the two Tables E and F.
As before, three trials were run on each day, but the last trial, on April 1,
was not finished on account of an accident preventing me getting the correct
weight of the water discharged by the brake. There are, consequently, only
eleven trials in the tables. The radiation constant for these trials worked
out to 8-16.
The mean value of K, given by the whole eleven trials, was 778'14, which
is lower than the two means for the separate series in Table F, on account
of the inclusion of the light trial No. 45, which does not appear in
Table F.
This new value of K, viz. 778'1 4, did not agree so closely with the former
one of 777-85 as we had hoped, and, after reducing the last two series of
712 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
trials, I devoted all my time to the checking of the whole of the apparatus
anew.
It was a consequence of this stringent supervision of every separate part
that the small errors in the 25-lb. weights, already noticed, were discovered
(par. 27).
51. Calculation showed that this error might account for the discrepancy
observed, and so it was decided to run a fresh series of trials with the weights
so arranged that no error could appear on their account.
In order to have no known outstanding errors whatever, I made a small
rectangular trough, fitted with a drain-pipe, by means of which all leakage
from the stuffing-box was caught.
52. A series of fifteen trials, numbered 54 to 68 inclusive, was ac-
cordingly made, beginning on June 29, 1896. Owing, no doubt, to the
long rest which the apparatus had had since Easter, a number of accidents
were met with which completely spoiled the whole series.
The lagging of the brake was very damp when the series was begun,
and, on account of the bursting of the various rubber-pipe connections, it
did not thoroughly dry during the whole course of this series of trials.
For these reasons the results are not tabulated.
53. After remedying all the defects which had developed in the previous
week's running I made two fresh series of six trials each between July 7
and 10 inclusive.
No further accidents occurred and the results were in every way satis-
factory.
These are shown in Tables G and H.
The radiation constant worked out at R — 7'98.
The mean value of K, given by the two series, was
K = 777-85,
which happens to be exactly the same as obtained previously from Series II.,
III., VI., and VII.
54. This last lot of trials afforded no explanation of the small difference
(778-14-777-85)
= 0-3 ft.-lb. nearly,
which occurred between the results given by the 1200 — 600 ft.-lb. determina-
tions and the 1200—400 ft.-lb. determinations respectively.
66]
ON THE MECHANICAL EQUIVALENT OF HEAT.
713
The difference, of course, may be due to terminal errors, which, I think,
have been mainly responsible throughout for the small discrepancies found
to occur between individual determinations. It is more likely, however,
that the small quantity of water dealt with in the 400 ft.-lb. trials, and
the consequent greater effect of the oscillations of the brake on the mean
moment, may have introduced some error into these lightly-loaded trials.
Further, some slight bias may have been given to the Series, Nos. VIII. and
IX., by the long rest caused by the Easter Vacation, between trials 47
and 48.
55. In the annexed table I give the mean value of the work done and
of the heat generated in the heavy and light jacketed trials respectively,
against which no known sensible error can be placed.
Trials
Numbers
Mean work
per trial
Mean heat
per trial
Heavy trials
(13, 17,
48, 49,
18,
50,
19,
72,
20,
73,
35, 36, 37,
74, 75, 76
38,
and
39, 46, 47,
77)
134,337,403
172,685
U'_;ht trials :
(14, 15,
1 I, If.
16,
51
21,
52
22,
53
23, 33, 34,
69, 70, 71,
40,
78,
41, 42, 43,
79 and 80)
61,355,503
78,867
D
fferences . .
72,981,900
93,818
and dividing the mean difference of work by the mean difference of heat
we have
# = 777-91.
This mean value of K deduced from the experiments requires correcting
on a few counts, which are due to the method of working. These will be
dealt with later.
56. The table given on page 714 illustrates the almost perfect manner in
which losses of heat were eliminated on the mean result, by the method
adopted throughout the investigation of always working on the differences
of the quantities of work done and heat generated in a pair of trials.
A value of K can be obtained by dividing the difference of work in
Column 3 by the uncorrected difference of heat in Column 4. This
operation gives
K = 773-06.
The various corrections which this number requires are as follows : —
I. Correction due to difference in number of revolutions of shaft between
light and heavy trials.
714
ON THE MECHANICAL EQUIVALENT OF HEAT.
[66
Since the difference in the number of revolutions is only 15, this cor-
rection, as previously indicated, when dealing with the balance of the brake,
will be zero (par. 29).
Heat
generated,
Differ-
ence of
Differ-
No. of
revolu-
tions of
shaft
Work done
less
losses
due to
terminals,
Loss of
heat by
leakage
of water
Terminal
correc-
tions
temper-
ature
between
stuffing-
temper-
ature
between
conduction,
&c.
box and
bearing
and air
Means for 21
17,817
134,337,403
171,510
4
-1
-3-9
140-5
accepted
heavy trials
Means for 23
17,832
61,355,503
77,710
1
-7
-5-4
141-5
accepted
light trials
Differences . . .
-15
72,981,900
93,800
3
6
1-5
-1-0
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
II. Correction due to loss by leakage of water from the brake.
3
This correction amounts to — ^ ^.^ = — 0-000032.
"o,800
III. Correction due to terminal differences of temperature of the brake.
This correction amounts to —
6
93,800
= - 0-000064.
IV. Correction due to loss of heat by conduction along the shaft.
1*5 x 12
This correction amounts to - -s/cr = "" 0'000192.
93,800
V. Correction due to loss of heat by radiation.
Assuming 9 for the value of the radiation constant, this becomes
9
93,800
= + 0000096.
The total correction factor is therefore (1 - 0'000192), which gives as
before
K= 777-91.
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 715
Corrections to the Mean Value of K given by the Experiments.
I. Length of Brake Lever.
57. In dealing with the calibration of the measurements of the brake
(par. 28), I have already mentioned that the value of K given by the
experiments would require a correction factor of (1+ 0*00042).
//. Salts Dissolved in the Manchester Water.
58. Professor Dixon kindly furnished Professor Reynolds with the results
of a number of analyses of the town's water made during the College session,
1894 — 95. The dissolved salts were
Common Salt, 14'4) .... ,.
H> milligrammes per litre;
Calcium Carbonate, 27 -7 j
therefore the proportion of salts by weight is 0'00t)0421. Taking their
specific heat at 0*2, we get for the correction factor required, due to the
lowering of the specific heat of the water,
1 +(1 - 0-2) x 0-0000421 = (1 + 0-00003).
///. Air Dissolved in the Water Used. — ^Part I., par. 43.)
59. Being rain water it probably contained about 2^ per cent, by volume
of dissolved air. As affecting the specific heat of the water, this air would
not have of itself any sensible influence.
It did, however, influence the resulting final temperature, as it was most
probably all boiled out of the water, and the bubbles of expelled air would
all be saturated with water vapour at a temperature of 212°, which vapour
could not be formed without extracting its latent heat from the surrounding
water.
I made some experiments in December, 1896, with the object of deter-
mining the actual volume occupied by the bubbles of mixed air and water
vapour under the conditions obtaining in the trials. The pressure on the
water in the discharge-pipe was 10 inches of mercury very nearly.
The method adopted was as follows : —
I put a depth of about two inches of mercury into the bottom of a strong
bolt-head flask, and above the mercury I poured in 1^ Ibs. of water. This
filled the flask nearly to the brim. A rubber stopper, through which passed
a glass tube, was then pressed into the neck of the flask, the glass tube bring
of such a length that the insertion of the stopper displaced mercury only up
716
ON THE MECHANICAL EQUIVALENT OF HEAT.
[66
the tube, care being taken that no bubbles of air were included under the
stopper. The stopper was then firmly tied into the neck, and the flask was
hung inside a large glass beaker, which was then filled with water to a depth
which covered the top of the rubber stopper.
One end of a piece of strong rubber tube was then fastened on the glass
tube protruding from the flask, while its other end was fixed to the vessel
shown at A, which was open to the atmosphere.
>0
Fig. 11.
Mercury was poured into the glass funnel at A, and it was raised till
there was a solid column of mercury from the bottom of the flask to the
surface in A. The water in the beaker was then heated by a Bunseri flame till it
boiled. This boiling was continued during a whole day, the water in the beaker
being replenished as required. By adjusting the level of the free surface of
the mercury at A, any required pressure could be put on the vapour column
which formed over the water in the flask neck and displaced some of the
mercury from the bottom. Also, by suddenly raising the pressure, the vapour
was compressed and cold mercury flowed down into the flask, condensing the
vapour in the neck as it descended. By this means the water in the flask
could be made to boil briskly for a few moments now and then, so as to
facilitate the escape of the air. At the close of the day the levels of mercury
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 7 17
and water were adjusted so as to give the requisite pressure on the vapour
column. The length of this column was then measured, and knowing the
diameters of the flask neck and tube, it was easy to calculate the volume of
vapour.
This was 2*2 cubic inches.
If this be reduced to a temperature of 32° and atmospheric pressure, the
proportion of air by volume appears to be 1'6 per cent.
This number is considerably less than the 2'5 per cent, already mentioned,
but as it was determined under conditions which approximated closely to
those which held in the main trials, it was used in the calculation of the
correction given below.
The weight of water vapour at a temperature of 212° per cubic foot
= 0-03797 Ib.
Therefore the correction due to the loss of the latent heat necessary to
evaporate this weight of water, is, relatively to the 180 thermal units
generated per Ib. of water discharged by the brake,
4 2-2 0-03797 x 966
5XI728X 180
The correction factor is therefore (1 — 0'00021).
IV. Reduction oft/te Weighings to Vacuo. — (Part I., par. 41.)
60. Taking the density of water
= 62-425,
and of air at 32° Fahr.
= 0-08073,
and also assuming 70° Fahr. as the mean temperature of the engine-room
(luring the trials, the correction factor becomes
i
1 - 0-08073 Xx. = 1-0-00120.
In the calculation of this factor it must be borne in mind that the density of
the air causes errors of equal magnitude in the measurement of both work
and heat on account of the alteration of apparent density of the cast-iron
weights used on the brake and on the lever of the weighing machine.
V. Varying Specific Heat of the Water. — (Part I., par. 51.)
61. According to Regnault the mean specific heat of water between
freezing and boiling points is 1*005, assuming the specific heat unity at the
718
ON THE MECHANICAL EQUIVALENT OF HEAT.
[66
lower temperature. If his formula for the specific heat be correct, then a
correction factor of (1 — 0'00006) is necessary to make the value of K derived
from the trials represent this mean specific heat. This factor is introduced
because it was not strictly the whole range of temperature between freezing
and boiling points which was dealt with in the trials, for the cold water sup-
plied to the brake had various temperatures ranging from 327° to 34'3°.
This correction would only just affect the second decimal place, and in con-
sideration of the uncertainty that exists as to the exact value of the specific
heat of water at any temperature, I do not propose to use a correction factor
on this account.
VI. Corrections due to the Fall in Pressure between the Supply and
Discharge Pipes.
62. From observations taken on October 1st, 1896, I determined the
pressure on the thermometer in the supply pipe to be : —
In the 1200 ft.-lb. trials 15 inches of mercury.
„ 600 „ „ 11
„ 400 „ „ 9-7 „
I have already stated that the pressure on the thermometer in the
discharge pipe was 1T3 feet of water in all trials.
From these varying pressures two corrections are obtained as follows : —
(a) ELEVATION of Temperature Readings by the Pressure on the Ther-
mometers.
1200 ft.-lbs.
600 ft.-lbs.
400 ft.-lbs.
Pressure on thermometer bulb in
supply pipe in inches of mercury
15-0
11-0
9-7
Consequent elevation in readings of
temperature (0°-0072 per inch)
0°-108
0°-0792
0"-0698
Pressure in discharge pipe in feet
of water
11-3
11-3
11-3
Consequent elevation in readings of
discharge temperature (0°'0066
per inch of mercury)
0°-066
0°-0(J6
0°-06G
Percentage correction to heat ob-
tained
0-042
0-013
0-004
1-8
= 0-0233
1-8
= 0-0072
1-8
= 0-0022
6G]
ON THE MECHANICAL EQUIVALENT OF HEAT.
719
If we now confine our attention to the combination of 1200 and 600 ft.-lb.
trials, the relative correction to the difference of heat is
0-000233- jx 0-000072 = ().ooom
i.e., the correction factor to K on this account is
(1-0-000394).
Considering next the 1200 — 400 ft.-lb. determinations, the relative cor-
rection to the difference of heat is
_ 0.000339
which makes the correction factor
(1 - 0-000339).
On the mean value of K deduced from the trials, I propose to make
this factor
(1 - 0-00037).
63. (6) GENERATION of Heat in the Water on account of the Loss of
available Head between the Supply and Discharge Pipes. (Part I., par. 53.)
1200 ft.-lbs.
600 ft.-lbs.
400 ft.-lbs.
Head in supply pipe in feet of
water
17-0
12-45
10-98
Loss of head before reaching the
discharge pipe in feet
5-7
1-15
-0-32
Correction required by the work
given in the tables per cent.
5-7
1-15
-0-32
1-8x777
] -8 x 777
1-8x777
=0-0041
= 0-0008
= -0-0002
Therefore the correction factors required are —
(a) For the 1200—600 ft.-lb. determinations
1 + 0-000041-^x0-000008 =
2
08) For the 1200—400 ft.-lb. determinations
1 + 0-0000*1 -1x0-000002 = (
This factor also I propose to give the value
(1 + 0-00007),
when applied to the mean value of K deduced from all the trials.
720
ON THE MECHANICAL EQUIVALENT OF HEAT.
[66
VII. Correction due to the manner of Engagement of the Revolution Counter
with the Engine Shaft — (Part I., par. 34.)
64. The spindle of the counter carried a wire pin parallel with the axis
of revolution, which pin was driven by another carried by, and passing at
right angles through, the axis of the spindle making connection with the
engine shaft.
The mean chance was therefore that at every engagement of the counter
with the shaft one-fourth of a revolution would be lost by the instrument,
while on disengaging the counter stopped the instant it was withdrawn.
The work in every trial should therefore be increased to compensate for
this loss.
The number of revolutions was approximately 1 8,000.
The correction factor is therefore
1
1 +
= (1+0-00001).
72,000
65. A summary of these corrections is appended.
Cause of correction
Magnitude and sign
+
-
I.
II.
III.
IV.
V.
VI.
VII.
Length of lever
0-00042
0-00003
Neglected
0-00007
o-ooooi
0-00021
0-00120
0-00037
Dissolved salts
Dissolved air
Weight of atmosphere
Varying specific heat of water
(a) Effect of pressure on thermometers
(b) Loss of head in the water
Engagement of revolution counter
Totals
0-00053
0-00178
Therefore the final correction factor is
(1 - 0-00125).
66. Applying this correction factor to the value obtained from the
experiments, we get for the value of the mean specific heat of water
between freezing and boiling points, expressed in mechanical units, at
Manchester,
777-91(1 -0-00125),
776-94.
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 721
APPENDIX.
Although no part of this research, it may be interesting to notice that
reduced to the latitude of Greenwich this becomes
777-07,
and reduced to latitude 45° at sea-level
777-53.
Expressed in metre-grammes and the centigrade unit of heat this last
value becomes
426-58.
The value of g being
980-63,
we have for the mean value of the specific heat of water between 0° and
100° C., expressed in absolute C.G.s. units,
41,832,000 ergs.
Making use of Regnault's formula for the specific heat of water at
different temperatures, this would give the mechanical equivalent of the
heat required to raise 1 Ib. of water at 60°*5 Fahr. through 1° Fahr. at
Manchester as
773-74 ft.-lbs.,
and taking water at 32° Fahr., this gives
773-07 ft.-lbs.
Similarly expressing the result in absolute C.G.s. units, we have for the
mechanical equivalent of the heat necessary to raise 1 gramme of water
through 1° C. in latitude 45° and at sea-level
(a) From a temperature 15°'8 C 41,660,000 ergs.
(6) 0°C 41,624,000 ergs.
O. K. II.
46
722
ON THE MECHANICAL EQUIVALENT OF HEAT. [66
TABLE A. — SHOWING THE PRELIMINARY
Heavy trials. Moment, 1200 ft.-lbs.
Light trials.
a
C3 SC 01
si°s
-3
K
•4
jj
&5S
111
aj
Deter-
s
o
£ 5*°
S-2-o
_ '. -
g
,0
mination
8
TJ^at
S
o
^ H ft
9
a
Heat
number
&
a
Work done
JLJ.tr a>u
generated
o
le
o fe
|||
a> *»
gJa.JS
a
3
Work done
gene-
rated
H
1
E
<D
si % *
<u -° §
£ 2 M
1 g-S
'(H
H
H
9 J* _§
•5 ^ _§
•S ^ 9
s
ft""
a~
Series
Numbe
r /.
I.
1 134,201,602
167,191
+ 11
...
139-3
2
68,310,950
82,626
II.
4
138,446,542
172,957
- 63
...
140-5
3
68,182,773
83,090
III.
6
135,935,775
169,686
+ 31
137-6
7
67,926,419
82,432
IV.
9
136,063,953
169,859
- 10
138-9
8
68,096,065
82,725
V.
10
136,674,680
170,573
- 13
+ '9-4
( + "2-4)
140-6
11
68,084,755
82,497
Series Number II.
12
133,628,584
169,519
- 40
+ 10-1
144-4
Co
mbined with
trial 14
VI.
13
135,392,907
172,591
+ 12
+ 9-3
(+"2-3)
143-8
14
67,933,958
86,054
VII.
17
135,098,853
172,408
+ 6
- 3-9
(-10-9)
140-2
16
67,677,604
85,819
15
67,658,754
85,737
Series Number III.
VIII.
18
133,734,142
170,604
- 69
+ 6-3
(- 0-7)
141-0
21
66,580,557
84,173
IX.
19
133,892,479
170,867
+ 63
+ 1-1
(- 5-9)
140-9
22
67,142,275
85,012
X.
20
135,332,588
172,666
- 29
+ 0-3
(- 6-7)
140-8
23
66,765,283
84,703
Series Number IV.
24
139,870,565
178,183
+ 104
+ 2-1
- 6
141-7
25
139,448,444
177,847
+ 40
- 1-4
- 7-4
139-1
26
140,073,809
178,984
- 40
- 3-1
-10-2
139-3
Series Number V.
XI.
XII.
XIII.
30
31
32
134,073,435
134,623,843
135,257,190
171,054
171,793
172,618
- 12
- 12
+ 7-9
- 6-3
- 4-0
- 1-3
-11-7
-10-3
145-8
145-4
141-4
27
28
29
67,353,391
67,146,045
67,315,692
85,344
85,147
85,406
Series Number VI.
XIV.
XV.
35
36
134,744,481
135,702,040
171,995
173,226
+ 6
- 6
- 2-9
- 3-1
- 9
- 9
144-9
143-4
33
34
67,692,684
66,765,283
85,724
84,625
Series Number VII.
XVI.
XVII.
XVIII.
37
38
39
134,819,879
135,151,632
134,895,277
172,059
172,550
172,250
- 6
- 17
+ 8-9
+ 1-1
- 0-3
- 0-3
- 4-4
- 5-9
145-9
144-3
144-8
40
41
42
67,703,993
67,112,116
67,130,965
85,555
85,135
85,316
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 723
REDUCTION OF TRIALS, 1 TO 42 INCLUSIVE.
Moment, 600 ft.-lbs.
Differences
a
i
s
0
a
e8 SC tr
*• a rr
S o g
g JS
a
gj t£ rf]
gbo
rT 00
eS JS
^
J
S,SE2
H _S *B
o-£ 7s
O.J
5
ali 1
a; a a,
« 2
5.S
M-(
O
o
S 5 "°
S 5-=
a
§
- — -—
a* s -S
a
E
~ X ~
S • h
^ "oo *i
E
O
a &
CM ^ >"
— Q
Heat
B
Q
*" a "
a)
"3
O
°o | g«
° S O
"o sj
Work done
gene-
0
"sl|
O oj O
o|
"5
0 -S TJ
<D ^~"
o .g -a
S » t,
rated
T|
a, s — •
o -** ^
fe
I
a £ c
h ,
g^'S
1
a v "
2-° «
a S a
2^ §
<U e8
c3
3
E
B
83 * ^
JB § S
sa * *°
E
1 2 *
oj g x
<1J g ^3
S
H
Q-5-5
|2J
g5 i
H .
gSJ
g 5 J
g - eS
1
CH
137-4
65,890,662
84,565
+ 11
779-1
-'e
140-7
70,263,769
89,867
- 57
...
782-4
-91
140-8
68,009,356
87,254
+ 122
• . .
778-3
-29
139-1
67,967,888
87,134
+ 19
779-9
-11
- 3-3
(-10-3)
141-1
68,589,925
88,076
_ 2
...
( + 12-7)
...
778-8
787-4
- 5
- ii-2
(-18'2N
142
67,458,949
86,537
+ T?
( + 20-5)
779-4
+ 55
-10-9
(-17-9)
140
67,421,249
86,589
- 49
(+ 7)
...
779-1
-22
- 1-9
(- 8-9)
140-8
- 5
+ 3
- 2-6
145-3
67,153,585
86,431
- 64
(+ 1-9)
777-:.
-38
- 1-9
- 7-9
144-3
66,750,204
85,855
+ 101
(+ 2)
776-6
-55
- 2
- 7-7
144-4
68,667,306
87,963
+ 26
(+ 1)
779-3
- 5
- 2-6
-ll-l
85,710
- 7
1 !('S
778-5
-11-8
18-3
144-2
<;7'l77J!>s
86,646
- 17
4 <;<;
77N-!)
-16
- 6-2
-13-7
140-4
67,941,498
87^18
+ 16
...
4- 3-4
...
7789
- ">
+ 2-1
- 5-1 1I.V7
67,051,797
8(!,27 1
+ 61
- 3-9
776-7
'... + 0-9
- 5-0
1 1T2
68,936,767
88,601
- 6
- 4-0
...
778-1
-27 +24-9
+ 11-6
1 HJ-.-i
6 <,1. 1 .ri,ss(j
86,604
+ 21O
-11-9
775-7
-16
+ I'M
1-7
1 !•'! "
68,039,516
s7^ 1 1 :•
ii-'.i
2-7
...
77H- 1
-21
-16-6
I i:;-i
67,764,312
86,934
+ 21-0
+ 107
77'.r:>,
46—2
724
ON THE MECHANICAL EQUIVALENT OF HEAT.
[66
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726
ON THE MECHANICAL EQUIVALENT OF HEAT.
TABLE I).
[66
Determin-
ation No.
Trial No.
Work
Difference
of Work
Heat
(corrected)
Difference
of Heat
K
8
eries No. I.
I.
1
134,201,612
172,366
2
68,310,950
65,890,662
87,567
84,799
777-02
III.
6
135,935,775
174,818
7
67,926,419
68,009,356
87,407
87,411
778-04
IV.
9
136,063,953
174,998
8
68,096,065
67,967,888
87,699
87,299
778-56
V.
10
136,674,680
175,772
11
68,084,755
68,589,925
87,563
88,209
777-58
Mean value = 77 7 -81.
Series No. II.
VI.
13
135,392,907
173,973
14
67,933,958
67,458,949
87,156
86,817
777-02
VII.
17
135,098,853
...
173,591
16
67,677,604
67,421,249
86,965
86,626
778-3
Mean value = 777*66.
Series No. III.
VIII.
18
133,734,142
171,843
21
66,580,557
67,153,585
85,493
86,350
777-69
IX.
19
133,892,479
172,174
22
67,142,275
66,750,204
86,225
85,949
776-63
X.
20
135,332,588
173,871
23
66,765,283
68,567,305
85,903
87,968 779-46
Mean value = 7 77 '94.
Series No. I7.
XL
30
134,073,435
172,386
27
67,353,391
66,720,044
86,600
85,786
777-75
XII.
31
134,623,843
172,998
28
67,146,045
67,477,798
86,277
86,721
778-1
XIII.
32
135,257,190
173,813
29
67,315,692
67,941,498
86,536
87,277
778-46
Mean value = 778'1.
.
Series No. VI.
XIV.
35
134,744,481
173,245
33
67,692,684
67,051,797
86,967
86,278
777-16
XV.
36
135,702,040
174,450
34
66,765,283
68,936,757
85,910
88,540
778-59
Mean value = 777 '89.
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XVI.
37
134,819,879
173,410
40
67,703,993
67,115,886
87,034
86,376
777-02
XYIL
38
135,151,632
173,826
41
67,112,116
68,039,516
86,433
87,393
778-55
XVIII.
39
134,895,277
173,530
.
42
67,130,965
67,764,312
86,431
87,099
778-01
Mean value = 777'86.
66]
ON THE MECHANICAL EQUIVALENT OF HEAT.
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ON THE MECHANICAL EQUIVALENT OF HEAT.
[66
TABLE F.
Determin-
ation No.
Trial No.
Work
Difference
of Work
Heat
(corrected)
Difference
of Heat
K
Ser
:es No. VIII.
XIX.
XX.
46
43
47
44
135,688,500
45,133,482
135,641,722
45,251,606
90,555,018
90,390,116
174,477
58,093
174,344
58,231
116,384
116,113
778-07
778-47
Mean value = 778-27.
Series No. IX.
XXI.
XXII.
XXIII.
48
51
49
52
50
53
133,719,062
45,261,660
135,965,935
44,784,136
132,708,724
45,035,464
88,457,402
91,181,799
87,673,260
171,686
58,038
174,773
57,539
170,532
57,919
113,648
117,234
112,613
778-35
777-78
778-54
Mean value = 778'22.
66]
ON THE MECHANICAL EQUIVALENT OF HEAT.
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ON THE MECHANICAL EQUIVALENT OF HEAT.
TABLE H.
[66
Determin-
ation No.
Trial No.
Work
Difference
of Work
Heat
(corrected)
Difference
of Heat
K
8t
Ties No. X.
XXIV.
72
130,522,170
167,728
69
66,154,556
64,367,614
84,987
82,741
777-95
XXV.
73
132,158,316
169,980
70
67,130,965
65,027,351
86,278
83,702
776-89
XXVI.
74
133,734,142
171,921
71
68,216,702
65,517,440
87,757
84,164
778-44
Mean value = 777'74.
Series No. XI.
XXVII.
75
132,165,855
169,863
78
65,732,325
66,433,530
84,424
85,439
777-56
XXVIII.
76
134,646,463
173,106
79
66,358,132
68,288,331
85,336
87,770
778-03
XXIX.
77
135,370,287
174,071
80
67,458,948
67,911,339
86,790
87,281
778-07
Mean value = 777'88.
DESCRIPTION OF THE PLATES.
(See end of Volume.}
PLATE 1.
From a photograph in 1888. Is a front view of the triple expansion engines
(100 H.-P.) and brakes, as they existed in the engineering laboratory, Owens College,
before any modifications for the determination of the equivalent. The engine-shafts are
disconnected from each other, and are working on three separate brakes. In the trials
the three large pulleys (5 feet in diameter) were removed with the brakes on the high-
pressure and intermediate engines, and the engine-shafts coupled by intermediate shafts,
the work being all absorbed by the brake on the low-pressure engine — seen, on the right
hand of the plate, overhanging the last bearing of the brake-shaft. On this shaft are two
heavy 3-feet pulleys, which served as fly-wheels during the trials.
It was the facilities afforded by this brake and its appurtenances (§11) that suggested
the research and rendered it possible : and, although the method of admitting the water
and air to the brake was necessarily modified in the experiments, the brake remained
essentially the same. Part of the trials was made with the brake uncovered, as seen
in this plate ; and it was after the brake was covered that the subsequent photographs
were taken.
The vertical pipe supplying the town's water from the service tank to the brake,
with the hand-cock and the automatic inlet-cock above, leading through the bowed pipe and
flexible indiarubber tube to the inlet passage over the bush of the brake, are seen on
the immediate right. Immediately on the left and a little behind and lower, is another
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 731
bowed pipe leading from the top of the brake, with a gap in it; this is the air passage
leading through the vanes to the centres of the vortex chambers, to secure atmospheric
pressure there. The suspended and riding loads on the lever, the dash-pot, the front stop
on which the lever rests (not being at work), are also seen. The hand wheel for adjusting
the height of the lever when at work, the linkage connecting the automatic inlet and
outlet-cocks with each other and with the front stop, together with the outlet-cock, the
receptacle for waste, and the drip-can for the water escaping from the front bush, can be
traced, though they are obscure in this plate.
Up high on the photograph is seen a shaft with two large pulleys ; these are for
connecting the separate engine-shafts by l>elts and rope (seen), and have no place in the
trials. But the bright shaft immediately below, seen as driven by a rope pulley from
behind the wall of the engine-roorn, is the line shaft driven by the separate engine, always
running, which afforded most important facilities for the research.
PLATE 2.
From a photograph, 1896. Also shows a front view of the engine-roorn, but taken
more to the right ; it includes only the low-pressure engine. It shows a general front view
of the appliances in the condition in which they were during the final experiments, as well
as some of the standing appliances not included in Plate 3.
Low down, immediately on the right, is the front of the weighing-machine, with the
tank resting on it ; and immediately behind this, against the wall, are seen the mercury
balances for the pressures of water in the mains; also the town's main to the service tank
(out of sight on the right), in front of which is the 3-inch quadruple turbine which drives
the (H-inch) quintuple centrifugal pump (out of view, behind the tank) supplying the
brake through the ice-cooler (§ 20). On the left of the tank, and passing through its
cover, is the water-switch ; and over this is the nozzle of a vertical pipe, straight almost to
the roof, then horizontal, with an open vertical branch, to form an air-gap, then down
again into the lower of the two horizontal pipes ; this is the stand-pipe on the outlet from
the condenser, for securing pressure in the final thermometer chamber (§ 22). The upper
of the two horizontal pipes is the water-jacketed outflow pipe or "condenser," which passes
to the end of the room, and returns as the lower horizontal pipe to the stand-pipe.
Immediately on the left of the plate, standing on the floor, is the frame for the hand-
brake (§ 30). Besides the appliances mentioned, as seen, in this plate, nearly all the
appliances arc seen in front view; but many are better seen in the following plates, though
this plate affords the best view of the general arrangement, and the best idea of the
circumstances under which the observations were made. The passage between the brake
and the 3-inch pipe supplying condensing water to the engine afforded the only jx>st of
observation for the counter, thermometers, speed-gauge, and pressure-gauges. The centri-
fugal speed-gauge, with its scale, is seen rising vertically from behind the small pressure-
gauge on the brake.
PLATE 3.
This is a nearer and simplified front view of the more special appliances shown in
Plate 4. Proceeding f'n>m the right is the switch and outlet nozzle from the condenser,
with the water flowing into the tank over the thermometer. From the switch may be
traced the linkage forming the automatic connection of the switch with the counter,
immediately in front of the covered bush of the brake. Supi>orted by the original
supply pij)e to the brake (the hand-cock being shut) is seen the new inlet pipe from
732 ON THE MECHANICAL EQUIVALENT OF HEAT. [66
the ice-cooler, behind the brake. The pipe, rising on the right from behind the brake,
passes a branch to the by-channels leading to the bushes (not seen) and a branch to the
large pressure-gauge, then to the regulator ; thence the water flows upwards past the bulb
of the inlet thermometer, some of it passing up through the glass thermometer chamber,
and so to waste through the small pipe at the top, but the main stream passing through
the covered horizontal branch, and down the flexible indiarubber pipe into the brake. On
the top of the brake is seen the new air-passage, of flexible indiarubber, leading to the
vessel in which is the artificial atmosphere, which is connected with the large mercury-
gauge on the left, also with the syringe. The automatic outflow cock is clearly seen under
the brake, also the curved flexible pipe, covered with cotton-wool, which receives the
water from the outflow cock, leading to the fixed pipe behind the regulator, also covered,
in which is the bulb of the outflow thermometer, and immediately over this the glass
thermometer chamber, with its indiarubber continuation leading back into the main out-
flow channel which rises up behind the inlet thermometer chamber, till it turns at right
angles into the condenser. Behind and on the left of the brake are seen protruding the
stems of the thermometers for measuring the difference of temperature in the stuffing-box
and the near bearing. Of the two bottles standing on the floor, that on the left is
collecting the leakage from the stuffing-box, and the other the leakage caught in the
indiarubber bag enclosing the automatic outflow cock.
PLATE 4.
This is a back view. On the left, close in front of the tank on the weighing-machine,
over which is the condenser leading to the switch, is seen the 1^-inch quintuple centrifugal
pump, with its driving gear and the pipe supplying it from the service tank. On the
other side of the 3-inch pipe for condensing water for the engines, and partly behind it, is
seen the pipe leading from the pump up and along behind the 3-inch pipe, then down
again into the ice-tank (on the extreme right of the plate) ; through this it passes in a coil,
emerging from the cover again as the covered pipe rising obliquely to the regulator and
inlet thermometer chambers (not seen), with the branch to the pressure-gauge. The small
horizontal branch coming through from beneath the pressure-gauge, continued by the
covered indiarubber pipe, passing behind the vortex vessel of the speed-gauge to the
stuffing-box, is one of the by-paths taking ice-cold water to the bushes ; that on the left
is behind the brake. The outlet thermometer chamber, with its indiarubber continuation
to the main outflow channel into the condenser, is also clear ; as are also the belt and
pulley driving the paddle in the ice-tank.
PLATE 5.
This is again a back view, but taken so as to show the appliances up to the end of the
engine-room, not seen in the previous plates. In the middle front is seen the 6-inch
quadruple centrifugal pump in circuit, with the rising 4-inch main from the lower tank to
the tank in the tower (§ 3), together with the belt from the line shaft by which this pump
is driven. Immediately on the left of this plate, standing on a bench, is the end of the
3-inch quadruple vortex turbine, driven by water from the tower, and driving by a cord
the 1^-inch quintuple centrifugal pump. The standard, the lever, and the large riding
weight of the weighing-machine, with the tank behind, are completely in view ; and over
these again appears the condenser for cooling the effluent water, passing to the end of
the room and returning underneath to the stand-pipe and thence to the switch.
66] ON THE MECHANICAL EQUIVALENT OF HEAT. 733
PLATE 6.
This is from a photograph of the apparatus for correcting the high temperature th<Tn,o-
meter. On the table is the barometer, and to the right is the vapour chamber, in which
the thermometer is immersed through the cork on the top as far as to leave the top of the
mercury visible. The escape passage and regulator are seen on the right. The pips
leading from the top is the connection of the vapour chamber with the lower mercury
chamber in the barometer. This, after passing through the flask, receives by the branch
(seen) a slight current of air from the pressure reservoir, with the top of which it is
connected by a restricted pipe, so that the current is so slow that the resistance is
negligible, though sufficient to prevent the vapour passing to the barometer ; the pressure
of air in the reservoir is shown by the large mercury-gauge, and is maintained by
occasional pumping with the syringe seen in connection. The nozzle on the barometer,
to which the air-passage is connected, leads into the cast-iron bottle which forms the
mercury chamber, above the surface of the mercury. The level of this surface is observed
through the circular windows, of which that which is in front is shown to the left of the
axis of the barometer, above the nozzle. Immediately above this window is seen the
c\ lindrical brass curtain, which screws on to the neck of the bottle, by which the light
through the windows over the mercury can be eclipsed. Attached to this curtain, and
co-axial with it, is the outer brass tube extending up to the gap, with a vertical scale
attached reaching past the gap. Behind the vertical scale, and screwed into the tube on
the lower curtain, is a tube screwed throughout its length, and having two parallel slots,
as windows, some 5 inches long, through which the upper limb of the mercury may be
observed. From the top of this windowed tube downward is screwed the cap, the lower
limb of which forms a cylindrical curtain for eclipsing the light over the upper limb of the
mercury (§ 48).
67.
ON THE SLIPPERINESS OF ICE.
[From the Forty-third Volume of the " Memoirs and Proceedings of the
Manchester Literary and Philosophical Society." Session 1898 — 9.]
(Received and read February 7th, 1899.)
THE slipperiness of ice is, and has been, one of the most noticeable,
interesting, and important circumstances under which we live, as well as one
of the commonest. Ice is not the only slippery thing in the world, but it is
the only one of all the solid substances which, in the condition nature has
left them on the surface of the earth, possesses the property of perfect
slipperiness. This being so, and being commonly known to be so, it is
certainly remarkable that, whatever may be the reason, there appears to have
been little or no curiosity as to the physical significance of the unique
property which ice possesses. Speaking for myself this is simply explained ;
ice was slippery when I was born, I never knew it otherwise, and, to put it
shortly, it was slippery because it was ice, whereas it now seems to me that,
of all the secrets nature has concealed by her method of deadening curiosity
by leaving them exposed, in this her method has been the most successful.
The cause of my ultimately discovering the secret, unsought by me, was
an accident, though brought about by another line of research. The other
sources of perfect slipperiness are complex ; a smooth solid surface covered
by a viscous fluid, as a well-greased board, is perfectly slippery just as ice is,
which fact had been taken for granted much in the same way as the slipperi-
ness of ice, neither more nor less.
That surfaces of machines would not slip over each other without grease
was well known and followed out, but the physical significance of the
action was apparently not questioned until, in 1884, Mr Beauchamp Tower1,
1 Proc. Inst. M. E., Nov. 1883 and Jan. 1884.
67] ON THE SLIPPERINESS OF ICE. 735
while making experiments as to the resistance of a railway journal, accident-
ally came across a fact of very striking significance.
In this experiment, instead of using an axle, Mr Tower used an overhang-
ing shaft driven by a steam-engine, the shaft being supported on bearings in
the usual manner. The overhanging portion of the shaft was turned to the
same shape as one of the journals of a railway wheel, four inches in diameter
and six inches long. On this journal the ordinary axle-box was suspended,
the load to correspond with the proportion of the weight of a loaded truck
being suspended from the axle-box underneath the shaft. The axle-box had
the usual brass wearing-piece, and the provision for lubrication was, as usual,
an oil or grease cup communicating through a vertical oil-hole, so that the
oil might descend by gravitation through the brass on to the surface of the
journal, and thence escape, after being used, to the ground. This was in the
first instance, but, after experimenting in this way, Mr Tower proceeded to
find what would be the effect on the resistance if, instead of allowing the oil
or grease to escape freely from underneath the journal, the whole under side
of the journal was encased in a vessel, so as to form a bath of oil in which
the journal would be completely covered.
In commencing these experiments with the bath, Mr Tower noticed with
surprise that, although the oil in the bath did not cover the top of the brass
when the journal was at rest, when in motion the oil escaped upward against
gravity through the oil-hole, and as this was inconvenient, tending to empty
the bath, he drove a plug of wood into the hole and tried again, when to his
still greater surprise he found that the oil forced out the wooden plug. This
led him to fit a pressure gauge to the hole ; this immediately rose to the top
of its scale, 200 Ibs. per square inch. Then, realising that he had before him
evidence of aii action in lubrication until then unsuspected, Mr Tower
turned his attention to its experimental investigation, finding that when the
journal was run at 400 revolutions a minute, the pressure on the square inch
indicated on the gauge was somewhere about 3/2 of the pressure necessary,
if distributed over the whule horizontal area of the section of the bearing,
to sustain the load. The pressure in the oil-hole would be 600 Ibs. per
square inch when the total load was 9,600 Ibs., whence, as the area of the
horizontal section was 24 square inches, the mean intensity of pressure
would be 400 Ibs. This, however, was only when the speed of the journal
was greater than a certain limit depending on the load ; when the speed
diminished below this limit, the pressure on the gauge fell to any degree
below that necessary to sustain the load. But this was not all. When the
speed was such as to sustain the load, the friction was 1 in 400, but when
running slow the friction reached 1 in 3, or the journal seized the brass.
Taking these two things together, it made clear the fact which lnul
never been surmised before, that the <i<-finn <>f lubrication conxixinl in the
actual separation of the sol i< I miri'm-ps by a film of fluid of finite thickness.
736 ON THE SL1PPERINESS OF ICE. [67
These discoveries of Mr Tower excited great interest at the time, and,
being myself occupied in the study of fluid motion, I was induced to under-
take the theoretical analysis of Mr Tower's experimental results, from which,
after two years' work, I was able to publish a complete theory of lubrication1,
showing that not only in the case of the oil-bath, when the thickness of the
separating film of oil was about 2/l,OOOth of an inch, but in cases of ordinary
lubrication where the thickness of the film is less than '0001 of an inch, the
surfaces are separated by a complete film.
This is very strikingly indicated by a rarely shown but simple experiment.
Two cylindrical hard steel gauges, male and female, one inch in diameter,
made to gauge to within 1 /20,000th of an inch will not pass one into the other,
if wiped as clean as possible of all oil, without the use of great pressure or of
a mallet. If oiled and kept moving they can be easily passed one into the
other. But should the motion be arrested for a second, they seize and can
only be separated by the mallet, which shows that a film of oil less than the
l/20,000th of an inch is sufficient to sustain perfect slipperiness, while the
least contact destroys this property.
My research also led to the recognition that the property on which the
lubricating action depends is the viscosity of the fluid, and that all fluids are
lubricants, provided they are not corrosive. Air lubricates, as is shown by
the floating of one true surface plate on another with perfect slipperiness.
Now water had, at the time, not been recognised as a lubricant ; its viscosity
is from 200 to 400 times less than oil, but from my research it appeared that
it is a lubricant in proportion to its viscosity.
All this is now matter of history, and its bearing on the slipperiness
of ice may not as yet be clear. But it has a fundamental bearing never-
theless.
It was about 1886, while I had this subject of lubrication very fresh in
my mind, that I was, for some reason, using a common soldering-iron, and
was in the act of testing the copper point of the hot iron to see if it was hot
enough to melt the solder, when, from some cause or another, instead of
merely touching the block gently with the point of the copper, I must have
pushed the sloping edge obliquely and somewhat roughly on to the flat top
of the block, fou, to my surprise, instead of melting a little pock in the
surface, the square-edged side of the copper slipped without friction right
along the face of the solder. It was a perfectly casual accident, but, under
the circumstances, it caused me a sense of mental shock, as I instantly
recognised the analogy to the skate.
The barely hot enough, parallel sharp edge of the copper, pressed and
pushed forward on the block, was just able to melt the immediate surface,
which completely lubricated the iron on the solder beneath. The then well-
known property of the lowering of the melting point of ice under pressure at
1 Phil. Trans. 1880, Part I., pp. 157—234, p. 228 in this volume.
67] ON THE SLIPPKRINESS OF ICE. 737
once presented itself; the shock was the result of the instantaneous reflection
that I had never before thought of considering why ice was slippery.
On trying to remember whether I had ever heard of any attempt to
explain the slipperiness of ice in any way — for I felt at the moment as
though everyone was laughing at me — I found that I could not recall any
mention of the subject. And then, in self-extenuation, I reflected that
water was not recognised as a lubricant, so that even James Thomson himself,
or his brother, Lord Kelvin, might have failed to realize that the melting of
the ice under the pressure of the skate would lubricate the moving skate,
and rendered the ice slippery to any hard body pressed against it. I also
reflected, that had not my mind been full of the circumstances of lubrication,
including the lubricating properties of all fluids, I should not have recognised
in the slipping of the hot iron the action of the lubricant, and that, even
if I had, I should not have attributed like properties to melted ice.
Of course, this evidence as to the cause of the slipperiness was altogether
one-sided, and it was still open for ice to have other properties which would
account for the slipping besides the property of melting under pressure, and
it was at once plain that to render the evidence complete it was necessary to
show that, under circumstances of temperature and pressure such that the
pressure was nowhere sufficient to melt the ice, the property of perfect
slipperiness of ice did not exist.
Looking carefully into the matter from the theoretical side, with Lord
Kelvin's determination of the laws of the melting point, 0'014° F. for each
additional atmosphere, it appeared that taking a weight of 140 Ibs., and an
area of 1 '4/10 (=1/7) square inch, a man skating would melt ice at 31° F.
with a skate-bearing of T4/10 square inch, while to melt ice at a temperature
of 22° F. the bearing must be reduced 1 '4/100 (= 1/70) square inch. That
is, the ice at 22° F. would have to be able to sustain a pressure up to
10,000 Ibs. on the square inch. That ice should stand such pressure at first
o. R. ii. " 47
738 ON THE SLIPPERINESS OF ICE. [67
sight seems unlikely, but then our general impression as to the hardness
of ice is derived from ice at or near its melting point.
That this theory admits of experimental verification is certain, but such
experiments only become possible when the general surroundings are at
a temperature of 22° F.
Tt was this consideration which caused me, in the first instance, to delay
any publication of the facts I observed until there came a frost sufficient for
my purpose. There have been frosts of sufficient extent when my prepara-
tions were not ready, and my preparations have been ready when there were
no frosts ; until, at last, my patience has given way and I have determined
to wait no longer. In taking this decision, however, I have been greatly
influenced by my general observations on the effect of the temperature
on the ease of skating, and on the liability to slip. I notice that without
great care you cannot walk on ice at 31^° in leather boots without nails,
whereas you can walk safely with boots and somewhat blunt nails under the
same circumstances; with a temperature of 27° you can walk with leather
boots almost as safely as on any polished floor, while with somewhat blunt
nails it is very unsafe to walk on uneven ice.
On ice near 32° skaters find no resistance however slowly they may move,
while on hard ice it is necessary to move quickly, or the skates seize, showing
that the ice melts under the edge, but owing to the small area of the
lubricating surface, the lubricant is squeezed out rapidly, thus destroying the
lubrication below certain speeds, as in Mr Tower's experiment.
But the circumstance that has most confirmed me in the view that the
slipperiness of ice is due to the lubrication afforded by the melting under
pressure is a casual but emphatic statement made by Dr Nansen, in his book
on Greenland, that at the low temperatures he there encountered the ice
completely lost its slipperiness.
INDEX.
Analyzers, harmonic, 519
Balancing of machines, 17
Boiling of water at ordinary temperatures,
578
Boundary conditions of fluid motion, 132
Brake, hydraulic, 353
Colour bands, use of, in fluid motion, 524
Criterion of steady motion, Theory of, 561
Critical velocity of water, 51, 535
Currents, action on beds of rivers and
estuaries, 326, 380, 410, 482
Diagram, indicator, 163, 368
Dilatancy of media, 203, 217
Dissipation function, 544
Dryness of steam, 591
Dynamic similarity, 321, 380, 410
Dynamics of oscillations, 8, 25, 35, 41
Dynamo, stresses in, 28, 44
Eddies in water, 51, 153
Energy, directed and undirected, 138
„ storage of, 39
„ transmission of, 106
Engine trials, 336
Equations of motion of viscous fluids, 132,
258, 544
Errors of steam-engine indicator, 163
Estuaries, regime of, 326, 380, 410, 482
Flow of gases, 311
Fluid motion, dynamical theory of, 535
„ use of colour bands in, 51,
158, 524
Friction of lubricated bearings, 228
„ in reciprocatory motion, 41
Gases, flow of, 311
Gravitation, possible explanation of, 203, 217
Harmonic analyzers, 519
„ motion, 25
Heat, mechanical equivalent of, 601
Hydrodynamics, equations of, 132, 258
Hydraulic brake, 353
Ice, slipperiness of, 734
Indicator, errors of steam-engine, 163
„ diagrams, to combine, 368
Inertia, forces due to, 1, 28
Isochronous vibration, 25
Lifeboats, qualities of, 321
Limits to speed, 1
Lubrication, theory of, 228
Mechanical equivalent of heat, 601
Media, dilatancy of, 203, 217
Model estuaries, 326, 380, 410, 482
Motion of water, two manners of, 51, 153,
524
Reservoirs of energy, 39
Resistance of water, law of, 51
Rivers and estuaries, regime of, 326, 380,
410, 482
Saturated steam, dryness of, 591
Speed, limits to, 1
Stability of motion in water, 51
740
INDEX.
Steam-engine indicator, errors of, 163
„ stresses in, 28, 44
„ trials, 336
Stresses in reciprocatory and rotary mo-
tion, 28
Theory of lubrication, 228
„ motion of viscous fluids, 535
Thermodynamics, lecture on, 138
Tides, action on beds of rivers and
estuaries, 326, 380, 410, 482
Transmission of energy, 106
Vibrations of structures, 12
„ isochronous, 12
Viscosity, character of, 235
„ of olive oil, 238
Viscous fluids, dynamical theory of, 535
Vortices in water, 524
Water boiling at ordinary temperatures, 578
Water, critical velocity of, 51, 524
„ eddies in, 51, 153, 524
Waves, action on beds of rivers and estu-
aries, 326, 380, 410, 482
END OF VOLUME II.
CAMDBIDGE : PRINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS.
Plate 1
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Plate 3
Plate 4
Plate 5
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