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HYDRAULICS
BY
F. C. LEA,
B.Sa (London Engineering),
8SNI0B WmrWOBTH 8CH0LAB; ASSOC. B. COL. 80.; A. M.INST. 0. £. ;
TELVORD PBIZBMAM; LBCTUBEB IK APPLIED MECHANICS AND ENOINEEBINO DESIGN
m THE CITT AND GUILDS OF LONDON CBNTBAL TECHNICAL COLLEGE.
LONDON
EDWARD ARNOLD
41 & 43, MADDOX STREET, BOND STREET, W.
1908
[All Rights rexerrad]
THE NEW YORK
ASTOR, LffNOX AMD
TIL^EN FOuS'OATlONa.
J
XJSTHEN the author undertook fioma time ago to writ^ thi
IT worlE, it was under the iiopressionj which irapression wa
shared by many teachers, that a book was required by Engineering
etndents dealing with the subject of Hydraulics in a wider sense
th^D that covered by existing tejtt books. In addition tho author
' ' ' ^ thoagh several excellent text books were in existence,
,r amount of experimental re&earch carried out during the
last 10 or lt5 years, very little of which has been done in this
eotantry, on the subject of the flow of water, had not received the
atteBtiun it deserved. The great developments in turbines and
ceotrifagal pumps also mt^'ited some notice*
An attempt has been made to embody the results of the latest
r€>9earcha^ in the book, and to give sufficient details to indicate
the methods used in obtaining these resuhs, especially in those
cables where such information and the references thereto, are
hkely to prove of value to those desirous of carrying out ex
perimeTits on the flow of water.
Perfiapa in no branch of Applied Science is it more difficult
to coordinate results and express them by general formulae than
in Hydraulics. Practical Engineers engaged in the design of
WEter channels frequently complain of the large differences they
obtain in the calculated dimensions of such channels by using
thp ff»rmnlae put forward by different authorities. Before any
can be used with assurance it is necessary to have son
Mi.i...4ge of the data used hi determining the empirical con^
wmniM in the formula. For this reason a little attention Jms been
giv^n to the historical development of the formulae for determining
the flow in pipes and channels, and some particulars of the data
from which the constants were determined are given. In thia
respect the loganthmic analysis of experimental data, especially
in Chapter YI, tngetlier with the plottings of Fig. 114 and the
references to experiments, will it is hoped be of assistance to
IV PREFACE
cngineerB in enabling them to choose the coefficients saitable to
given circujiiatances, and it is further hoped that the methods of
analysis given will be educational and useful to studentSj and
h^jlpful ill the interpretation of experiments.
The chapter on the flow of w^ater in pipes is an*anged so that a.
student who reads as far as section 93 should be able to solve
a large number of problems on flow of water in pipes, without
further readings At the end of the chapter the formulae derived
in the chapter are Bummarisedj and various kinds of practical
problems solved, and arithmetical examples worked out. In the
chapter on flow in channels the student who reads to section 119,
and then sections 124 and 129 should be able to foHow the
problems at the end of the chapter, and to work the examples.
Chapter YIII enables the student who is desirous of studying
the elementary theory of the impact of water on vanes, and of
turbines, to do so apart from the details of turbines, and the more
practical problems that arise in connection with their design.
The principles of construction of the various types of turbines
are illuBtiated in Chapter IX by diagrams of the simpler and
older types, as well as by drawings of the more complicated
modern turbines. The dravdngs have been made to scale, and
in particular cases sufficient dimensions are given to enable the
student acquainted with the principles of machine design to
design a turbine. The author believes the analysis given of the
form of the vanes for mixed flow turbines and also for parallel
flow turbines is new.
The subject of centrifugal pumps is treated somewhat fully,
because of the complaint the author has often heard of the
difficulty engineers and students have in determining what the
performance of a centrifugal pump is likely to be under varying
conditions* The method of analysis of the losses at entrance and
exit as given in the text, the author believes, is due to Professor
Unwin, and he willingly acknowledges his obligation to him,
Tlie general formula given in article 237 is believed to be new,
and the examples given of its application in sections 235, etc,,
shuw that by such an equation, w^hich may be called the character
istic equation for the pump, the performance of the pump under
varying conditions can be approximately determined*
The effects of inertia forces in plunger pumps and the effect of
air vessels in diminishing these forces are only imperfectly treated,
as no attempt is made to deal with the variations of pressure in
the air vessel Sufficient attention is however given to the
snbject to emphasise the importance of it, and it is probably
tr^ted as fully as is desirable, considered from a practical
PREFACE V
engineeriiig standpoint. The analysis of section 260, although
too refined for practical purposes, is of vahitj to the student in
ihat, neglecting losses which cannot very well be determined, it
enables htm to realise how the energy given as velocity head to
the vrat*;r both in the cylinder and in the suction pipe is recovered
before the end of the stroke is reached. The examples given of
** Hydraulic Machines " have been chosen as types* and no attempt
himB been made to introduce veiy special kinds of machines* Tlie
antbor has had a wide experience of this cla^ of machinery^ and
lie think« the examples illustrate sufficiently the principles and
prBCtice of the design of such machines.
The last two chapters have been introduced in the hope that
ibrnf will be of assistance to Umversity studentii^ and to candidates
fofT the Institution of Civil Engineers examinations*
if r Fronde's experiments, on the frictional resistance of boards
\mg through water, are considered in Chapter XII simply in
I ir relationship to the resistance of ships^ and no attempt has
n made, as is frequently done, to use them to determine so
ciiilJed taw*s of fluid friction for water flowing in pipes and
cbanneb.
The author harfUy dares to hope that in the large amount of
arithmetical work involved in the exercises given, mistakes will
liMt have crept in^ and he ^vill be grateful if those discovering
miMtake<t will kindly point them out.
The author wishes to express his sincereat thanks to his
friend, Mr W. A. Taylor, Wh,Sc., A.K.CS., for his kindness in
reading proofs, and for many valuable suggestions, and also to
Mr W, Hewsunj B.8c., who has kindly read through some of the
proofs.
Ty the following firms the author is ander great obligation for
the ready way in which they acceded to Ins request for information;
Messrs Kscher, Wj^ss and Co, of Zurich for drawings of
torbtnes and for loan of block of turbine filteri
Messrs Piccai*d, IHctet and Co. of Geneva for di^wings of
turbinas*
Messrs Worthington and Co* for drawings of centrifugal
pctm^j^^ and fur loan of block.
Meagre fielding and Piatt of Gloucester for drawings of
BceixmvlBtoT.
Messrs Tangye of Birmingham for drawings of pumps.
&li*^rs Glen Held and Kennedy of Kilmarnock for drawijigs of
tneter and for loan of blocks.
Messrs G, W. Kent of London for description and loan of
blockjt of Vt*niuri meter recording gear*
1
VI PREFACE
Messrs W. and L. E. Gurley of Troy, N.Y., U.S.A. for loan
of block of current meter.
Messrs Holden and Brooke of Manchester for drawing of
Leinert meter.
Messrs W. H. Bailey and Co. of Manchester for drawing of
hydraulic ram.
Messrs Armstrong, Whitworth and Co. for drawings of crane
valves.
Messrs Davy of Sheffield for loan of block of forging press.
F. C. LEA.
Gbntral Tbchnical GOLLSaE,
November, 1907.
CONTENTS.
CHAPTER I.
FLUIDS AT REST.
Introdnction. Fluids and their properties. Compressible and incom
pressible fluids. Density and specific gravity. Hydrostatics. Intensity
of pressure. The pressure at a point in a fluid is the same in aU directions.
The pressure on any horizontal plane in a fluid must be constant. Fluids
at rest with free surface horizontal. Pressure measured in feet of water.
Pressure head. Piezometer tubes. The barometer. The differential gauge.
Transmission of fluid pressure. Total or whole pressure. Centre of
pressure. Diagram of pressure on a plane area. Examples . Page 1
CHAPTER II.
FLOATING BODIES.
Conditions of equilibrium. Principle of Archimedes. Centre of
buoyancy. Condition of stability of equilibrium. Small displacements.
Metacentre. Stability of rectangular pontoon. Stability of floating vessel
containing water. Stability of floating body wholly immersed in water.
Floating docks. Stability of floating dock. Examples . . Page 21
CHAPTER III.
FLUIDS IN MOTION.
i^teady motion. Stream line motion. Definitions relating to flow of
^*ter. Energy per pound of water passing any section in a stream line.
^niottilli's theorem. Venturi meter. Steering of canal boats. Extension
^ Bernouilli^s theorem. Examples Page 37
Vlll
CONTENTS
CHAPTER IV.
FLOW OF WATER THROUGH OEIFICES AND OVER WEIRS.
Velocity of clischargc? frgni aii oriUce. Coefficient of contraction for
sharp edged orifice. Coefficient of Telocity for ehaj^ edged orifice. Bazin'st
experiments on a sharp edged orifice. Distribution of Telocity in tlie plane
of the orifice. Pressure in the plajie of tlie orifice. Coefficient of discharge
Efi^ect of fjiipprcBBed con traction on the coefficient of discharge. The form
oi the jet from sharpedged orifices. Large orifices. Drowned orifices.
Partially drowned orifice. Velocity of approadx. Coefficient of resistance,
Sndden enlargement of a current of water. Sudden contraction of a
current of water. Loss of head due to sharp* edged entrance into a pipe or
mouthpiece. MouUipieceB. Borda*8 mouthpiece. Conical mouthpieces
and nozzleu. Flow througli orifices and mouth pieces under constant
pressure. Time of emptying a tank or reservoir. Notches and weirs.
Bectangulai* sharp edged weir. Derivation of the weir formula from that
of a large orifice* Thomson'^ principle of similarity* Discliarge tlirongh
a trianglar notch by the principle of similarity. Diaeharge through a
rectangular weir by the principle of similarity. Rectangular w^eir with
end contractions, Bazin's formula for the discharge of a wx*ir. Bazin*s
and tlio CorneO experiments on weirs. Velocity of approach. Influence ei
the height of the weir sill above the beil of tlie stream on the contractioii,
Disdiarge of a weir when ttie air is not fi*eely admitted beneath the nappe.
Form of the nappe. Depressed nappe. Adhering nappes. Drovnied or
wetted nappes. Instability of the form of the nappe. Drowned weirs with
sharp crests* Yei'tical weirs of small thickness. Depressed and wetted
nappes for flatcrested weirs. Drowned nappes for flatcrested weirs. Wide
flatcreated weirs. Flow over dams. Form of weir for accurate gau
Boussinesq'a theory of the discharge over a weir* Determining by
proximation the discliarge of a weir, when the velocity of approach 1?"
unknown. Time required to lower the water in a reservoir a given dbtanoe
by means of a weir. Examples ....... Page 60
CHAPTER V.
FLOW THROUGH PIPEK.
Resistances to the motion of a fluid in a pii>e. Loss of head by friction*
Head lost at the entrance to tlie pipe. Hydrauhc gradient and virtual
slope* Determination of the loss of head due to friction. Reynold'a
apparatus. Equation of flow in a pipe of uniform diameter and determi
nation of the head lost due to friction. Hydrauhc mean dex>th. Empiriod
CONTENTS
IX
LtUae for lo^ at head due to friction, Formola of Darcy, Variatioa
ol C m the formula v — Cs^mi with service, Gangtdllet and Kutter*s
foo^itilA. BeynoUrsi experiroents and the logarithmic formula. Critical
f«locit>% Critical yelocity by the method of colour baade. Law of
frictional resistance for velocities above tlie critical velocity* The de*
terminAlidii ol the values of C given in Table XII. Variation of k, in the
lulft i = It*, with Hm diameter, Oriiieifiin of experiments. Piezometer
fittings. Effect of temperature on the velocity of flow, Lo«s of head due
to bands and elbows. Variations of the velocity at the cross section of a
liadrical pipe. Head necessary to give the meaa velocity v^ to the
in Uie pipe. Practical problcniB. Velocity of flow in pipeB, Trans
snteioil of power along pipes by hydraulic pressure. The hmiting diameter
«l cwai iitm pipes. Pressures on pii>e bends. Pressure on a plate in a pipe
filled With flowing water. Pressure on a cylinder. Examples , Page 112
^Bfosmi
^■fittJl!
CHAPTER VI.
FLOW IN OPEN CHANNELS.
Vamly of the forma of channels. Steady motion in uniform channeli*
^ for the flow when the motion is uniform in a channel of uniform
land slope. Formula of Chezy, Fonnulae of Prony and Eytelwein.
i1a €d Darcy and Bajdn, GangniUet and K utter *8 formula. Bazin's
bi. Variations of the coefficient C. Logarithmic formula for flow In
Approximate formula for the flow in earth channels. Diwtribu
L of velocity in tiie cross section of open ohannek, Fomi of the curve
I velocitiets on a vertical section. The dopes of channels and the velocities
rallown] in them. Sections of a(uo<liictft and sewerH. Siphons forming
pan of a^edncte. The best form of channel. Depth of flow in a circular
diasne] for maximum velocity and maximum chscharge. Curves of velocity
ttud discharge for a channel. Apphcations of the formulae. Problems.
Dpleci ,..., Page 178
CHAPTER VIL
QAtrOiNG THE FLOW OF WATER
[ the flow of water by weighing. Meters. Measuring the flow
of an orifice. JleaJHiiring the flow in oi^en channels. Surface
Doable floats. Bod floats). The current meter, Pitot tube. Cali^
bvmtiim of Pitot tubes. Gauging by a weir. The hook gauge. Oaujring
h# (low in pipes ; Venturi meter. Deacon 'u waste water meter, Kennedy's
Oanging the flow of streams by chemical means. Examples
Page 2H
CONTENTS
CHAPTER YllL
IMPACT OF WATER OK VAlflTS.
Deflnitioii of ?ector. Sum of two vectors. Resultant of two Telocitiee,
Difference of two sectors. Impulse of water on vanes* Relative velocity*
Befinition of relative velocity as a vector. To find the presstire on a
moving vane, and the rate of doing work. Impact of water on a vane
when the directions of motion of the vane and jet are not paralleL
Conditions which the vanes of hydra viltc machines should fiatisfy*
De&nition of angular m omen torn. Change of angnl&r moment nm. Two
important principles. Work tlone on a series of vanea filled to a wheel
expressed in terms of the velocities of whirl of the water entering and
leaving the wheel. Carved vanes. Pel ton whecL Force tending to move
a vessel from which water is issuing through an orifice. The propulsion
of ships by water jets. Examples . , « . > . Pa^fe 261
CHAPTEE IX
WATER WHEELS AND TURBINES.
Overehot water wheeb, Brea4^t wheel Sagehien wheels. Impulse
wheelB. Poncelet wheel. Turbines. Reaction turbines. Outward flow
turbines. Losses of head due to frictional and other resistances in outward
flow turbines. Some actual outward flow turbines. Inward flow turbiueo.
Some actual inwarrl flow turbines. The best peripheral velocity for
inward and outward flow ttirbines, Exi^erimental determination of the
beat peripheral vekjcity for inward and outward flow turbines. Value of e
Yv
to be used in the formula — =^H. The ratio of the velocity of wliirl Y to
the velocity of the inlet periphery t\ The velocity with which water
leaves a turbine. Bernouilli's equations for inward and outward flow
turbinei neglecting friction. Bemouilli*s eixuations for the inward and
outward flow turbinea including friction. Turbine to develope a given
horse power. Parallel or axial flow turbinea. Regulation of the flow to
parallei flow turbines, Bernouilli's equations for axial flow turbii
Mixed flow turbinea. Cone turbine. Effect of changing the direction
the guide blade > when altering the flow of inward flow and mixed flow
turbines. Effect of diminishing the flow thi^ough turbinea on the velocity
of exit. Kegulatiou of the flow by means ^ cylindrical gates. The Bwain
ptte. The form of the wheel vanes between the inlet and outlet of
turbines* The limiting head for a single stage reaction tnrbine. Series
or multiple stage reaction turbines. Impulse turbines. The form of tlie
vanes for impulse turbines, neglecting ifriction. Triangles of velocity for
an axial flow impulse turbine considering friction. Impulse turbine lor
Mgh head, Pelton wheel. Oil pressure governor or regulator. Water
pressure regulators for impulse turbines. Hammer blow in a long turbine
supply pipe. Examples *,,.,,.< Page 283
J
CONTENTS
CHAPTER X,
PUMPS.
Cesitrifiig&l &nd tor&me pimips. Starting centrifugal or torbme pnmps.
I of the v^nes of centrifugal pumps. Work done on the water by the
^ifbeel* lUMo ol velocity of whirl to peripheral velocity. The kinetic energy
of ihm water &t exit from the wbeeL Gross lift of a centrifugal punip^
£fficsietiides of a centrifugEl pump. Experimental deteimlnatioti of the
dficieocy of a centrifiigal pump. Desi^ of pump to give a discharge Q,
Tlie centiiftigal head impressed on the water by the wheel. Head velocity
citiiFe ef & centrifugal pump at zero discbarge. Variation of the discbarge
of & oenfcrifngal pomp with tlio head when the speed ia kept constant*
BemomUi's equations applied to centrifugal pumps. Ltoeses in centrifugal
Variation of the bead with diHcbarge and with the speed of a
I pump. The effect of the variation of the centrifugal head and
the low hf friction on the discharge of a pump. The c^ect of the diminu
ol ibe oentiifugal head and tlie increase of the friction head as the
ft, (m the velocity. Di^targe curve at constant head» Special
ftnmiigeniOBta for converting tbe velocity bead — , with which the water
laftves ttie wheel., into pressure bead. Tmbino pumps. Losses in tbe
flpltml aiam^ of centrifugal pumps* General equation for a oentrifngal
»r^ i The T jpii t.ing height to which a single wheel centrifugal pump caa ]
lie laed to raiae water. The suctioa of a centrifugal pump. Series or
mulsiti&age ttirfaine pumps^ Advanta^^es of ceotrifngal pumps. Pump
«Urfwiig into a long pipe line. Parallel flow turbine pump. Inward flow
tBrMoe pnmp* Reciprocating pumps Coeflicient of discharge of the
{mmp. Slip^ Diagram of work done by tbe pump* The accelerations
ol the pomp plunger and tbe water in tlie suction pipe. The effect of '
oioeelonition of tbe plonger on tbe pressure in Uie cylinder during the
sncttoii steoike. Accelerating forces in the delivery pipe. Variation of
IHiiiiiirr in tbe eylinder due to friction. Air vessel on the suction pipe.
Air TCBBcl on the delivery pipe. Separation daring the suction stroke.
K^sliTe slip, Heimration in tbe delivery pipe, Diagram of work done i
ooiyddering tbe variable qnantity of water in the cylinder. Head lost at
tbe ww^&OMk talve. Variation of tbe pressure in bydrauhc motors due to
iaciriia incces. Worked examples. High pressure plunger pump, Tangye
Oeplex fiiisii. Tbe hydraulic ram. Lifting water by comprised air.
Kxaifitiipa Page 392
CHAPTER XI,
HYDRAULIC MACHI!?ES.
Jomta tmd packings used in hydraulic work. The acciimulator. Dif
feffpiiftl oocamnk.tor. Air accumulator. Intenaifiers. Steam intensiiiera.,
HydeMiUo forging preis. Hydraulic cranes. Double power crane
Bjdrsnlie crane valveSi Hydraubc press. Hydraulic riveter. Brother*
hood Odd Bigg hydraulic engines. Examples . , . . Page 4B^
Xll CONTENTS
CHAPTER XII.
RESISTANCE TO THE MOTION OF BODIES IN WATER.
Froude*8 experiments on the resistance of thin boards. Stream line
theory of the resistance offered to motion of bodies in water. Determination
of the resistance of a ship from that of the model. Examples . Page 507
CHAPTER XIII.
STREAM LINE MOTION.
Hele Shaw's experiments. Curved stream line motion. Scouring of
river banks at bends ...•••... Page 517
Answers to Examples • . Page 521
Index Page 525
HYDEAULICS.
CHAPTER I.
FLUIDS AT REST.
1. IntrodnctioiL
The science of Hydraulics in its limited sense as originally
understood^ had for its object the consideration of the laws
regulating the flow of water in channels, but it has come to
have a wider significance, and it now embraces, in addition, the
study of the principles involved in the pumping of water and other
fluids and their application to the working of different kinds of
machines.
The practice of conveying water along artificially constructed
channels for irrigation and domestic purposes dates back into
it antiquity. The Egyptians constructed transit canals for
purposes, as early as 3000 B.C., and works for the better
of the waters of the Nile were carried out at an even
According to Josephus, the gardens of Solomon
utiful by fountains and other water works. Tlie
some of which were constructed more than
among the " wonders of the world," and
\ Athens is partially supplied with water by
bt constructed probably some centuries before
ydraulics, however, may be said to have oi
ce at the end of the seventeenth century wh
osophers was drawn to the problems involv
of the fountains, which came into considerable
idscape gardens, and which, according t
great beauty and refreshment." The fou'
Torricelli and Marriott from the exper'
from flie theoretical, side. TThe experip
of Marriott to determine the discharge ^ ^ ™*
fees in the sides of tanks and through shr ... ,
° / cubic inches.
L. B.
The Aqueducts of Rome. FroDtinus, translajf 85*9 cubic feet.
1—2
2 HYDRAUUGS
mark the first attempts to determine the laws regulating the
flow of water, and Torricelli's famous theorem may be said to
be the foundation of modem Hydraulics. But, as shown in the
chapter on the flow of water in pipes, it was not until a century
later that any serious attempt was made to give expression to the
laws regulating the flow in long pipes and channels, and firacti
cally the whole of the knowledge we now possess has been
acquired during the last century. Simple machines for the
utilisation of the power of natural streams have been made for
many centuries, examples of which are to be found in an interest
ing work Hydrostatika and Hydrauliks written in English by
Stephen Swetzer in 1729, but it has been reserved to the workers
of the nineteenth century to develop(» all kinds of hydraulic
machinery, and to discover the principles involved in their correct
design. Poncelet's enunciation of the correct principles which
should regulate the design of the "floats" or buckets of wafer
wheels, and Foumeyron's application of the triangle of velocities
to the design of turbines, marked a distinct advance, but it must
be admitted that the enormous development of this class of
macliinery, and the very high standard of efficiency obtained, is
the outcome, not of theoretical deductions, but of experience,
and the careful, scientific interpretation of the results of
experiments.
2. Fluids and their properties.
The name fluid is given, in general, to a body which offers
very small resistance to deformation, and which takes the shape
of the body with which it is in contact.
If a solid body rests upon a horizontal plane, a force is required
to move the body over the plane, or to overcome the friction
between the body and the plane. If the plane is very smooth
the force may be veiy small, and if we conceive the plane to be
perfectly smooth the smallest imaginable force would move the
body.
If in a fluid, a horizontal plane be imagined separating the
'^lid into two parts, the force necessary to cause the upper
' slide over the lower will be very small indeed, and
however small, applied to the fluid above the plane
' to it, Avill cause motion, or in other words will cause
of the fluid.
a very thin plate be immersed in the fluid in any
te can be made to separate the fluid into two
'^tion to the plate of an infinitesimal force,
!t fluid this force would be zero.
FLUIM AT REST 3
luide found in nature are not perfect and are
cosity but when they are at rest the conditions
equilibritiin can be obtained, with sufficient accuracy, on
aasumption that they are perfect fluids, and that therefore
^tangential stresses can exist along any plane in a fluid.
Its branch of the study of fluids is called Hydrostatics; when
laws of movement of fluids are consideredj as in Hydraulics,
5)8 tangential, or firictional forces have to be taken into
ideration.
1
Compressible and inooDELpresslble fluidB.
There are two kinds of fluids, gases and liquids, or those which
are easily compressed, and those which are compressed with
* "" * V, The amount by which the volumes of the latter are
ftir a very large variation in the pressure is so small that
practical problems this variation is entirely neglected, and
bey are therefore considered as incompressible fluids.
In this volume only incorapreasible fluids are considered, and .
Attention is eonflned, almost entirely, to the one fluid, water. ^H
4* Density and specific gravity.
The deusity uf any substance is the weight of unit volume at
he standard temperature and pressure.
f specific gravity of any substance at any temperature and
Bnpe id the ratio of the weight of unit volume to the weight
tmit volume of pure water at the standard temperature and
ttxre.
The variation of the volume of liquid fluids, with the pressure,
as eluted above, is negligible, and the variation due to changes of
tempfn^lnrc, such as are ordinarily met with, is so small, that in
practical problems it is unnecessary to take it into account.
In tht? case of water, the presence of salta in solution is of
,ter importance in determining the density than variation
temperature, as will be seen by comparing the densities of sea
Vator and pure water given in the foUowing table.
■f t€
TABLE I.
Useful data.
One cnbio foot of water at 391' F. weighs 62425 lbs.
60* F. „ 6286 „
One oablo foot of average sea water at 60° F. weighs 64 Ibe*
0De gallon of pure water at 60* F, weighs 10 lbs.
One gallon of ptue water bas a volome of 277*25 cubic mches*
One kai ol pure water at 60" F. has a volume of 85*9 cubic feet.
1—2
HTDRA.ULICB
Table of dendtiea of pure water.
Temperatnn
^rees Fahxenheii
Demdty
82
09967
89*1
1000000
M)
OiW978
60
099906
80
099664
104
0il9288
From the above it will be seen that in practical {irobleiiis it
will be KufHciently near to take the weight of one cubic foot of
fresh water as 62*4 lbs., one gallon as 10 ponnds, 6124 gallons in a
cubic foot, and one cubic foot of sea water as 64 pounds.
5. Hydrostatics.
A knowledge of the principles of hydrostatics is very helpful
in approaching the subject of hydraulics, and in the wider sense
in which the latter word is now used it may be said to include the
former. It is, therefore, advisable to consider the laws of fluids
at rest.
There are two cases to consider. First, fluids at rest under the
action of gravity, and second, those cases in which the fluids are
at rest, or are moving very slowly, and are contained in closed
vessels in which pressures of any magnitude act upon the fluid,
as, for instance, in hydraulic lifts and presses.
6. Intensity of pressure.
The intensity of pressure at any point in a fluid is the pressure
exerted upon unit area, if the pressure on the unit area is uniform
and is exerted at the same rate as at the point.
Consider any little element of area a, about a point in the fluid,
and upon which the pressure is uniform.
If P is the total pressure on a, the Intensity of Pressure p, is then
P
or when P and a are indefinitely diminished,
^ap
^ da'
7. The pressure at any point in a fluid is the same in all
directions.
It has been stated above that when a fluid is at rest its resist
ance to lateral deformation is practically zero and that on any
plane in the fluid tangential stresses cannot exist. From this
experimental fact it follows that the pressure at any point in the
fluid is the same in all directions.
FLUIDS AT BEST
Consider a smaU wedge ABC, Pig, 1^ floating immersed in a
II aid at rest.
Sine© there caimot be a tangential
reag on any of the planes AB, BC, or AC,
tf prussurad on them must be normaL
Lei p, pi and ps be the intenaitiea of
tires on these planes respectively,
H© weight of the wedge will be very
fftnal] and may he neglected*
An thtt wedge m in equihbriiim under the forces acting on
three faces, the resolved components of the force acting on
;C in the directions of p and pi mnst balance the forces acting
AB and BC respectively*
Therefore p, . AC cos  p , AB,
d psACsin^^PaBC,
But AB = ACcose,
And BC^ACsin^*
Thert*fore p = pj = p, ,
8. The pressure on any horizontal plane in a fluid must
eonfilant.
Confiider a small cylinder of a fluid joining any two points A
nd B OQ the same horissontal plane in the fluid.
Since there can be no tangential forces acting on the cylinder
purallel to the axiB^ the cylinder must be in equilibrium under the
pressures on the ends A and B of the cylinder, and since these
re of equal area, the pressure must be the same at each end of
he cylinder*
9. Fluids at rest, with the free surface horizontal,
'Hie pres.sure per unit area at any depth h below the free
of a finid due to the weight of the fluid is equal to the
b.t of a column of fluid of height k and of unit sectional area.
Let the pressure per unit area acting on the surface of the
Sold be p lbs. If the fluid is in a closed vessel, the pressure p may^
bave any assigned value, but if the free surface is exposed to tin
aCiQO&phere, p will be the atmospheric pressure.
klf a small open tube AB, of length h^ and cross sectional area a,
placed in the fluid, the weight per unit volume of which is
Ib&, with its axis vertical, and its uppor end A coincident with
9 0tif£ftC0 of the fluid, the weight of fluid in the cylinder uiUBt be
ip.apA Ibfls. The, pressure acting on the end A of the colunm
is pa Ib^.
a
HTDRAUUCS
Since there cannot be any force acting on the colnmn parallel
to the eddes of the tube, the force of imiA lbs. + pa lbs. most be
kept in equilibrium by the pressure of the external fluid acting on
the fluid in the cylinder at the end B.
The pressure per unit area at B, therefore,
= ^ = (wh + p) lbs.
The pressure per unit area, therefore, due to the weight of the
fluid only is tch lbs.
In the case of water, w may be taken as 62*40 lbs. per cubic
foot and the pressure per sq. foot at a depth of h feet is, therefore,
62*40% lbs., and per sq. inch '433b lbs.
It should be noted that the pressure is independent of the form
of the vessel, and simply depends upon the vertical depth of the
point considered below the surface of the fluid. This can be
illustrated by the different vessels shown in Fig. 2. If these
were all connected together by means of a pipe, the fluid when
at rest would stand at the same level in all of them, and on any
horizontal plane AB the pressure would be the same.
TUJ
.^/
Pressure an the Pixuie A^^wh lbs persq Foot,
Fig. 2.
If now the various vessels were sealed from each other
by closing suitable valves, and the pipe taken away without
disturbing the level CD in any case, the intensity of pressure on
AB would remain unaltered, and would be, in all cases, equal
to wh.
Example, In a condenser containing air and water, the pressure of the air is
2 lbs. per sq. inch absolute. Find the pressure per sq. foot at a point 8 feet below
the free surface of the water.
I> = 2xl44 + dx62*4
=475*2 lbs. per sq. foot.
FLUIDS AT REST
10. PreiBiires meaBiired in feet of water. Fressure head.
It is oonvenient in hydrostatics and hydraulics to express the
inreesore at any point in a fluid in feet of the fluid instead of pounds
per sq. foot or sq. inch. It follows from the previous section that
if the pressure per sq. foot is p lbs. the equivalent pressure in feet
of water, or the pressure head, is A = ^ ft. and for any other fluid
w
having a specific gravity p, the pressure per sq. foot for a head
h of the fluid is p = w.p.hy or A = — .
top
IL Piesometer tubes.
The pressure in a pipe or other vessel can conveniently be
measured by fixing a tube in the pipe and noting the height to
which the water rises in the tube.
Such a tube is called a pressure, or piezometer, tube.
The tube need not be made straight but may be bent into any
form and carried, within reasonable limits, any distance horizon
tally.
The vertical rise h of the water will be always
where p is the pressure per sq. foot in the pipe.
If instead of water, a liquid of specific gravity p is used the
height h to which the liquid will rise in the tube is
w .p
Example. A tube having one end open to the atmosphere is fitted into a pipe
containing water at a pressure of 10 lbs. per sq. inch above the atmosphere. Find
the height to which the water will rise in the tube.
The ^*ater will rise to such a height that the pressure at the end of the tabc in
the pipe due to the column of water will be 10 lbs. per sq. inch.
Therefore
^ 10x144 ^„^^ ^ ,
h= =2308 feet.
12. The barometer.
The method of determining the atmospheric
pressure by means of the barometer can now be
understood.
If a tube about 3 feet long closed at one end be
completely filled with mercury, Fig. 3, and then
turned into a vertical position with its open end
in a vessel containing mercury, the liquid in the
tube falls until the length h of the column is about
30 inches above the surface of the mercury in the
vessel.
Fig. 3.
HYDRAULICS
Since the pressure p on the top of the mercury is now zero, iYm^
pressure per unit area acting on the section of the tube, level witX^
the surface of the mercury in the vessel, must be equal to ih^
weight of a column of mercury of height h.
The specific gravity of the mercury is 13*596 at the standard
temperature and pressure, and therefore the atmospheric pressure
per sq. inch, pa, is,
80" X 13596 X 624 ,.^„ . ,
Pa= ^2 X 144 ~ ' • P®^ ^* "^
Expressed in feet of water,
14*7 X 141
A = i> I— = 3392 feet.
62 4
This is so near to 34 feet that for the standard atmospheric
pressure this value will be taken throughout this book.
A similar tube can be conveniently used for measoring low
pressures, lighter liquids being used when a more sensitive gauge
is required.
13. The differential gauge.
A more convenient arrangement for measuring pressures^ and
one of considerable utility in many hydraulic exx>eriment8, iB
known as the differential gauge.
Let ABCD, Fig. 4, be a simple U tube
containing in the lower part some fluid of
known density.
If the two limbs of the tube are open to
the atmosphere the two surfaces of the fluid
will bo in the same horizontal plane.
If, however, into the limbs of the tube a
lighter fluid, which does not mix with the
lower fluid, be poured until it rises to C in
one tube and to D in the other, the two
surfaces of the lower fluid ^vill now be at
different levels.
Let B and E be the common surfaces of
the two fluids, h being their difference of
level, and hi and th the heights of the free
surfaces of the lighter fluid above E and B respectively.
Let p be the pressure of the atmosphere per unit area, and d
and di the densities of the lower and upper fluids respectively.
Then, since upon the horizontal plane AB the fluid pressure must
be constant,
p + rfjfes = p + djii + dh,
d\ (Jh — hi) = dh.
Fig. 4.
FLUIDS AT REST
9
li now^ instead of the two limbs of the U tube being open to
»tmcMipha?e, they are ooimected by tubes to closed vessels in
ich the pressures are pi and p^ pounds per sq* foot respectively^
d ^1 and fh are the vertical lengths of the colnnms of fluid above
ind B respectively^ then
Pt^di^kt =Pi + di^hi + d.hf
p^ Pi^d,kdi{fhhi).
kik application of snch a tube to determine the difference of
fmuTe at two pointa in a pipe containing flowing water is shown
I % 88, page 116.
Fluidk generally used in such U ttihee. In hydraulic experiments
f Bpper part of the tube is filled with water, and therefore the
lid b the lower part must have a greater density than water.
fWthe difference of pressure is fairly large, mercury is generally
tli8$peci£c gravity of which ia 13'596. When the difference
' pemife is small, the height k is difficult to measure iv^th
m that if this form of gauge is to be used, it is desirable
i»pli«e the mercury by a lighter liquid. Carbon bisulphide
s Ijeen used but its action is sluggish and the meniscus between
and the water is not always well defined,
jth)bi*imne ^v^ good results^ its prin
'1*1 fault being that the falling meniscus
^ DOt very quickly assume a definite
ape.
Hf inverted air gauge, A mure sen
^*^ gauge can be made by inverting a
^ and enclosing in the upper part
**rt&iii quantity of air as in the tube
Be, Fig. 5.
I^ the pressure at D in the limb DF
Pi poimds per square foot, equivalent
;*liead hi of the fluid in the lower part
^ gauge, and at A in the limb AK let
* Iffmure be p%^ equivalent to a head h%.
W A bt* the difference of level of G and C*
Tfien if CGH contains air^ and the weight of the air be
[beted, being very small, the pressure at C muBt equal the
at G I and since in a fluid the pressure on any horizontal
constant the pressure at C is equal to the pressure at D,
the pressure at A equal to the pressure at B. Again the
at G is equal to the pressure at K.
Th^tiefore hth^ht^
Pfptpw.h,
H
».^
iBlCJ
^ F
u
Fig. &.
v.i uy using, in tli«
upper part of tlie tube, an oil liglitei
than water instead of air, as sliowii
in Fig. 6.
Let pi and pj be the pressures in
the two limbs of the tube on a given
horizontal plane AB, hi and }h being
the equivalent heads of water. The
oil in the bent tube will then take up
some such position as shown, the
plane AD being supposed to coincide
with the lower surface C.
Then^ since upon any horizontal
plane in a homogeneous fluid the
pressure must be constant, the pres
■es at G and H are equal and also
>se at D and C.
Let pi be the specific gravity of
) water, and p of the oil.
m Pihiph = Pi(hi'h),
i irefore fc (pip) =Pi (^fei)
(Pi  p)
)stituting for hi and h^ the values
FLUIDS AT EEST
thi^t either kerosene, gasoline, or sperm oil gave excellent results,
h\x% flperm oil waa too sluggish in its action for rapid work.
Kei?«fieiia gave the be^t results.
nkmperaiure coeffictent of the inverhd oil gauge, Dnhke the
inverted far gaog^ the oil gauge has a considerable temperature
oc^^rfficietit, aa will be seen from the table of specific gravities at
viLriotis t^'Uipera tares of water and the kerosene and gasoline used
br >Villiams, HubbeU and Fenkell
In this Cable the specific gravity of water is taken as unity
Wat^
T
40 ! 60
100092
100
EefQseiie
I'OOOO *9041
40 60 100
■7955 *7879 '7725
Gft«oliii6
40 60 80
*72147 71587 '70547
jf^^ caltbrafwn of ike hiverted oil gauge. Messrs Williams,
i^ >il>beU and Fenkell have adopted an ingenious method of
<^^lihniting the oil gauge. This will readily be understood on
'**f»PBiioe to Fig. 6
*The difference of level of E and F clearly gives the difference
^ liand acting on the plane AD in feet of water, and this from
<^^^tion (1) equals M?l:ip) ,
Pi
"VFatar is put into AE and FD so that the surfaces E and F
*''^*^ <m the same levels the common surfaces of the oil and the
*^^ter abo being on the 'same level, this level being aiero for the
^^^ Water is then run out of FD until the surface F is
f^^djy I inch below E and a reading for h taken. The surface F
^ ^^in lowered 1 inch and a reading of h taken* This process
* csontinued tmtfl F is lowered as far as convenient, and then
irt^ wati*r in EA is drawn out in a similar manner. When E
^^itrl p are again level the oil in the gauge should read zero.
14. Tranamission of fluid pressure.
If an external pressure be applied at any point in a fluid, it ia
^^nsmifcted equally in all direc
^u^B through the whole mass.
T^UJ is proved experimentally
h means of a simple apparatus
^ch aa shown in Fig. 7,
I^ lire P is exerted upon
>^' , on Q of a sq inches Fig. 7
12 HYDRAULICS j
P
area, the pressure per unit area p = — , and the piston at B on the
same level as Q, which has an area A, can be made to lift a load W
P
equal to A — ; or the pressure per sq. inch at R is equal to the
pressure at Q. The piston at R is assumed to be on the same level
as Q so as to eliminate the consideration of the small differences of
pressure due to the weight of the fluid.
If a pressure gauge is fitted on the connecting pipe at any
point, and p is so large that the pressure due to the weight of the
fluid may be neglected, it will be found that the intensity of
pressure is p. This result could have been anticipated from that
of section 8.
Upon this simple principle depends the fact that enormous
forces can be exerted by means of hydraulic pressure.
If the piston at Q is of small area, while that at R is large,
then, since the pressure per sq. inch is constant throughout the
fluid,
W__A
P "a'
or a very large force W can be overcome by the application of
a small force P. A very large mechanical advantage is thus
obtained.
It should be clearly understood that the rate of doing work
at W, neglecting any losses, is equal to that at P, the distance
moved through by W being to that moved through by P in
the ratio of P to W, or in the ratio of a to A.
Example, A pomp ram has a stroke of 3 inches and a diameter of 1 inch. The
pump supplies water to a lift which has a ram of 5 inches diameter. The force
driving the pomp ram is 1500 lbs. Neglecting all losses dne to friction etc.,
determine the weight lifted, the work done in raising it 5 feet, and the nnmber
of strokes made by the pomp while raising the weight.
Area of the pump ram ='7854 sq. inch.
Area of the lift ram = 19*6 sq. inches.
Therefore W = 1?1J^= 37,500 lbs.
Work done = 37,600 x 5 = 187,500 ft. lbs.
Let N equal the number of strokes of the pump ram.
Then N x A x 1500 lbs. = 187,600 ft. lbs.
N= 600 strokes.
or ^ or whole pressure.
From (2) it leessure acting on a surface is the sum of all the
different from thacting on the surface. K the surface is plane all
differences of presiLel, and the whole pressure is the sum of these
•Frou
PLUIJDS AT REST
13
Ijet any sarfece, which need not be a planej be immerBed
a finid. Let A be the area of the wetted aiirface, and k the
head at the centre of gravity of the area. If the area
is inLHiersed in a fluid the presisure on the surface of which m zero,
tiie free snrface of the fluid will be at a height h above the centre
of gravity of the area* In the case of the area being immeraed in
a flnidf the surface of which is exposed to a pressure p, and below
vliick the depth of the centre of gravity of the area is ih^ then
w
If the area exposed to the fluid pressure is one face of a body^
the opposite faee of which m tsxposed to the atmospheric pT*e&aure,
in the case of the side of a tank containing water, or the
dam of Fig, 14, or a valve closing the end of a pipe as
Fig. 8 the pressure due to the
itmosphere is the same on the two
■fec^ts and therefore may be neglected.
Let IP be the weight of a cubic
foot of the fluid. Then, the whole
preesure on the area is
If the surface is in a horizontal
plane the theorem is obviously true,
fliDce the intensity of pressure is eon
slant and equals w , K
In general, imagine the surface.
Fig % divided into a large number of small areas a> ai, o^ ... *
Let « be the depth below the free surface FS^ of any element
of area a ; the pressure on this element = tc . a; , a.
Tlie whole pressure P ^ Sw, ac * a.
But tr is constant, and the sum of the moments of the elements
[of the area about any axis equals the moment of the whole area*
fal>ont the same axis, therefore
5iE , fi = A . ftj
juid P = W7 . A . A.
16* Centre of pressure.
The centre of pressurt? of any plane
snrfo^re acted upon by a fluid is the
point of action of the resultant pressure
acting upon the surface.
J^^ of ihs centre of presswre. Let
DBC, Fig. 9, be any plane surface
exposed to fluid pressure.
* Em %eitbook8 on Me^haoiei.
Fig. 8.
B(
ffi
C
Fig. 9.
14 HYDRAULICS
Let A be the area, and h the pressure head at the centre of
gravity of the surface, or if FS is the free surface of the fluid, h is
the depth below FS of the centre of gravity.
Then, the whole pressure
F = w.A.h.
Let X be the depth of the centre of pressure.
Imagine the surface, as before, divided into a number of small
areas a, ai, Os, ... etc.
The pressure on any element a
= w .a.x,
and P = ^wax.
Taking moments about FS,
P.X= (wax* + waia>i*+ ...)
= Siooa;',
or X =
wAh
Ah '
When the area is in a vertical plane, which intersects the
surface of the water in FS, Soa?* is the "second moment" of the
area about the axis FS, or what is sometimes called the moment
of inertia of the area about this axis.
Therefore, the depth of the centre of pressure of a vertical
area below the free surface of the fluid
moment of inertia of the area about an axis in its own plane
_ ' and in the free surface
~" area x the depth of the centre of gravity '
or, if I is the moment of inertia,
Areao A.h
^^ t of Inertia about any axis. Calling I© the Moment
Thereforebout an axis through the centre of gravity, and I the
_ ,  nertia about any axis parallel to the axis through the
Work done .. , . ,. . i. jl. .l
» ^, , xty and at a distance h from it.
Let N equal ti*^ t t a 7 •
Then I = Io^AA«.
*ea is a rectangle breadth 6 and depth d.
P=ir.6.d./i,
or ' or 6^
From(2)itieess. "12+^^'*'
different from thacti. ^+wfc«
differences of presdel, X=
Proc.
hdh
FLUIDS AT REST
16
If the free sorfiuie of the water is level with the upper edge of the rectangle,
,= , mod X = .^
2'
is a drole of radios B.
X=
^+xR.*.
TSFT
If the top of the circle is jast in the free sorfaoe or A=B,
X = 1R.
TABLE n.
Table of Moments of Inertia of areas.
Form of area
Moment of inertia abont
an axle AB through the
C. of G. of the section
Rectangle
JlJ
A^'
Triangle
k''''
Circle
1©°
64
Semicircle
rj^
About the axis AB
8
Paarabola
i
>
16
HYDRAUUCS
17. Diagram of pressure on a plane area.
If a diagram be drawn showing the intensity of pressure on
a plane area at any depth, the whole pressure is equal to the volume
of the solid thus formed, and the centre of pressure of the area is
found by drawing a line through the centre
of gravity of this solid perpendicular to the
area.
For a rectangular area ABCD, having the
side AB in the surface of the water, the
diagram of pressure is AEFCB, Fig. 10. The
volume of AEFCB is the whole pressure and
equals ^bcPw, b being the width and d the
depth of the area.
Since the rectangle is of constant width,
the diagram of pressure may be represented
by the triangle BCF, Fig. 11, the resultant pressure acting
through its centre of gravity, and therefore at id from the surface.
^''' ""s. 12 and 13.
h g f^
ow cucL.
Fig. 12.
For a vertical circle the diagram of pressure is as shown in
The intensity of pressure ah on any strip at a
o» oovii }i^ ig ^h^^ The whole pressure is the volume of the truncated
^Sk'er efkh and the centre of pressure is found by drawing a
'T)endicular to the circle, through the centre of gravity
^^^<uncated cylinder.
Work done
Let N eqaali
Then
or ' Oi
From (2) it lee^o)
different from thaci
differences of presde.
♦ IVoc Pig. 18.
^,
FLUIDS AT REST
It
Another, and fretiuently a very convenient method of dpfcer
tniniiig^ the depth of the centre of pressure, when the whole of the
«x*ea is itt some dietanco below the surface of the water, is to
eoiiisider the pressure on the area as made up of a uniform pressure 1
over the whole surface, and ii pressure of variable intensity.
Take again, as an example^ the vertical circle the diagrams of
pressure for which are shown in Figs. 12 and 13.
At any depth h the intensity of pressure on the strip ad is
wh„  wHa ^ y^fh '
The pressure on any strip ad is^ therefor*?, made up of a
4' ' presssure per unit area wtiA and a variable pressure wJh ;
ii : ^vhole pressure is equal to tlie volume of the cylinder efghj
tSg. 12, together with the circular wedge fkg.
The wedge fkg is equal to the whole pressure on a vertical
circle, the tangent k> which is in the free surface of the water and
equals *r * A . ^ , and the centre of gravity of this wedge will he at
the same vertical distance from the centre of the circle as the
centre of pressure when the circle touches the surface* The whole
pressure P umy be supposed therefore to be the resultant of two
pttndlel forces Pi and Pa acting through the centres of gravity of
the cylinder efght and of the circular wedge fkg respectively, the
ma^tudes of P^ and Pj being the volumes of the cylinder and
the wedge respectively*
Tu find the centre of pressure on the circle AB it is only
iieoeeeary to find the resultant of two parallel forces
Pi^A.whj, and P^^w.Al^jt
<jt irhicli P, act** at the centra c, and P, at a point ct which is at
a distance from A of  r.
£iai^. A tsJMonry dam, Fig* 11,
jM«i hetirtit of 80 fe*t tmm the founda
Ml aa4 tbe »&ter tmi^ is ineliJaed At
*^^K^tm 10 the vcTiical ; find the whole
f^*o»* m the fkce tlue to the water per
•*^»iiilii ijf the dftm^ «nd the oehtr^ of
y*^*Oft, «h«i3 th* Wfttcr Harf*o« t«* levpl
^*4ie tii) of lh« imm, Thu atmo
*9^ pnmam m^ be »etlecicd<
^^ vJlole pnsmme will be the force
J^J^ to flff •rtura tii«} iUtti, Biace Uie
"•**w»Ul oQiufMJiii^iii *yf I 111 prefisute
« A» 4«f to ^ wiU be
IJ«wit* of .c LiieBfittre on
^ ^*^ " !.■ the prsMure
•&*!»« a ijuimiii. jiiiii the miaufiiiy
U proportional to tUa ^eplli,
RUtAereSfttLajit tAtitst
on tlie /jasi^^ IIBoftd arfs
oJt tJie poirU^ E,
Fig, 14.
in.
^■ii
18
HYDRAULICS
the diagram of pressure on the face AB will be the triangle ABC, BC being equal
to wd and perpendicular to AB.
The centre of pressure is at the centre of gravity of the pressure diagram and is,
therefore, at  the height of the dam from the base.
The whole pressure acts perpendicular to AB, and is equal to the area ABC
= i iccP X sec 10° per foot width
= i . 624 X 6400 X 1054 =r 20540 lbs.
Combining P with W, the weight of the dam, the resultant thrust R on the base
and its point of intersection £ with the base is determined.
Example. A vertical flap valve closes the end of a pipe 2 feet diameter ; the
pressure at the centre of the pipe is equal to a head of 8 feet of water. To determine
the whole pressure on the valve and the centre of pressure. The atmospherie
pressure may be neglected.
The whole pressure P = irirR^ . 8'
= 624.x. 8 = 1670 lbs.
Depth of the centre of pressure.
The moment of inertia about the free surface, which is 8 feet above the centre
of the valve, is
82
Therefore
1= — +irR*''.
X = ^**^=8'(^"
IT . O
That is, } inch below the centre of the valve.
The diagram of pressure is a truncated cylinder efkh. Figs. 12 and 13, ef and hk
being the intensities of pressure at the top and bottom of the valve respectively.
Example. The end of a pontoon which floats in sea water is as shown in Fig. 15.
The level WL of the water is also shown. Find the whole pressure on the end of
the pontoon and the centre of pressure.
Fig. 15.
The whole pressure on BE
= 64 lbs. X lO' X 45' x 225' = 6480 lbs.
The depth of the centre of pressure of BE is
1.46=8'.
The whole pressure on each of the rectangles above the quadrants
=ii>. 6 = 320 lbs.,
and the depth of the centre of pressure is  feet.
Tlie two quadrants, since they are symmetrically placed about the vertieal
centre line, may be taken together to form a semicircle. Let d be the distance
below the centre of the semicircle of any element of area, the distance of the
element below the surface being h^ .
FLUIDS AT REST
19
i the mteBsitT of preisare Jil deplh h^
wlM>le prcsAiire on the lemicirde Le P =
I
Hp lemlciyeto «1i«d the diameter \& la ^be eurfnee of tb^ waler.
^TW diateiioe of the t^niu of gmvitj of & s^toitiirde from the oentre of tho
2
2' * thi whole presstire
Ik
'^^ =S01B* + 42 66R>^ 1256 4 666 lbs.
I Tb# ^etAb of the oentni of pteaBUre c>f tb@ semieir^^Ie when t!i« sorfiaoe of the
P=wfB*+
virB> 4B
li ml the oenire at the cir^
ptefiBU
X^=
8
,R
75^
s
16
M7'>
^^■■^^MiM^the whrrk pressure ort th« 9emi«!ircle is the tium of two Forces,
^^^HHHHHH6 Ibi^.^ acU at th^ ceutre of gr&vity, ar At & diKtance of 3 06' from
■iPBiCBBff 0( r>66 Ibi^. acta ut u diati^tice of 3^47' from AD.
T^tfi taklti^ momenta al»out AD thi^ product of the preBSiire on the whole area
isiCc* the (t«.'nth of lii& ceiitr«* of pressure is equal to the moiuetittt of i^ll th« forced
OB the afv», a boat AI), The depth of this centre of priMiiF& in, therefore,
6%m Ibi. X ^' + 320 Ibi. X 2 X 1' + 1266 lbs. x 306 + S66 Ibw, x 3^47'
^a*9a feet.
64ti€ + ti40'f 1355 + 666
EXAMPLES.
l\ A f«AMfiBl*^ ^&o^ 1^ ^^^ l<^^g* ^ ^^^ wide, a^d 5 leet deep is
with ni^
F'ind ih« iota) prassure on an end and side of the tank.
Finil the total presj^ure and the centre of preaaure, on a vertical
e, eLrctdiir in fortn, 2 feet in diameter, the centre of which is 4 feet
Ibe sorface of tlje water. [M. S. T. Cambridge, 1901,]
A oiaaonry dam vertical on the water side supports water of
I feet depth. Find the pressure per square foot at depths of 20 feet and
. iitiBi the Miirface ; also the total pressni^ on 1 foot length of Uie dam,
0) A dodc gate is hinged horiEontallj at the hottoni and Bttpported in
Im ^viiicai pcMttion hj horiacontal chains at the top.
i of ^te 45 feet, width 80 ft. Depth of water at ope siJe of the
[ And 20 feet on the other side. Find the tension in the diains.
cr weigbfl ©4 poitada per cubic foot,
II ui<erctir>' ia ISJ times as heavy as water, find the height of a
com^sponduig to a pressure of 100 lbs. per square inch,
, sirai^t pipe 6 inches diameter has a rightangled bend connected
^ the end of ilie bend being closed by a flange.
\ oaDtaifis water at a pressure of 700 !bs, per sq. inch. Determine
I In ttie bults at both ends of tlie elbow,
2—2
^^
20
HYDRAULICS
(7) The end of a dock caisson is as shown in Fig. 16 and the water
level is AB.
Determine the whole pressure and the centre of pressure.
Ut — 44) H
Fig. 16.
(8) An U tube contains oil having a specific gravity of 1*1 in the lower
part of the tube. Above the oil in one limb is one foot of water, and above
the other 2 feet. Find the difference of level of the oil in the two limbs.
(9) A pressure gauge, for use in a stokehold, is made of a glass U tabs
with enlarged ends, one of which is exposed to the pressure in the stokehold
and the other connected to the outside air. The gauge is filled with water
on one side, and oil having a specific gravity of 0*95 on the other — ^the
surface of separation being in the tube below the enlarged ends. If the
area of the enlarged end is fifty times that of the tube, how many inches of
water pressure in the stokehold correspond to a displacement of one inch
in the surface of separation ? [Lond. Un. 1906.]
(10) An inverted oil gauge has its upper U filled with oil having a
specific gravity of 0*7955 and the lower part of the gauge is filled with
water. The two limbs are then connected to two different points on a pipe
in which there is flowing water.
Find the difference of the pressure at the two points in the pipe when
the difference of level of the oil surfaces in the limbs of the U is
15 inches.
(11) An opening in a reservoir dam is closed by a plate 8 feet sqnare^
which is hinged at the upper horizontal edge ; the plate is inclined at an
angle of 60"* to the horizontal, and its top edge is 12 feet below the suifaoe
of the water. If this plate is opened by means of a chain attached to the
centre of the lower edge, find the necessary pull in the chain if its line of
action makes an angle of 45° with the plate. The weight of the plate ia
400 pounds. [Lond. Un. 1905.]
(12) The width of a lock is 20 feet and it is closed by two gates at each
end, each gate being 12' long.
If the gates are closed and the water stands 16' above the bottom on one
side and 4' on the other side, find the magnitude and position of the resultant
pressure on each gate, and the pressure between the gates. Show also that
the reaction at the hinges is equal to the pressure between the gates. One
cubic foot of water=625 lbs. [Lond. Un. 1905.]
CHAPTER 11.
FLOATING BODIES.
IB. ConditionB of equilibxium.
Wlieii a body floats in a fluid the surface of the body in
contact ^th the fluid is subject to hydrostatic pressures, the
intensity of pressure on any element of the surface depend
ing upon its depth below the surface. The resultant of the
vertical components of these hydrostatic forces is called the
buoyancy, and its magnitude must be exactly equal to the weight
ol the body, for if not the body will either rise or sink. Again
the horizontal components of these hydrostatic forces must
be in equilibrium amongst themselves, otherwise the body will
have a lateral movement.
The position of equilibrium for a floating body is obtained
when (a) the buoyancy is exactly equal to the weight of the
body, and (b) the vertical forces — the weight and the buoyancy —
act in the same vertical line, or in other words, in such a way as
to produce no couple tending to make the body rotate.
Let G, Fig. 17, be the centre of gravity of a floating ship and
BK, which does not pass through G, the line of action of the
resultant of the vertical buoyancy forces. Since the buoyancy
Fig. 17.
Fig. 18.
must equal the weight of the ship, there are two parallel forces
each equal to W acting through G and along BK respectively,
and these form a couple of magnitude Wx, which tends to bring
the ship into the position shown in Fig. 18, that is, so that BK
22
HYDRAULICS
passes through G. The above condition (6) can therefore only be
realised, when the resultant of the buoyancy forces passes through
the centre of gravity of the body. K, however, the body is
displaced from this position of equilibrium, as for example a ship
at sea would be when made to roll by wave motions, there will
generally be a couple, as in Fig. 17, acting upon the body, which
should in all cases tend to restore the body to its position of
equilibrium. Consequently the floating body will oscillate about
its equilibrium position and it is then said to be in stable eqm
librium. On the other hand, if when the body is given a small
displacement from the position of equilibrium, the vertical force*
act in such a way as to cause a couple tending to increase the
displacement, the equilibrium is said to be unstable.
The problems connected with floating bodies acted upon by
forces due to gravity and the hydrostatic pressures 011I7,
resolve themselves therefore into two,
(a) To find the position of equilibrium of the body.
(b) To find whether the equilibrium is stable.
19. Principle of ArchimedeB.
When a body floats freely in a fluid the weight of the body i»
equal to the weight of the fluid displaced.
Since the weight of the body is equal to the resultant of the
vertical hydrostatic pressures, or to the buoyancy, this principle
will be proved, if the weight of the water displaced is shown to ^
equal to the buoyancy.
Let ABC, Fig. 19, be a body floating in equilibrium, AC bei^^
n the surface of the fluid.
thv
16 L
(11
which «
angle of i
of the wal
centre of th.
action make(»
400 pounds. [
(12) Thewi.
end, each gate be.
sideand^d'ontheoa^^*^' element ab of the surface, of area a and
pressoie on each gattPf the element being inclined at any angle 6 to
the reaction at the hinen, if to is the weight of unit volume of the
cubic foot of water«62 sure on the area a is wha, and the vertical
sure is seen to be wha cos 0,
Fig. 19.
FLOATIJfO BODIES
23
IniRgine now a vertical cylinder standing on this area^ the top
^ which is in the snrface AC.
le horizontal sectional area of this cylinder m a cos^, the
is ha cos and the weight of the water filling this volume
fshaccmB^ and is, therefore, equal to the buoyancy on the
stmilar cylinders be imagined on all the little elements
I area which make up the whole immersed surface, the total
iome of these cylinders is the volume of the water displaced,
L ihe total buoyancy is, therefore, the weight of this displaced
li the body is wholly immersed as in
iy is supposed to he mmle up of small
ffrticttl cylinders intersecting the surface of
j\\e hAj in the elements of area ah and ab\
ch are inclined to the horizontal at angles
^ and having areas a and aj resi^ctively,
w vertical component of the pressure on ah
111] be itha cos and on a'^' wnll be whiaj cos ^.
« C(M§ must equal aj cos ^, each being ^^^^ ^*''
I to the horizontal section of the small cylinder. The whole
is therefore
^wha cos  Swh^at cos <^j
fiBafam equal to the weight of the water displaced*
h this case if the fluid be assumed to be of conBtant density
^i th weight of the body as equal to the weight of the Huid
r^ tie same volume, the body will float at any depth. The
F«ligtt*^ increase in the weight of the body would cause it to
(«iik njitil it reached the bottom of the vessel containing the Huid,
^litfe a very small diminution of its weight or increase in ita
^*^iiiii(? would cause it to rise immediately to the surface. It
wcruld cle?fcrly be practically impossible to maintain such a body
I *n t'qiiilibrium, by endeavouring to adjust the weight of the body,
%piimping out* or letting in» water, as has been attempted in a
ni type of submarine boat. In recent submarines the lowering
'^liringof the boat are controlled by vertical screw propellers,
20. Centre of buoyancy.
^irict* the bunyancy on any element of area is the weight of
Tierrical cylinder of the fluid above this area, and that the
\ ancy is the sum of the weights of all these cylinders, it
■llowH, that the resultant of the buoyancy forces rami
1 through the eentre of gravity of the water displaced, and this
Bt is, therefore, called the Centre of Buoyancy.
24
HYDRAUUCS
21. Condition of stability of equilibrium.
Let AND, Fig. 21, be the section made by a vertical plane
containing G the centre of gravity and B the centre of buoytincy
of a floating vessel, AD being the surface of the fluid when the
centre of gravity and centre of buoyancy are in the same vertical
line.
Fig. 21.
Fig. 92.
Let the vessel be heeled over about a horizonal axis, PE being J
now the fluid surface, and let Bi be the new centre of buoyancy, J
the above vertical sectional plane being taken to contain G, B,
and Bi. Draw BiM, the vertical through Bi, intersecting the Kne
GB in M. Then, if M is above G the couple W . x will tend to
restore the ship to its original position of equilibrium, but if M i&^
below G, as in Fig. 22, the couple will tend to cause a furthe^^
displacement, and the ship will either topple over, or will heel ova^^
into a new position of equilibrium.
In designing ships it is necessary that, for even large displac^^.
ments such as may be caused by the rolling of the vessel, tkr^^
point M shall be above G. To determine M, it is necessary "fc^
determine G and the centres of buoyancy for the two positiox:ii
of the floating body. This in many cases is a long and somewH^
tedious operation.
22. Small displacements. Metacentre.
When the angular displacement is small the point M is call^^
the Metacentre, and the distance of M from G can be calculated.
Assume the angular displacement in Fig. 21 to be small an(^
equal to 0.
Then, since theVolume displacement is constant the volume of
the wedge CDE muk equal CAF, or in Fig. 23, CiCaDE must equal
CiCAF.
25
Let Gi and Gj be the centres of gravity of the wedges C1C2AF
and CiCsDE reflectively.
df D
Fig. 23.
The heeling of the ship has the effect of moving a mass of
witer equal to either of these wedges from Gi to G2, and this
movement causes the centre of gravity of the whole water
di^Iaced to move from B to Bj.
Let Z be the horizontal distance between Gi and Gj, when FE
is horizontal, and S the perpendicular distance from B to BiM.
Let V be the total volume displacement, v the volume of the
^wredge and tc the weight of unit volume of the fluid.
Then tr.v.Z = 'M7. V. S
= w .Y . BM . sin 0,
Or, ance ^ is small, ='m?.V.BM.^ (1).
The restoring couple is
tr.V.HG = ir.V.GM.^
= ii?.V.(BMBG)^
=^w.v,Zw.Y.BG,e (2).
Bat If . v . Z = twice the sum of the moments about the axis
CiCijOf all the elements such as acdh which make up the wedge
CCiDE.
Taking ah as ar, hf is x6, and if ac is SZ, the volume of the
element is ia^^.aZ.
The centre of gravity of the element is at fa; from C1C2.
3
Therefi
:ore
tr . r . Z = 2w
7?dl
p
.(3).
But, ^ is the Second Moment or Moment of Inertia of the
element of area aceb about C2C1, and 2 / .5  is, therefore, the
.'0 'J
Moment of Inertia I of the waterplane area AC1DC2 about C1C2
Therefore w ,v ,Z = w ,1.0 (4).
26 HYDRAULICS
The restoring couple is then
wI0w.Y.BG.e.
If this 18 positive, the equilibrium is stable, but if negative it is
unstable.
Again since from (1)
wv.Z = w.Y.BU.e,
therefore w.Y.BM.0 = wI0y
and BM = ;5 (5),
If BM is greater than BG the equilibrium is stable, if less than
BG it is unstable, and the body will heel over until a new position
of equilibrium is reached. If BG is equal to BM the equilibrium
is said to be neutral.
The distance GM is called the Metacentric Height, and varies
in various classes of ships from a small negative value to a positive
value of 4 or 5 feet.
When the metacentric height is negative the ship heels until
it finds a position of stable equilibrium. This heeling can be
corrected by ballasting.
ExampU, A ship has a displacement of 15,400 toDS, and a dranght of 27*5 feet.
The height of the centre of buoyancy from the bottom of the keel is 15 feet.
The moment of inertia of the horizontal section of the ship at the water line
is 9,400,000 feet« units.
Determine the position of the centre of gravity that the metacentric height shall
not be less than 4 feet in sea water.
M00^000x64
16,400x2240
= 171 feet.
Height of metacentre from the bottom of the keel is, therefore, 32*1 feet.
As long as the centre of gravity is not higher than 0*6 feet above the anrfiace of
the water, the metacentric height is more than 4 feet.
23. Stability of a rectangular pontoon.
Let RFJS, Fig. 24, be the section of the pontoon and G its
centre of gravity.
Let VE be the surface of the water when the sides of the
pontoon are vertical, and AL the surface of the water when the
pontoon is given an angle of heel 0,
Then, since the weight of water displaced equals the weight of
the pontoon, the area AFJL is equal to the area VFJE.
Let B be the centre of buoyancy for the vertical position,
B being the centre of area of VFJE, and Bi the centre of buoyancy
for the new position, Bi* being the centre of area of AFJL. Then
the line joining BGr must be perpendicular to the surface VE and
* In the Fig., B, is not the centre of area of AFJL, as, for the sake of cleamesi^
it is farther removed from B than it actually should be.
FLOATING BODIES
27
s the direction in which the buoyancy force acts when the sides
){ the pontoon are vertical, and BiM perpendicular to AL is the
direction in iwrliich the buoyancy force acts when the pontoon is
heeled over ttrough the angle 0. M is the metacentre.
Fig. 24.
The forces acting on the pontoon in its new position are, W the
weight of the pontoon acting vertically through G and an equal and
parallel buoyancy force W through Bj .
There is, therefore, a couple, W • HG, tending to restore the
pontoon to its vertical position.
If the line BiH were to the right of the vertical through G, or
in other words the point M was below G, the pontoon would be in
unstable equilibrium.
The new centre of buoyancy Bi can be found in several ways.
The following is probably the simplest.
The figure AFJL is formed by moving the triangle, or really
the wedgeshaped piece CEL to CVA, and therefore it may be
imagined that a volume of water equal to the volume of this wedge
is moved from G2 to Gi . This will cause the centre of buoyancy
to move parallel to GiGj to a new position Bi, such that
BBi X weight of pontoon = G1G2 x weight of water in CEL.
Let b be half the breadth of the pontoon,
I the length,
D the depth of displacement for the upright position,
d the length LE, or AV,
and w the weight of a cubic foot of water.
Then, the weight of the pontoon
W = 2b.B.l.w
and the weight of the wedge CLE = ^ ^ I .w.
28 H7DRAUUCS
Therefore BBi.26.D = 
2
and BBi = tjz GriGg.
Besolving BBi and GiGj, which are parallel to each other, along
and perpendicular to BM respectively,
d r. ^ d /2^A bd 6»tan^
B.Q = ^G.K=^(26) =
3D 3D '
A TiT>Tin G^ _hd d _ iT _ ^'tan'tf
and ^*^^»'^G.K"3D26~6D" 6D '
To find the distance of the point Mfrom G and the value of the
restoring couple. Since BiM is perpendicular to AL and BM to
YE, the angle BMBi equals 6.
Therefore QM = B,Q cot« = ^ cottf = ^.
Let z be the distance of the centre of gravity G from C.
Then QG = QC z = BCBQ a
_D 5'tan*g
2 6D '•
Therefore
rnur rnur nr ^* D^fc'tan'tf^
And since HG = GM sin 0,
the righting couple,
w Tin w • af^' D . 5'tan *g . \
The distance of the metacentre from the point B, is
QM + QB = B,Q cot« + ^^^
3D "^ 6D •
When is small, the term containing tan*tf is negligible, and
This result can be obtained from formula (4) given in
section 22.
I for the rectangle is j\l (25)» = ilb\ and V = 2bl>l.
Therefore BM = ^.
If BG is known, the metacentric height can now be found.
FLOATING BODIES
29
Example, A pontoon hM a displaoemeDt of 200 tons. Its length is 50 feet.
The centre of gravity is 1 foot above the centre of area of the oross section. Find
tbe fareadth and depth of the pontoon so that for an angular displacement of 10 degrees
the metaoentre shall not be less than 8 feet from the centre of gravity, and the free
board shall not be less than 2 feet.
BeCerring to Fig. 24, G is the centre of gravity of the pontoon and O is the
eeotre of the eroes section BJ.
Then, GO =1 foot,
Fo=2 feet,
GM=3 feet.
Ijet D be the depth of displacement. Then
D X 26 X 624 x 50 lbs. =200 tons x 2240 lbs.
Therefore D6=715 (1).
The height of the centre of bnoyancy B above the bottom of pontoon is
BT=JD.
Since the freeboard is to be 2 feet,
OT=4(D2).
Then B0 = 1' and BG=2'.
Therefore BM=5'.
Bnt BM = QM + BQ
__^ 6»tan«g
■"8D"*" 6D •'
(2).
Moltiplying namerator and denominator by 6, and substitating from equation (1)
6» &»tan«^ ^,
=6,
from which
therefore
and
2146 ^ 429
6»(2 + (176)«)=5x429,
6=101 ft..
D = 71 ft.,
The breadth B= 202 ft.)
,, depth
3 = 202 ft.)
= 7lft.
Ans,
24. Stability of a floating vessel containing water.
If a vessel contains water with a free surface, as for instance
the compartments of a floating dock, such as is described on page
31, the surface of the water in these compartments will remain
horizontal as the vessel heels over, and the centre of gravity of
the water in any compartment will change its position in such
a way as to increase the angular displacement of the vessel.
In considering the stability
of such vessels, therefore, the
taming moments due to the jj ^^^'""""' Q
water in the vessel must be
taken into account. ^
As a simple case consider E
the rectangular vessel. Fig. 2o, H ^ \ jj^
which, when its axis is vertical,
floats with the plane AB in the ^^8 ^5.
■€^
K
D
30
HYDRAULICS
surface of the fluid, DE being the surface of the fluid in the
vessel.
When the vessel is heeled through an angle ^, the surface of
fluid in the vessel is KH.
The effect has been, therefore, to move the wedge of fluid OEH
to ODK, and the turning couple due to this movement is tu . t? .Z,
V being the volume of either wedge and Z the distance between
the centre of gravity of the wedges.
If 2b is the width of the vessel and I its length, v is^l tan 6,
Z is 6 tan ^, and the turning couple is w ft' I tan* 0.
If is small wvZ is equal to wI0y I being the moment of inertia
of the water surface KH about an axis through 0, as shown in
section 22.
For the same width and length of water surface in the
compartment, the turning couple is the same wherever the
compartment is situated, for the centre of gravity of the wedge
OHE, Fig. 26, is moved by the same amount in all cases.
If, therefore, there are free fluid surfaces in the floating vessel,
for any small angle of heel ^, the tipplingmoment due to these
surfaces is 'S.wIOy I being in all cases the moment of inertia of the
fluid surface about its own axis of oscillation, or the axis through
the centre of gravity of the surface.
E
H
04
pH
D
Fig. 26.
Fig. 27.
25. Stability of a floating body wholly immersed.
It has already been shown that a floating body wholly im
mersed in a fluid, as far as vertical motions are concerned, can
only with great difficulty be maintained in equilibrium.
If further the body is made to roll through a small angle, the
equilibrium will be unstable unless the centre of gravity of the
body is below the centre of buoyancy. This will be seen at once
on reference to Fig. 27. . Since the body is wholly immersed the
centre of buoyancy cannot change its position on the body itself,
as however it rolls the centre of buoyancy must be the centre of
gravity of the displaced water, and this is not altered in form by
FLOATING BODIES
31
any movement. of the body. K, therefore, G is above B and the
body be given a small angular displacement to the right say, G
will move to the right relative to B and the couple will not restore
the body to its position of equilibrium.
On the other hand, if G is below B, the couple will act so as to
brin^ the body to its position of equilibrium.
26. Floatiiig docks.
Figs. 28 and 29 show a diagrammatic outline of the pontoons
forming a floating dock, and in the section is shown the outline of
a ship on the dock.
Fig. 29.
To dock a ship, the dock is sunk to a sufficient depth by
admitting water into compartments formed in the pontoons, and the
ship is brought into position over the centre of the dock.
Water is then pumped from the pontoon cliarubers, and the
dock in consequence rises until the ship just rests on the keel
blocks of the dock. As more water is pumped from the pontoons
the dock rises with the ship, which may thus be lifted clear of
the water.
Let Gi be the centre of gravity of the ship, Ga of tlie dock and its
water ballast and G the centre of gra\4ty of the dock and the
ship.
The position of the centre of giavity of the dock >W11 vary
32 HTDRAULICS
relative to the bottom of the dock, as water is pumped from the
pontoons.
As the dock is raised care must be taken that the metacentre
is above G or the dock will " list."
Suppose the ship and dock are rising and that WL is the
water line.
Let Ba be the centre of buoyancy of the dock and Bi of the
portion of the ship still below the water line.
Then if Vi and V2 are the volume displacements below
the water line of the ship and dock respectively, the centre of
buoyancy B of the whole water displaced divides BsBj, so that
BBi_Va
BBa"Vx
The centre of gravity G of the dock and the ship divides GiGt
in the inverse ratios of their weights.
As the dock rises the centre of gravity G of the dock and the
ship must be on the vertical through B, and water must be
pumped from the pontoons so as to fulfil this condition and as
nearly as possible to keep the deck of the dock horizontal.
The centre of gravity Gi of the ship is fixed, while the centre of
buoyancy of the ship Bi changes its position as the ship is raised.
The centre of buoyancy Ba of the dock will also be changing,
but as the submerged part of the dock is symmetrical about its
centre lines, Ba will only move vertically. As stated above, B
must always lie on the line joining Bi and Ba, and as G is to be
vertically above B, the centre of gravity Ga and the weight of
the pontoon must be altered by taking water from the various
compartments in such a way as to fulfil this condition.
Quantity of water to he 'pumped from the pontoons in raising the
doch. Let V be the volume displacement of the dock in its lowest
position, Vo the volume displacement in its highest position. To
raise the dock mthout a ship in it the volume of the water to be
pumped from the pontoons is V  Vo .
If, when the dock is in its highest position, a weight W is put
on to the dock, the dock will sink, and a further volume of water
W . .
— cubic feet will be required to be taken from the pontoons to
raise the dock again to its highest position.
To raise the dock, therefore, and the ship, a total quantity of
water
w
cubic feet will have to be taken from the pontoons.
FLOATING BODIES
33
Ifft^vlf, K lUmtmg dock ms fibotrtt di mentioned m Fir. 2B is made np of s
not! 540 feet long x OG fc«l wide x 1475 feet deep, two aide pontooDt
gx 13 ft»et wide y 4^8 feet de«p, the bottom of these poutoonB being ^
"<.*vf Ihc bcittoin of the dock^ aud two side chambers on the top of thm I
poatooti 447 feel long by 8 feet deep imd 2 feet wide at the top and 8 fe«l ftt 
. itoiD. The kf^l blocks may be taken as 4 feet deep.
I h« dock is Ui lift » sbip of 15,400 ttiris dij^plncetneul and 27' 6" dratlght.
l)ttj^nriiine the amouDt of water that mu^t be pumped from the dock, to raJBe
ifi tm that the deck of the lowest pontoon is m the water surface.
»:ji the xbip jnst t^kes to the keel bloeks on the dock^ the bottom of the
atw * im 27*5' ^ 14 '75' + 4' = 46 '25 feet below the water line,
Tht voloiik^ displaovment of the dock is then I
1476 * MO ^ 96 ^' 2 ^ 4425 X X3 X aSO + 447 X 8 X 5' = 1,3K7,6(H) cabic feet, I
Tb« foltime of dock diaplacemeat when ih© deck is just awash m I
540 X 96 M 1475 + 2 k 3B0 x 13' >^ ( l4*7o  2) = H90,OC)O mhm feel. I
The TolTim« djsplacemeat of the ahip ii I
15.400 >c 2240 e.rt^^ t , ,
^7 = 640, 000 ctibio feet ,
ASid tlik ttfOftlt Ihe weight of the ship in cubic feet.
of tb« ^1,000 citbic feel di»placemeni wheu the ship is olear of the water^
&S1 /Wll eotiic feel J!i therefore reiiTtired to support Ihe dock aloDe.
'I riiply lei tninm the dock through 31*5 feet the amount of wat^r to l^e pumped ia
Teretkcs i>f the diaplacemenlH^ and is, tlierefon*, 347i*K]0 cubic feet.
"'^ lh# ehip with the dock an additiooal ^40,000 cubits feet must be
':7aD the pontoous*
:%) q^tAolitjt Iherefore, to be taken from the pontoons from the time the
kkem to iht ked blocks lo when the pontoon deck is in the HurCaoe of the
^ 887.600 cubic feet ^ 25,380 tons, ,
27. Stability of the floatiBg dock.
As some (jI the compartments of tlie dock are partially filled
w^iiierT it in necesHaiy, in considering the stability, to take
• 'f the tipplirigmoments eauaed by the movement of the
^ r.ce of the water in these compartments.
kL Ii G i» the centre of gra^nty of the dock and ship on the
" ih^^c^ B the centre of bnoyancyj I the moment of inertia of the
n of the ship and dock by the waterplane abont the axis of ]
ti, and Ii, Is etc. the moments of inertia of the water
in the compartments about their axes of oscillation^ the
n^ moment when the dock receives a amaU angle of
ifW ti? (Yi + Va) BO^ t£vi9 (I, + 1, +...). I
llir moment of inertia of the waterplane section YarieAl
eoiu»iclerahly ub the dock ib raised, and the stability varies
When tht; i*hip is immersed in the water, I is equal to the
jnamenT of inertia of the horizontal section of the ship at the
ter fiarfAC«», together with the moment of inertia of the
■ ' ection of the side pontoons, about the axis of
t a
li^Mlktflki
34 HTDRAUUCS
When the tops of the keel blocks are just above the SDrEaea:
of the water, the waterplane is only that of the side pontoon^,
and I has its minimum value. If the dock is Lshaped as
Fig. 30, which is a very convenient form
for some purposes, the stability when
the tops of the keel blocks come to the
surface simply depends upon the moment
of inertia of the area AB about an axis
through the centre of AB. This critical
point can, however, be eliminated by
ii
fitting an air box, shown dotted, on the p. ^
outer end of the bottom pontoon, the
top of which is slightly higher than the top of the keel blockB*
Example. To find the height of the metacentre above the centre of baoymoflj if "
the dock of Fig. 28 when
{a\ the ship just takes to the keel blocks,
{h) the keel is just clear of the water,
\c) the pontoon deck is just above the water.
Take the moment of inertia of the horizontal section of the ship at tht
water line as 9,400,000 ft> units, and assume that the ship is symmetricallj
placed on the dock, and that the dock deck is horizontal. The horizontal distanee
between the centres of the side tanks is 111 ft.
(a) Total moment of inertia of the horizontal section is
9, 400,000 4 2 (380 x 1 3' x 66 S' 4 tV x 380 x 13») = 9,400,000 + 30,430,000 + 139,000.
The volume of displacement
=640,000 + 1,237,600 cubic feet.
The height of the metacentre above the centre of buoyancy is therefore
39,968,000 «... .
^^=1,932,000=^^^^^*
(h) When the keel is just clear of the water the moment of inertia is
30,569.000.
The volume displacement is
540 X 96 X 1475 + 380 x 2 x 13 x (1475 + 42)
= 930,000 cubic feet.
Therefore BM = 32*8 feet.
(c) When the pontoon deck is just above the surface of the water,
I = 30,569,000 + iV x 5 40' X 96»
y.. =70,269,000.
W "^he volume displacement is 890,000 cubic feet.
— CUL .
W '^fow BM= 798 feet.
raise the (X^^ of ^^^ centre of buoyancy above the bottom of the dock can be
rn ^ ^p ' finding the centre' of buoyancy of each of the parts of the dock, and
lo raise » ^ jjj jjj^ water, and then taking moments about any axis,
water To find the height h of the centre of buoyancy of the dock and
^ ship just comes on the keel blocks,
oyancy for the ship is at 15 feet above the bottom of the keel,
vancy of the bottom pontoon is at 7*375' from the bottom.
I. r i. ^^^\. ^ »» side pontoons „ 24125' ,,
cubic feet Will have ,, „ chambers ., i794' ,;
IXOATUIQ BODIES
35
\ tnometits &boat the bottom of the dock
h (540,000 + 437,000 + 765,0004 35,760)
= 540.000 X 33*75 + 765,000 x 7'375
+ 437,00O X *i4'i25 i 3S,760 X 17 "95,
A 19 7 feet,
{^) ihe metiio^iitre is^ tberefi^re, 40*3' above the bottom of the dock. If
«si]iT« of griL^i^ of Ibe dcHsk &ad sUip u kuowa the metacentrio height
EXAMPLES.
A «titp ^bcn fully loaded has a total burden of 10,000 tons. Find
i displacemeDt in aea water.
The aides d a ship are vertical near the water lino and the area of
sittl section at the water line ia 22,000 aq. feet. The total weight
r sJiip IB 10,000 tons when it leaves the river dock,
tbe diffei^iice in draught in the dock and at sea after the weight
F ship ii&a been rednoed by consiimption of ooaU etc., by 1500 totis,
> 8 be Ibe difference in draught,
I c K 22.000= the difference in volume displacement
10,000 ic 2340 _ 6500x 2240
" 6248 64
=61S0 cnbic feet.
^■i*2'rSf6et
sS 34 inches.
Tlie moment of inertia of the section at the water line of a boat
foot* tmita; the weight of the boat is 11 '5 tons.
I the height of the metacentre above tlie centre of buoyancy.
{%} k ship has a total displacement of 15.000 tons and a draught of
1 the ship is lifted by a floating dock so that the depth of the bottom
\ ked is 16*5 feet, tlie centre of buoyancy is 10 feet from the bottom of
^kael aad tlie displacement is 9000 tons.
f nwiiDeait of inertia of Uie waterplane is 7t600,000 foot* units.
Iiorueoiital section ol the dock* at tlie plane 16*5 feet above the
of ttw keel^ consists of two rectangles 380 feet k 11 feet, the distance
i td the «enlre lines of the rectangles b^ing 114 feot
The TobtBie displacement of tlie dock at this level is 1,244,000 cubic feet.
Tbeciotra of buoyancy for the dock alone is 24*75 feet below the surface
f water.
line ia] The centre of buoyancy for the whole ship and the dock.
Tlw height of tlie metacentre above the centre of buoyancy.
h
fV
A rectangulai' pontoon 60 feet long is to have a displacement of
% fre^^ board of not leas than 3 feet, and the metacentre in not to
than S feet above the centre of gravity when the angle of heel
The centre of gravity coincides with the centre ol tigure.
tbe vddUi and depth ol the i>ontooii.
3—2
36 HYDRAULICS
(6) A rectangular pontoon 24 feet wide, 50 feet long and 14 feet deep»
has a displacement of 180 tons.
A vertical diaphragm divides the pontoon longitudinally into two
compartments eadi 12 feet wide and 50 feet long. In the lower part
of each of these compartments there is water ballast, the 8iir&<» of the
water being free to move.
Determine the position of the centre of gravity of the pontoon that it
may be stable for small displacements.
(7) Define "metacentric height** and show how to obtain it graphicallj
or otherwise. A ship of 16,000 tons displacement is 600 feet long, 60 leel
beam, and 26 feet draught. A coefficient of ^ may be taken in the moment
of inertia term instead of ^ to allow for the waterline section not beii^
a rectangle. The depth of the centre of buoyancy from the water line »
10 feet. Find the height of the metacentre above the water line and
determine the position of the centre of gravity to give a metacentric hei^
of 18 inches. [Lond. Un. 1906.]
(8) The total weight of a fully loaded ship is 5000 tons, the water line
encloses an area of 9000 square feet, and the sides of the ship are vertical
at the water line. The ship was loaded in fresh water. Find the changB
in the depth of immersion after the ship has been sufficiently long at sea to
bum 500 tons of coal.
Weight of 1 cubic foot of fresh water 62} lbs.
Weight of 1 cubic foot of salt water 64 lbs.
CHAPTER III.
FLUIDS IN MOTION.
28. Steady motion.
The xnotioii of a fluid is said to be steady or permanent, when
the particles which succeed each other at any point whatever
have the same density and velocity, and are subjected to the same
pressure.
In practice it is probably very seldom that such a condition of
flow is absolutely realised, as even in the case of the water flowing
steadily along a pipe or channel, except at very low velocities, the
velocities of succeeding particles of water which arrive at any
point in the channel, are, as will be shown later, not the same
either in magnitude or direction.
For practical purposes, however, it is convenient to assume
that if the rate at which a fluid is passing through any finite area
is constant, then at all points in the area the motion is steady.
For example, if a section of a stream be taken at right angles
to the direction of flow of the stream, and the mean rate at which
water flows through this section is constant, it is convenient
to assume that at any point in the section, the velocity always
remains constant both in magnitude and direction, although the
velocity at different points may not be the same.
Mean velocity. The mean velocity through the section, or the
mean velocity of the stream, is equal to the quantity of flow per
unit time divided by the area of the section.
29. Stream line motion.
The particles of a fluid are generally regarded as flowing along
definite paths, or, in other words, the fluid may be supposed to
flow in threadlike filaments, and when the motion is steady these
filaments may be supposed to be fixed in position.
In a pipe or channel of constant section, the filaments are
generally supposed to be parallel to the sides of the channel.
38
HYDRAULICS
An
DaUuiLlLe^
Fig. 81.
30. Definitions relating to flow of water.
Pressure head. The pressure head at a point in a fluid at rest
has been defined as the vertical distance of the point from the free
surface of the fluid, and is equal to — , where p is the pressure per
sq. foot and w is weight per cubic foot of
the fluid. Similarly, the pressure head at
any point in a moving fluid at which the
pressure is p lbs. per sq. foot, is  feet,
w
and if a vertical tube, called a piezometer
tube. Fig. 31, be inserted in the fluid, it
will rise in the tube to a height h, which
equals the pressure head above the atmo
spheric pressure. K p is the pressure per
sq. foot, above the atmospheric pressure,
h= —, but if p is the absolute pressure per —
sq. foot, and px the atmospheric pressure,
W W
Velocity head. If through a small area around the point B,
the velocity of the fluid is v feet per second, the velocity head is
rt , g being the acceleration due to gravity in feet iter second per
second.
Position head. If the point B is at a height z feet above any
convenient datum level, the position head of the fluid at B above
the given datum is said to be z feet.
31. Energy per pound of water passing any section in
a stream line.
The total amount of work that can be obtained from every
pound of water passing the point B, Fig. 31, assuming it can fall to
the datum level and that no energy is lost, is
2 + ^' + 2 ft. lbs.
w 2g
Proof. Work available due to pressure head. That the work
which can be done by the pressure head per pound is ~ foot
pounds can be sho'WTi as follows.
Imagine a piston fitting into the end of a small tube of cross
sectional area a, in which the pressure is h feet of wator as in
I
FLUIDS IN MOTION 39
g. 32, And let a small quantity ^Q cubic feet of water enter the
ibe and move the piston through a small dis
knce 6as.
Then dQ^a.dx.
The iwork done on the piston as it enters
Nrill he
w.h.a.dx = w .hdQA
But the ^weight of 5Q cubic feet iaw.dQ pounds, ^'8. 82.
and the ^wrork done per pound is, therefore, A, or — foot pounds.
A. pressure head h is therefore equivalent to h foot pounds of
energy ijer ix>und of water.
Work available due to velocity. When a body falls through
a height h feet, the work done on the body by gravity is h foot
pounds i)er pound. It is shown in books on mechanics that if the
body is allowed to fall freely, that is without resistance, the
velocity the body acquires in feet per second is
V = J2ghy
And since no resistance is offered to the motion, the whole of
the work done on the body has been utilised in giving kinetic
energy to it, and therefore the kinetic energy per pound is ^ ~ •
In the case of the fluid moving with velocity v, an amount of
energy equal to ^y foot pounds per pound is therefore available
before the velocity is destroyed.
Work available due to position. If a weight of one pound
falls through the height z the work done on it by gravity will be
z foot pounds, and, therefore, if the fluid is at a height z feet above
any datum, as for example, water at a given height above the
sea level, the available energy on allowing the fluid to fall to
the datum level is z foot pounds per pound.
32. Bemouilli's theorem.
In a steady moving stream of an incompressible fluid in which
the particles of fluid are moving in stream lines, and there is no
loss by friction or other causes
p v^
w 2g
is constant for all sections of the stream. This is a most important
theorem and should be carefully studied by the reader.
40
HYDRAULICS
It has been shown in the last paragraph that this expression
represents the total amount of energy per pound of water flowing
through any section of a stream, and since, between any two
points in the stream no energy is lost, by the principle of the
conservation of energy it can at once be inferred that this
expression must be constant for all sections of a steady flowing
stream. A more general proof is as follows.
Let DE, Fig. 33, be the path of a particle of the fluid.
Fig. 33.
Imagine a small tube to be surrounding DE, and let the flow
in this be steady, and let the sectional area of the tube be so small
that the velocity through any section normal to DE is uniform.
Then the amount of fluid that flows in at D through the area
AB equals the amount that flows out at E through the area CF.
Let Pj) and Vd, and p^ and ve be the pressures and velocities at
D and E respectively, and A and a the corresponding areas of the
tube.
Let z be the height of D above some datimi and Zi the height
of E.
Then, if a quantity of fluid ABAiBi equal to 3Q enters *at D,
and a similar quantity CFCiFi leaves at E, in a time dtj the
velocity at D is
Vj,=
AuV
and the velocity at E is
Ve =
aot'
The kinetic energy of the quantity of fluid uQ entering at D
FLUIDS IN MOTION 41
and tJiat of tlie liquid leaving at E
Since tlie flow in the tabe is steady, the kinetic energy of the
portion ABGF does not alter, and therefore the increase of the
kinetic energy of the quantity dQ
The work done by gravity is the same as if ABBiAi fell to
CFFiCi and therefore equals
The total pressure on the area AB is po • A, and the work done
at X> in time ot
= PdAi?d9^=Pd9Q,
and the work done by the pressure at E in time t
But the gain of kinetic energy must equal the work done, and
therefore
^^ . (ve' V) = wdq (z  2,) + Pd ^QPeC'Q.
From which
2g 2g w w^
or ^«%2?^^^ = V+e5 + ;, = constant.
2g w 2g w
From this theorem it is seen that, if at points in a steady
moving stream, a vertical ordinate equal to the velocity head plus
the pressure head is erected, the upper extremities of these
ordinates will be in the same horizontal plane, at a height H
equal to — + s~ "•" ^ above the datum level.
^ ttj 2g
Mr Froude* has given some very beautiful experimental illus
trations of this theorem.
In Fig. 34 water is taken from a tank or reservoir in which
the water is maintained at a constant level by an inflowing
stream, through a pipe of variable diameter fitted >vith tubes
at various points. Since the pipe is short it may be supposed to
be frictionless. If the end of the pipe is closed the water will rise
in all the tubes to the same level as the water in the reservoir, but
if the end C is opened, water will flow through the pipe and the
water surfaces in the tubes will be found to be at different levels.
* British Assoc. Report 1875.
42
HYDRAULICS
The quantity of water flowing per second through the pipe can be
measured, and the velocities at A, B, and C can be found by
dividing this quantity by the crosssectional areas of the pipe at
these points.
Fig. 34.
If to the head of water in the tubes at A and B the ordinates
S 2
1^ and g^ be added respectively, the upper extremities of these
ordinates will be practically on the same level and nearly level
with the surface of the water in the reservoir, the small difference
being due to frictional and other losses of energy.
At C the pressure is equal to the atmospheric pressure, and
neglecting friction in the pipe, the whole of the work done by
gravity on any water leaving the pipe while it falls from the
surface of the water in the reservoir through the height H, which
is H ft. lbs. per pound, is utilised in giving velocity of motion to
the water, and, as will be seen later, in setting up internal motions.
Neglecting these resistances.
Due to the neglected losses, the actual velocity measured will be
less than vq as calculated from this ^nation.
If at any point D in the pipe, the sectional area is less thaai the
area at C, the velocity will be greater than Vc, and the pressure
will be less than the atmospheric pressure.
If V is the velocity at any section of the pipe, which is supposed
to be horizontal, the absolute pressure head at that section is
" " " ^2g w^'2g 2g'
w w
Pa being the atmospheric pressure at the surface of the water in
the reservoir.
At D the velocity Vd is greater than Vo and therefore p© is less
FLUIDS IN MOTION
43
than p.. If coloured water be put into the vessel B, it will rise in
the tube DE to a height
w
w
If the area at the section is so small, that p becomes negative, the
fluid will be in tension, and discontinuity of flow will take place.
If the fluid is water which has been exposed to the atmosphere
and which consequently contains gases in solution, these gases
will escape from the water if the pressure becomes less than the
tension of the dissolved gases, and there will be discontinuity even
before the pressure becomes zero.
Figs. 35 and 36 show two of Froude's illustrations of the
theorem.
Fig. 86.
Fig. 36.
At the section B, Fig. 36, the pressure head is h^ and the
velocity head is
= /l/lB = H.
If a is the section of the pipe at A, and a^ at B, since there
is continuity of flow.
and
If now a is made so that
2g
K
the pressure head /ia becomes equal to the atmospheric pressure,
and the pii)e can be divided at A, as shown in the figure.
Professor Osborne Reynolds devised an interesting experiment,
to show that when the velocity is high, the pressure is small.
He allowed water to flow through a tube  inch diameter
under a high pressure, the tube being diminished at one section to
01)5 inch diameter.
£4
W
2ff
= 2?
w
2ff
w 2*7'
H =
Pi"
2ff
If 'Ua is equal to Vi, pa is theoretically equal to pi, but there is
vays in practice a slight loss of head in the meter, the difference
 Pa being equal to this loss of head.
• Tramactiom Am,S.C,E,, 1887.
"1
HYDRAULICS
At this diminished section, the velocity was very high and the
assure fell so low that the water boiled and made a hissing
jse.
33. Venturi meter.
An application of Bemouilli's theorem is found in the Venturi
ter, as invented by Mr Clemens Herschel*. The meter takes
name from an Italian philosopher who in the last decade of the
h. century made experiments upon the flow of water through
lical pipes. In its usual form the Venturi meter consists of two
mcated conical pipes connected together by a short cylindrical
►e called the throat, as shown in Figs. 37 and 38. The meter is
erted horizontally in a line of piping, the diameter of the large
is of the frustra being equal to that of the pipe.
Piezometer tubes or other pressure gauges are connected to
) throat and to one or both of the large ends of the cones.
Let a be the area of the throat.  ^
Let ai be the area of the pipe or the large end of the cone
A.
Let Oa be the area of the pipe or the large end of the cone
C.
Let p be the pressure head at the throat.
Let pi be the pressure head at the upstream gauge A.
Let Pa be the pressure head at the downstream gauge C.
Let H and Hi be the differences of pressure head at the throafe
i large ends A and C of the cone respectively, or
w w^ . ^
d H. = ^2.
w w
Let Q be the flow through the meter in cubic feet per sec.
Let V be the velocity through the throat.
Let Vi be the velocity at the upstream large end of cone A.
Let Va be the velocity at the downstream large end of cone 0.
Then, assuming Bemouilli's theorem, and neglecting friction,
PiOTDS IN MOTION
46
HYDRAULICS
The velocity t? is — , and i?i is —
Oi
Therefore Q^ (^,  ^,) = 2flf . H,
and
Q =
aai
Joa  d^
V2^TH.
Due to friction, and eddy motions that may be set up in the
meter, the discharge is slightly less than this theoretical value, or
4 being a coefficient which has to be determined by experiment.
For a meter having a diameter of 25*5 inches at the throat and
54 inches at the large end of the cone, Herschel found the
following values for fc, given in Table III, so that the coefficient
varies but little for a large variation of H.
TABLE m.
Herschel
Coker
Hfeet
h
in cu. ft.
k
1
2
6
12
18
28
•995
•992
•985
•9785
977
•970
0418
0319
•0254
•0185
•0096
•0084
•9494
•9587
•9572
•9920
12021
18588
Professor Coker*, from careful experiments on an exceedingly
well designed small Venturi meter. Fig. 38, the area of the throat
of which was '014411 sq. feet, found that for small flows the
coefficient was very variable as shown in Table III.
These results show, as pointed out by Professor Coker from an
analysis of his own and Herschel's experiments on meters of
various sizes, that in large Venturi meters, the discharge is very
approximately proportional to the square root of the head, but for
small meters it only follows this law for high heads, and for low
heads they require special calibration.
Example. A Venturi meter having a diameter at the throat of 86 inohes is
inserted in a 9 foot diameter pipe.
The pressure head at the throat gauge is 20 feet of water and at the pipe gauge
is 26 feet.
* Canadian Society of Civil Engineers, 1902.
FLUIDS IN MOTION
47
?xiid the diaeharge, and the Telocity of flow through the throat.
The area of the pipe is 63*5 sq. feet.
throat 706
The differenee in prewore bead at the two gauges is 6 feet.
^ , ^ 636 X 706 ,
Therefore Q= =====: ^2x32*2x6
= 137 c. ft. per second.
The Tdoeity of flow in the pipe is 2*15 ft. per sec.
„ „ through the throat is 19*4 ft. per sec.
34. Steering of canal boats.
An interesting application of Bemouilli*8 theorem is to show
the effect of speed and position on the steering of a canal boat.
l\Tien a boat is moved at a high velocity along a narrow
and shallow canal, the boat tends to leave behind it a hollow
which is filled by the water rushing past the boat as shown
in Figs. 39 and 40, while immediately in front of the boat the
impact of the bow on the still water causes an increase in the
pressure and the water is " piled up '' or is at a higher level than
the still water, and what is called a bow wave is formed.
Fig. 39.
Fig. 41.
A
Fij?. 40.
Let it be assumed that the water moves past the boat in
stream lines.
If vertical sections are taken at E and F, and the points E and
F are on the same horizontal line, by Bemouilli's theorem
w 2g w 2g'
At E the water is practically at rest, and therefore Vs is
zero, and
w w 2g'
The surface at E will therefore be higher than at F.
48 HTDRAUUCS
Wlien the boat is at tlie centre of the canal the stream lines on
both sides of the boat will have the same velocity, but if the boat
is nearer to one bank than the other, as shown in the figures, the
velocity vr of the stream lines between the boat and the neaarer
bank, Fig. 41, will be higher than the velocity Vr on the other
side; But for each side of the boat
w w 2g w 2g '
And since vr is greater than vv, the pressure head pf is
greater than pn or in other words the surface of the water at
the right side D of the boat will be higher than on the left side R
The greater pressure on the right side D tends to push the
boat towards the left bank A, and at high speeds considerably
increases the diflSculty of steering.
This difficulty is diminished if the canal is made sufficiently
deep, so that flow can readily take place underneath the boat.
35. Extension of Bemooilli's theorem.
In deducing this theorem it has been assumed that the fluid
is a perfect fluid moving with steady motion and that there are no
losses of energy, by friction of the surfaces with which the fluid
may be in contact, or by the relative motion of consecutive ele
ments of the fluid, or due to internal motions of the fluid.
In actual cases the value of
p v^
w 2g
diminishes as the motion proceeds.
If hf is the loss of head, or loss of energy per pound of fluid,
between any two given points A and B in the stream, then more
generally
£^ + i.%,, = 2B^LV,3.;,, (1).
w 2g "^ w 2g
Tht EXAMPLES.
. ^ "^ The diameter of the throat of a Venturi meter is  inch, and of
various S.^ which it is connected IJ inches. The discharge through the
approxima) minutes was found to be 814 gaUons.
small metenerence in pressure head at the two gauges was 49 feet,
heads they ree coefficient of discharge.
Example. A V^i meter has a diameter of 4 ft. in the large part and
inserted in a 9 foot oat. With water flowing through it, the pressure head is
is 26 feer^^^^"^ ^^^^ P*^ ^^^ ^"^ ^' ** ^® ^i^oai. Find the velocity in the
1 discharge through the meter. Coefficient of
Cat.
FLUIDS IN MOTION
49
A pipe AB, 100 ft loiig^ has an inclinatton ot 1 in 5. The haad dae
iliie pre^xne at A is 45 tt^ the velocity Is 3 ft. per aecondf ajid the section
kf the pipe is 3 sq, ft. Find the head due ta the preasure at B, where the
is 1 sq. ft. Take A as the lower end of the pipe.
(4^ The section pipe d a pump h laid at an inclinatioii of 1 in 5, and
is pnnaped ihrongb it at 6 ft. per fiecond. Suppose the air in the
>ter is disengaged if the pressure ialls to more than 10 lbs. below
^eric pceesare. Then deduce the greatest practicable length of
ptpa Friction neglected.
Wsler is delivered to an Inwardflow turbine under a head of 100 feet
IX j. The pressure just outside the wheel ia 25 lbs. per
jliy gan^e.
Fihe velocity with which the water approaches the wheel* Friction
(0^ A sh£»rt conical pipe varying in diameter from 4^ 6'' at the large end
feel at tiie small end forms part of ei horizontal water main. Tlie
! head at the large end is found to be 1 00 feet, and at the small end
[ the disohaige through the pipe. Coefficient of discharge nnity^
Three cubic l^t of water per second flow along a pipe which as it
in diameter from 6 inches to 12 inches. In 50 feet the pipe
ieei. Pne to rarions causes there is a leas of head of 4 feet*
(aj the lo6S ol energy in fcxjt ponndfi par minute^ and in horse
id the difference in pressure head at the two points 50 feet apart.
eqnmtioii 1, section 35*)
9i A boriso&tal pipe in whidi the sections vary gradually has sections
1 10 9go«r« feet* I square foot, and 10 square feet at sections A, B, and G*
haad at A i% 100 feet, and the yelocity S feet per second.,
head and velocity at B.
tliftt in another case the difference of the pressure heads at A
, B It* 2 f«?et. Find the velocity at A,
A Ventnri meter in a water main consists of a pipe converging to
fcod enlarging again gradually. The section of main is 9 sq. ft.
of UuToat 1 sq. ft. The difference of presstire in the main and
ibroAi is 12 feet of water* Find the discharge of the main per hour*
If If tbe inlet area of a Venturi meter is n times the throat area, and
Ijp mm the ^lelodty and preB«!.ure at the throat, and the inlet pressure
thmt Up Asd mp are observed, t^ can be found.
CHAPTER IV.
FLOW OF WATER THROUGH ORIFICES AND
OVER WEIRS.
36. Flow of fluids through orifices.
The general theory of the discharge of fluids through orifices,
as for example the flow of steam and air, presents considerable
difficulties, and is somewhat outside the scope of this treatise.
Attention is, therefore, confined to the problem of detennining the
quantity of water which flows through a given orifice in a given
time, and some of the phenomena connected therewith.
In what follows, it is assumed that the density of the fluid k
constant, the effect of small changes of temperature and pressure j
in altering the density being thus neglected. !
Consider a vessel. Fig. 42, filled with water, the free surEace of
which is maintained at a constant level ; in the lower part of tbe
vessel there is an orifice AB.
Fig. 42.
Let it be assumed that although water flows into the vessel ^
as to maintain a constant head, the vessel is so large that at soi^^
surface CD, the velocity of flow is zero.
Imagine the water ir^ the vessel to be divided into a number ^
stream lines, and consider any stream line EF.
Let the velocities at £3 and F be Ve and t^p, the pressure heads
h^ and h^ and the positio^ heads above some datum, z^ and 9ft j
respectively.
FLOW THBOUGH ORIFICES
61
Then, applying Bemouilli's theorem to the stream line EF,
If tf is zero, then
1 t^E* T V9
217
= fcp~fcB + 2ji— 2£,
But from the figure it is seen that
is equal to A, and therefore
Ve
=fc,
or
2g
Since Ae is the pressure head at E, the water would rise in
a tabe having its end open at E, a height /^e> and h may thus
be called — ^following Thomson — the fall of "free level for the
point B."
, At some section GK near to the orifice the stream lines are all
practically normal to the section, and the pressure head will be
equal to the atmospheric pressure ; and if the orifice is small the fall
of free level for all the stream lines is H, the distance of the centre
of the section GK below the free surface of the water. If the
orifice is circular and sharpedged, as in Figs. 44 and 45, the section
GK is at a distance, from the plane of the orifice, about equal to
its radius. For vertical orifices, and small horizontal orifices,
H may be taken as equal to the distance of the centre of the
orifice below the free surface.
The theoretical velocity of flow through the small section GK
is, therefore, the same for all the stream lines, and equal to the
velocity which a body will acquire, in falling, in a vacuum,
through a height, equal to the depth of the centre of the orifice
below the free surface of the water in the vessel.
The above is Thomson's proof of Torricelli's theorem, which
was discovered experimentally, by him, about
the middle of the 17th century.
The theorem is proved experimentally as
follows.
If the aperture is turned upwards, as in
Fig. 43, it is found that the water rises
nearly to the level of the water in the vessel,
and it is inferred, that if the resistance of the
air and of the orifice could be eliminated, the
jet would rise exactly to the level of the
surface of the water in the vessel.
1!
1^
'
Fig. 43.
4—2
52
HTDBAUU08
Other experimentB deflcribed on pages 54— S6, also bIiow thal^
with carefully constracted orifices, the mean velocity throogh the
orifice differs from ^2gB, by a very small quantify. ;
37. Coeffloient of contraotion for aharpedged orifice.
If an orifice is cut In the flat side, or in the bottom of a veeBe^.
and has a sharp edge, as shown in Figs. 41 and 45, the stream Knfli;
set up in the water approach the orifice in all directions, as shovi
in the figure, and the directions of flow of the particles of wato^ [
except very near the centre, are not normal to the plane of Am ^
orifice, but they converge, producing a contraction of the jet \
Fig. 44.
Fig. 46.
At a small distance from the orifice the stream lines become
practically parallel, but the cross sectional area of the jet is
considerably less than the area of the orifice.
If w is the area of the jet at this section and a the area of the
orifice the ratio  is called the coefficient of contraction and may
a
be denoted by c. Weisbach states, that for a circular orifice, the
jet has a minimum area at a distance from the orifice slightly less
than the radius of the orifice, and defines the coefficient of
contraction as this area divided by the area of the orifice. For a
circular orifice he gives to c the value 0*64. Recent careful
measurements of the sections of jets from horizontal and vertical
sharpedged circular and rectangular orifices, by Bazin, the
results of some of which are shown in Table IV, show, however,
that the section of the jet diminishes continuously and in fact hss
no minimum value. Whether a minimum occurs for square orifice^
is doubtful.
The diminution in section for a greater distance than tha*
given by Weisbach is to be expected, for, as the jet moves aw»y
from the orifice the centre of the jet falls, and the theoretical
velocity becomes J2g (R + y),y being the vertical distance betweeO
the centre of the orifice and the centre of the jet.
FLOW THROUGH ORIFICES
5a
M a Muall distance away from the orifice, however, the stream
iiueaitpe prscdcally parallel, and very h'ttb error is introduced in
tile coeieieut of contraction by measuring the stream near the
Poncelet and Lesbros in 1828 fotmdj for an orifice '20 m. square,
I a minimom section of the jet at a distance of *3 uu from the orifice
ianti Bt lim section c was '.56^3. M. Bazin^ in discussing these
IiwdI^ remarks tlmt at distances greater than 0*3 m, the section
j becoinf» very difficult to measure^ and althongh the vein appears
jto expand, the sides become hollow, and it is uncertain whether
[tlieur^ is really diminiBhed,
Compkii* eontractimi. The maxinnim contraction of the jet
aitef place when the orifice is sharp edged and is well removed
Mm Ike Hides and bottom of the vesseL In this case the contrac
on i» said to be complete* Experiments show, that for complete
»&tfaeti(>n the dista^nce from the orifice to the sides or bottom of
p reesel should not be lea® than one and a half to twice the least
ijter of the orifice.
fipMe or supprmmd contractimi. An example of incom
tmtraction is shown in Fig. 46, the lower edge of the
Jar orifice being made level with the bottom of the vessel.
time effect is produced by pla^ring a horizontal plate in
VB»el level with tlie bottom of the orifice. The stream
ii rlie lower part of the orifice are normal to its plane
'the contraction at the lower edge is consequently suppressedp
Fig, a.
Similarly, if the width of a rectangular orifice is made equal
^?6selj or the orifice ahcd m pro\'ided with side walls
J he side or lateral contraction is suppressed. In any
ttippressed contraction the discharge is increased, but, as
later, the discharge coeificient umy vary more than
n traction is complete. To suppress the contraction
ly» the orifice must be made of such a form that the
liiie& biM^otne parallel at the orifice and normal to its plane.
54
HYDRAULICS
Experimental deiermination of c. The section of the stream
from a circular orifice can be obtained with considerable accu
racy by the apparatus shoT^Ti in Fig, 40, which consists of a
ring having four radial set
screws of fine pitch, Tlie
screws are adjusted until the
points thereof touch the jet.
M. Bazin has recently lised an
octagonal frame v^ith twenty
four set screws^ all radiating
to a common centre, to deter
mine the form of the section
of jets from various kinds of
orifices.
The screws were adjusted
until they just touched the jet. The frame was then placed upcm
a sheet of paper and the positions of the ends of the screwi
marked upon the paper. The forms of the 8^:stions could theo
be obtained, and the areaa measured with considerable accuracy.
Some of the results obtained are shown in Table IV and also in
the section on the form of the liquid vein,
38. Coefflcient of velocity for sliarpedged orifice.
The theoretifai velocity through the contracted section is, m
shown in section 36, equal to V2^H, but the actual velocity
Vi is slightly less than this due to friction at the orifice. The
ratio — = 4 is called the coeflicient of velocity. ^
Eayperimental determination of k, Tliere are two methods
adopted for determining k experimentally.
First method. The velocity is determined by measuring the
discharge in a given time under a given head, and the cross
sectional area w of the jet* as explained in the last paragraph, is
also obtained. Then, if Vi is the actual velocity, and Q the
discharge per second,
and
Jc^^
Second method. An orifice, Fig, 50, is formed in the aide of a
vessel and water alloweil to fiow from it. The water after leaving
the orifice flows in a parabolic curve. Above the orific e is fixed
a horizontal scale on which is a slider carrying a vertical scale, i
to the bottom of which is clamped a bent piece of wire, w ^tli a sharp 1
i
FLOW THROUGH ORIFICES
55
point The vertical scale can be adjusted so that the point touches
the upper or lower surface of the jet, and the horizontal and vertical
distances of any point in the axis of the jet from the centre of the
orifice can thus be obtained.
Fig. 50.
Assume the orifice is vertical, and let Vi be the horizontal
velocity of flow. At a time t seconds after a particle has passed
the orifice, the distance it has moved horizontally is
X = Vit (1).
The vertical distance is
y = hgt' (2).
Therefore y = \g —%
and . = V
The theoretical velocity of flow is
Therefore h = J= = ^ ?— .
j2gK 2'JyB.
h is better to take two values of x and y so as to make
allowance for the plane of the orifice not being exactly perpen
dicular.
B the orifice has its plane inclined at an angle to the
^^cal, the horizontal component of the velocity is Vi cos and
^e verrical component Vi sin 9.
At a time t seconds after a particle has passed the orifice, the
horizontal movement from the orifice is,
X = Vi COS Ot (1),
and the vertical movement is,
y = v,smet + yt^ (2).
After a time ii seconds ah = ViCos^^ (3),
yi = ViamOti + igt{' (4).
56 HYDRAULICS
Substituting the value of t from (1) in (2) and U from (3)
in (4),
y=''*^^*^2i#^ ®'
and, y, = «,tan« + 2^,^ (6).
From (5), 2V^.^W1
Substituting for Vi' in (6),
i^^e^yi^^^^ (8).
Having calculated tan 6^ sec can be found from mathematics^
tables, and from (7) Vi can be calculated. Then
^ sl2gR'
39. Bazin's experiments on a sharpedged orifice.
In Table IV are given values of A; as obtained by Bazin from
experiments on vertical and horizontal sharpedged orifices, for
various values of the head.
The section of the jet at various distances from the orifice waa
carefully measured by the apparatus described above, and the
actual discharge per second was determined by noting the time
taken to fill a vessel of known capacity.
The mean velocity through any section was then
Q being the discharge per second and A the area of the section.
The fall of free level for the various sections was different, and
allowance is made for this in calculating the coefficient h in the
fourth column.
Let y be the vertical distance of the centre of any section
below the centre of the orifice ; then the fall of free level for that
section is H + y and the theoretical velocity is
V2sr(H + y).
The coefficients given in column 3 were determined by dividing
the actual mean velocity through different sections of the jet by
J2gT3.y the theoretical velocity at the centre of the orifice.
Those in column 4 were found by dividing the actual mean
velocity through the section by \/2g (H + y), the theoretical
velocity at any section of the jet.
The coefficient of column 3 increases as the section is taken
further from the jet, and in nearly all cases is greater than unity.
\
FLOW THBOUGH ORIFICES
57
TABLE IV.
Sharp^dged Orifices Contraction Complete.
Table showing the ratio of the area of the jet to the area of
the orifice at definite distances from the orifice, and the ratio of
the mean velocity in the section to >/2grH and to J2g.(R + y)f
H being the head at the centre of the orifice and y the vertical
distance of the centre of the section of the jet from the centre of
the orifice.
Vertical circular orifice 0*20 m. ('656 feet) diameter, H = '990 m.
(3248 feet).
Coefficient of discharge m, by actual measurement of the flow is
m = 5977*.
Mean Velocity
>/2</(H + y)
= k
I^tanoe of the Beetion
^ni the plane of the
orifice in metres
(K)8
013
017
0235
0335
0516
Area of Jet
Area of Orifice
=c
•6079
•6971
•5951
•6904
•5830
•5690
Mean Velocity
0983
1001
1004
1012
10^5
1050
•998
•999
1003
1^007
1^010
Horizontal circular orifice 0*20 m. ('656 feet) diameter,
fl = 975m. (3198 feet).
0075
0093
0110
0128
0145
0163
m = 0*6035.
06003
05939
05824
05734
05658
05597
1005
1016
1036
1053
1067
1078
0968
0971
0982
0990
0996
0998
Vertical orifice '20 m. (656 feet) square, H = '953 m. (3126 feet).
m = 0'6066.
0151
06052
1002
•997
0175
06029
1006
1000
0210
05970
1016
1007
0248
05930
1023
1010
0302
05798
1046
1027
0350
05788
1049
1024
The real value of the coefficient for the various sections is
lowever that given in column 4.
For the horizontal orifice, for every section, it is less than
inity, but for the vertical orifice it is greater than unity.
Bazin's results confirm those of Lesbros and Poncelet, who in
See section 42.
58 HTDRAULICS
1828 found that the actual velocity through the contracted section
of the jet, even when account was taken of the centre of the
section of the jet being below the centre of the orifice, waa
■^ greater than the theoretical value.
This result appears at first to contradict the principle of the
conservation of energy, and Bemouilli's theorem. 1
It should however be noted that the vertical dimensions of the !
orifice are not small compared with the head, and the explanation
of the apparent anomaly is no doubt principally to be found in the
fact that the initial velocities in the different horiuzontal filaments
of the jet are different.
Theoretically the velocity in the lower ps^rt of the jet is greater
than J2g (R + y), and in the upper part less than J2g (H + y).
Suppose for instance a section of a jet, the centre of which is
1 metre below the free surface, and assume that all the filaments
have a velocity corresponding to the depth below the free surface,
and normal to the section. This is equivalent to assuming that
the pressure in the section of the jet is constant, which is probably
not true.
Let the jet be issuing from a square orifice of '2 m. ("656 feet)
side, and assume the coefficient of contraction is '6, and for
simplicity that the section of the jet is square.
Then the side of the jet is '1549 metres.
The theoretical velocity at the centre is \/2gr, and the discharge
assuming this velocity for the whole section is
'6 X '04 X J2g = '024 J2g cubic metres.
The actual discharge, on the above assumption, through any
horizontal filament of thickness dh, and depth fe, is
oQ = 01549xdAxN/2^,
and the total discharge is
/lOTTS
Q = 01549^2^ h^dh
^ ^ y9225
se^.. = '0241 n/2^.
'e theoretical discharge, taking account of the varying heads
The co?^®' r004 times the discharge calculated on the assumption
jn^ , head is increased this diiierence dimmisnes, and when
^ J ®. CTeater than 5 times the depth of the orifice, is very
Those m cc
velocity througi^ ^jg^^ agrees very approximately with that given
velocity at any se. gquare orifice, where the value of k is given bs
The coefficient .
further from the jet,
rum THROUGH ORIFICES
59
Uris partly then, ejq^lams the anomalous values of k^ but it
UPO< be Ifxiked upon as a complete explanation*
B^ < tual jet are not. exactly those assumed,
H)» fry normal to the plane of the section is
EtttUly much more complicated than here Essumed.
iAm Baein further points out, it m probable that, in jets like
pn^m the square urifiee, which, as will be seen later when the
jbof the jet i« considered, are sabject to considerable deformation,
^diverfent^e of m^me of the iilaments gives rise to pressures less
BQ that of the atmosphere.
^k litteinpted to demonstrate thist experimen tally , and
Hti> ', Fig, 150, registered pressures less than that of the
Imosphen*; but he doubts the reliability of the results, and
Mte out the extreme difficulty of satisfactorily determining the
^pix* in the jet,
^^imt Uie inequality of the velocity of the filaments is the
tmmrf CHUt^e, receives support from the fact that for the
^vnoQlal orifice, discharging downwards, the coefficient k is
luili difhtly less than unity. In this case, in any horizontal
lolimj below tlie orifice, the head is the same for all the stream
1166, fti>d the velocity of the filaments is practically constant.
np: '*^" ^ of velocity is never less than '96, bo that the loss
^P'' rnal friction of the liquid is very small,
40. Distribtitioii of velocity in the plane of the orifice.
Btttn has examined the distribution of the velocity in the
ioois sections of the jet by means of a fine Pi tot tube (see
^ 215). In the plane of the orifice a minimum velocity
teow, which for vertical orifices is just aliove the centre, but at a
4kiaiice from the orifice the minimum velocity is at the top
F()r orifices having complete contraction Bazin found the
toimaiii velocity to be '62 to '64 n/2^H, and for the rectaTigular
ttifioft, frith lateml contraction suppressed, 0'69 v^2^H.
Ab the ilii^tance from the plane of the orifice increases, the
riucities in the transverse section of the jets from horizontal
rtfices, rapidly become uniform throughout the transverse section*
For vertical orifices, the velocities below the centre of the jet
1*^ greater than those in the upper part.
41. Presstire in the plane of the orifice.
M* Lagerjelm stated in 1826 that if a vertical tnbe open at
was placed with its lower end near the centre, and not
uiy below the plane of the inner edge of a horizontal
I
I
\
60 HYDRAULICS
orifice made in the bottom of a large reservoir, the water rose in
the tube to a height equal to that of the water in the reservoir,
that is the pressure at the centre of the orifice is equal to the head
over the orifice even when flow is taking place.
M. Bazin has recently repeated this exx)eriment and founds
that the water in the tube did not rise to the level of the water in
the reservoir.
If Lagerjelm's statement were correct it would follow that the
velocity at the centre of the orifice must be zero, which again does
not agree with the results of Bazin's experiments quoted above.
42. Coefficient of discharge.
The discharge per second from an orifice, is clearly the area
of the jet at the contracted section GK multiplied by the mean
velocity through this section, and is therefore,
Q=c.fc.aN^H.
Or, calling m the coefficient of discharge,
This coeflScient m is equal to the product c . A;. It is the only
coefficient required in practical problems and fortunately it can
be more easily determined than the other two coefficients c and h.
Experimental determination of the coefficient of discharge.
The most satisfactory method of determining the coefficient of
discharge of orifices is to measure the volume, or the weight of
water, discharged under a given head in a known time.
The coefficients quoted in the Tables from M. Bazin*, were
determined by finding accurately the time required to fill a vessel
of known capacity.
The coefficient of discharge m, has been determined with
a great degree of accuracy for sharpedged orifices, by Poncelet
and Lesbrost, WeisbachJ, Bazin and others §. In Table IV
Bazin's values for m are given.
The values as given in Tables V and VI may be taken as
representative of the best experiments.
For vertical, circular and square orifices, and for a head of
about 3 feet above the centre of the orifice, Mr Hamilton Smith
junr. II, deduces the values of m given in Table VI.
♦ Annales dfs Pont$ et ChausBies^ October, 1888.
t Flow through Vertical Orijices,
X Mechanics of Engineering.
§ Experiments upon the Contraction of the Liquid Vein. Bazin translated bj
Trautwine.
II Tfw. Flow of Water through Orijices and over Weirs and through open CondmUi '
and Pipes, Hamilton Smith, junr., 1886.
FLOW THBOUOH ORIFICES
61
TABLE V.
SiMriiiieDtar
Partioulara of orifice
Coefficient of
discharge m
Baxm
Pbooelet and
Leabitn
Bam '
n
n
" 1
Vertioal aquaie orifice side of square 06662 ft.
» »» t« 9,
Vertical Rectangular orifice 666 ft. high x 2624
ft wide with side contraction suppressed
Vertical circular orifice 0*6662 ft. diameter
Horizontal
08281
0606
0606
0627
0698
06086
06068
TABLE VI.
Cvrcvlar orifices.
DiuKterof
infti
0O197
I
0627
0O296
0^17
0089
0611
0O492
0606
00984
0608
I
0164
0600
0328
0699
06662
0698
09848
0597
Square orifices.
IT
Side^of^uare ^^^ I ^y^^^ \ ^^^
0681
0612
0607
0197 ! 06906 I 09843
0605 I 0604 I 0603
I !
TABLE VIL
' Table showing coefficients of discharge for square and rect
angular orifices as determined by Poncelet and Lesbros.
1
HeftJ of water '
Width of orifice 6502 feet
Width (»f orifice
1 908 feet i
ilwre the top ,
of the orifice :
Depth of orifice in feel
•
in feet
0328
0656
•0984
•1640
•607
•3287
6562
•0656
6562
•0328
701
•660
•630
•0656
694
659
634
•615
596
572
•643
1312
683
•658
•640
•623
•603
582
•642 ]
595
•2624
670
656
•638
•629
•610
589
•640 1
601
•3937
663
•653
•636
•630
•612
593
•638 1
603
•6562
655
•648
•633
•630
•615
598
•635 1
605
1640 1
642
•638
•630
•627
•617
604
•630 1
607
3281
632
633
628
•626
•615
605
•626 1
605
4921
616
619
620
•620
•611
602
•623 1
602
6562
611
612
612
•613
•607
601
•620 ,
602
9^*48
609
•610
608
•606
•603
601
•615 1
601
62 HTDRAUUCT
The heads for which Bazin determined the coefficientB
Tables IV and V varied only from 2"6 to 3"3 feet, but, as wilJ
seen from Table YII, deduced from reaults given by Poiicelet
Lesbros* in their cla^cal work, when the variation of head is
small, the coefficienta for rectangular and square orifices vary
considerably with the head.
43. Effect of suppressed contraction on the coeffideni
of discharge*
Sharp^ged orifice. When &ome part of the contraction of &
transverse section of a jet issuing from an orifice is suppressed^
the cross sectional area of the jet can only be obtained witk
diflSculty.
The coefficient of ditwharge can, however^ be easily obtained,
as before, by determining the discharge in a given time. Ttfr^
most complete and accurate experimenta on the effect of contrac
tion are those of Lesbros, some of the results of which are quoted
in Table VIII. The coefficient is most constant for square or
rectangular orificee when the lateral contraction is suppressed. The
reason being, that whatever the head, the %'ariation in the section
of the jet i^ confined to the top and bottom of the orifice, the
width of the stream remaining constant, and therefore in a greater
part of the transverse section the stream lines are normal to the
plane of the orifice*
According to Bid one, if iP is the fraction of the periphery of a
sharpedged orifice upon which the contraction is suppressed, and
m the coefficient of discharge when the contraction is complete,
then the coefficient for incomplete contraction is,
Wi = m (1 + "IStc),
for rectangular orifices, and
for circular orifices,
Bidone's formulae give result* agreeing fairly well with
Lesbros' experiments.
His formulae are, howeverj unsatisfactory when x approaches
unity, as in that case mi should be nearly unity.
If the form of the formula is preserved, and m taken ae '606^
for mi to be unity it would require to have the valne, i
jrii  m (1 + '65ir), ^
For accurate measurements, either orifices with perfect a
traction or, if possible, rectangular or square orifices with
lateral contraction completely suppressed, should be used. It ^
* Experiences hydrauliquee tur Us lois de Vicoulement de Veau h
^eSf etc., 1882. Ponoelet and Lesbros.
FLOW THK017OH ORIFICES
6S
Qf be neceeeaiy ki caljbmte the orifice for variotis heads,
W as shown above the coefficient for the latter kind is more
likely b] be constant.
TABLE Vni,
Table showing the effect of soppressing the contraction on the
efiidetit of discharge. Lesbroa *,
Square vertical orifica 0"65d feet square.
1 R«v4 ,j walCT
Side con
Contrfttitiot]
Contraction
Ppet
Sh&rp^^dged
tr&otjon
Buppres^d al
HnppFeased at
the hmet tmd
Bide edgen
t ■ ■ mfiee
Bappreseed
the lower edge
1
mm
0572
0599
01640
i>585
0*6S1
0^608
09281
0592
0'631
0615
0*6562
0596
0*632
0621
0*708
, imo
06(^
0^631
0623
0680
W81
0606
(>628
0624
0676
4^931
0602
0*627
0*624
0'672
fl^62
0601
0^626
0619
0668
^^§&
0^1
0624
0*614
0^665 '
Fig* 51, Section of Jet from
cireokr orifice.
W, Tlie fonn of tlie jet from sharpedged orifices.
From a circular orifice the Jet emerges like a cylindrical rod
iirf retaimt a form nearly cylindrical for some distance from the
hg. 51 3how8 three sections of a jet from a vertical circular
[(►nfice u varying distances from the
e, m given by M, Bazin,
The flow from square orifices m
Dpanied by an interesting and
c«riou« phenomenon called the in
of the jet.
At a vitry small distance from
llhH tjrifice the section becomes as
Y^ifmi in Fif* 52, The aides of the
t ant concave and the comers* are
ve sections. The
lies octagonal as in
\ and afterwards takes the form of a square with concave
[and rounded comers, the diagonals of the square being
cuhir to the sides of the orifice, Fig. 54»
DOO
Figs. 62 — 54. Section of jet froni
aqtiare orifice*
* Kjp^fimefiU h§drmUiqueM tur Ut loi$ d^ ricmdent^nt de Vtau^
^?':riare orifice
^« tlxe coe
.j^T GO ^"^
c^^^
64
HTDRAUUCS
45. Large orifioea
Table VII shows very clearly that if the depth of a vertical orifice
is not small compared with the head, the coefficient of discharge
varies very considerably with the head, and in the discussion of
the coeflScient of velocity fc, it has already been shown that the
distribution of velocity in jets issuing from such orifices is not
uniform. As the jet moves through a large orifice the stream
lines are not normal to its plane, but at some section of the stream
very near to the orifice they are practically normal.
If now it is assumed that the pressure is constant and equal to
the atmospheric pressure and that the shape of this section is
known, the discharge through it can be calculated.
Rectangular orifice. Let efgh, Fig. 55, be the section by a
vertical plane EF of the stream issuing from a vertical rectangular
orifice. Let the crest E of the stream be at a depth h below
the free surface of the water in the vessel and the under edge
F at a depth h^.
Fig. 66.
At any depth h, since the pressure is assumed constant in the
section, the fall of free level is h, and the velocity of flow through
the strip of width dh is therefore, kJ2gh, and the discharge is
kbJ2ghdh.
If & be assumed constant for all the filaments the total discharge
in cubic feet per second is
Q = kJ2^j\hidh = f J2gkb (hi* ho*).
Here at once a difficulty is met with. The dimensions ^, hi
and b cannot easily be determined, and experiment shows tJial
they vary with the head of water over the orifice, and that they
cannot therefore be written as fractions of Ho, Hi, and B.
FLOW THBOUQH ORIFICES
65
By replacing %«, &i and b by Ho, Hi and B an empirical
formnla of the same form is obtained wbicli, by introducing a
coefficient c, can be made to agree with experiments. Then
Q = 5c^^.B(HltHo*),
or replacing c by n,
Q = nV^.B(Hi»Ho») (1).
The coefficient n varies with the head Ho, and for any orifice
the simpler formnla
Q=m.a.^J2gR (2),
a being the area of the orifice and H the head at the centre,
can be used with equal confidence, for if n is known for the
particular orifice for yarious values of Ho, m will also be known.
From Table YII probable values of m for any large sharp
edged rectangular orifices can be interpolated.
Rectangular slvices. If the lower edge of a sluice opening is
some distance above the bottom of the channel the discharge
through it will be {vractically the same as through a sharpedged
orifice, but if it is flush with the bottom of the channel, the
contraction at this edge is suppressed and the coefficient of
discharge will be slightly greater as shown in Table VIII.
46. Drowned orifices.
"WTien an orifice is submerged as in Fig. 56 and the water in
the upstream tank or reservoir is moving so slowly that its velocity
may be neglected, the head causing velocity of flow through any
filament is equal to the difference of the up and downstream
levels. Let H be the difference of level of the water on the two
sides of the orifice.
Fig. 56.
L. H.
66
HYDRAULICS
■h^^hf
Consider any stream line FE which passes through the orifice
at B. The pressure head at E is equal to Aj, the depth of E below
the downstream level. If then at F the velocity is zero,
29[
or Ve = ^2g (h  A,)
or taking a coefficient of velocity k
VE = kJ2g.B.,
which, since H is constant, is the same for all filaments of the
orifice.
If the coefficient of contraction is c the whole discharge through
the orifice is then
Q = ckas/2gR
= m.a, J2gK.
47. Partially drowned orifice.
If the orifice is partially drowned, as in
Fig. 57, the discharge may be considered in
two parts. Through the upper part AC the
discharge, using (2) section 45, is
and through the lower part BC
Q2=mi.ai.N/2sr.Hi.
48. Velocity of approach.
BS^=
I
Fig. 57.
It is of interest to consider the efEect of the
water approaching an orifice having what is
called a velocity of approach, which will be equal to the velocity
of the water in the stream above the orifice.
In Fig. 56 let the water at F approaching the drowned orifice
have a velocity Vi.
Bemouilh's equation for the stream line drawn is then
2g
+ h,:
J.^'"'
■'^^2^'
and Ve=>/2^(r^^) ,
which is again constant for all filaments of the orifice.
Then Q = m.a.N^.(H+gy.
SUDDEN ENLABOEMEirr OF A STREAM 67
49. Effecit of velocity of approach on the discharge
throng a large reetangnlar orifice.
If the water approaching the large orifice, Fig. 55, has
a velocity of approach t?i, Bemonilli's equation for the stream hne
passing throug^h the strip at depth hy will be
w 2g w 2gr'
Pm being the atmospheric pressure, or putting in a coefficient of
velocity,
The discharge through the orifice is now,
50. CoefELcient of resistance.
In connection with the flow through orifices, and hydraulic
plant generally, the term " coefficient of resistance " is frequently
nsed. Two meanings have been attached to the term. Some
times it is defined as the ratio of the head lost in a hydraulic
system to the effective head, and sometimes as the ratio of the
head lost to the total head available. According to the latter
method, if H is the total head available and h/ the head lost,
the coefficient of resistance is
Cr jj.
51. Sudden enlargement of a current of water.
It seems reasonable to proceed from the consideration of flow
throagh orifices to that of the flow through mouthpieces, but
before doing so it is desirable that the effect of a sudden
enlargement of a stream should be considered.
Suppose for simplicity that a pipe as
in Fig. 58 is suddenly enlarged, and that
there is a continuous sinuous flow along
the pipe. (See section 284.)
I At the enlargement of the pipe, the
J stream suddenly enlarges, and, as shown
L in the figure, in the comers of the large
pipe it may be assumed that eddy motions p. gg
are ^t up which cause a loss of energy.
5—2
68
HYDRA ULICS
Ck>nsider two sections oa and dd at each a distance from bh
that the flow is steady.
Then, the total head at dd equab the total head at oa minaii
the loss of head between oa and dd^ orH hm the loss of head doe
to shock, then
Va Vj Pd Vd 1
w 2g w 2g
Let A« and A^ be the area at oa and dd respectively*
Since the flow past oa a ' ^ ^^^^ ^^
Then, assuming that each
velocity Vo, and r^ at dd^ tlie m'
which passes aa in unit time i^ <
of the water that passes dd is
t
nt of fluid at aa has
iin of the quantity of waterl
w
 Afli'rt'i and the tDOmentmii
9 '
the momentum of a mass of M ide moving with a velo
V feet per second being Mt* pounds feet.
The change of tnonientum is therefore,
The forces acting on the water between tm and dd to produce
this change of moment um, are
p<tAa acting on aa, p^A^ acting on dd^
and, if p is the mean pressure per unit area on the annular riu^
hhy an additional force p{Ad A^)^
There is considerable doubt as to what is the magnitude of the
pressure p, but it is generally assiuwed that it is equal to pa, for
the following reason.
The water in the enlarged portion of the pipe may be looked
upon as divided into two parts, the one part having a motion of
translation, while the other part, which is in contact with the
annular ring, is practically at rest* (See section 284 J
K this assumption is correct, then it is to be expected that the
pressure throughout this atill water will be practically eqtial at all
points and in all directions, and must be equal to the pressure ia
the stream at the section 65, or the pressure p is equal to p«.
Therefore
PcjA<  Pa (Ad  A«) paAa = 'M?— (V«Vd),
KkVa
9
from which (pd  po) A4 = to (t?o — t?d) ;
SUDDEN ENLARGEMENT OF A STREAM 69
and aince A«t7« = A^rVd,
9
Adding ^ to both sides of the equation and separating
2Sr
~ into two parts.
vr
or & the loss of head dae to shock is eqaal to
According to St Venant this quantity should be increased by
1 Vd^
wi amount equal to 5 nZ > ^^^ this correction is so small that as
& nile it can be neglected.
52. Sudden contraotion of a current of water.
Suppose a pipe partially closed by means of a diaphragm as in
Fig. 59.
Afi the stream approaches the diaphragm
""■which is supposed to be sharpedged —
it contracts in a similar way to the stream
passing through an orifice on the side of
a vessel, so that the minimum cross sec
tional area of the flow will be less than the Fig. 59.
area of the orifice.
The loss of head due to this contraction, or due to passing
through the orifice is small, as seen in section 39, but due to
the sodden enlargement of the stream to fill the pipe again, there
is a considerable loss of head.
Let A be the area of the pipe and a of the orifice, and let c be
the coefficient of contraction at the orifice.
Then the area of the stream at the contracted section is ca, and,
therefore, the loss of head due to shock
2g
70 HTDRAULICS
If the pipe simply diminishes in diameter as in Fig. 58, the
section of the stream enlarges from the contracted area oa to fill j
the pipe of area a, therefore the loss of head in this case is
"'iil^)' «•
Or making St Venant correction
^m')'^i} «)•
Valtte of thecoefficient c. The mean valneof cfor aaharpedged
circular orifice is, as seen in Table IV, about 0*6, and this maj be
taken as the coefiicient of contraction in this formula.
Substituting this value in equation (1) the loss of head k
found to be ^— , and in equation (2), g— ,v being the velodUy iB J
0*5o^
the small pipe. It may be taken therefore as i^a'' ^^^'*'^^]
experiments are required before a correct value can be assigned.
53. Loss of head due to sharpedged entrance Into a pipi^
or mouthpiece.
When water enters a pipe or mouthpiece from a vessel throogk
a sharpedged entrance, as in Fig. 61, there is first a contraction, and
then an enlargement, as in the second case considered in section 52.
The loss of head may be, therefore, taken as approximately 5
and this agrees with the experimental value of — ^ given by
Weisbach.
This value is probably too high for small pipes and too low far
large pipes*.
54. Mouthpieces.
If an orifice is provided with a short pipe or mouthpiece, througfc
which the liquid can flow, the discharge may be very differem.'^
from that of a sharpedged orifice, the difference depending npo*
the length and form of the mouthpiece. If the orifice is cylindric^i^
as shown in Fig. 60, being sharp at the inner edge, and so shor^
that the stream after converging at the inner edge clears th^
outer edge, it behaves as a sharpedged orifice.
Short external cylindrical mouthpieces. If the mouthpiece 1J
cylindrical as ABFE, Fig. 61, having a sharp edge at AB anC3
a length of from one and a half to twice its diameter, the je^*^
* See M. Bazin, Exp€rience» nouvelUs sur la distribution des vite$se» dam^
le$ tuyaux.
FLOW THROUGH MOUTHPIECES
71
contracts to CD, and then expands .to fill the pipe, so that at EF
it discharges foil bore, and the coefficient of contraction is then
onitF. Experiment shows, that the coefficient of discharge is
— B
Fig. 60.
Fig. 61.
from 080 to 0"85, the coefiicient diminishing with the diameter
of the tube. The coefficient of contraction being unity, the
coefficients of velocity and discharge are equal. Good mean
^iies, according to Weisbach, are 0*815 for cylindrical tubes,
w»d 0*819 for tubes of prismatic form.
These coefficients agree with those determined on the assump
^on that the only head lost in the mouthpiece is that due to
sndden enlargement, and is
05i;"
^ being the velocity of discharge at EF.
Applying Bernouilli's theorem to the sections CD and EF, and
taking into account the loss of head of p— , and pa as the atmo
spheric pressure,
w 2g w 2g 2g w^
or 7^^ — = H.
Therefore
2g
v^ = '66 X 2gR
md i? = 812N/2^H.
The area of the jet at EF is a, and therefore, the discharge
er second is
a.v = '8V2as/2gR.
Or m, the coefficient of discharge, is 0*812.
The pressure head at the section CD. Taking the area at CD
s 0606 the area at EF,
rcD = l'66t7.
72 HYDRAULICS
Therefore S^ = S^ . ^'  22l^ Pa _ lf^\
or the pressure at C is less than the atmospheric pressure.
K a pipe be attached to the mouthpiece, as in Fig. 61, and tt^
lower end dipped in water, the water should rise to a height of abo^c
— s — feet above the water in the vessel.
55. Borda's mouthpiece.
A short cylindrical mouthpiece projecting into the vessel, as ^B
Fig. 62, is called a Borda's mouthpiece, arid is of interest, as tfcr^
coefficient of discharge upon certain assumptions can be readiK
calculated. Let the mouthpiece be so short
that the jet issuing at EF falls clear of GH.
The orifice projecting into the liquid has
the effect of keeping the liquid in contact
with the face AD practically at rest, and
at all points on it except the area EF the
hydrostatic pressure will, therefore, simply "
depend upon the depth below the free ^^'
surface AB. Imagine the mouthpiece produced to meet tli0
face EC in the area IK. Then the hydrostatic pressure on AID,
neglecting EF, will be equal to the hydrostatic pressure on BC,
neglecting IK.
Again, BC is far enough away from EF to assume that the
pressure upon it follows the hydrostatic law.
The hydrostatic pressure on IK, therefore, is the force whicl
gives momentum to the water escaping through the orifice, ove^
comes the pressure on EF, and the resistance of the mouthpiece.
Let H be the depth of the centre of the orifice below the free
surface and p the atmospheric pressure. Neglecting frictional
resistances, the velocity of flow v, through the orifice, is j2glEL.
Let a be the area of the orifice and cu the area of the transverse
section of the jet. The discharge per second will be it; . w J2gK lbs.
The hydrostatic pressure on IK is
pa + wdK lbs.
The hydrostatic pressure on EF is pa lbs.
The momentum given to the issuing water per second, is
M = .o,.2^H.
Therefore pa + — o> 2gK = pa + wdR,
and (I) = ia.
FLOW THROUGH MOUTHPIECES 73
The coefficient of contraction is then, in this case, equal to
one half.
Experiments by Borda and others, show that this result is
justified, the experimental coefiicient being slightly greater
than \.
56. Conical monthpieoes and nozzles.
These are either convergent as in Fig. 63, or divergent as in
Fig. 64.
j=^
Fig. 63. Fig. 64.
Calling the diameter of the mouthpiece the diameter at the
ontlet, a divergent tube gives a less, and a convergent
tabe a greater discharge than a cylindrical tube of the
®^ diameter.
Experiments show that the maximum discharge for a
convergent tube is obtained when the angle of the cone
is from 12 to 13^ degrees, and it is then 0*94 . a . J2gh,
K instead of making the convergent mouthpiece conical,
te sides are curved as in Fig. 65, so that it follows as
ear as possible the natural form of the stream lines, the
^efficient of discharge may, with high heads, approxi
ate very nearly to unity.
Weisbach*, using the method described on page 55
determine the velocity of flow, obtained, for this
>nthpiece, the following values of k. Since the mouth p. gg
?ce discharges full the coefficients of velocity k and
icharge m are practically equal.
Head in feet
066
164
1148
558
888
k and m
•959
•967
•975
•994
•994
According to Freeman t, the firehose nozzle shown in Fig. 66
3 a coefficient of velocity of *977.
• Mechanics of Engineering.
t TramaetionM Am, Soc. C.E., Vol. xxi.
74 HYDRAULICS
If the mouthpiece is first made convergent^ and then divergent,
Fig. 66.
as in Fig. 67, the divergence being sufficiently gradual for the
stream lines to remain in contact with the tube, the coefficient of
contraction is unity and there is but a
small loss of head. The velocity of efflux
from EP is then nearly equal to >/2gfH
and the discharge is m . a . N/2grH, a being
the area of EP, dnd the coefficient m
approximates to unity.
It would appear, that the discharge
could be increased indefinitely by length
ening the divergent part of the tube and
thus increasing a, but as the length increases, the velocity
decreases due to the friction of the sides of the tube, and farther,
as the discharge increases, the velocity through the contracted
section CD increases, and the pressure head at CD consequently
falls.
Calhng Pa the atmospheric pressure, pi the pressure at CSD>
and Vi the velocity at CD, then
w
and
2g w
w w 2g
If s^ is greater than H + — , 2^ becomes negative.
As pointed out, however, in connection with Proude's apparatoBi
page 43, if continuity is to be maintained, the pressure cannot be
negative, and in reaUty, if water is the fluid, it cannot be lew
than 7 the atmospheric pressure, due to the separation of the air
from the water. The velocity Vi cannot, therefore, be incr
indefinitely.
FLOW THROUGH MOUTHPIECES
75
Assmnin^ th© pressure can J list become zero, a,nd taking the
eric preamre as equivalent to a head of 34 tL of water, the
mm possible Yelocitjr, is
Ur=^%/2ff CH + 34ft.)
id the tnaiciTnnm ratio of the area of EF to CD is
^A^
S4ft
H
Praclically^ tlie maximtim value of vi may be taken as
and the maximnin ratio of EF to CD as
v/^.
The maxim HID discharge is
y
^
Tlie ratio gWen of EF to CD may be taken as the masdnaum
beiween the area of a pipe and the throat of a Venturi meter
[lo be used in the pipe.
57. Flow tliroiigh orifices and mouthpieces under constant
freasure.
The head of water causing flow through an orifice may be
produced \yf a pomp or other mechanical means, and the discharge
iB^ tdce place into a vessel, such as the condenser of a steam
, in which the pressure is less than that of the atmosphere.
example, suppose water m be discharged from a cylinder
I a vemel B, Fig, 6H^ through
It orifice or moathpiece by means
U iMtoEi loaded with P lbs., and
fft^preettre per sq. foot in B
Lpt the area of the piston be
A ijuare feet. Let h be the height
of the water in the cylinder above
*fc« ORiti« of the orifice and fh of
^ water in the vesael B. The
tlworetical effective head forcing water through the orifice may
be writlm
Fig, 68.
Aw
w
76 HTDRAITLICS
If P is large K and h will generally bo negligible.
At the orifice the pressure head la K^^t a^id therefore for
w
any stream line through the orifice, it there is no friction^
2g w Aw
The actual velocity will be less than u, due to frictioG, and Kl]
IB a coefiicient of velocity, the velocity is then
and the discharge i^i Qm.a*J2gH.
In practical examples the cylinder and the vessel will geneiaDy
be connected by a short pipe, for which the coefficient of velocity
will depend upon the length.
If it is only a few feet long the principal loss of head will be
at the entrance to the pipe, and the coefficient of discharge will
probably vary between 0*65 and 0'85,
The effect of lengthening the pipe mil be understood aft'er tbd
chapter on flow through pipes has been read.
Example. Water is di^ehait^ed from a pamp inta & condenaer in ^hioh Hift
pressure is 3 lbs. per aq. inch Ibroui^li a f^ort pipe S^ inchea diameter.
The pressure in the puiup i& 2Q iba. per kij. tneh.
Find the dischargi^ into the condenser, takinR the coefficieut of discharge 0'7S*
The effective head is
^^20x144 3jcU4
as k^
624
section v^.
=39 a feet.
falls.
f Q=?i>v '7M54
^^ V . /
' i** '
jjij4:^4i y. 3li*2 cubic feet per sec
i
Calling J =1'84 cubic ft. per sec.
and Vi the Vi
e of emptying a tank or reservoir.
reservoir to have a sharpedged horizontal orifice
It is required to find the time taken to empty
and
UjS vf the horizontal section of the reservoir at any
If ^ is greater t^ ^^^^ ^^ ^ gq f^^^ ^^^ ^1^^ ^^^ ^j ^^
As pointed out,ho>^ let the ratio ~ be sufficiently large that the
page 43, if continuity . ? , , .
negative, and in reaUt? the reservoir may be neglected,
than i the atmospheric J* *he water is at any height h above the
from the water. The veV Ao^s through the orifice m any time dt
indefinitely.
FLOW THROUGH MOUTHPIECES 77
The amoimt dh by which the surface of water in the reservoir
falls in tKe time dt is
j_ maJ2ghd t
A.
;.. Aoh
or or =   . — . .
ma v2gA*
The time for the water to fall from a height H to Hi is
^^ f^ Adh_ ^ i_ r« Adh
J ^tma ^gh a J2g J h, mh^ '
If A is constant, and m is assumed constant, the time required
for the surface to fall from a height H to Hi above the orifice is
_ . Adh
ma ^2g •
^Vc^Hn/Hi),
ma J2g
and the time to empty the vessel is
^ ^ 2 A n/H
mu \/2g*
or is equal to twice the time required for the same volume of
wwter to leave the vessel under a constant head H.
Time of emptying a lock with vertical drowned sluice. Let the
'water in the lock when the sluice is closed be at a height H,
Fig^. 56, above the downstream level.
Then the time required is that necessary to reduce the level in
the lock by an amount H.
"When the flow is taking place, let x be the height of the water
Ij^ the lock at any instant above the downstream water.
\jet A be the sectional area of the lock, at the level of the
^vrater in the lock, a the area of the sluice, and m its coefficient of
discharge.
The discharge through the sluice in time dt is
cQ = m . a >/2gx . ot.
If da? is the distance the surface falls in the lock in time cty then
Adx = m^ J2gxdt,
Aox
or ^^ =" " /?r 1 •
m^ y/2gx^
\
To redace the level by an amount H,
f ^ Adx
ma *J2gxi
H
.J > : ni: of
: .■■t±;:tn: cf
": iTri A is
r
• i :•:&.; ^f 40 a To
iischai^e 10 eft
FLOW THROUGH ORIFICES AND MOUTHPIECES 79
(5) A jet 3 in. diameter at the orifice rises vertically 50 ft. Find its
at 25 ft. above the orifice.
(6^) An orifice 1 sq. ft. in area discharges 18 c. ft. per second under a
head of 9 ft. Assuming coefficient of velocity =0*98, find coefficient of
ooatraction.
{7) The pressure in the pump cylinder of a fireengine is 14,400 lbs.
per sq. ft.; assuming the resistance of the valves, hose, and nozzle is such
that the coefficient of resistance is 0*5, find the velocity of discharge, and
the height to which the jet will rise.
<8) The pressure in the hose of a fireengine is 100 lbs. per sq. inch;
the jet rises to a height of 150 ft. Find tlie coefficient of velocity.
<9) A horizontal jet issues under a head of 9 ft. At 6 ft. from the
orifice it lias faUen vertically 15 ins. Find the coefficient of velocity.
i 10) Required the coefficient of resistance corresponding to a coefficient
of velocity =097.
Ill) A fluid of one quarter the density of water is discharged from a
Tessel in which the pressure is 50 lbs. per sq. in. (absolute) into the
fttmosphere where the pressure is 15 lbs. per sq. in. Find the velocity of
d&Mharge.
(12) Find the diameter of a circular orifice to discharge 2000 c. ft. per
hour, under a head of 6 ft. Coefficient of discharge 0*60.
(18 1 A cylindrical cistern contains water 16 ft. deep, and is 1 sq. ft. in
cross section. On opening an orifice of 1 sq. in. in the bottom, the water
lervel fell 7 ft. in one minute. Find the cocfficicut of discliarge.
(14) A miner*s inch is defined to be the discharge through an orifice in
a vertical plane of 1 sq. in. area, under an average head of 6J ins. Find
the supply of water per hour in gallons. Coefficient of discharge 0*62.
(15 1 A veJisel fitted with a piston of 12 sq. ft. area discharges water
imder a head of 10 ft. What weight placed on the piston would double the
jmte of discharge?
(16) An orifice 2 inches square discharges under a bead of ICX) feet
1*888 cubic feet per second. Taking the coefficient of velocity at 0*97, find
the coefficient of contraction.
(17) Find the discharge per minute from a circular orifice 1 inch
diaiiieter, under a constant pressure of 84 lbs. per sq. inch, taking 0*60 as
the coefficient of discharge.
(18) The plunger of a fireengine pump of one quarter of a sq. ft. in
ana is driven by a force of 9542 lbs. and the jet is observed to rise to a
hdigjit of 150 feet. Find the coefficient of resistance of the apparatus.
(19) An orifice 8 feet wide and 2 feet deep has 12 feet head of water
■bofe Hb centre on the upstream side, and the backwater on the other
ie is at the level of the centre of the orifice. Find the discharge if
fli.»i«CH$2.
80
HYDRA tTLlCS
(20) Tan e. ft, of watat per second flow thfQu^ a pipe of 1 aq. ft. Bxe^
which suddenly enlarges to 4 sq. ft. area. Taking the pressure at 100 lbs;
par sq. ft. in the mmaUar part of the pipe, find (1) the head lost in aho
(*2j the preaaure in the larger part, (3) the work expended in forcing
water through the enlargement.
(21) A pipe of 3" diameter ia suddenly enlarged to 5'^ diameter* A \
tuba containing marcnry is connected to two points, one on eaeh side of
enlargement, at points where the flow ia steady. Find the differenee :
level in the two Linabs of the U when water flows at the rate of 2 c, ft»
second from tlie small to the large section and i^ice versd. The sp
gravity of mercury is IB'6, Lond» Un,
(22) A pipe is suddenly enlarged from 2 indies in diameter to
inches in diameter. Water flows tlirough these two pipes from the i
to the larger, and the discharge from the end of the higger pipe is twd
gaDons per second. Find:—
(a) The loss of head, and gain of pressure head, at the
ment.
(b) The ratio of head lost to velocity bead In small pipe.
(28 J The head and tail water of a vertical sided lock differ in levdl
12 ft The area of the lock bai^in is TOO sq. ft. Find the time of emp^flQf I
the lock, through a sluice of 5 sq. ft. area, with a coefficient 0"6, Tli»1
sluice discharges helow tail water level.
(24 1 A tank X200 sq. ft. in area discharges through an orifice 1 a^ilt J
in area. Calculate the time retj^aired to lower the level in the tank frodi j
60 ft. to 26 ft. ahove the orifice. Coefficient of discharge 0"6.
(26) A verticalsided lock is 65 ft. long and 18 ft, wide. Lift li5 H ]
Find the area of a sluice helow tail water to empty tlie lock in 5 minute!. 
Coefficient 0'6*
(26) A reservoir has a bottom width of 100 feet and a length of 3S0 J
feet.
The sides of the reservoir are vertical.
The reservoir ia connected to a second reservoir of the same dimensioM I
by means of a pipe 2 feet diameter. The surface of the water in tlie &til
reservoir is 17 feet above tliat in the other. The pii>6 is Iwlow the su
of the water in both reservoirs. Find the time taken for the water in 1
two reservoirs to become level* Coefficient of discharge 0*8.
59* Notches and Weirs.
Wben the sides of an orifice are
prod need J so that they extend be
yond the free surface of the water,
as in Figs. 69 and 70, it is called a
notch,
Notches are generally made tri
angular or rectangular as shown
in the fignres and are largely used
for gauging the flow of water*
Fig. 09* Triangukt KoKfh,
FLOW OYER WEIRS
81
For example, if the flow of a small stream is required, a dam is
ooQstrticted scroti t^e stream and the wate^r allowed to pasg
tliTotigli a noteli cut in a board or metal plate *
Fig. 70, B£eta.»gulftr Notch.
They can conveniently be used for measuring the compensation
[irater to be supplied from collecting reservoirSj and also to gauge
f fiupplj of water to water wheels and turbines.
The tenn weir is a name givon to a structure used to dam np
I a strnun and over which the water flows.
The conditiomi of flow are practically the same as through ■
a rectangtilar notchj and hence such notches are generally caUedl
ifreiri, and in wlmt folio wft ihe latter term only is used. The top
of the weir corresponds to the horisfiontal edge of the notch and is
called the sill of the weir.
r ijf water flowing over a weir or through a notch is
g* V 1 lied the vein, sheetj or nappe*
The shape of the nappe depends upon the form of the sill and
sid«6 of the weir, tlie height of the sill above the bottom of the
oi>€ti^«ra channel J the width of the upstream channel, and the
c^iDitrocd/m of the channel into which the nappe falls.
The effect of the form of the Bill and of the downstream,
lehaimiel will be considered later, butj for the present, attentioi
J will lie confined to weirs with sharp edges, and to those iu which
[the air has free access under the nappe so that it detaches itself
ply from the weir as shown iu Fig, 70,
60, Rectangular sharpedged weir.
If the crest and sides of the weir are made sharpedged^ aa
^wn in Fig. 70, and the weir i^ narrower than the approaching
",and the sill some distance above the bed of the stream,
at the sill and at the ddes, contraction similar to that at
» abarp^ged orifice,
Tbr imrface of the water m it approaches the weir falls, taking J
eitmid form, so that the thickness K, Fig. 70, of the vein over^
weir, id teea than H, the height, above the sill, of the water at
t.a. 6
&ni&iULIC8
some distance from the weir. The height H, whicjh is ealledlfl
head over the weir, should be carefully measured at such a distaoM
from itj that the water surface has not commenced to cumaj
Fteley and Steams state, that this distance should be equal tol
2^ times the height of the weir above the bed of the stream^ 1
For the present, let it be assumed that at the point where H if J
measured the water is at rest. In actual cases the water will I
always have some velocity, and the effect of thi^ velocity will haFel
to be considered later, H may be called the still water head oyer I
the weir, and in aU the fonnulae following it has this meaning, I
Side cofUraclwi, According to Fteley and Stearns the amount I
by which the stream is contracted when the weir is sharpedg^ I
is from 0*06 to 0'12H at each side, and Francis obtained a mean of I
O'lH. A wide weir may be divided into several baj^B by parti 1
tions, and thert* may then be mt>re than two contractions, at each 1
of which the effective width of the weir will be diminished, if 1
Francis' value be taken, by 0*1 H. I
If L is the total width of a rectangLilar weir and N the mimber I
of contractions, the effective ^^'idth l^ Fig. 70, is then, I
(LO'lN). I
When L is very long the lateral contraction may be neglect^* I
Suppresdo7i of the cmitractton. The side contraction can be I
completrely suppressed by mating the approaching channel with I
vertical sides and of the same width as the weir, as was done fof I
the orifice shown in Fig. 47. The width of the stream ia then I
equal to the \\^dth of the sill. I
61. Derivation of the v^eir formula f^om that of a liidtJ
orifice. I
If in the formula for large orifices, p, 64, /i& is made equal te
zero and for the effective \\idth of the stream the length I ill
substituted for b^ and k is unity, the formula becomes I
Q = t^2gJ.h^ ....a), j
If instead of hi the head H, Fig, 70, is substituted^ ondl
a coefficient H introduced, I
The actual width I is retained instead of L, to make allowBan
for the end contraction which as explained above is equal to 0*13
for each contraction, I
If the width of the approaching channel is made equal to thi
width of the weir I is equal to L, J
With N contractions I = (L  OIN), I
and QC^.(L01N)H*. I
If C is given & mean value of 0*625, I
Q = 333CL0aN)H* ..,,,..(2). I
FLOW OVER WEXBS
83
wellknown formula deduced by Prancb* from^
of €jcperiments on sharpedged w^eirs,
itila, as an empirical one, is approximately cMjrrect and'j
Me \*alne^ for the discharge,
Ki^thcid of obtaining it from that for large orifices is,
Lipen to very serious objection, as the velocity at F on
EP, Fig* 70, is clearly not equal to zero, neither is the
flaw at the surface perpendicnlar t^ the section EF,
[»re«i&ttre on EF, as will be underst*:jod later (section 83)
(Tkely to be constant,
Ilhc* directions and the velocities of the stream lines are
irorn those through a section taken near a sharpedged
[seen by comparing tho thickness of the jet in the two 
li the Ci:»effictent of discharge,
%e tthiirpedged oritic^ ^vith mde contractions suppressed,
of the thicknees of the jet Uj the depth of the orifice is not
afferent fr*itn tlie coefficient of discharge, being about 625,
llicknetos EF of the nappe of the weir is very nearly 0'78Hj
lie coefficient of discharge is practically 0'625, and the
[is therefore V2A times the coefficient of discharge,
pars therefore, that although the assumptions made in
the t!ow through an orifice may be justifiable, pro\nding
[above the top of the orifice is not very small ^ yet when
riies xero, the assumptions are not approximately true.
Igles which the stream lines make with the plane of EF
I very different from 90 degrees, so that it would appear,
?rror principally arises fa^om the asaamption that the
bronghout the section is uniform.
f«ir special cases has carefully measured the fall of the
i* and the tliickness EF, and if the assumptions of constant
•e and stream lines perpendicular to EF are madei the
•ge through EF can Vie calculated.
r, example, the height of the point E above the sill of the
ae of Baziu'S experiments was 0'n2H and the tliickness
HBU. The fall of the point F is, therefore, 0*1 08H.
constant pressure in the section, the discharge per foot
He weir is, then,
Mmm
^f V2^,Hi {C888}*^C108)^l
• Lowall, BudrnMe Mi^erimtm, New York, 186S.
6—2
84
HYDRAULICS
The actual discharge per foot width, by experiment, was
g = 0433 n/2^.H*,
so that the calculation gives the discharge 1*228 greater than the
actual, which is approximately the ratio of the thickness EF to
the thickness of the stream from a sharpedged orifice having
a depth H. The assumption of constant pressure is, therefore,
quite erroneous.
62. Thomson's principle of similarity.
" When a frictionless liquid flows out of similar and similarly
placed orifices in similar vessels in which the same kind of liquid
is at similar heights, the stream lines in the different flows are
similar in form, the velocities at similar points are proportional to
the square roots of the linear dimensions, and since the areas of
the stream lines are proportional to the squares of the linear
dimensions, the discharges are proportional to the linear dimensions
raised to the power of 4*"
Let A and B, Figs. 71 and 72, he exactly similar vessels with
similar orifices, and let all the dimensions of A be n times those
of B. Let c and Ci be similarly situated areas on similar stream
lines.
Fig. 71.
Fig. 72.
Then, since the dimensions of A are n times those of B, the
fall of free level at c is n times that at Ci. Let v be the velocity
at c and Vi at Ci.
Then, since it has been shown (page 36) that the velocity in
any stream line is proportional to the square root of the &I1 of
free level,
.'. V : Vi :: vn : 1.
Again the area at c is n^ times the area at Ci and, therefore,
the discharge through c ^ . .
the discharge through Ci ~ ^ s/n^n ,
which proves the principle.
* British Association Reports 1858 and 1876.
FLOW OVER WEIRS
S5
63. Discharge throngh a triangular notch by the
principle of similarity.
Lies AI>C\ Figs, 73 and 74, b© a triangular notch.
^gr
rr^
^=^y^ —
4 '*'•
yV^ —
D
Fig. 74.
the depth of the flow through the notch at one time be H
and at another n , H.
Stippfj^^ the area of the stream in the two cases to be divided
into the same number of horissontal elements, such as ab and aibi ,
Then clearly the thickness of ab will be n times the thickness
of Oi^w
Let di&i be at a distance m from the apex B, and ah at a
distance m^; then the width of ab is clearly n times the width
The area of ab will therefore ben* times the area of ^i&i ,
Again, the head above ab is n tiniea the head above ai6i and
therefore the velocity thraugh ah will be \/n times the velocity
through a^bj and the discharge through ab will be n' times
that through ai&i.
More generally Thomson expresses this as follows :
** If tm'o triangular notches, similar in form, have water flowing
through them at different depths, but with similar passages of
apfiroach, the cross sections of the jets at the notches may be
limitarly divided into the same number of elements of area, and
the area^ of corresponding elements will be proportional to the
aquaree of the Hnea! dimensions of the cross sections, or pro
portionai to the squares of the heads.*^
Ab the depth h of each element can be expressed as a fraction
€f the head H, the velocities through these elements are propor
tional to the stjuare root of the head, and, therefore, the discharge
ift propiirtional to H*.
~ Therefore Q a«o H^
being a coefficient which has to be determined by experiment.
Prom experiments with a sharpedged notch having an angle
the rertex of 90 degrees, he found C to be practically constant
'jM baads and equal to 2*635. Tlien
Q2 635.H* .,,,..(3),
Mmta
86 HTDRAULICS
64. Flow tbrongh a triangular notch.
The flow through a triangalar notch is frequently given as
Q=Aw>/27.BHt.
in which B is the top width of the notch and n an experimental coefficient.
It is deduced as follows:
Let ADC, Fig. 74, be the triangular notch, H being the still water bead over
the apex, and B the width at a height H above the apex. At any depth h the
width b of the strip a, 6, is r^ —  •
If the velocity through this strip is assumed to be v=kj2gh^ the width of the
stream through a^fii , — — , and the thickness dh, the discharge tbrongh it is
The section of the jet just outside the orifice is really less than the ares EFD.
The width of the stream through any strip a,&j is less than a^bj, the surface is lovff
than EF, and the apex of the jet is some distance above B.
The diminution of the width of a^b^ has been allowed for by the coefficient c, vA
the diminution of depth might approximately be allowed for by integrating between
^=0 and ^=:H, and introducing a third coefficient c^.
Then Q = kecy^ /' ^B(Hfe) ^ — ^^
=TVcCifcN/2^BH*,
Replacing eeJt by n
Q=T^.nV27.BHt (4).
Calling the angle ADC, 6,
B=2Htan,
and Q=^^tn^/2^.tan.H*.
When 6 is 90 degrees, B is equal to 2H, and
Taking a mean value for n of 0*617
Q= 2636. H*,
which agrees with Thomson's formula for a rightangled notch.
The result is the same as obtained by the method of similarity, but the method
of reasoning is open to very serious objection, as at no section of the jet are all th0
stream lines normal to the section, and A; cannot therefore be constant. The
assumption that the velocity through any strip is proportional to Jh is also open
to objection, as the pressure throughout the section can hardly be uniform.
65. Discharge through a rectangoiar weir by the
principle of similarity.
The discharge through a rectangular weir can also be obtained
by the principle of similarity.
FLOW OVSB WEIRS
87
Consider two rectangular weirs each of length L, Figs. 75
and 76, and let the head over the sill be H in the one case and
Hi, or nH, in the other. Assume the approaching channel to be
of such a form that it does not materially alter the flow in either
case.
K L ^
^
Fig. 75.
B
Fig. 76.
To simplify the problem let the weirs be fitted with sides
projecting up stream so that there is no side contraction.
Then, if each of the weirs be divided into any number of equal
XiartB the flow through each of these parts in any one of the weirs
will be the same.
Supxx)se the first weir to be divided into N equal parts. K
N H
then, the second weir is divided into J^ equal parts, the parts
±±1
in the second weir will be exactly similar to those of the first.
By the principle of similarity, the discharge through each of
the parts in the first weir will be to the discharge in the second
H*
as — ; , and the total discharge through the first weir is to the
Hi*
discharge through the second as
N.H^ H^_ 1
N.H .Hi^ Hi^ n^'
Hi
Instead of two separate weirs the two cases may refer to the
same weir, and the discharge for any head H is, therefore, pro
pjrtional to H^ ; and since the flow is proportional to L
Q = C.L.H^
in which C is a coefficient which should be constant.
66. Rectangular weir with end contractions.
If the width of the channel as it approaches the weir is greater
than the width of the weir, contraction takes place at each side,
and the effectual width of the stream or nappe is diminished ; the
amount by which the stream is contracted is practically inde
pendent of the width and is a constant fraction of H, as explained
above, or is equal to &H, k being about 0*1
88 HTDRAULTCS I
Let the total width of each weir be now divided into three I
parts, the width of ea4?h end part being equal to » . fc , H, The I
width of the end parts of the transTerae section of the stream will ]
each be i7i — l)k . H, and the width of central part L 2«tH. I
The flow through the central part of the weir will be equal to I
Qi = C(L2tim)Hi 1
Now, whatever the head on the weir, the end pieces of the I
stream, since the width is (n — 1) JcK and fe is a conBtsjit, will he I
similar figures, and, therefore, the flow through them can be I
expressed as I
Qa = 2C,(nl)fcHHl I
The total flow is, therefore, I
Q = C (L  2rifcH) H^ + 2C» (n  1) fcHH^ I
If now Ci is assumed equal to C I
Q = C(L2^H)Hl 1
If instead of two there are N contractions, due to the weir]
being divided into several bays by posts or partitions, the formula ]
becomes I
Q = C(LNOa.H)Hi I
This is Francis* formula, and by Thomson's theory it is thual
shown to be rational* I
67 Bazin's* formula for the discharge of a weir, ]
The discharge through a weir with no side contraction may be I
written _ J
or Q = mL%/2^.H, ^J
the coefficient m being equal to 7=. . ^H
v2^ ^1
Taking Francis* value for C as 3'33, m is then 0'415.
From experiments on sharpcrested weirs with no side
traction Bazin deduced for mt the value
m ^ 0'405 ^ ■□■ .
In Table IX, and Fig. 77, are shoviTi Bazin'ti values for m foi
different heads, and also those obtained by Rafter at Cornell upon
a weir similar to that used by Bazin^ the maximum head in thi
Cornell experiments being much greater than that in Baxin':
experiments. In Fig* 77 are also shown several values of m, lu
calculated by the anthorj from Francis' experimental data*
t " Eiperimeots oa flow over Weira^" Am.8.C.E^ Td. xx^n.
FLOW OVEK WEIRS
TABLE IX.
89
Valueti of the coeflScient m in the f onmda Q = mL ^2g H^*
W^ir* aharpcrestedj 6 '56 feet wide with free oyerfall and lateraX
Dotitrmctioti suppressed, H being the atill water head over the weir,
>r ihe nuiafliirted head A* corrected for irelocity of approach.
Bazin,
bileet
a164
0^328
0*656
0'984
1*312
164
1968
fit
0448
0483
0421
0417
0414
0412
0409
m
tHOS^"^.
Rafter.
^^H
He&d in
f«et
m
C
^^^^
0^1
04286
3437
^^^H
0*5
04230
8392
^^^H
1*0
0*4174
3348
^^^^H
l'&
04136
3317
^^^H
M
04106
8293
^^^H
S^
0'4094
3283
^^^^1
8^
0*4094
sm^
^^^H
S'5
0*4099
8288
^^^H
40
0*4112
8^298
^^^H
4'fi
04125
8808
^^^f
m
04188
8815
^^^^
§'&
04135
8816
H
e0
04136
3317
W. Bazta's and the Cornell experiments on weirs*
Bazin's experinient« were made on a weirt 6'56 feet long
faATiiig the approaching channel the same width as the weir, so
lluit the lateral contractions were suppressed, and the discharge
was meftsiired by noting the time taken to fill a concrete trench of
Iehovth capacity.
The bead over the weir was measured by means of the hook
lESOge, page 249. Side chambers wei^e constructed and connected
to the channel by means of circular pipes 0*1 nu diameter.
The water in the chambers was very steady, and its level
cotaUd therefore be accurately ganged. The gauges were placed
h metree from the weir. Tht^ maximum head over the weir in
Bazin% experiments was however only 2 feet.
Tlic experiments for higher heads at Cornell University were
tuaile on a weir of practically the same width as fiassin's, 6'53 feet,
fher conditioDii being made as nearly the aame as possible;
Liiaximum head on tlie weir was 6 feet.
* Sed page 90.
i Mnnalm 4e$ Font* tl Vkau9s^f*, p. 445, Yq], ii. Ig91.
J
00
HYDRAULICS
The results of these experiments, Fig, 77, show that the
cciefficient m diminishes and then increases, having a minimmn
value when H is between 2'5 feet and 3 feet.
"I 3 3 ^
JfiKzO' oitffxajmt atryes ^r Sharp ^d^mt H'oLrs
A FnMfuxa* " (Deduced by ths ajuUh^r}
Fig, 77.
It is doubtful, however, although the experiments were made
with great care and skill, whether at high heads the deduced
coefficients are absolutely reliable*
To measure the iiead over the weir a 1 inch galvanised pipe
with holes Jinch diameter and opening downwards^ 6 inchea
apart, was laid acrosus the channel To this pipe were connected
\ inch pipes pfissing through the weir to a convenient point beloir
the weir where they couki be connected to the gauges by rubber
tubing. The gauges were glas^ tubes J inch diameter mounted
on a frame, the height of the water being read on a seal*
graduated to zmm* spaces.
69, Velocity of approach.
It should be clearly understood that in the formula given, it^
has been assumed, in giving values to the coefficient m that H is
the height, above the sill of the weir, of the still water surface.
A
FLOW OVER WEIRS 91
In actual cases the water where the head is measured will have
ome velocity, and due to this, the discharge over the weir will be
acreased.
If Q is the actual discharge over a weir, and A is the area of
he upstream channel approaching the weir, the mean velocity in
he channel is t? = ? .
A
There have been a number of methods suggested to take into
iccouit this velocity of approach, the best perhaps being that
idopted by Hamilton Smith, and Bazin,
This consists in considering the equivalent still water head H,
Dfver the weir, as equal to
a being a coefficient determined by experiment, and h the
measured head*
The discharge is then
<i.mj2ih(h*^' (5),
<3...L(<..f)y2,(A.g).
CLV
Expanding (5), and remembering that ^7 is generally a small
quantity,
The velocity v depends upon the discharge Q to be determined
and is equal to ^ •
Therefore Q = ^LA>/2^ ( 1 +  2£0 ^^^•
From five sets of experiments, the height of the weir above the
bottom of the channel being different for each set, Bazin found
the mean value of a to be 1*66.
This form of the formula, however, is not convenient for use,
since the unknown Q appears upon both sides of the equation.
If, however, the discharge Q is expressed as
Q = nL ^2gh, h,
the coefficient n for any weir can be found by measuring Q and h.
It will clearly be different from the coefficient m, since for m
to be used h has to be corrected.
From his experimental results Bazin calculated n for various
heads, some of which are shown in Table X.
..,u).
:iC .:
.:t' :o the
fr: above
FLOW OVER WEIRS 93
coefficient to determine Q, h most first be corrected, or Q
calculated from formula 9.
Baher in determining the values of m from the Cornell ex
periments, increased the observed head ^ by o; only, instead of
by 166^.
2g
Fteley and Steams*, from their researches on the flow over
weirs, f onnd the correction necessary for velocity of approach to
be from
145 to 15^.
Hamilton Smith t adopts for weirs with end contractions
sappressed the valaes
133 to 140^,
and for a weir with two end contractions,
11 to 1251^.
TABLE X.
Coefficients n and m as calculated by Bazin from the formulae
Q= nL'J2gh^
and Q = mLN/^H^
h being the head actually measured and H the head corrected for
velocity of approach.
Head
h in feet
Height of sill
p in feet
Coefficient
n
Coefficient
m
0164
0656
6*560
0458
0448
0448
0984
0656
6560
0500
0421
0417
1640
0656
6560
0500
0421
04118
An example is now taken illustrating the method of deducing
ilie coefficients n and m from the result of an experiment, and the
difference between them for a special case.
EaoM^U. In one of Bazin*8 experiments the width of the weir and the
■ffMiMiiliiiHj, channel were hoth 656 feet. The depth of the channel approaching
mm wmr mminrnd at a point 2 metres up stream from the weir was 7544 feet and
hflad menBiired over the weir, which may be denoted by A, was 0984 feet. The
~ diaehazge was 21*8 cubic ft. per second.
♦ TranMoetiom Am,S,C.E,, Vol. xn.
t Hydrauliet,
94
HTBRAULICS
The Telooitj at the sectiou whsre h was me&Bnrfld» and which may be called Ihe I
Tilooitj of approach was, therefore,
Q _ 2ia
^ ~ 7'54i X B 56' ' 7 544 jc eW
= 0^44 feet jier aeccmd.
If now the formala for discharge be written
and n ia qaleuLated from this formula by Bubstftuimg the known valaea of]
Q, L and h
n = 0*421.
Correeting h for veiooitj of approacb.
2ff
Then
from whtoh
= •9888,
918
'emj2g.^dms
= 0415.
It will sieem froin Table X that when the h«ip;hi p of the dll of the weir abore
the stream bed is amall compared with tba head, Ihe dilFerenoe majr be much '
larger than for thia eiample.
When the head is 164 feet and larger than p, the coeffiolent n ia eighteen]
per oent. greater than m. In auoh eases failnTe to eoireet the eoefficient wUJ letil
to considerable inaccDracy,
70. Influence of the height of the weir slU above the bed
of the stream on the contraction.
The nearer the sill is to the bottom of the streamj the less the
contraction at the sill, and if the depth is small compared with H, ]
the diminution on the contraction may conBiderably affect the]
flow.
When the sill was Vlb feet above the bottom of a channel, I
of the same width as the weir, Bazin found the ratio ^ (Fig. 85)
to be 0^097, and %vhen it was S70 feet, to be 0'112. For greater ]
heights than these the mean value of ^ was OlS.
71. Discharge of a weir when the air is not fireelj '
admitted beneath the nappe. Form of the nappe.
Francis in the Lowell experiments, found that, by making the
width of the channel below the weir equal to the width of the
weir, and thus preventing free access of air to the underside of the
nappe J the discharge was increased. Bazin*, in the experiments 
already referred to, has investigated very fully the effect upon
the discharge and upon the form of the nappe, of restricting the
free passage of the air below the nappe. He finds, that when the
flow is sufficient to prevent the air getting under the nappe, it niay
assume one of three distinct forms, and that the discharge for
* Annalev d^t FmU ti Chau4s€ea, 1891 and 1896.
L=>
ite^
FLOW OVXB WEIBS
95
I them may be 28 per cent, greater tlian when the air is
admitted, or the nappe is ''free.'' Which of these three
the nappe assomes and the amount by which the discharge
ater than for the ''free nappe/' depends largely upon the
over the weir, and also upon the height of the weir above
ater in the downstream channel.
18 phenomenon is, however, very complex, the form of the
' for any head depending to a very large extent upon
ler the head has been decreasing, or increasing, and for a
head may possibly have any one of the three forms, so that
jscharge is very uncertain. M. Bazin distinguishes the forms
^pe as follows :
) Free nappe. Air under nappe at atmospheric pressure,
70 and 78.
) Depressed nappe enclosing a limited volume of air at a
ire less than that of the atmosphere. Fig. 79.
) Adhering nappe. No air enclosed and the nappe adher
the downstream &ce of the weir. Fig. 80. The nappe in this
oay take any one of several forms.
Tcpafauam£l\ !
Fig. 79. Fig. 80.
Drowned or wetted nappe, Fig. 81. No air enclosed but
ppe encloses a mass of turbulent water which does not move
le nappe, and which is said to wet the nappe.
Fig. 81.
96
ETBEA0LICS
72* Depressed nappe.
The air below the nappe being at less than the atmofipherie"
pressure the excess pressure on the top of the nappe causes it to^
be depressed* There is also a rise of water in the downst
channel under the nappe.
The discharge is slightly greater than for a free nappe. On i
weir 2*46 feet above the bottom of the upstream channel ^ thej
nappe was depressed for heads below 0'77 feet, and at this head
the coefficient of discharge was 1*08 wii, mi being the absolute
coefficient for the free nappe,
73* Adhering nappes.
As the head for tliis weir approached 0*77 feet the air
rapidly expelledj and the nappe became vertical as in Fig, 80, ita
surface having a corrugated appearance* The coefficient of di
charge changed from r08 Wi to r28TOi This large change ia
the coefficient of discharge caused the head over the weir to fall
to 0'69 feet, but the nappe still adhered to the weir,
74. Browned or wetted nappes.
As the head was further increased, and approached 0'97 fe
the nappe came away from tlie weir face, assuming the drowned
form, and the coefficient suddenly fell to 119 mi. As the he
was further increased the coefficient diminished^ becoming V12\
when the head was above IS feet.
The drowned nappes are more stable than the other two, bu
whereas for the depressed and adliering nappes the discharge
not affected by the depth of wat^r in the downstream channd,
the height of the water may influence the flow of the dro^
nappe. If when the drowned nappe falls into the down streanl
the rise of the water takes place at a distance from the foot of thai
nappe^ Fig, 81 j the height of the downstream water does not affe
the flow. On the other hand if the rise encloses the foot of tia
nappe. Fig. 82, the discharge is affected. Let K^ be the differenc
Fig. as.
FLOW OVER WEIRS 97
erf level of the siJI of the weir and the water below tho weir. The
coefficieiit of discharge in the first ease iB independent of h^ bat is
dependent upon p the height of the sill above the head of the up
stream channel, and is
m^ = m, ^0878 + 0128 Y.. ..(11).
Bazin funnd that the drowned nappe could not be formed if h
is lees than 0*4 p and, therefore, r cannot be greater than 2'5,
Snbatitnting for f»i its value ■
£rom (10) page 92
w», = 0"470 + 0TO75^' .,....„... (12).
In the second case the coefficient depends upon ^, and ia,
m^mi(im + 0'my(^om)^ „,(13),
for iprhich, with a safficient degree of approximation, may be
nbptitated the simpler formula,
m, = mj(r05+M5^) ...,....(14).
The limiting value of ?% is 1*2 m^ for if hi becomes greater
ths^ h the nappe is no longer drowned.
Further, the rise can only enclose the foot of the nappe when
h^ i» lesa than Q p k). Ay h^ passes this value the rise is pushed
duwn arream away from the foot of the nappe and the coefficient
chAHgr^e to that of the preceding case.
fB* Instability of the form of the nappe,
' ul at wliich the form of nappe changes depends upon
^rh* i*' head is increasing or diminishing, and the depressed
ajid adiiering nappes are very nnstable^ an accidental admission
'jr other mterference causing rapid change in their form,
r, the adhering nappe is only formed under special circum
, and as the air is expelled the depressed nappe generally
,1.1 ^. r^ directly to the drowned form,
tf, therefore, the air is not freely admitted below the nappe
iven head is very uncertain and the discharge
L with any great degree of assurance.
With the weir 2 46 feet above the bed of the channel and 6 56
' '* i obtained for the same head of 0*656 feet, the four
^ ; : . , the coefficients of discharge being as follows ;
98 fiTDEAULlCS
Free nappe, 0*433
Depre^ed nappe, 0*460
Drowned nappe, level of water down stream
O'^l feet below the crest of the weir, 0"4OT
Nai>pe aflhering to downstream ince, 0'554
The dii?charge for tliis weir w^hile the head was kept oonstant,
thus vari<.Hi 26 per cent.
76. Drowned weirs with sharp crests*.
When the surface of the water n stream is higher than t^
sill of the weir, as in Fig. 83, the w s said to be drowned
T
r
: ^^ —
Fig, 83.
*
Bazin gives a formula for deducing the coefficients for such a
weir from those for the sharpedged weirs with a free nappe, w^bicli
in its simplest form is,
ITIe
bH'^'^^/'n^] "^'.
^
Aa being t!ie height of the downstream water above the sill of
the weir, k the head actually measured abov^e the weir, p the
height of the sill above the upstream channel, and mi the
coefficient ((10), p, 02) for a sharpedged weir. This oxpre^isioti
gives the same value within 1 or 2 per cent, as the formulae (13)
and (14).
Example. The heiid ovex a mnr is I foot, and tbe height of the sill abova
npstream channel Ia fj feet. The leugth is ti Seet &iid the eurface of the
in the down stream channel is 6 inches alM>vi) the Rill, Find the diachajge.
From formnla (10)i pa^e f^2, the eoeffici«nt fitj for a b harpedged weir with tim
nappe is
* AttempU have been made to esprega the diiieharge over a drowned wdx
equivalent to thnt throagh a drowned orifice of an area equal to LA^, under a hi
hha* together with a disobarge over a weir of length L when the head ia h^hm.
I^e dUscharge is then
n^2ghh^{hh^)i + m^lj{hh^)^,
n and m being coefficients.
^m FLOW O^EB WEIRS 99
^^^K» m,== '4215 [1^(1 +031} 0761]
^^^^^ ^3440,
I Thai Q = *a44>^V2^.1*
I =22i)S eubio ft* }ser second.
I 77. Vertical weirs of small thickness.
I likhtt^ail of makiiig the sill of a weir sharpedged, it niaj^
I bave a flat sill of thickness c. This will frequently be the ease in
nbetict^ the weir being constructed of timbers of uniform width
placed mvi* upon the other. The conditions of flow for these weirs
I My be v^ry dilferent from those of a sharpedged weir,
I I'he Ttappes of such weirs present two distinct forms, according
I as the WHter is in contact with th€+ crent of tlie weir, or becomes
I tWtat'liwl at the upstream edge and leaps over the crest without
I *<itjcliiTig the downstream edge. In the second case the discharge
I » tile hHtne as if the weir were sharpedged. When the head k
I 'jrt'er the weir is more than 2c this condition is realised, and may
I *»^tftiii when h pa?+ses c. Between these two value** the nappe is
I t«»coTuHtian of unstable equilibrium; when k is less than fc the
I oapjx' adheres to the sill, and the coefficient of discharge is
I m^ = mi (070 + 0*185^),
V *3^ oxtemal perturbation such as tlie entrance of air or the
F ^'^'^^r 1,1 a floating body causing the detachment.
\ li^ppL* adheres between ic and 2c the coefBcient m^ varies
I ^lik Mnii to r07iwi, bat if it is free the coefficient w^^mi.
I " n«D B = Jc, m^ is 79mi . If therefore the coefficients for a
I *wtfpetiin?d weir are used it is clear the error may be con
HfnabliC
HP^ formula for iWis gives approximately correct results when
*fc^ width of the sill m great, from 3 to 7 feet for example.
If the up{^tream edge of tlie weir is rounded the discharge is
iDCi^.Hed. Ilic di&charge* for a weir having a crest 6'bG feet
^di\ when the upstream edge was rountk?d to a radius of 4 inches,
•^ mcreaeed by 14 per cent., and that of a weir 2'624 feet wide
ky 12 per cent.
The roofiding of the comers^ due to wear, of timber weirs of
dmienmonsi to a radius o£ 1 inch or less, will, therefore,
IpcI the flow eonsiderahly.
78. Depressed and wetted nappes for flatcrested weirs.
The nappes of weirs having tiat sills may be depre^ed, and
ijf become drowned as for sharpedged weirs,
• dnmla du Pmtt et Ch&u»»iti, Val u. 18^.
7—2
416V02
100 HYDRAULICS
The coefficient of discharge for the depressed nappes, whether
the nappe leaps over the crest or adheres to it, is practically the
same as for the free nappes, being slightly less for low heads and
becomes greater as the head increases. In this respect they differ
from the sharpcrested weirs, the coefficients for which are always
greater for the depressed nappes than for the free nappes.
79. Drowned nappes for flatcrested weirs.
As long as the nappe adheres to the sill the coefficient m may
be taken the same as when the nappe is free, or
When the nappe is free from the sill and becomes drowned,
the same formula
77io = mi(o878+0128),
as for sharpcrested weirs with drowned nappes, may be used.
For a given limiting value of the head h these two formulae give
the same value of m©. When the head is less than this limiting
value, the former formula should be used. It gives values of m
slightly too small, but the error is never more than 3 to 4 per cent
When the head is greater than the limiting value, the second
formula should be used. The error in this case may be as
great as 8 per cent.
80. Wide flatcrested weirs.
When the sill is very wide the surface of the water fali^
towards the weir, but the stream lines, as they pass over the weimr^
are practically parallel to the top of the weir.
Let H be the height of the still water surface, and h the depbii
of the water over the weir, Fig. 84.
T
L'
Fig. 84.
Then, assuming that the pressure throughout the section of tb^
nappe is atmospheric, the velocity of any stream line is
v=J2g{B."h),
and if L is the length of the weir, the discharge is
Q = ^Lhs/W^) (16).
FLOW OVER WBIBS 101
For the flow to be pennanent (see page 106) Q must be a
naximom for a given value of h, or ^ must equal zero.
Therefore
From which 2(Hfe)fe = 0,
ind h = f H.
Substituting for h in (16)
= 0385L n/^ . H = 308L VH . H.
The actual discharge will be a little less than this due to
friction on the sill, etc.
Bazin found for a flatcrested weir 6*56 feet wide the coefficient
m was 0373, or C = 2991.
Lesbros' exx>eriment8 on weirs sufficiently wide to approximate
to the conditions assumed, gave '35 for the value of the co
efficient w.
In Table XI the coefficient C for such weirs varies from 2'66
to 310.
81. Flow over dams.
Weirs of various forms, M. Bazin has experimentally investi
?^^rf the flow over weirs having (a) sharp crests and (6) flat
crests, the up and downstream faces, instead of both being vertical,
(1) vertical on the downstream face and inclined on the
"Pstream face,
(2) vertical on the upstream face and inclined on the down
^ream face,
(3) inclined on both the up and downstream faces,
^^d (c) weirs of special sections.
The coefficients vary very considerably from those for sharp
c^ted vertical weirs, and also for the various kinds of weirs.
^^fficients are given in Table XI for a few cases, to show the
^^essity of the care to be exercised in choosing the coefficient for
^^y weir, and the errors that may ensue by careless evaluation of
the coefficient of discharge.
For a full account of these experiments and the coefficients
^otained, the reader is referred to Bazin's* original papers, or to
Rafter's t paper, in which also will be found the results of experi
• Afmaie* de$ Fonts et Chau*s4ett, 1898.
t Traruactions of the Am.S.C.E., Vol. XLiv., 1900.
102
HYDRAULICS
TABLE XI. ^^
Values of the coefficient C in the formula Q = CL . fc% for weirs
of the sections shown, for various values of the observed head A.
Bazva.
Section of
Head iu feet
weir
03
06
10
13
20
30
40
60
60
V31S
2*66
266
2*90
8*10
1
1
i
861
8*80
401
8*91
y^
^
s
402
416
418
416
<
\s
846
867
849
8*86
8*80
*<
^^^^C
8*46
8*59
8*68
gc:^^
808
808
819
8*22
FLOW OVER WEIBS
103
TABLE XI (continued).
Baain.
Section of
Head in feet
weir
08
06
10
13
20
30
40
60
60
— ^ '€S
810
827
805
878
890
878
i^Y^
T^
276
862
Section of
weir
— J3^
Rafter.
Head in feet
03
06 10 13 20 30 40 50 60
885
814
295
868
842
816
888
852
827
877
861
845
868
866
856
870
866
871
864
87]
86£
861
866
86^
104
ByDRACTLlCS
inents made at Cornell University on the discharge of weirs, sitaiJar
to those used by Bazin and for heads higher than lie used, and
also weirs of sections approximating more closely to tho&e oi
existing masonry dams, nsed as weirs* From Bazin's and Rafter's
experimentSj curves of discharge for varying heads for some of
these actual weirs have been dra\vn up,
82. Form of weir for accurate gauging.
The uncertainty attaching itself to the correction to be applied
to the measured head for velocity of approach, and the difficulty
of making proper allowance for the imperfect contraction at the
sides and at the sill, when the sill is near the bed of tlie channel
and is not sharpedged, and the inetabihty of the nappe and
uncertainty of the form for any given head when the admission of
air below the nappe is imperfect, make it desirable that as far as
possible, when accurate gaugings are reqniredj the %veir should
comply with the following four conditions, aa laid down by
Bas&in,
(1) The sill of the weir must ba made as high as possible
above the bed of tlie stream,
(2) Unless the weir is long compared with the head, the
lateral contraction should be suppressed by making the channel
approaching the weir with vertical sides and of the same width as
the weir.
(3) The sill of the weir must be made sharpcrested.
(4) Free access of air to the sides and under the imppe of
the weir must be ensured.
83. Boussinesq's* theory of the discharge over a weir.
As stated above, if air is freely admitted below the nappe of
a weir there is a contraction of the stream at the sharp edge of the
sillj and also due to the falling curved surface.
If the top of the sill is well removed from the bottom of thd
channel, the amount by which the arched under side of the nappe
is raised above the sill of the weir is assumed by Boussinesq— And
this assumption has been verified by Bazin's expeinments—to be
some fraction of the head H on the weir.
Let CDj Fig, 85, be the section of the vein at wliich the
maximum nBe of the bottom of the vein occurs above the sill, and
let e be the height of D above S.
Let it be assumed that through the section CD the stream
lines are moving in curved paths normal to the section^ and that
they have a common centre of cur^^ature 0.
* Comptf* Eendm, XSSI ftjad 1889.
FLOW OYER WEIRS
105
Let H be the height of the snr&ce of the water up stream
ove the silL Ijet R be the radius of the stream line at any
bt E in CD at a height x above S, and Bi and Bs the radii of
rvatore at D and C respectively. Let V, Vi and Vj be the
locities at E, D, and C respectively.
Fig. 85.
Consider the equilibrium of any element of fluid at the point
E, the thickness of which is 8R and the horizontal area is a. If w
is the weight of unit volume, the weight of the element is w . a3R.
Since the element is moving in a circle of radius R the centri
V8R
fngal force acting on the element is wa — ^~ lbs.
gix
The force acting on the element due to gravity is iraSR lbs.
Let p be the pressure per unit area on the lower face of the
element and p + Sp on the upper face.
Then, equating the upward and downward forces,
/ ^ \ ST> . 'M?aV'SR
(p + op) a + tcaoK = pa+ ^ — .
gti
From which ^dR=l^^ ^^^
Assuming now that Bemouilli's theorem is applicable to the
ream line at EF,
w 2g
Differentisitingy and remembering H is constant,
, dp YdV
w
= 0,
ldp_ ^ YdY
■w dx g.dx
IM ffTTntjfcrxics
fa
xR
vrv
tSt. '
iW* ■
V,iK = 0
trakos of die
the
pressme
is
= *ig HU
"^n = T 3i, JBBt. 3, i^m die ignre is (B, + * — «), therefore,
Kt
'^^''^ZJZ, (2).
'^^^^'(s;^.)'^
= ^ix 3* Rt I
Bi^jre
n.K
=2? ^^ B.Ioe^=g^ (3).
^'^ 1 at TOW .»v«r ii? WOT is permanent, the thickness K <rf
:t» :»fK» auBC iiUit&gf :t&ik£. iu due for the given head H the
'*Sr ^HiuatttflBt ^w iuwi*Ter cui only take place if each
'flSff ^'v^'xvtt i>F Jft^ tai? maximnm velocity possible to
:at« iliuneiLGs will be accelerated; and
^•i  .;^v«t iteiciMciQi^ n* tan:icii>tst^ «• k therefore a minimnm, or
vc *r .^r*««> ^^^ "C ^ ^iK xbscfurae is a maximnm. That is, when
^ '::feto<^<ktK^ A ^litt J* wmtMi J6 a fimction of fc,, the valae of
^ ^NiUv.u "iiat^ 4 ^ ^^^^ni ^iin^ can be determined by differ
A.
Therefore, A« = (H  e) (1  »«),
Ri = w(l + n)(H~e).
Sabstitatm£^ this value of Bi in the expression for Q,
Q=V2^.(H~6)*(n + n»)logi
FLOW OVER WEIRS 107
n'
lAich, smce Q is a Tnaximnin when ^ = 0, and A is a function
% is a maxiiniiin when p = 0.
Bifferentiatincr &i^d equating to zero^
(l+2n;
D solution of which gives
(l+2n)log(l + n) = 0,
TV
n = 04685,
and tlierefore, Q = 05216 ■J2g(a e)*
= 05216>/^(lg)*H*
= 05216 (lg)*s^.H*
• Ae coefficient m being equal to
^ 05216 (l  g)*.
M. Bazin has found by actual measurement, that the mean
i^^ iralne for ^^ when the height of the weir is at considerable
< disfcanoe from the bottom of the channel, is 0*13.
Then, (lg)* = 0812,
and m = 0'423.
It will be seen on reference to Fig. 77, that this value is very
near to the mean value of m as given by Francis and Bazin, and
the Cornell experiments. Giving to g the value 32*2,
Q = 3'39 H* per foot length of the weir.
If the length of the weir is L feet and there are no end con
tractions the total discharge is
Q = 339L.Ht,
and if there are N contractions
Q = 339(LN01H)Hi
108 HYDRAULICS
The coefficient 3*39 agrees remarkably well with the mean
value of C obtained from experiment.
The value of a theory must be measured by the closeness of
the results of experience with those given by the theory, and in
this respect Boussinesq's theory is the most satisfactory, as it not
only, in common with the other theories, shows that the flow is
proportional to H*, but also determines the value of the
constant C.
84. Solving for Q, by approximation, when the veloeitj
of approach is nnknown.
A simple method of determining the discharge over a wdr
when the velocity of approach is unknown, is, by approximation,
as follows.
Let A be the crosssectional area of the channel.
First find an approximation to Q, without correcting for
velocity of approach, from the formula
Q = mLh J2gh,
The approximate velocity of approach is, then.
and H is approximately
«=!.
A nearer approximation to Q can then be obtained by sub
stituting H for fe, and if necessary a second value for v can be
found and a still nearer approximation to H.
In practical problems this is, however, hardly necessary.
Example, A weir without end contractions has a length of 16 feet The beal
as measured on the weir is 2 feet and the depth of the channel of approaeh belov
the sill of the weir is 10 feet. Find the discharge.
„»=0405 + ^5^=.4099.
Therefore C=3«28.
Approximately, Q=328 2^.16
=r 148 cubic feet per second.
The velocity v = — — ^ = '77 ft. per sec. ,
and 1^=. 0147 feet.
A second approximation to Q is, therefore,
Q = 328 (20147)^16
= 150 cubic feet per second.
A third value for Q can be obtained, but the approximation is snffioieiiily Mtf
for all practical purposes.
In thin case the error in neglecting the velocity of approaoh altogether, ii
probably less than the error involved in taking m as 0*4099.
PLOW OVER WEIRS 109
85. Time required to lower the water in a reservoir a
given distance hy means of a weir.
A reservoir has a weir of length L feet made in one of its sides,
and kavin^ its siU H feet below the original level of the water in
tihe reservoir.
It is reqnired to find the time necessary for the water to fall to
m level H« feet above the sill of the weir. It is assumed that the
area of the reservoir is so large that the velocity of the water as
it approaches the weir may be neglected.
When the snr&ce of the water is at any height h above the sill
the flow in a time dt ia
Let A be the area of the water surface at this level and dh the
distance the surface falls in time dt.
Then, CUfidt = Adh,
The time required for the surface to fall (HHo) feet is,
therefore,
^^1 (^ Adh
The coefficient C may be supposed constant and equal to 3'34.
If then A is constant
^2A/_1 1_\
CLWilo n/H/
To lower the level to the sill of the weir, Ho must be made
equal to and t is then infinite.
That is, on the assumptions made, the surface of the water
never could be reduced to the level of the sill of the weir. The
time taken is not actually infinite as the water in the reservoir is
not really at rest, but has a small velocity in the direction of the
^Bveir, which causes the time of emptying to be less than that
^ven by the above formula. But although the actual time is
not infinite, it is nevertheless very great.
9.x
When Ho is iH, t =
WTien EU is ^H, t =
CLn/H'
6A
clVh'
So that it takes three times as long for the water to fall from
\n to iVH as from H to iH.
110 HYDRAULICS
Example 1. A reservoir has an area of 60,000 sq. yards. A weir 10 feet long
has its sill 2 feet below the surface. Find the time required to redaoe the level of
the water 1' 11".
H,=A'. H = 2'.
Therefore t=^^^^^ (Sid 0109),
o'o4 . lU
2.640.000 ^
334.10 '^
= 89,000 sees.
= 247 hours.
So that, neglecting velocity of approach, there will be only one inoh of
the weir after 24 hours.
Example 2. To find in the last example the dischaige from the reeerfoir in
15 hours.
2. A
Therefore 54 000=p^^^ ( l=  ^) .
C.LV^Ho V2/
From which jEQ = 0'i21,
Ho=0176 feet.
The discharge is, therefore,
(20176) 540,000 cubic feet
= 984,960 cubic feet.
EXAMPLES.
(1) A vreir is 100 feet long and the head is 9 inches. Find the disdiaigs
in c. ft. per minute. C = 3'34.
(2) The discharge through a sharpedged rectangular weir is 500
gallons per minute, and the still vrater head is 2^ inches. Find the effective
length of the vreir. m = 43.
(3) A vreir is 15 feet long and the head over the crest is 15 incheii
Find the discharge. If the velocity of approach to this weir were 5 feel
per second, what would be the discharge ?
(4) Deduce an expression for the discharge through a rightao^fid
triangular notch. If the head over apex of notch is 12 ins., find the
discharge in c. ft. per sec.
(5) A rectangular weir is to discharge 10,000,000 gallons per di^
(1 gallon =10 lbs.), with a normal head of 15 ins. Find the length at tilt !
weir. Choose a coefficient, stating for what kind of weir it is applioaUai
or take the coefficient C as 8*38.
(6) What is the advantage in gauging, of using a weir without end
contractions?
(7) Deduce Francis' formula by means of the Thomson principle d
similarity.
Apply the formula to calculate the discharge over a weir 10 feet widr
under a head of 1*2 feet, assuming one end contraction, and neglecting thi
effect of the velocity of approach.
I
FLOW OVER WEIRS
111
W
(S> A ratnfAtl of fy inch per hoar i» discharged from a catohment area
ai 5 aqua^re mtlea. Find tbe still water head when this Tolutue flowB ot^^er
a weir mill free oTerfftll 30 feet m longib^ (x»nstract^ in m% bays, each
^ iee4 iride, tulriiyg 0*415 as Basin's coefficdent.
A dteSrici of 6500 acxes (1 acreB43,560 aq. ft.) drains into a large
I neserroir. The ma^iuum late at which rain falls in the dii^trict ia
jl ins^ is 24 hoars. Whan r&in talk after tbe reservoir is full^ the water
x^aii^es lo be discharged over a weir or byewanh which has its crest at
tbe ordinary top water level of the reservoir. Find tVie length of such a
m^mtMT fof tlie abore reaervoir^ under the condition that the water in the
seMSrroir aball never rin^ more than 18 ins. above it^ top water level
Tlie top of tlie weif may be Bopposed flat and abotit 18 inches wide
(10> Compare rectangular and V notches in regard to aocnraey and
^amTenieDae when there is considerable variation in tl)e flow.
In ft rectangnl&r notch 50^' wide the still water surface level is 15'' above
iba BilL
If the same quantity of water Bowed over a rightangled V notch, what
^vroiiltl be the height of the still water surface above the apex ?
Lf the channels are narrow how would you correct for velocity of
jkpproAch tti each case? Lton. Ua. 1906.
ll 1 \ Tbe heaviest daily record of rainfall for a catchment area was
if!Hi.iid to be 4*2^ mlUion gallonB. Assuming twothirds of the rain to reach
^^iotage retnervoir and to ftass over the waste weir, find the length of
•mi erf the wa&te weir* so that the water shall never riae more than two
f[0^ above the fiOL
(12) A weir is 300 yards long. What Ib the disebarge when the liead
hk 4 feoi t Take Basdn's coeMcieut
m="40SH , — ,
(iSf Saipoae tbe water approaches the weir m the last question in a
g b att Pf*' f$* 6'' ^e#f> and nOO yarda wide. Find by approximation the dis
cbaqfe. t*kiitg into acoouut tlie velocity of approach.
(14^ The area of the water surface of a re^rvoir is 20,000 square
ymt^m Ffa>d Uie ttme reiu]ru(l for tbe surface to fall one foot, when the
wmier diadiuiges over a sbarp^edged weir 5 feet long and the original head
crrrr tbe weir is 2 feet,
1 15> Find, from the following data, the horsepower available is a given
wsterCaU:
Available height of fall 120 feet.
A tTHtangitlar notch above the fall, 10 feet long, is used to measure
%km qoantity of water, and the mean head over the notch la found to be
tDicbefet when tlie velocity of approach at the point where ttie head
i m 100 feet per miniita Lon. Un. 1905.
CHAPTER V.
FLOW THROUGH PIPES.
86. ResiBtances to the motion of a fluid in a pipe.
When a fluid is made to flow through a pipe, certain resistances
are set up which oppose the motion, and energy is consequently
dissipated. Energy is lost, by friction, due to the relative motion
of the water and the pipe, by sudden enlargements or contractions
of the pipe, by sudden changes of direction, as at bends, and by
obstacles, such as valves which interfere with the free flow of the
fluid.
It will be necessary to consider these causes of the loss of
energy in detail.
L088 of head. Before proceeding to do so, however, the student
should be reminded that instead of loss of energy it is convenient
to speak of the loss of head.
It has been shown on page 39 that the work that can be
obtained from a pound of water, at a height z above datum,
moving with a velocity v feet per second, and at a pressure head
* is — + s + « foot pounds.
w^ w 2g ^
If now water flows along a pipe and, due to any cause, K foot
pounds of work are lost per pound, the available head is clearly
diminished by an amount h.
In Fig. 86 water is supposed to be flowing from a tank throngli
a pipe of uniform diameter and of considerable length, the end B
being open to the atmosphere.
Fig. 86. Lobs of head by friction in a pipe.
FLOW THROUGH PIPES 113
Let *^ l>e the head due to the atmospheric pressure.
Then if there were no resistances and assuming stream line
flow, Bemonilli's equation for the point B is
w 2g w*
from which st = ZpZb = H,
or rB= V2grH.
The whole head H above the point B has therefore been
utilised to give the kinetic energy to the water leaving the pipe at
B. Experiment would show, however, that the mean velocity of
the water would have some value v less than Vb, and the kinetic
energy would be oZ«
Ahead /, = ___ = H2^
ha3 therefore been lost in the pipe.
By carefully measuring H, the diameter of the pipe d, and the
discharge Q in a given time, the loss of head h can be determined.
For "7^'
and therefore A = H ^*;
The head h clearly includes all causes of loss of head, which,
in this case, are loss at the entrance of the pipe and loss by
friction.
87. Loss of head by friction.
Suppose tubes 1, 2, 3 are fitted into the pipe AB, Fig. 86, at
equal distance apart, and with their lower ends flush with the inside
of the pipe, and the direction of the tube perpendicular to the
direction of flow. If flow is prevented by closing the end B of the
pipe, the water would rise in all the tubes to the level of the water
in the reservoir.
Further, if the flow is regulated at B by a valve so that the
mean velocity through the pipe is v feet per second, a permanent
regime being established, and the pipe is entirely full, the mean
velocity at all points along the pipe will be the same ; and there
fore, if between the tank and the point B there were no resistances
offered to the motion* and it be assumed that all the particles
L.H. 8
114 HYDRAULICS
have a velocity equal to the mean velocity, the water would again
rise in all the tubes to the same height, but now lower than the
surface of the water in the tank by an amount equal to sr
It is found by experiment, however, that the water does nofc i
rise to the same height in the three tubes, but is lower in 2 thaa j
in 1 and in 3 than in 2 as shown in the figure. As the fluid movai \
along the pipe there is, therefore, a loss of head.
The difference of level fh of the water in the tubes 1 and 2 iB
called the head lost by friction in the length of pipe 12. In any .
length I of the pipe the loss of head is h.
This head is not wholly lost simply by the relative movement
of the water and the surface of the pipe, as if the water wotb
a solid body sliding along the pipe, but is really the sum of tiia
losses of energy, by friction along the surface, and due to relatifft
motions in the mass of water.
It vrill be shown later that, as the water flows along the pipe^ ;
there is relative motion between consecutive filaments in the jripe^
and that, when the velocity is above a certain amount, the waler
has a sinuous motion along the pipe. Some portion of this headk
is therefore lost, by the relative motion of the filaments of water,
and by the eddy motions which take place in the mass of fliB
water.
When the pipe is uniform the loss of head is proportional
to the length of the pipe, and the line CB, drawn through the tops
of the columns of water in the tubes and called the hydraulic
gradient, is a straight line.
It should be noted that along CB the pressure is equal to that
of the atmosphere.
88. Head lost at the entrance to the pipe.
For a point B just inside the pipe, Bernouilli's equation is
— + j^ + head lost at entrance to the pipe = Aa + ~ ,
2? being the absolute pressure head at E.
The head lost at entrance has been shown on page 70 to be
about P5 — , and therefore,
2g
w w 2g
That is, the point C on the hydraulic gradient vertically above
l'5v'
E, is ^ — below the surface FD.
' 2g
FLOW THKOUQH PIPES
115
If the pipe is bellmoathed, there will be no head lost at entrance,
id the point C is a distance equal to ^ below the surface.
89. Hydraulic gradient and virtual slope.
The line CB joining the tops of the columns of water in the
nhe, is called the hydraulic gradient, and the angle i which it
Doakes with the horizontal is called the slope of the hydraulic
gradient, or the virtual slope. The angle i is generally small, and
mi may be taken therefore equal to i, so that j = ^«
In what follows the virtual slope j is denoted by i.
More generally the hydraulic gradient may be defined as the
line, the vertical distance between which and the centre of the
pipe gives the pressure head at that point in the pipe. This line
will only be a straight line between any two points of the pipe,
wben the head is lost uniformly along the pipe.
If the pressure head is measured above the atmospheric
fffessure, the hydraulic gradient in Fig. 87 is AD, but if above
lero, AiDi is the hydraulic gradient, the vertical distance between
LD and AiDi being equal to
ressuru per sq. inch.
w
, Pa being the atmospheric
Fig. 87. Pipe rising above the Hydraulic Gradient.
If the pipe rises above the hydraulic gradient AD, as in Fig. 87,
c* pressure in the pipe at C will be less than that of the atmosphere
r a head equal to CE. If the pipe is perfectly airtight it will
:t a> a siphon and the discharge for a given length of pipe will
jt }je altered. But if a tube open to the atmosphere be fitted at
8—2
116
HYDRAULICS
the highest point, the pressure at C is equal to the atmospheric
pressure, and the hydraulic gradient will be now AC, and the flow
will be diminished, as the available head to overcome the resSst*
ances between B and C, and to give velocity to the water, will only
be CF, and the part of the pipe CD will not be kept full.
In practice, although the pipe is closed to the atmosphere, yeA
air will tend to accumulate and spoil the siphon action.
As long as the point C is below the level of the water in tihs
reservoir, water will flow along the pipe, but any accumulation ol
air at C tends to diminish the flow. In an ordinary pipe line it it
desirable, therefore, that no point in the pipe should be allowed to
rise above the hydraulic gradient.
90. Determination of the loss of head due to frietton.
Reynolds' apparatus.
Fig. 88 shows the apparatus as used by Professor Reynolds* for
determining the loss of head by friction in a pipe.
Fig. 88. BeyuoldB' apparatus for determining loss of head by friction in a pipe^
A horizontal pipe AB, 16 feet long, was connected to the water
main, a suitable regulating device being inserted between tlie
main and the pipe.
At two points 5 feet apart near the end B, and thus at a distance
sufiiciently removed from the point at which the water entered
the pipe, that any initial eddy motions might be destroyed and i
steady regime established, two holes of about 1 mm. diameter wero
pierced into the pipe for the purpose of gauging the pressoie, it
these points of the pipe.
Short tubes were soldered to the pipe, so that the hofc
communicated with these tubes, and these were connected 1
* PhiU TratiB. 1883, or Vol. n. Scienti/ic Papen^ Bejmolds.
rnSianil
FLOW THROUGH PIPES
111
ibber pipes ki the limbs of a siphon gauge G^ made of glass]
and which contained mercuiy or bisulphide of carbon. Sealetl
wi?re fixed behiiKl tbt* tubes ^y that the height of the columns
in each limb of the gauge could be n?ad.
For i'ery small differences of level a cathetometer was used*,
'""' < made to flow through the pipe, the diffeiience in
' columns in the two limbs of the siphon measured
presfure at the two points A and B of tlie pipeJ
ouM iiiM fiM.i'Mied the lo&s of head due to friction. ■
If * is the specific gravity of the liquid, and H the difference
in ht'tght of the columns, the loss of head due to friction in feet oil
waitir is A^H (^1). I
The quantity of water fl(»wing in a time t was obtained bil
actual measttTemeEit in a graduated flask .
Calling r th© mean velocity in the pipe in feet per eecondi Q
the dischafg^ in cubic feet per secondi and d the diameter of thai
{ripe m f00l, J
The lo» of head at different velocities was carefully measured^
tlia law connecting head lost in a given length of pipe, with
T«lodty» determined*
Hie resolU obtained by Reynolds, and others^ using tins
of expt^rimeniing, will be refened to later.
Bqmation of flow in a pipe of uniform diameter
detenaination of the head lost due to friction.
Lei ci he the length of a ginall element of pipe of uniform
r. Fig. m
A
Fig. 81).
I,, rea of the transverse section be uj, P the length of
lu . jutact of the water and the surface on this section, or
wetutl perimeter, n the inclination of the pipe, p the pressme
' unit area on AB, iind pdp the pressure on CD*
• p. 268, Vol. I, ScieniifU Papif*^ Bejnoldf.
118 HYDRAULICS
Let V be the mean velocity of the fluid, Q the flow in cubic
feet per second, and w the weight of one cubic foot of the fluid.
The work done by gravity as the fluid flows from AB to CD
= Qw .dzio,v .w .dz.
The work done on ABCD by the pressure acting upon the area
AB
= p.io.v it. lbs. per sec.
The work done by the pressure acting upon CD against the
flow
= (p — dp) . ci> . 17 f t. lbs. per sec.
The frictional force opposing the motion is proportional to the
area of the wetted surface and is equal to F . P . oZ, where P is some
coefficient which must be determined by experiment and is the
frictional force per unit area. The work done by friction per aecX
is, therefore, F . P . 5Z . v.
The velocity being constant, the velocity head is the same ttt
both sections, and therefore, applying the principle of the con
servation of energy,
p.tt},v + <o,v.w.dz= (p dp) CO . i; + F . P . 3Z . «.
Therefore w . w . 92? = 82) . w + F . P . 3Z,
, dp F.P.dZ
or dz = —  + .
Integrating this equation between the limits of z and Zi, p and
Pi being the corresponding pressures, and I the length of the pipe^
z
Zi
w
w
F.PZ
W ci»
w
+ z
=£1
w
+ Zl +
FP I
W CD*
Therefore,
FPZ
The quantity — is equal to A of equation (1), page 52, and it
the loss of head due to friction. The head lost by friction it
therefore proportional to the area of the wetted surface of the pipe
PZ, and inversely proportional to the cross sectional area of the
pipe and to the density of the fluid.
92. Hydraulic mean depth.
The quantity p is called the hydraulic radius, or the hydranlio
mean depth.
If then this quantity is denoted by m, the head h lost hf
friction, is
w .m
FLOW THBOtJOH PIPES
119
Thio quantity F, which hns been oalled above the friction per
imit mr^A, is found by experiments to vaiy with the density^
^imoom^^ and velocity c»f the flaid, and v%nth the diameter and
lOO^hneas cif the internal mirface of the pipe.
In Hydraulics, the fluid considered is water, and any variationB
in dtrnnity or viscosity, doe to changes of temperature^ are generally
nieGfliiribl^, F^ therefore, may be taken as proportional to the
«^ * <r to the weight w per cubic foot, to the roughness of the
; , . . i as some function, /{v) of the mean velocity, and f{d) of
rh** «liaaEM»ter of the pipe,
Then,
j^^pfM/m
m
in which expresion ^ may be called the coefficient of friction.
li will be seen later^ that the mean velocity v is different from
the Ttdathe Telocity u of the water and the surface of the pipe,
arMi it probably would be better to express F as a function of «, '
1 itftelf probably varies? with the roughness of the pipe and]
^\__ :uer circumstances, and cannot directly be deteruiined, it,
mnplifictd matters to express F, and thus A, as a function of i?.
93. Empirical formulae for loss of head due to friction,
Tha difficulty of con^ctly determining the exact value of
/(t) f{d)^ has led to the use of empirical formulae, which have
poporcd of great practical service, to express the head h m terms of
the trelodty and the dimenMons of the pipe.
Tbe fiiniplest formula assumes that the friction simply varies as
tlif* nqaare of the velocity, and is independent of the diameter of
ape, or f{v) fid) = at^.
Then,
I
or writiiig gi for a,
m
i^*^
'^"C^
•(1),
.(2),
frcmi which is deduced the wellknown Chezy formula,
v = C ^/m.j^
or 1?  C ^Jmt.
Another form in which formnla (1) ia often found is
120
HYDRA ITLICS
or since wi = 7 for a circular pipe full of water,
t 4,f.vH
^~2g.d
.(3),
in which for a of (1) is substituted
L
29 •
The quantity 2g was introduced by Weisbach so that h is
expressed in terms of the velocity head.
Adopting either of these forms, the values of the coefficients C
and / are determined from experiments on various classes of pipes.
It should be noticed that C = a/ ? .
Values of these constants are shown in Tables XTT to XTV for
different kinds and diameters of pipes and different velocities.
TABLE Xn.
Values of C in the formula v = G Jrni for new and old castiron
pipes.
New oast
iron pipes
Old oaafc.
iron pipes
Velocities in ft. per second
1
3
6
10
1
3
6
10
Diameter of pipe
8"
95
98
100
102
68
68
71
78
6"
96
101
104
106
69
74
77
79
9"
98
105
109
112
78
78
80
84
12"
100
108
112
117
77
82
85
88
16"
102
110
117
122
81
86
89
91
18"
105
112
119
125
86
91
94
97
24"
111
120
126
131
92
98
101
104
30"
118
126
131
186
98
108
106
109
86"
124
181
136
140
108
108
111
114
42"
180
186
140
144
106
111
114
117
48"
185
141
145
148
106
112
115
118
60"
142
147
150
162
For method of determining the values of C given in the tables,
see page 102.
On reference to these tables, it will be seen, that C and / are
by no means constant, but vary very considerably for different
kinds of pipes, and for different values of the velocity in any
given pipe.
FLOW THROUGH PIPES
121
The fact that C varies with the velocity, and the diameter of
the pipe, suggests that the coefficient C is itself some function of
the velocity of flow, and of the diameter of the pipe, and that
/i/(r) f{d) does not, therefore, equal at;'.
TABLE Xni.
Values of / in the formula
, _4/t;»J
1
New eastiron pipes
Old castiron pipes
VeloeitiMm
ft.p«rieeoDd
1
3
6
10
1
8
6
10
DiAiiLofpipe
8"
•0071
0067
•0064
•0062
•0152
•0189
•0128
•0122
6"
•007
•0068
•006
•0057
•0135
•0117
•0108
•0103
9^
•0067
•0058
•0055
•0051
•0122
•0105
•010
•0092
IT
0064
•0056
•0051
•0048
•0108
•0096
•0089
•0084
15"
0062
•0058
•0048
•0048
•0099
•0087
•0081
•0078
1 18"
•0058
•0051
•0045
•0041
•0087
•0078
•0073
•0069
1 24"
•0058
•0045
•0040
•0037
•0076
•0067
•0063
•0060
1 W
•0046
•0040
•0087
•0035
•0067
•0061
•0067
•0066
86"
•0042
•0087
•0085
•0038
•0061
•0056
•0062
•0060
1 42"
0068
•0085
•0038
•0081
•0058
•0052
•005
•0048
1 48"
0086
•0082
•0081
•0029
•0067
•0061
•0049
•0046
60"
0032
•0030
•0029
•0028
TABLE XIV.
Valaes of C in the formula vC Jmi for steel riveted pipes.
Velocities in ft per second
1
8
5
10
Diameter of pipe
8"
81
86
89
92
11"
92
102
107
115
111"
15^
93
99
102
105
109
112
114
117
88"
113
113
113
118
42"
102
106
108
111
48"
105
105
105
105
72"*
110
110
111
111
72"
93
101
105
110
108"
114
109
106
104
See pages 124 and 137.
122 HYDRAULICS
94. Formula of Daroy.
In 1857 Darcy* published an account of a series of experiments
on flow of water in pipes, previous to the publication of which, it
had been assumed by most writers that the friction and consequently
the constant C was independent of the nature of the wetted surface
of the pipe (see page 232). He, however, showed by experimenis
upon pipes of various diameters and of different materiah^
including wrought iron, sheet iron covered with bitumen, lead,
glass, and new and old castiron, that the condition of the internal
surface was Of considerable importance and that the resistance was
by no means independent of it.
He also investigated the influence of the diameter of the pipe
upon the resistance. The results of his experiments he expressed
by assuming the coefficient a in the formula
T al 2
m
Q
was of the form a = o + 
r '
r being the radius of the pipe.
For new castiron, and wroughtiron pipes of the same
roughness, Darcy's values of a and P when transferred to English
units are,
a = 0000077,
/J = 0000003235.
For old castiron pipes Darcy proposed to double these values.
Substituting the diameter d for the radius r, and doubling ft for
new pipes,
;.=(ooooo77.«:«o^)^
= 000000647 (12^1) ^^
or "^^^^V i2dTl^^ (*)
^^^s/mri^' (5)
Substituting for m its value 2 > ^^^ multiplying and dividing
by 29,
\'M'^mW^ w
For old castiron pipes,
;.=oooooi294(i^)^
\ a / m
«'"('^is)Ss o
2g
* Reeherehea ExpirimentaUt.
^^^^1 FLOW THROUGH PIPES 123
To!^ v = 27S^^^^^i (8)
I ^^\^l^l^^ t»>'
■ As the atudant cannot possibly retain, witliout unnecessary
Biboiff, Tallies of / sod C for different diameters it is convenient
■to Temember the rim pie fonnSj
I for Dew pipes, and
■ for old pipes.
H According to Darcy, therefore, the coefficient C in the Chezy
■ fontiala varies only with the diameter and roughness of the pipe.
I The %*mlae8 of C afi calcolated from his experimental results, for
■ t^m? of the pipes, were practically constant for all velocitieBj and
■Jiotubty for those pipes which had a comparatively rough internal
^■picef hut for smooth pipes, the value of C varied from 10 to
^Hper ct*nL for the siame pipe as the velocity changed. The
Experiments of other workers show the same results,
V aption that ^f{v)f(d) = av^ in which a is made to
Tii^ : h the diameter and roughness, or in other words, the
aoiunptian that h is proportional to ij" is therefore not in general
stilted by experiments.
As stated above, the formulae given must be taken as
^npiriea], and though by the introduction of suitable
they can be made to agree with any particular experi
even set of experiroents, yet none of them probably
rprt!!Si^s truly the laws of fluid friction.
The formula of Chezy by its simplicity has found favour, and
I is likely, that for aome time to come^ it i^411 continue to be used,
in the form t? = C vmi, or in its modified f onu
^ 2gd '
In making calculations, values of C orf, which most nearly suit
ly given case, can be taken from the tables,
00. Yariatioii of C in the formula v = C /mi with aervioe.
It should be clearly borne in mind, however, that the dis
diaf^ging capacity of a pipe may be considerably diminished after
m few yeanf* service.
Dwtjy'B re«u]t« show that the loss of head in an old pipe may
doable that in a new one, or since the velocity v is taken as
^hm doi
124
HYDRAULICS
proportional to the square root of A, the discharge of the old pif
for the Bame head will be ^ time^ that of the new pipe, or aboafj
30 per cent. less.
An experiment by Sherman* on a 3dinch castiron main sh(
that after one year's gervice the discharge was dmiinishei
23 iiev cent., but a second year's service did not make any furthi?r
altc^ration.
Experiments by Kuichlingt on a 36inch castiron main showed
that the discharge during four years diminished 36 per cent* while
experiments by Fitzgerald t on a castiron main, coated with tar,
which had been in use for 16 years, showed that cleaning inci^eased
the discharge by nearly 40 per cent, Fitzgerald also found tlat
the discharge of the Sudbury aqueduct diminished 10 per ceni. in
one year due to accumulation of slime.
The experiments of Marx, Wing, and Hoskins on a 72inc)i stt'el
main, when new, and after two years' service, showed that ther#
had been a change in the condition of the internal surface of tfiB
pipe, and that the disscharge had diminished by 10 per cent, at lo^
velocities and about 5 per cent, at the higher velocities.
If, therefore, in calculations for pipes, values of C or /are used
for new pipes, it will in most cases be advisable to make the pipe
of such a size that it will discharge under the given bead at least
from 10 to 30 per cent, more than the calculated value.
97* Ganguiliet and Kiitter*s formula.
Granguillet and Katter endeavoured to determine a form foi
the coefficient C in the Chezy formula v = G ^mi, applicable
to all forms of channels, and in which C is made a function of the
virtual slope i, and also of the diameter of the pipe.
They gave C the value,
1811 000281
416
C = ^
1,(410^2:29^) «
This formula is very cumbersome to use^ and the value of th
coefficient of roughness n for different cases is uncertain. Tabk
have however been prepai*ed which considerably facilitate the
of the formula.
* Tram. Am^SX.E. Vol xjurv, p, 85.
t Tram. Am.S.CE, Vol. eliv, p. 56.
t Tram. Am.S.C.E. ToL xw¥, p* 87.
g Sea Table No. KIV,
FLOW THROirOH PIPES Iti
Folties of H in Ganguilht ami Kutler*s formula.
Wood pip€*s ^ '01, may be as high as "015.
Castiron and sttnA pipes = *011, „ „ 02*
Glaj^ eartheuwar© = *013*
&8* Keynolds' experiments and the logarithmic formtila.
The formulae for \om of head due to friction previously givea
iiave all teen founded upon a probable law of %*ariation of h
"Wiik V, bat oo rational basis for the assumptions has been adduced.
It has been ^tjited in Bection 93, that on the assumption that h
vam with 17^, the ci:^etticient C in the formula
Cy/
h
ilitolf a ftinction of the \'elocity.
Tilt? experiments and deductions of Reynolds, and of lati^r
workers, throw considerable light upon this subject, and show that
i i» proportional to t'**, where n is an index which for very small
^locities*— as previously shown by Poiseuille by experiments on
CBfiillary tubes — is equal to unity, and for higher velocities may
We jk variable vahte, which in many cases approximates to 2.
} '^ experLments marked a decifled advaneej in showing
^1"  ily that the roughness of the wetted surface has an
fiffisct upon the loss due to friction^ so Heynolds' work marked
her step in showing that the index w depends upon the state
intt'mal surface, being generally greater the rougher the
I
student will be better able to follow Reynolds, by a brief 
tion of one of his experiments,
n Table XV are shown the results of an experiment made 
f Reynolds with apparatus as illustrated in Fig. 88.
In calumiis 1 and 5 are shown the experimental values
= J, and « peKpjectively,
The curves. Fig. 90, were obtained by plotting v as abscissae
kd t 88 ordinates.
For velocities up to 1*347 feet per second, the points lie very close
A straight line and i m simply proijortional to the velocity, or
i = hv „, (IIX
bein^ a coefficient for this particular pipe.
Above 2 feet per second j the points he very near to a continuous^
I the equation to which is
iitu** ,.........(12).
• Phil Tmns, 1863,
126
HYDRAULICS
Taking logarithms,
log i = log h + nlog t;.
The curve, Fig. 90 a, was determined by plotting log i as
ordinate and logv as abscissae. Reynolds calls the lines of this
figure .the logarithmic homologues.
Calling logi, y, and log v, a?, the equation has the form
y = ki 7WJ,
which is an equation to a straight line, the inclination of which to
the axis of x is
^ = tan""^7i,
or n = tan 0,
Further, when » = 0, y = fc, so that the value of Jc can reardily be
found as the ordinate of the line when x or logv = 0, that is,
when v = 1.
Up to a velocity of 1*37 feet per second, the points lie near to
a line inclined at 45 degrees to the axis of v, and therefore, n is
unity, or as stated above, i = kv.
The ordinate when v is 6qual to unity is 0*038, so that for the
first part of the curve k = '038, and i = '0381?.
FLOW THROUGH PIPES
127
Above the velocity of 2 feet i)er second the points lie about
ft second strai^Ht line, the inclination of which to the axis of t; is
» = tanU70.
Therefore log i = 1*70 log v + k.
The ordiixate when v equals 1 is 0*042, so that
fc = 0042,
and t = 0042t;i^.
Fig. 90 a. Logarithmic plottings of t and v to determine the index n in
the formula for pipes, t = A;i;*^.
In the table are given values of i as determined experimentally
and as calculated from the equation i = k, v^.
The quantities in the two columns agree within 3 per cent.
128
HTDRAUUCS
Lead Pipe.
TABLE XV.
Experiment on Resistance in Pipes.
Diameter 0*242". Water from Mancliester Main.
Slope
"1
k
n
Velocity ft per
second
Experimental value
Calculated from
•0086
•0092
•038
289
•0172 •
•0172
•038
•451
•0258
•0261
•038
•690
•0346
•0347
•038
•914
•0480
•0421
•038
1109
•0516
•0512
038
1849
•0602
• •>
1482
•0682
...
1573
•0861
...
1671
•1033
...
1775
•1206
...
1857
•1378
•1362
•042
170
1987
•1714
•1610
•042
170
2208
•3014
•2944
•042
170
8141
•4306
•4207
•042
1^70
898
•8185
•8017
•042
170
566
1021
1033
•042
170
657
1438
1476
•042
170
811
2455
2^404
•042
170
1079
8^274
3206
•042
1^70
1279
3^878
3899
•042
170
1429
Note. To make the columns shorter, only part of Beynolds' results are ghcs.
99. Critical velocity.
^t appears, from Reynolds' experiment, that up to a certun
 which is called the Critical Velocity, the loss of head li
which is a)nal to v, but above this velocity there is a definite change
the axis ow connecting i and t;.
'cperiments upon pipes of different diameters and the
Qj, variable temperatures, Reynolds found that the critical
hich was taken as the point of intersection of the two
Further, ^^^^ag
found as the _ '0388?
when t? = 1. ^* " D '
Up to a velooeing
a line inclined at p_. 1 ,^qx
unity, or as stated a 1 +00336T + •0000221P ^^^^'
The ordinate wheature in degrees centigrade and D the diameter
first part of the curve.
^
¥upw through pipes
129
Critical velocity by the method of colour bands.
exisftnice of the rrirical velocity has been beautiftill}'
by Reynolds, by tho metliod of colour baTid^^ and his
 r^ slIs*j explain why there is a sudden change in the law
^ I and r.
Water was drawn through tubes (Figs. 91 and 92), ont of
^rigp gla?^ tank in which the tnbcg were immersedj and in
th ihi* water had been allowe<l to come to rest, arrangements
made %B shown in tlie figure so that a streak or streaks of
liy coloured water entered the tubes with the clear water,"
^
3
Fig. 91.
^
Fig. 92,
' Tfsults were a^ follows :—
" (I) When the velocities were sufficiently low, the streak
<iliiur extended in a beautiful straight line through the tube*'
**{2> A« the velocity was increa^^d by small stages, at
r pc»int in the rube, always at a considerable distance from the
jimjjrt«ha)ed intake, the colour band would all at once mix up
the Huri'MnTnlitig water, and fill the rest of the tube with
»if coltHired water" (Fig. 92),
ij* Httdden change takes place at the cntical velocity.
such a change takes place is also shown by the appaiatus
Bjstrated in Fig. ^S'f when the critical velocity is reached there is
liiknt disturbance of the meruury in the U tube.
iJij therefore, a definite and sudden change in the con
of fl€>w. For vehjcities below the critical velocity, the flow
to the tubes, or is ** Stream Line" flow, but after the
Jocity ha.s been passed, the motion parallel to the tube is
Qinpaniecl liy eddy motions, which cause a definite change to
plute iu the law of resistance.
Banie^i and Coker* have determined the critical velocity by
l^<iting the sudden change of temperature of the water when its
ttduii changes. They have also found that the critical velocity,
determined by noting the velocity at which streamline flowr
PrxK4t4imji> *>/ iht Hmjat Socuqf, Vol. lxMV. 1904; Phil, TranMctioni,
Tliat
There
130
HYDRAULICS
breaks up into eddies, is a much more variable quantity tlian
that determined from the points of interBection of the two \mm
as in Fig* 90* In the former case the critical velocity depends
upon the oondition of the water in the tank, and when it k
perfectly at rest the stream lines may be maintained at much
higher velocities than those given by the formula of Reynolds.
If the water is not perfectly at rest, the results obtained by both
methods agree with the formula.
Barnes and Cokor have called the critical velocity obtained by
the method of colour bands the upper limit, and that obtained by
the interiiiection of the logarithmic homologues the lower critical
velocity* The first gives the veloc^ity at which water flowing from
rest in streamline motion break.s up into eddy motion, while the
second gives the velocity at which water that is initially dii^turbed
persists in flowing with eddy motions throughout a long pipe, or
in other words the velocity ib too high to allow stream line^ to be
formed.
That the motion of the water in large conduits is in a similar
condition of motion is shown by the experiment of Mr G* H,
Benzenberg* on the discharge through a sewer 12 feet in diameter*
2534 ft,, long.
In order to measure the velocity of water in the sewer, red
eosine dissolved in water was suddenly injected into the sewer,
and the time for the coloured water to reach the outlet half a
mile away was noted. The colour was readily perceived and it
was found that it was never distributed over a length of more than
Q feet. As >vill be seen by reference to section 130, the velocities
of translation of the particles on any cross section at any instant
are very difFerentj and if the motion were streatn line the colour
must have been spread gut over a much greater length.
101. Law of factional resistance for velocities above
critical velocity*
As seen from Reynolds' formula, the critical velocity ea
for very small pipes is so very low that it is only nece&sary S^
practical hydraulics to consider the law of frictional resistance for
velocities above the critical velocity*
For any particular pipe,
and it remains to determine k and n»
From the plottings of the results of his own and Dmr
* Tramaeiiom Am.SM,E, 18t3 and alto Fn>eeeMttfft Am,S.€.E.^ VoL m.rm*
FLOW THROUGH PIPES
131
experiments, Beynolds found that the law of resistance " for all
pipes and all velocities" could be expressed as
t 
Transposing,
, /BD \«
B'D'.u'.P'
t =
and
4=
AP.D*
B"P"
.(14).
.(15),
A D"*
D is diameter of pipe, A and B are constants, and P is obtained
from formula (13).
Taking' the temperature in degrees centigrade and the metre
as nnit length,
A = 67,700,000,
B = 396,
1
P =
or
in w
t =
y=.
1 + 00361 + 000221T"
B'.t)'.F» y.v*
67,700,000 D»»" D»»
B»P^"
.(16),
^^^ ' 67,700,000
Values of y tchen the temperature is 10' C.
n
7
176
186
196
200
0000266
0000888
0000687
0000704
\
The values for A and B, as given by Reynolds, are, however,
only applicable to clean pipes, and later experiments show that
alt^oagli
it is doubtful whether
p = 3  w,
as given by Reynolds, is correct.
Value of n. For smooth pipes n appears to be nearly I'lo.
Reynolds found the mean value of n for lead pipes was 1*723.
Saph and Schoder*, in an elaborate series of experiments
carried out at Cornell University, have determined for smooth
* TranBoeticnt of the American Society of Civil Engineers, May, 1903. See
czerdte 31, psge 172.
9—2
132 HYDRAULICS
brass pipes a mean value for n of 1*75. Coker and Clements
found that n for a brass pipe *3779 inches diameter was 1*731. In
column 5 of Table XVI are given values of n, some taken from
Saph and Schoder's paper, and others as determined by the
author by logarithmic plotting of a large number of experiments.
It will be seen that n varies very considerably for pipes of
different materials, and depends upon the condition of the surface
of a given material, as is seen very clearly from Nos. 3 and 4.
The value for n in No. 3 is 1*72, while for No. 4, which is the
same pipe after two years' service, the value of n is 1*93. The
internal surface had no doubt become coated with a deposit of
some kind.
Even very small differences in the condition of the surfafCe,
such as cannot be seen by the unaided eye, make a considerable
difference in the value of 7i, as is seen by reference to the xdines
for galvanised pipes, as given by Saph and Schoder. For large
pipes of riveted steel, riveted wrought iron, and cast iron, the
value of n approximates to 2.
The method, of plotting the logarithms of i and v determined
by experiment, allows of experimental errors being corrected
without difficulty and with considerable assurance.
102. The determination of the values of C given in
Table XH.
The method of logarithmic plotting has been employed for
determining the values of C given in Table XII.
If values of C are calculated by the substitution of the
experimental values of v and i in the formula
s/mi
many of the results are apparently inconsistent with each other
due to experimental errors.
The values of C in the table were, therefore, determined as
follows.
Since i = kv^
and in the Chezy formula
v = C J mi,
. mC
or 1 = — 3,
therefore —pa = fry*
and 2 log C = 2 log v  (log m + log A; + n log t?) (17).
The index n and the coefficient k were determined for a
number of castiron pipes.
FLOW THROUGH PIPES
Valoes of C for i.'tJodties fnim 1 to 10 were calculated. Carvi
w<T\^ tlieTi plf»tt^^ for diffei^nt velocities, having C as oMinai
atid dianieters as abscisisae, and the values given in the table we;
Ai?<iaced from the curves.
The vBlneB of C so interpolated diifer very conaiderably,
fOiue cases, from the es(>t*riinental values* The diffieulties
&Umding the accurate deterii*ination of i and v are very great,
and tbe %*alties of C, for any given pipe, as calculated by substi
titing in the Chezy formula the losses of head in friction and th
vt^liiciries as determined in the experiments, were frequent!;
inamsiMeiit wHth each father.
As, for example, in the pipe of 3*22 ins. diameter given i
Tible XTI which was one of Darcy's pipes, the variation of C a
calculated from h and r given by Darcy m from 78'8 to 100.
Chi plotting log ft and logr and correcting the readings a
tiw they all lie on one line and recalculating C the variation wa
^ ' r > be only from ^5*9 to 101.
■tar correotioni^ liave been made in other cases.
Thv author thinks this procedure is justified by the fact that
mauy of the best experiments do not show any such inconsistencies.
An attempt to draw up an interpolated table for riveted pipes
mti not 8atisfjictor>% The author has therefore in Table XI Y
givtm the values of C as calculated by formula (17), for %^arioua
Vidoctties, and the iliameters of the pipes actually experimentec
ap»kn. If curves are plotted from the values of C given ii
Tntde XIV, It will be seen that, except for low velocities, th<
es are not continuous, and, until further experimental evidenc<
i^ i^irthcoming for riveted pipes, the engineer must be content
with ch*X)sing values of C, which most nearly coincide, as far
be can judge, with th© case he is considering.
103. Vwiation of k, in the formula i = kv", with the
diameter.
It has beeii shown in section 98 how the value of fc, for
iveo pipe, can be obtiiined by the logarithmic plotting of t and t?;
In Ttihle XVI, are given values of A\ as determined by the
authcirt by plotting the results of different experiments, Saph
 hoder found that for smooth harddrawn Inass pipes
us siases n varied between 173 and 1*77, the mean value
being 1'75. ■
By plotting logff tm abscissae and log ft as ordinates, as in'
Fig* I'ti, for these bniss piX'S the points He nearly in a straight line
which hoA aJi inelination & with the axis of d, such that
tan^P25
134
HYDRAULICS
and the equation to the line is, therefore,
\ogh = logyp\ogd,
where p = 1*26,
and log y = log Jc
when d = 1.
From the figure
y = 0000296 per foot length of pipe.
•05
03.
(m
EqucUwntu liive
Zo€f. 'k^Log m  12SLog d '
dz 03 Of 06 08 ho W^ '3
Log di
Fig. 93. Logarithmio plottings of fc and d, to determine the index p in the fonnul
. 7 . r*
On the same figure are plotted logd and log A;, as deduce
from experiments on lead and glass pipes by various workers. I
will be seen that all the points lie very close to the same line.
For smooth pipes, therefore, and for velocities above tb
critical velocity, the loss of head due to friction is given by
% =
_yv
~ d^
the mean value for y being 0*000296, for n, 1*75, and for p l'2o.
From which, v = 104i""d'^',
or log V = 2017 + 0572 log i + 0*715 log d.
FLOW THROUGH PIPES 135
The value of p in this formula agrees with that given by
Reynolds in his formnla
Pnrfessor Unwin* in 1886, by an examination of experiments
on castiron pipes, deduced the formula, for smooth castiron
pipes,
•0004t?^*
* =
cf
and for rough pipes, % = ^^ •
M. Flamantt in 1892 examined carefully the exi)eriment8
available on flow in pipes and proposed the formula,
for all classes of pipes, and suggested for y the following values :
Lead pipes ^
Glass „ [ •000236 to 00028,
Wroughtiron (smooth) J
Castiron new '000336,
„ „ in service •000417.
If the student plots from Table XVI, log d as ordinates, and
log" A as abscissae, it will be found, that the points all lie bqtween
two straight lines the equations to which are
log k = log 00069  125 log d,
and log & = log '00028  125 log d.
Further, the points for any class of pipes not only He between
these two lines, but also lie about some line nearly parallel to
these lines. So that p is not very different from r25.
From the table, n is seen to vary from 1*70 to 208.
A general formula is thus obtained,
, 00028 to •00069t;^'^^^^^/
The variations in y, n, and p are, however, too great to admit
of the formula being useful for practical purposes.
For new castiron pipes,
, 000296 to 0004181?^^ '«^*^Z
fe = ^ •
If the pipes are lined with bitumen the smaller values of y and
T< may be taken.
* Industries, 1SS6.
t AimaUs des Pouts et Chauss^et, 1892, Vol. ii.
136
HYDRAULICS
For new, steel, riveted pipes,
^_' 0004to'00054i;^"^'«'Z
^ d}^
Fig. 94 shows tlie result of plotting logfe and logd for all
the pipes in Table XVI having a value of n between 1*92 and 1*94.
They are seen to lie very close to a line having a slope of 1^,
and the ordinate of which, when d is 1 foot, is '000364.
, •000364r^"*'Z
Therefore
d}''
or r = 59i««d"'
very approximately expresses the law of resistance for particular
pipes of wood, new cast iron, cleaned cast iron, and galvanised
iron.
Fig. 94.
Taking a pipe 1 foot diameter and the velocity as 3 feet per
second, the value of i obtained by this formula agrees with that
from Darcy's formula for clear castiron pipes within 1 per cent.
Use of the logarithmic formula for 'practical calculations. A
very serious difficulty arises in the use of the logarithmic
formula, as to what value to give to n for any given case, and
consequently it has for practical purposes very little advantage
over the older and simpler formula of Chezy.
TABLE XVI.
1
Diameter
(in int.)
Velocity in
Value of n ^
/alue of k
enter 1
Kind of pipe
fk per seo.
from to
in formula 1
n formula
le ^
Wood
44
846 — 4416
178
0001254
«»
54
228 — 468
175
000088
skissV
w
725
1 ~ 4
172
000061
M
726
1 — 66
198
000048
witcham
Riveted
8
1^88
00245
oith
Wrought
11
181
000516
iron or steel
11}
1^90
000470
ff
16
194
000270
ding
»»
88
•506— 1254
2^0
000099
^
»»
42
210  499
198
00011
»»
48
2  6 (?)
20
000090
^ipg)
»»
72
1—4
199
000055
99
72
1 — 66
1^85
000077
2bel
»9
108
1 — 45
208
000036
cy
Cast iron
8*22
•289— 1071
197
00156
new
689
•48 — 158
197
00079
tf
744
•678—1617
1^956
00062
ff
12
1779
000823
iZHfl
}f
1625
1^858
000214
pe
»»
165
248 — 809
1^80
000267
«f
1968
188 — 87
1^84
00022
nan
»i
36
4 — 7
2*
000062
ns
»»
48
1248 828
1^92
0000567
FenkeU
30
2
00003
ZJ
Cast iron
14136
•167— 2077
199
0098
old and
31296
•403 3747
194
0085
tuberculated
9575
1007—1258
198
0009
aan
^1
20
271 — 511
?i
36
11 — 45
2
000105
raid
48
1.176— 3533
204
0(KX)83
♦>
48
1135— 3412
200
00(X)85
:y
Castiron
14328
371— 369
185
(K)41
old pipes
31536
633— 50
197
(K)lHo
cleaned
1168
8 —10368
20
(KK);J75
raid
yj
48
367 — 56
202
0(K)082
"
48
•895— 7245
194
()(KX)59
:y
Sheet iron
1055
098— 8225
176
(K)74
,^
324
•3281278
181
00154
,*,'
772
5911972
178
(X)059
112
1296— 1052
181
00039
Gas
48
113— 392
183
0278
155
•205— 8521
186
191
00418
0072
Sclioder
Galvanised
364
196
0352
•494
191
0181
,,
623
186
0182
•824
180 1
0095
,^
1048
193
•(K)82
Hard drawn
15 pipes
175
•00025 to
brass
up to 184
0(K)85
.Ids
/
Lead
1782
•55 1
161 1
1761 \
0\^e>
1^783 \
•0^^^
138
HTDBAUUCS
TABLE XVn.
Showing reasonable values of y, and n, for pipes of TarioDf
kinds, in the formula,
Takes for j
1 1
i^
«
y
i
Clean caBtiron pipes
Old cast iron pipes
Riveted pipes
Galvanieod pipes
Sheetiron pipes cover
ed with bitumen
Clean wood pipes
BrasR and lead pipes
00029 to 000418
'00O47 to '00069
'00040 to *00054
00035 to 00045
'00030 to *00088
'00056 to '00063
1*80 to 1*97
1'94 to 2*04
193 to 2'08
1*80 to 196
l'76tol'81 ;
172 to 175 '
■00086
00060
•00050
'00040
00034
■O0060
*00080
2
t
vm
m
When further experiments have been performed on pipes, rf
which the state of the internal surfaces is accurately known, vA
special care taken to ensure that all the loss of head in a git®
length of pipe is due to friction only, more definiteness maybe
given to the values of y, n, and p.
Until such evidence is forthcoming the simple Chezy formnl*
may be used with almost as much confidence as the mow
complicated logarithmic formula, the values of C or/ being takea
from Tables XII — XIV. Or the formula h = fcr* may be used,
values of k and n being taken from Table XVI, which most nearly
fits the case for which the calculations are to be made.
I
104. Criticism of experiments.
The difficulty of differentiating the loss of head due to friction .
from other sources of loss, such as loss due to changes in direction,
change in the diameter of the pipe and other causes, as well as the
possibilities of error in experiments on long pipes of large diameter,
makes many experiments that have been performed of very little
value, and considerably increases the difficulty of arriving at
correct formulae.
The author has found in many cases, when log i and log d were
plott<3d, from the records of experiments, that, although the results
seemed consistent amongst themselves, yet compared with other
eYperiments, they seemed of little value.
\ for one ol Couplot^s* e3:periments on a lead and
ipe being &b low as 1*56, while the results of an
iSunpecint cm a custiroii pipe gave n as 2*5. In the
fe were a number uf bends in the pipe*
experiments for loss of head due to friction, it is
I the pipe should be of uniform diameter and aa
Uible between the points at which the pressure head
Further, special care Rhould be taken to ensure the^
I air, and that a perfectly steady flow m established
Fhere the pressure is taken*
Eometer fittings,
heme iuiixirtance that the
ktmections shall be made
JBference in the pressures
any two pcjints shall be
fiction^ and friction only,
points,
Hiiptes that there shall
^P^B to interfere with the
Se w*ater, and it h, there
bntial that all burrs sliall
hom the inside of the pipe,
bients un small pipits in
JT the best results are no
bd by cutting the pipe
irough at the connection
t'ig. 95, which illustrates
I connection use<l by Dv p gg
(u experiments cited on
lie two ends of the pipe ai*e not more than u^nr
^B the method adopted by Marxj Wing and Hoskins
riments on a 72^inch wooden pipe to ensure a correct
B pressure.
I? X was connected to the top of the pipe only while
jted at four points as shown.
(erences were observed in the readings of the t%vo
ji they thought were due to some accidental circum
big the gauge X only, as no change was obsoiTcd
of Y when the points of communication to Y were
of the cocki*,
utici, HamUton Smith. Jnnr.
iinfft fif the Ttntitufe of Vint Euffineertf 1855,
^Ita
140
HYDRAULICS
106. Effect of temperature on the velocity of flow.
Poiseuille found that by raising the temperature of the water
from 50** C. to 100' C. the discharge of capillary tubes was
doubled.
Fig. 96. Piezometer connections to a wooden pipe.
Reynolds* showed that for pipes of larger diameter, the effect
of changes of the temperature was very marked for velocities
below the critical velocity, but for velocities above the critical
velocity the eifect is comparatively small.
The reason for this is seen, at once, from an examination of
Reynolds'* formula. Above the critical velocity n does not differ
very much from 2, so that P"* is a small quantity compared >vith
its value when n is 1.
Saph and Schodert, for velocities above the critical velocity,
found that, as the temperature rises, the loss of head due to
friction decreases, but only in a small degree. For brass pipes of
small diameter, the correction at 60° F. was about 4 per cent, per
• Scientific Papers, Vol. ii.
+ See also Barnes and Coker, Proceeding* of the Royal Society ^ Vol. utx. 1904 ;
Coker and Clements, Transactions of the Royal Society, Vol. cci. Proceeding*
Am.S.C.E. Yol xxix.
FLOW THROUOH PIPES 141
degrees F. With galvanised pipes the correction appears to
le from 1 per cent, to 5 per cent, per 10 degrees F.
Since the head lost increases, as the temperature falls, the
lischarge for any given head diminishes with the temperature,
mt for practical purposes the correction is generally negligible.
107. Ij088 of head due to bends and elbows.
The loss of head due to bends and elbows in a long pipe is
generally so small compared with the loss of head due to friction
Q the straight part of the pipe, that it can be neglected, and
onsequently the experimental determination of this quantity has
lot received much attention.
Weisbach*, from experiments on a pipe 1^ inches diameter,
rith bends of various radii, expressed the loss of head as
*(«^*T)g.
being the radius of the pipe, B the radius of the bend on the
pntre line of the pipe and v the velocity of the water in feet per
?<rond. If the formula be written in the form
7 at?*
le table shows the values of a for different values of ^ .
r
R
•1 167
•2 250
•6 626
St Venantt has given as the loss of head h^ at a bend,
Ab = 001524 y^^=Ol 4 yi nearly.
being* the length of the bend measured on the centre line of the
?nd and d the diameter of the pipe.
ANTien the bend is a right angle
RVR 2VR'
When
d
= 1,
•5,
%
d
= 157,
Ml,
•702
iB =
■ '"g'
•111
29'
<
. • Ueehatue$ of Engineering,
i Compte* Rendtii, 1862.
143
HTTDRiUXICS
Kt^ent experiments by Williams, Hubbell and Fenkell^onc
iron inpes asphalted, hy Saph and &boder on brajss pipe.**,
oth* r* by Alexander t on wooden pipes, show that the loss of ]
in lK>nds, as in a straight pipe, can be expressed as
n Wing a variable for different kinds of pipes, while
Ar
y'Hr
y being a constant coeffieii lipe.
For the castircin pipes or and Penkell, y^ n, m, and f ♦
have approximately the {ollowm^ les.
Dia&Mter of pipe
ir
ir
mr
<€0«
1TB
im
im
^^^len r is 3 feet per second and p" is i, the bend being a righl
angle^ tlie lo^ of head as calculated by thia formula for difi'j
i2ineh pipe is \^ , and for ta© dUmch pipe ^  .
For the brass pipes of Saph and Schoder, 2 inches diametO'^
Alexander found,
and for vanitshecl wood pipes when ^ is less than 0'2,
and wlieii u is betn^een 0"2 and O'o,
11
He further found for \*amished wood pipes that^ a bend of
radias equal to 5 times the radius of the pipe gives the mini mum
loss + ^^'rf} '»*^'i f^>* ^*^ !<»si?^tance is equal to a straight pipe 3"3S
times the length of the bend.
Messrs Williams, Hubbell and Fenkell also state at the end
their elaborate paper, that a bend having a radius equal to :
• Proe. Amer. Soc, Civil EmgiHeen, Y6L xxm.
t Proc, Intt. Civil Emgineen, Vol. clix.
rLOW THROUGH TlfES
143
loffers hm remstanoe to the flow of water than those of
aditm. It should not be overlooked, howeverj that although
nf liead in a bend of radius equal to 2A diameterB of the
ess than for any other, it does not follow that the loss of
JT unit length of the pipe measured along its centre line
cniiiit&um value for bends of this radiuB.
Variations of the velocity at the cross section of a
rieal pipe*
(eriinents show that when water flows through conduits of
rm, the velocities are not the same at all points of any
turn aeetion, but decrease from the centre towards the
terence.
f first experimentsi to determine the law of the variation of
[ocity in cylindrical pipes were those of Darcy, the pipes
If in diameter frfjm 7"8 inches to 19 incites. A complete
tf of the exi>eriraents is to be found in his Recherchm
tnmiiales dan^ les iuyauit^
• Telocity was me^isured by means of a Pitot tube at five
on a vertical diamfetjer, and
suits plotted as iihown in
r.
ling V the velocity at the
of a pipe of radins R^ u the
Y at the circumference, tv
san velocity, y the vt^wity
distance r from the centre,
losB of head per unit
[the pipe, Darcy deduced the formulae
\ the unit is the metre the value of k is \\% and 20'4 when
ft IB the English foot.
'jBT earperiments commenced by Darcy and continued by
on the distribution of velocity in a semicircular channel,
rface of the water being maintained at the horizon tal
g" and in which it was assumed tho conditions were similar
^Ki a cylindrical pipe, showed that the velocity near the
^OT the pipe diminished much more rapidly than indicated
iormula of Darcy,
144 HYDRAUUCS
Bazin substituted therefore a new formula,
Yv = 38y/Ri(^J (1),
or since t7» « C Jmi = —j^ JTU
C'^ii)' ■■■ <^>
It was open to question, however, whether the conditions of flow
in a semicircular pipe are similar to those in a pipe discharging
full bore, and Bazin consequently carried out at Dijon*, experi
ments on the distribution of velocity in a cement pipe, 2'73 feel
diameter, the discharge through which was measured by means
of a weir, and the velocities at different points in the transverse
section by means of a Pitot tubet.
From these experiments Bazin concluded that both formulae (1)
and (2) were incorrect and deduced the three formulae
V..38^RS((0(iy*(g} (3),
V.^VBi{38(^)\4 9(g(l rigy (4),
Vr = N/Rt535{lyi95(j)'} (5),
the constants in these formulae being obtained from Bazin's by
changing the unit from 1 metre to the English foot.
Equation (5) is the equation to an ellipse to which the sides of
the pipes are not tangents but are nearly so, and this formula
gives values of v near to the surface of the pipe, which agree much
more nearly with the experimental values, than those given by
any of the other formulae.
Experiments of WilliamSy Hvhhell and FenkellX* An elaborate
series of experiments by these three workers have been carried out
to determine the distribution of velocity in pipes of various
diameters, Pitot tubes being used to determine the velocities.
The pipes at Detroit were of cast iron and had diameters of 12,
16, 30 and 42 inches respectively.
The Pitot tubes § were calibrated by preliminary experimentB
on the flow through brass tubes 2 inches diameter, the total
* ' * Memoire de TAcad^mie des Soiences de Paris, Becueil des SavantsEtraxigdiv,"
Vol. XXXII. 1897. Proc, Am,S.C,E. Vol. xxvii. p. 1042.
t See page 241.
X ** Experiments at Detroit, Mich., on the effect of carvatare on the flow of
water in pipes,*' Proc. Am.S.C.E. Vol. xxvu. p. 318.
§ See page 246.
FLOW THROUGH PIPES 145
discliarge being determined by weighing, and the mean velocity
thas determined. From the results of their experiments they
came to the conclusion that the curve of velocities should be an
ellix>8e to winch the sides of the pipe are tangents, and that the
velocity at the centre of the pipe V is TlOvm, t?» being the mean
velocity.
These results are consistent with those of Bazin. His experi
V
mental value for — for the cement pipe was 1*1675, and if the
constant "95, in formula (5), be made equal to 1, the velocity curve
be<x>me8 an ellipse to which the walls of the pipe are tangents.
V
The ratio — can be determined from any of Bazin's formulae.
Substituting —^ for >/Ri in (1), (3), (4) or (5), the value of
V at radius r can be expressed by any one of them as
'^#/(S)
Then, since the flow past any section in unit time is VmyfR\ and
that the flow is also equal to
2'7rrdr . v,
f
therefore v^^W = 2t J^v  5^"/(^)jrdr (6).
(t\ ^lA***
g^j , its value ^ from equation (1), and
integrating.
;; = ^"c (7),
and by substitution oifi^j from equation (4),
l'*v <«.
V
so that the ratio — is not very different when deduced from the
simple formula (2) or the more complicated formula (4).
Wlien C has the values
C = 80, 100, 120,
from (8) ~ = 1287, 123, ri9.
The value of C, in the 30inch pipe referred to above, varied
between 109"6 and 123*4 for different lengths of the pipe, and
i^ H. 10
/:
146
HYDRAUUCS
the mean value was 116, so that there is a remarkable agreement
between the results of Bazin, and Williams, Hubbell and Fenkell.
The velocity at the 8v/rface of a pipe. Assuming that the
velocity curve is an ellipse to which
the sides of the pipe are tangents, as
in Fig. 98, and that V= ri9t?«, the
velocity at the surface of the pipe
can readily be determined.
Let u = the velocity at the surface
of the pipe and v the velocity at any
radius r.
Let the equation to the ellipse be
in which a? = v  u,
and b = y — u.
Then, if the semiellipse be revolved about its horizontal axis,
the volume swept out by it will be ^rR''6, and the volume of
discharge per second will be
r^
7rR'»t7^ = 27rrdr . i; = ttR' . u + §7rR«6,
J
itt + f X ri9r«,
and
u = '621v„,.
u
Using Bazin's elliptical formula, the values of — for
C = 80,
are
u
=•552, 642,
Vm
100, 120,
702.
The velocities, as above determined, give the velocity of
translation in a direction parallel to the pipe, but as shown by
Reynolds' experiments the particles of water may have a much
more complicated motion than here assumed.
109. Head necesisaky to give the mean velocity Vm to
the water in the pipe.
It is generally assumed that the head necessary to give a mean
V ^
velocity Vm to the water flowing in a pipe is ^, which would be
correct if all the particles of water had a common velocity Vm.
If, however, the form of the velocity curve is known, and on the
assumption that the water is moving in stream lines with definite
velocities parallel to the axis of the pipe, the actual head can
be determined by calculating the mean kinetic energy per lb. of
water flowing in the pipe, and this is slightly greater than ^ .
FLOW THBOUOH PIPES 147
As bef ore, let v be the velocity at radius r.
The kinetic energy of the quantity of water which flows past
any aectioii per second
w . 2vrdr . t; . ^r ,
2g'
ir being tlie ^vreight of 1 c. ft of water.
The kinetic energy per lb., therefore,
f^w.2irrdrv*
«io 2g
^^^
1 w . 2xr dn?
2g}oV
V2r. ^/r\l» ,
(9).
The
simplest value for /
(^ is that of Bazin's
formula (1)
above.
from
which
■(^F)
and
/G
D=^i"
Substituting these values and integrating, the kinetic energy
per 1^ is — , and when
C is 80, 100,
a is 112, 1076.
On the assumption that the velocity curve is an ellipse to which
the ijvalls of the pipe are tangents the integration is easy, and the
value of a is 1*047.
Using the other formulae of Bazin the calculations are tedious
and the values obtained differ but slightly from those given.
The head necessary to give a mean velocity Vm to the water in
the pipe may therefore be taken to be o" > t;he value of a being
ftbout 1'12. This value* agrees with the value of 1*12 for a,
obtained by M. Boussinesq, and with that of M. J. Delemer who
finds for a the value 11346.
IIO. Practical problems.
Before proceeding to show how the formulae relating to the
loss of head in pipes may be used for the solution of various
problems, it will be convenient to tabulate them.
* Flamant's Hydrauliqw.
10—2
148 hydraulics
Notation.
fc = los8 of head due to friction in a length Z of a straight pipe.
. i = the virtual slope = y .
t? = the mean velocity of flow in the pipe,
d = the diameter.
m = the hydraulic mean depth
A i*ftfli A fl
Fcyrmuh,!. h = ^=^.
This may be written y = 7^5— ,
or 1; = C J mi.
The values of C for castiron and steel pipes are shown in
Tables XII and XIV.
Formula 2. h = rf^ ,
f , . . 1
^ in this formula being equal to 7^ of formula (1).
Values of /are shown in Table XIII.
Either of these formulae can conveniently be used for
calculating fc, t?, or d when /, and Z, and any two of three
quantities fe, v, and d, are known.
Formula 3. As values of C and / cannot be remembered for
variable velocities and diameters, the formulae of Darcy are
convenient as giving results, in many cases, with sufficient
accuracy. For smooth clean castiron pipes
A = 005(l^^^,
or ^=m^j^M
For rough and dirty pipes
or « = i39y__V5f
=27«yi2^^
FLOW THROUGH PIPES 149
II d is the unknown, Darcy^s formulae can only be used to solve
for d by approximation. The coefficient ( 1 + t^ j is first neglected
and an approximate value of d determined. The coefficient can
then be obtained from this approximate value of d with a greater
degree of accuracy, and a new value of d can then be found, and
so on. (See examples.)
Formula 4. Known as the logarithmic formula.
h . y . r*
Values of y, n, and p are given on page 138.
By taking logarithms
log h = log y + n log v + log I — p log d,
from which h can be found if i, v, and d are known.
If A, Z, and d are known, by writing the formula as
n log V = log h  log I  log y + p log d,
T can be found.
If A, /, and v are known, d can be obtained from
p log d = log y + n log v + log I  log h.
This formula is a little more cumbersome to use than either (1) or
(2) but it has the advantage that y is constant for all velocities.
Formula 5. The head necessary to give a mean velocity v to
ri2i7'
the iJvater flowing along the pipe is about — ^ — , but it is generally
convenient and sufficiently accurate to take this head as ^ , as
was done in Fig. 87. Unless the pipe is short this quantity is
negligible compared with the friction head.
Formula 6.^ The loss of head at the sharpedged entrance to a
pij>e is about ^ and is generally negligible.
Formula 7. The loss of head due to a sudden enlargement in
a pipe where the velocity changes from Vi to rj is ^ .
^9
Formula 8. The loss of head at bends and elbows is a very
variable quantity. It can be expressed as equal to y in which
9
a varies from a very small quantity to unity.
ProbUm 1. The difference in level of the water in two reservoirs is h feet,
FUr. 99, And thej are connected by means of a straight pipe of length I and
diameter d ; to find the discharge through the pipe.
150
HYDRAULICS
Let Q be the number of eabic feet discharged per second. The head h is utilised
in giving velocity to the water and in overcoming resistance at the entrance to the
pipe and the frictional resistances.
Fig. 99. Pipe connecting two reservoirs.
Let V be the mean velocity of the water. The head necessary to give the water
this mean velocity may be taken as —= — , and to overcome the resistance at the
entrances
Then
•6t;«
^' 2g "^ 2g'^2g,d'
Or using in the expression for friction, the coefficient 0,
;i =0174v3+ 0078179+
= •025r2 +
4fa«
C^d
I .
If  is greater than 300 the head lost dae to friction is generally great compared
with the other quantities, and these may be neglected.
iflv^ _ 4Zt7«
~C^,d*
C /dh
Then
h = 
and
"2
As the velocity is not known, the coefficient C cannot be obtained from the
table, but an approximate value can be assumed, or Darcy*s value
C = 394
y
12d + l
for clean pipes,
C=278 a/toTi *^ *^® P^P® " ^*y»
and
can be talcen.
An approximation to v — which in many cases will be sufficiently near or will be
as near probably as the coefficient can be known — ^is thus obtained. From the
table a value of C for this velocity can be taken and a nearer approximation to
V determined.
Then
Q=^dM;.
The velocity can be deduced directly from the logarithmic formula A=^^,
provided y and n are known for the pipe.
FLOW THROUGH PIPES 151
The bydxmnlic gradient is EF.
At any point C distant x from A the pressore head  is eqaal to the distance
rtween the centre of the pipe and the hydranlio gradient. The pressure head
Lftt inside the end A of the pipe is Aa kz— « ^^^ ^^ ^^® ^^^ ^ ^^^ pressure head
last be eqaal to Ab. The hesd lost due to friction is h, which, neglecting the
oiall qnantity — ^ — , is eqoal to the difference of level of the water in the two
inks.
Example 1. A pipe 3 inehes diameter 200 ft. long connects two tanks, the
ifferenoe of lerel of the water in which is 10 feet, and the pressure is atmospheric.
find the discharge assuming the pipe dirty.
Using Darey'8 coefficient
17 = 278 ^y^ ^/SA^A"=69•5^/,i^
=3'8dft. per sec.
For a pipe 3 inches diameter, and this velocity, C from the table is about 69, so
iiat the approximation is sufficiently near.
^ ^ ,. 00064^1^/
Taking h= ^,^ ,
r=388 ft. per sec,
. oooei?*!
pves v=d'85 ft. per sec.
Example 2. A pipA 18 inches diameter brings water from a reservoir 100 feet
&bove datum. The total length of the pipe is 15,000 feet and the last 5000 feet
^re at the datum level. For tbis 5000 feet the water is drawn off by service pipes at
me Quifonn rate of 20 cubic feet per minute, per 500 feet length. Find the pressure
&t the end of the pipe.
The total quantity of flow per minute is
^ 6000x20 ^^ u .r .
= =200 cubic feet per minute.
Area of the pipe is 1*767 sq. feet.
The velocity in the first 10,000 feet iw, therefore,
200
The head lost due to friction in this length, is
4./^0^.2;888a
2i/.l5
In the last 5000 feet of the pipe the velocity varies uniformly. At a distance
1*888^
X feet from the end of the pipe the velocity is ^^ .
In a length dx the head lost due to friction is
4./. l'888».j'dg
2^.15.5000» '
and the total loss by friction is
.^/ll^??. /"^ .^_4/. (l888)« 6000
^~2^.1'5.6000»jo 2^.15 ' 3 *
The total head lost due to friction in the whole pipe is, therefore,
H=;r^. 18882 (10,000 + A«yui).
152 HYDRAULICS
Taking / as *0082, H = 14*3 feet.
Neglecting the velocity head and the loss of head at entrance, the pressure head
at the end of the pipe is (100  H) feet =857 feet
Problem 2. Diameter of pipe to give a given discharge.
Beqoired the diameter of a pipe of length I feet which will discharge Q onbie feet
per second between the two reservoirs of the last problem.
Let V be the mean velocity and d the diameter of the pipe.
Then v=^^ (1),
and ft=025t>»+^.
Therefore,
^/W A / .nos_i
Squaring and transposing,
^g 00406. (yd
If Hs long compared with d,
h
.(2).
A vs.
(8).
Since v and d are unknown G is unknown, and a value for C must be pro
visionally assumed.
Assume G is 100 for a new pipe and 80 for an old pipe, and solve equation (3)
for d.
From (1) find v, and from the tables find the value of G corresponding to the
values of d and v thus determined.
If G differs much from the assumed value, reoaloultfte d and v using this aeoond
value of C, and from the tables find a third value for G. This will generally be
found to be sufficiently near to the second value to make it unnecessary to oaloolate
d and v a third time.
The approximation, assuming the values of G in the tables are correct, can be
taken to any degree of accuracy, but as the values of G are uncertain it will not as
a rule be necessary to calculate more than two values of d.
Logarithmic formula. If the formula h = ^^^ be used, d can be found direct,
from
jp log d=n log V +log7+log £ log ^.
Example 3. Find the diameter of a steel riveted pipe, which will discharge
14 cubic feet per second, the loss of head by friction being 2 feet per mile. It is
assumed that the pipe has become dirty and that provisionally G = 110.
From equation (3)
^ 255.14 /6280
^=^[IoV^'
or ^^ log <i= log 1663,
therefore d = 308 feet.
For a thirtyeight inch pipe Euichling found G to be 113.
The assumption that G is 110 is nearly correct and the diameter may be taken
as 37 inches.
Using the logarithmic formula
^ •00045ri»Z
FLOW THROUGH PIPES
153
od sabetitating for v the yalne £
Tom whieh
515 log d == log 000i5  195 log 07854 + 1*95 log 14 + log 2640,
ind <i=d07 feet.
Short pipe. If the pipe Ib short so that the velooity head and the head lost at
Atruftee are not negligible oompared with the loss due to friction, the equation
•0406Q«d _ 65/Q*
^ h ^ C*h '
rhen a Talne is given to C, can be soWed graphically by plotting two carves
LZld
yi=
_ 0406<y
.d +
65«Q«
h ' Ch '
The point of intersection of the two corves will give the
HaxaetesT d.
It is however easier to solve by approximation in the
Dlloving manner.
Neglect the term in d and soNe as for a long pipe.
Chooee a new value for C corresponding to this ap
roximate diameter, and the velocity corresponding to it,
nd then plot three points on the cnrve y=d^t choosing
mines of d which are nearly eqnal to the caloolated value
f d, and two points of the straight line
•0406QSd
yi= r^ +
66/Q*
Fig. 100.
•6
The enrve y=(^ between the three points can easily
e drawn, as in Fig. 100, and where the straight line cuts
be cnrre, gives the required diameter.
KxamepU 4. One hundred and twenty cubic feet of water are to be taken
«T minate from a tank through a castiron pipe 100 feet long, having a square
diged entrance. The total head is 10 feet. Find the diameter of the pipe.
Neglecting the term in d and assuming G to be 100,
^=6M?4. = 026.
nd
Therefore
100.100.
(i=: 4819 feet.
2
j(4819)«
10"
= 109 ft. per sec.
From Table XIl, the value of C is seen to be about 106 for these values of
and r.
A second value for d" is
^,^65.100.4
.10
= •0233,
106*
rom which d= 476'.
The schedule shows the values of d' and y for values of d not very different
rom the calculated value, and taking C as i06.
d 4 5 6
d» 01024 08125 0776
yi 0297 0329
The line and curve plotted in Fig. 100, from this schedule, intersect at> for which
d= 498 feet.
154 HYDRAULICS
It is seen therefore that taking 106 as the Talae of C, oeglectiDg the term in i,
makes an error of 022' or 264".
This problem shows that when the ratio z is about 200, and the virtual slope k
even as great as ^, for all practical purposes, the friction head only need be oon
sidered. For smaller values of the ratio r the quantity *025t^ may become in
portant, but to what extent will depend upon the slope of the hydraulic gnuiieol.
The logarithmic formula may be used for short pipes but it is a little man
cumbersome.
Using the logarithmic formula to express the loss of head for short pipes witt
squareedged entrance,
•025Q» 7.Q».t
or d*»»^i«  •0406Q2d«*«^= V^t'^ •
When suitable values are given to y and n, this can be solved by plotting tin
two curves
and j,^=.0406(yd*»aT5+l:^^ ^
the intersection of the two curves giving the required value of d.
Problem 3. To find what the discharge between the reservoirs of problem (1)
would be, if for a given distance l^ the pipe ,
of diameter d is divided into two branches I i •
laid side by side having diameters d, and dj, k— J,^ — >W L — H
Fig. 101. I ^ ii _ t !
Assume all the head is lost in friction. A^ ^ fi ^ cL \ C.
Let Qi be the discharge in cubic feet. ^ *; "^ /^ *
Then, since both the branches BC and BD , j ^ V t
are connected at B and to the same reservoir, j ^^ ^  D
the head lost in friction must be the same in  j
BC as in BD, and if there were any number I* ^ *i
of branches connected at B the head lost in Yia, 101
them all would be the same.
The case is analogous to that of a conductor joining two points between iHikh
a definite difference of potential is maintained, the conductor being divided between
the points into several circuits in parallel.
The total head lost between the reservoirs is, therefore, the head lost in AB
together with the head lost in any one of the branches.
Let 17 be the velocity in AB, v^ in BC and Vj in BD.
Then vd^^v^d^^ + v^^ (1),
and the difference of level between the reservoirs
4?3r2 4/,V
C*d "^Ci^di '
I the same as i
iqual to Cg
C*d ^ Ci*di <^'
And since the head lost in BC is the same as in BD, therefore,
\ C,»d, C,«d, (')•
If proTirionally Ok be taken as equal to C,,
FLOW THROUGH PIPES
155
ThaeCore,
d,*+
''■A
.(4).
Fitni (2), V ean be found by sabstitatiiig for v^ Arom (4), and thus Q can
It dBiarmiiied.
If AB, BC, mnd CD are of the same diameter and 2^ is eqaal to 2,, then
ProfrinR 4. Pipes eonneeting three reservoin. As in Fig. 102, let three pipes
AB, EC, and BD, connect three reservoirs A, C, D, the level of the water in each
•f which remains constant.
Lei V,, V,, and v, be the velocities in AB, BG, and BD respectively, Q,, Qs,
md Q, Uie qoantities flowing along these pipes in cubic feet per sec., Z] , l^, and ^
h0 Vngtha of the pipes, and d^ , d, and d, their diameters.
Fig. 105^.
Jjgt x^, r,, and £, be the heights of the surfaces of the water in the reservoirs,
lad X, the height of the junction B above some datum.
Liet h^ be the pressure head at B.
/^ ffgyiTTM* all losses, other than those due to friction in the pipes, to be negligible.
Xhe head lost due to friction for the pipe AB is
tud for tlie pipe BC,
.(1),
.(2),
ihe upper or lower signs being taken, according as to whether the flow is from, or
Uywards, the reservoir G.
For the pipe BD the head lost is
(3).
Cs'di"^'"*" ""'''
Sinee the flow from A and G must equal the flow into D, or else the flow
from A most equal the quantity entering G and D, therefore,
or Vi'±r2^'»=rA' (4).
There are foor equations, from which four unknowns may be found, if it is
farther known which sign to take in equations (2) and (4). There are two cases to
eooaider.
156 HYDRAULICS
Ccue (a). Given the levels of the surfaces of the water in the reflerfoin aai
of the junction B, and the lengths and diameters of the pipes, to find the quaali^
flowing along each of the pipes.
To solve this problem, it is first necessary to obtain by trial, whether water floH
to, or from, the reservoir C.
First assume there is no flow along the pipe BC, that is, the pressure bead \ al
B is equal Xo z^Zq.
Q
Then from (1), substituting for v^ its value ^^f
<J.=t/^ »
from which an approximate value for Qj can be found. By solving (3) in the Mi
way, an approximate value for Q,, is.
Q,
=T\/'^^ (^
If Q, is found to be equal to Qj, the problem is solved ; but if Q. is greater te;
Qi , the assumed value for \ is too large, and if less, h^ is too smaU, for a diBur~
tion in the pressure head at B will clearly diminish Q, and increase Qj, and ^
also cause flow to take place from the reservoir C along GB. Increasbig te
pressure head at B will decrease Q, increase Q,, and cause flow from B to C.
This preliminary trial will settle the question of sign in equations (2) snd (A
and the four equations may be solved for the four unknowns, v^, v^, v^ and \. A
is better, however, to proceed by "trial and error."
The first trial shows whether it is necessary to increase or diminish h^ and :
values are, therefore, given to h^ until the calculated values of v, , v^ and r sati^^
equation (4).
Case (&). Given Q^, Q^, Q,, and the levels of the surfaces of the water i&
the reservoirs and of the junction B, to find the diameters of the pipes.
In this case, equation (4) must be satisfied by the given data, and, therelbn^
only three equations are given from which to calculate the four unknowns 6y
d], d^ and Iiq. For a definite solution a fourth equation must consequently n
fonnd, from some other condition. The further condition that may be taken ii
that the cost of the pipe lines shall be a minimum.
The cost of pipes is very nearly proportional to the product of the length tai
diameter, and if, therefore, Udi + l^+l^^ is made a minimum, the cost dL tbi
pipes will be as small as possible.
Differentiating, with respect to h^ , the condition for a minimnm is, that
^dh^^^dK^^dh, ^
Substituting in (1), (2) and (3) the values for 17^, r, and v,,
V
=_2l,
■■^'
dJ
4^
Q,
d^
differentiating and substituting in (7)
1
FXOW THBOUOH PIPES
157
Pottiiic the ^mlaes of Qi, Qa, and Q, in (1), (2), (8), and (8), there are four
mitaoM as before for four anlmown quantities.
U wUI be beUer however to solve by approximation.
Qiie some arbitrary valoe to say d,, and oalcolate Aq from equation (2).
Then calenlate cL and dm by putting h^ in (1) and (8), and substitute in
tMli0ll(8).
If this equation ie satisfied the problem is solved, but if not, assume a seoond
^ihw for dL and try again, and so on until such values of d,, dL, d. are obtained
«al (8) is satisfied.
In this, as in simpler ^stems, the pressure at any point in the pipes ought not
to hJl below the atmoepherie pressure.
flow through a pipe of constant diameter when the flow i$ diminishing at a
ms\f$ru rate. Let I be the length of the pipe and d its diameter.
Let h be the total loss of head in the pipe, the whole loss being assumed to be
^frietion.
Let Q be the number of oobio feet per second that enters the pipe at a section A,
mi. Q the number of cnbie feet that passes the section B, I feet from A, the
eaanti^ QQi being taken from the pipe, by branches, at a uniform rate of
QQ,
■ . * eoUe feet per foot.
I
Then, if the pipe is assumed to be continued on, it is seen from Fig. 103, that
M the rate of discharge per foot length of tlie
I is kepi oonstant, the whole of Q will be
1 in a length of pipe.
The diaeharge past any seetion, x feet from
CwiUbe
! lO.^
H
t^^B
M j^. .
The vekxity at the seetion is
Fig. 103.
4( QQ,)x
Aseoming that in an element of length dx the loss of head due to friction is
nd eabetituting for v^^ its value
d«
L^d»
4
le lose of head due to friction in the length I is
*dx
[^ / 4Q \»x»<
JLi'^ULd'y d'
_ 7 / 4Q Y
If Qj is zero, / is equal to L, and
» {L«*^'(L0*+*}
di«
_7_/4Q\»J_
n + l\rd^J di»'
The result is simplified by taking for dh the value
4v^dx
dh=
od asBTiming C constant.
Then
C«d
3T«C«d»
M,
r
FLOW THROUGH PIPES 159
IjbA Y be the ^eloci^ of the water in the pipe.
Then, nnoe there is oontinaitT of flow, v the velooity .with which the water
V.D"
BMB the noszle ie ^ .
The bead lost by friotion in the pipe is
2g.D~ 2gD^ '
r»
i Tbm kinetie energy of the jet per lb. of flow as it leaves the nozzle is ~ .
»«*" ^^=25 V^ V ) W'
fan vfaieh by transposing and taking the square root,
/ 2gT>^h \4
^[D^hm) ^'>
The weight of water which flows per second szjtP.v.w where 10 = the weight of
.(3).
% oobie foot of water.
T h erefore, the kinetie energy of the jet, is
t
Thia is a maximnm when j^=0>
neiefore
irir / 2gWi \*
~ d (I>»+4/W*)* (2^/iD»)* ll^^ (2pftD»)* (16/W») (D» + 4/W*)*=0...(4),
vlikh D» + 4/W*=12/W*,
D»=8/W\
t ^=^/87! (^)
If the nozzle id not circular but has an area a, then since in the circular nozzle
pf tiie same area
d'=a,
! ^y anbstituting the value of D' from (5) in (1) it is at once seen that, for
iMBmam kinetic energy, the head lost in friction is
[ f " •'■
I UnHem 7. Taking the same data as in problem 6, to find the area of the
^%tm^ thai the momentum of the issuing jet is a maximum.
I Dm momentum of the quantity of water Q which flows per second, as it leaves
lMi«Uif !_iE lbs. feet. The momentum M is, therefore,
9
9 4
lor v* from equation (1), problem 6,
160
HYDRAULICS
and
Differentiating, and equating to zero,
D»4/W*=0,
D»
4/r
If the nozzle has an area a.
d=
D» = 5/^»,
and
.=•392^:
Ti'
Sabstitating for D^ in eqoation (1) it is seen that when the momentom ii a
maximum half the head h is lost in friction.
Problem 6 has an important application, in determining the ratio of the mm
of the supply pipe to the orifice supplying water to a Pelton Wheel, while problem 7
gives the ratio, in order that the pressure exerted by the jet on a fixed plaa*
perpendicular to the jet should be a maximum.
Problem 8. Lom of head due to friction in a pipe, the diameter of wkieh vmUi
uniformly. Let the pipe be of length I and its diameter vary anifonnly from 4^
to d,.
Suppose the sides of the pipe produced until they meet in P, Fig. 104.
8. = ^.«. «=^
Then
r = ^ and S=
..av
8 + i do dodi
The diameter of the pipe at any distance x from the small end ia
~~ s •
The loss of head in a small element of length dx is ^^ , v being the veloeity
when the diameter is d.
Fig. 104.
If Q is the flow in cubic ft per second
t; = ^ = i Q
The total loss of head % in a length I is
^ 64Q». dx
ir»C»d»
64.Q«8»dj?
oir«C«di»(S+j;)»
16Q«.S»
(s* (S+OV'
Substituting the value of S from equation (1) the loss of head doe to
can be determined.
Problem 9. Pipe line comitting of a number ofpipei ojf different dtOMetefi. tk
practice only short conical pipes are used, as for mstance in the limbs of a VaBtn
meter. 
If it is desirable to diminish the diameter of a long pipe line, instead of ndq
a pipe the diameter of which varies uniformly with the length, the line is mads ^
of a number of parallel pipes of different diameters and lengths.
I
FLOW THBOUQH PIPES
161
tiel L« l^* t^ .«« he tiie leogth* and d^, d^^d^.,, the diiuneters retpeotiv^elj, of
Ibi w^timmi df ibid mpe,
1!be loial loss of B«a^ due lo bielioti^ if G b^ aaaiimdd oonatanti is
(?\d,^ d,^ d, '}
Tlie iTIiitifrtfr 4 of the pip«, which, for ihe mme total length , would ghe the
iMne diTJiTgn for the i&me lou of head due (o friotiou, can be found from the
Tlic leDftb Js
for the
oX ft pp^T c^f 4^ostant diameter D, which wiU give the lame
ae loe« of head by friction, is
■'"ii
+ — ^ + —
^PrtMam 10. Pipe aetinff at a tiptore. It U iome times i:ieoesBAry to t^ke a
pipe tiae otst aome obetraclion^ snch ae a hill, which neoe&Eitatea the pipe rising »
ttol onl^ »tiOTe the hydraolic gradient as in Fig. 87, but even above the origlEial
Ifffivl o(f tiM water in the reservoir from which the supply [fl derived »
Ii«t It be BQippoaed* as in Fig. 105, that water ie to be delivered from the resefvoir
3 to lb» fieaervoir C through the pipe BAG, which at the poiat A rificfi h^ feet above
Ihe iev^ af the lorlace of the water in the npper reeer voir.
Fig. 105.
X4»t atm iifftfenee is level of the s^rfaoes of the water in the xeservoiFB
I«et *. b« the prea^ni^ bead equivBlent to the atmospbcrio prcaatire*
To ttlan the flow in the pipe, it will he ncoeasary lo fill it by a pump or other
\j^ tl be a^eumed that the flow ia allowed to take place and is regulated ao that
^ i» ixyfitiiioo^ and the velocity i? is aa large as p<»seible.
aei^ecting the velocity heatl and reBistaoeee othef than that due to fVictionf
(1).
Xft asul il beilif the length and diameter of the pipe reepoctively^
Tkm h7draalie gradient is practleaUy the straight line DE.
Tb^cMVtieally II AF i^ made greater than k^^ which is about 34 feet^ the pfefeure
■t A \mtif^mf'^ nr^^ative and the flow will oeaa^«
Fcsrel>' nnot l»e made much greater than 25 feet.
X«jl«'' '''»*Tn vettKit^ poa*ibU in the ruinfj limb AB, to that tfu preuure
<^^§d mi A t^ini'ljiiit b^ UFO.
Lei e^m ^ ^^ telomty. Lei the datum level be the snr&ce of the water in C^
£^ B.
II
162
HYDRAUUC3
Then
fiai
Therefore
h^^hut^n —
20. d
+ H + ^.
H = ftB + »B + Ai*
.s/'
2g{h„.h,).d
1/«AB
.(»).
If the pre&sare head is not to be lees thaa 10 f«et of w^ter^
If i^m la loBB Ihan i>i the dischafge of tha siphon will be determined lij i
Umititig velociij, hhi] it will be nuce^arv to throttle tlie (jipe &t C by meiuis of I
valve, Ao tkB to keop the limb AC full and to keep the *' aiphoa *' from bf^mg bfbk«D. 1
In designing Htidh a siphon it lA, therefore, ned&ss&iy to determine whether ihi
flow through the pipe as a whole nnder a bead h^ is greater, or le^a Ibftn^ the flow
in the rising limb under a head h^ hi.
If AB is flhort, or A^ io imall that v^ is greater than t>, the head absorbed bf 
friction in A0 will be
2pd ^
If the end Oof the pipe U open to the atmosphere instead of heing ootmeotad I
a reserrotr, the total head available will be h^ tustead of h^.
111. Velocity of flow in pipes.
The mean velocity of flow in pipes is generally about 3 f©
per second, but in pipes supplying water to hydraulic machme
and in short pipes, it may be a^^ high as 10 feet per second.
If the velocity is high, the loss of head due to friction in lo
pipes becomes excessive, and the risk of broken pipes and valve
through attempts to rapidly cheek the flow, by the sudden clodi
of valves, or other causes, is considerahly increased.
On the other hand, if the velocity is too small, unless the wnti
is very free from suspended matter, sediment* tends to collt?ct i
the lower parts of the pipe, and farther, at low velocities it
probable that fresh water sponges and polyzoa will make thdd
abode on the surface of the pipe, and thus diminish it% \
capacity.
112, Transmission of power along pipes by hydraoHa
pressure.
Power can be transmitted hydraulically through a constderabll
distance, >vith very great efficiency, as at high pressures the
centage loss due to friction is small*
Let water be delivered into a pipe of diameter d feet under i
head of H feet, or pressure of p lbs, per aq, foot, for which
n
equivalent head is H =  feet.
* An interesting example of this is quoted on p.
Vol. XUT.
82 Trnw, JflUS^ai
FLOW THBOUOH PIPES 163
Let the velocity of flow be v feet per second, and the length of
the pipe Li feet.
The head lost due to friction is
^ 2g.d ^^^'
and the energy per ix)und available at the end of the pipe is,
therefore,
w 2gd
The efficiency is
B.h_. h
H "^ H
2gdB.'
The fraction of the giyen energy lost is
h
For a given pipe the eflSciency increases as the velocity
diiniiiishes.
If / and L are supposed to remain constant, the efficiency is
constant if jj^ is constant, and since v is generally fixed from
other conditions it may be supposed constant, and the efficiency
then increases as the product dR increases.
If W is the weight of water per second passing through the
jape^ the work put into the pipe is W . H foot lbs. per second, the
available work per second at the end of the pipe is W (H  fe), and
the horseiX)wer transmitted is
XiX
 560  550 ^^ '"'•
Since
W = 62%^d^v,
the horsepower
4 550 \^ 2gd J
= 089»d'H(lm).
From (1)
"*° 2gd '
therefore,
..1 VdmB.
and the horsepower
I
= 0357 ^^d*H*(lm).
11—2
164 HYDRAULICS
If p is the pressure per sq. incli
TT_ pl44
^^ 624 '
and the horsepower =1*24 a/ jj d*p* (1  m).
From this equation if m is given and L is known the diameter d
to transmit a given horsepower can be found, and if c2 is known the
longest length L that the loss shall not be greater than the given
fraction m can be found.
The cost of the pipe line before laying is proportional to its
weight, and the cost of laying approximately proportional to its
diameter.
If t is the thickness of the pipe in inches the weight per foot
length is 3T5irdt lbs., approximately.
Assuming the thickness of the pipe to be proportional to the
pressure, i.e. to the head H,
t = Jcp=JcK,
and the weight per foot may therefore be written
w = kid . H.
The initial cost of the pipe per foot will then be
C=fefeidH = K.d.H,
and since the cost of lajdng is approximately proportional to d,
the total cost per foot is
P = K.d.H + Kid.
And since the horsepower transmitted is
HP = 357 ^^ d*H» (1 m),
for a given horsepower and efficiency, the initial cost per horse
power including laying will be a minimum when
0357 ,y^ d*H* (1  m)
Jk
K.d.H + Kid
is a maximum.
In large works, docks, and goods yards, the hydraulic trans
mission of power to cranes, capstans, riveters and other machines
is largely used.
A common pressure at which water is supplied from the pumps
is 700 to 750 lbs. per sq. inch, but for special purposes, it is
sometimes as high as 3000 lbs. per sq. inch. These high preesnres
are, however, frequently obtained by using an intensifier (Ch, XI)
to raise the ordinary pressure of 700 lbs. to the pressure required.
FLOW THROUGH PIPES
165
The demand for hydraulic power for the working of liftB, etc.
1 led to the laying down of a network of mains in several of the
cities of Grieat Britain. In London a mean velocity of 4 feet
second is allowed in the mains and the presaur^ is 750 Ihs*
sq^. inch, In later installationSj pressures of 1100 lbs. per
inch are used.
113. The limiting diameter of castiron pipes.
The diameter d for a castiron pipe cannot be made very large
[ if the presscre is high.
If f> is the safe internal pressure per sq. inch, and s the safe
i^3^ees per sq. inch of the metal^ and r^ and r^ the internal and
external radii of the pipe^
p=
r,* + ri*
For a pressnre p = 1000 lbs. per aq. inch, and a stress a of
lbs. per sq. inch, Tt is 5' 65 inches when n is 4 inches, or the
ipe req aires to be 1*65 inches thick.
If, therefore, the internal diameter is greater than 8 inches, the
jpe becomefi Teiy thick indeed.
The largest castiron pipe used for this pressure is between
and ST internal diameter.
tTstng a nsanmam velocity of 5 feet per second, and a pipe
inches diameter, the ma^mum horsepower, neglecting friction,
it can be transmitted at 1000 lbs. per sq, inch by one pipe is
„^ 4418x1000x5
^*^^'^ 550^
400,
The following example shows that, if the pipe is 13,300 feet
15 per c^nt. of the power is lost and the maximum power
can be transmitted with this length of pipe is, therefore,
320 torsepower.
Steel mains are much more suitable for high pressures, as the
rkin^ stress may be as high as 7 tons per sq, inch. The greater
ity of the metal enables them to resist shock more readily
castiron pipes and slightly higher velocities can be used,
A pipe 15 inches diameter and  inch thick in which the
is 1000 lbs. per sq. inch, and the velocity 5 ft, per second,
to transmit 1600 horsepower,
E^x^tmpU. Power !■ InutBmitted ulong ^ c&st^iron m&in 7^ mcliea diameter at
A pwsmmrt of 1000 Iba. p«r ^, meli. The velooity of the wmUa u 5 feel per aeoond.
Fioil ib» tnAiiniiim 4istMio© tbe p&w^r can be tranBmitted so tldat the effleienQy
* Swing's Strength of Materktli,
4
166
therefore
Then
from which
HTDRAULICS
A = 0*l5 3ca3O0
4 X 0^0104 x_26^
2^ X 0626 •
345K&4x0^e25
S4fi' =
L =
00104 X 100
:^ 13,900 f^t
rest, the intensity of preasare
13^
£
t^
Fig, 106,
Pig^m
114. Pre&aures on pipe her
If a bent pipe contain a fluic
being the same in all directiot
the resultant force tending to mr
the pipe in any direction will
the pressure pernnitarea moltipl
by the projected area of the
on a plane perpendicular to
direction.
If one end of a rightangled
elbow, as in Fig. 106, be bolted to
a pipe full of water at a pressure p
pounds per sq. inch by gauge, and on the other end of the elboir
is bolted a flat cover, the tension in the bolts at A will be tie
same as in the bolts at B. The presi^ure on the cover B is clearly
'7854pcP, d being the diameter of the pipe in inches. If the elbow
be projected on to a vertical plane the projection of ACB is dmfc^
the projection of DEF is ahcfe. The resultant pressure on the
elbow in the direction of the arrow is, therefore, p . ahcd = '7S54jxf*.
If the cover H is removed, and water flows through the pipe
with a velocity v feet per second, the horizontal momentum ol the
water is destroyed and there is an additional force in the direction
of the arrow equal to '78'54irrfV,
When flow is taking place the vertical force tending to lift tbe
elbow or to shear the bolts at A is
If the elbow is less than a right
angle, as in Fig. 108, the total
tension in the bolts at A is
T = p (daehgc  aefgc)
+ 7854ii;dVcos^,
and since the area aehgcb is common to the two projected areas,
T = '1854<P(p'pcoQe'\wv'co&e).
PLOW THROUGH PIPES
167
[^Consider now a pipe bent m shown in Fig. 109, the limbs AA
and the water being supposed at rest»
direction AA ia
FF being fmrallel,
acting
total f Mrce
the i
P^p {degkea  asfgcb ^ dcgKea  aef*ge¥\
dh dearly is equal to 0.
V m^tead of the fluid being at rest it has a uniform
' ht* pressure must remain constantj and since there is no
of velocity tljere is no change of momentum^ and the re
tant pK?asttre in the direction paraUel to AA is still zero.
There is however a couple acting upon the bend tending to
it in a clockwise direction.
Let p and q be the centres of gravity of the two areas daekgc
1 a^fgch respectively, and m and « the centres of gravity of
dWeiig'c and aef*gch\
Through these points there are parallel forces acting BM shown
the arrows, and the couple iB
M — K  m7i  P » pq.
Tli^ ' • P*pg is also equal to the pressure on the semicircle
Ic II'. i by the distance between the centres of gravity of
k and efg^ and the couple P' . 7nn is equal to the pmssure on ad*c*
iltiplic^d by the distance between the centres of gra^nty of a'd't^
Ufg.
Tlu*n the resiUtant couple is the pressure on the semicircle efg
Inpbed by the distance between the centres of gravity of efg
u'r9.
of FF atid AA are on the same straight line the
as the force J becomes aero,
% can also be shown, by similar reasoningj that, as long as the
it4?r« at F and A are equal, the velocities at these aections
therefore equal, and the two ends A and F are in the same
ight line, the force and the couple are both zero, whatever the
of the pipe. If, therefore, as stated by Mr Fronde, " the
J
168 HYDRAULICS
two ends of a tortuous pipe are in the same straight line, there is
no tendency for the pipe to move."
115. Pressure on a plate in a pipe filled with flowing water.
The pressure on a plate in a pipe filled with fiowing water, with
its plane perpendicular to the direction of fiow, on certain assump
tions, can be determined.
Let PQ, Fig. 110, be a thin plate of area a and let the sectional
area of the pipe be A.
The stream as it passes the edge of
the plate will be contracted, and the
section of the stream on a plane gd will
be c(Aa), c being some coeflScient of
contraction.
It has been shown on page 52 that
for a sharpedged orifice the coeflBcient Fig. no.
of contraction is about 0*625, and when
part of the orifice is fitted with sides so that the contraction is
incomplete and the stream lines are in part directed perpendi
cular to the orifice, the coeflBcient of contraction is larger.
If a coeflBcient in this case of 0*66 is assumed, it will probably
be not far from the truth.
Let Vi be the velocity through the section gd and V the mean
velocity in the pipe.
The loss of head due to sudden enlargement from gd to ef is
2g '
Let the pressures at the sections db, gd, ef be p, p, and p, pounds
per square foot respectively.
Bemouilli's equations for the three sections are then,
w 2g w 2g ^^^,
and £..l^E,,^,(I^* (2).
Adding (1) and (2)
(V.V)'
The whole pressure on the plate in the direction of motion is then
(V.Y)'
\w wJ
F=(ppi).a = w.a. 2
FLOW THBOUQH PIPES
P ^ 4^a 5 nearly.
P =
116, Pre&aitre om a cylinder,
1£ inBtBfid of a thiii plate a cylinder be placed in the pif
Iwitli U« aads coincident with the axis of the pipe, Pig< 111, there
fare two anlargements of the section of the water.
As the stream passes the upstream edge of the cylinder, it
contracts to the section at ccl, and then enlarges to the section
ef. It again eniargi?s at the downstream end of the cylindc
I ftom the section a/ to the section gh.
«' C .^ J9
Fig. 111
l^t ^i, 1%, v»^ v^ be the velocities at ai, cd, ef and
lepectively, €4 and Vt being eqnah
Betw^eti cd and ^/ there is a loss of head
between e/and gh there is a loss of
2g •
The BemouiUi's eqnations for the sections are
w 2g w 2g'^' ' "
gk re
w 2g w 2g w 2g ^ '
w 2g w 2g 2g '
Miing (2> and (3),
«? 2g 2g '
(1),
.(2),
.(3).
ii^Mik
170 HTDRAtTLlCS
If the coefficient of oontractioti at cd ise, the area at cd
c
A
Then v^= — > v' " v ^nd i'>=^^^.
Therefore
atid the pressure on the cylinder is
EXAMPLES,
(1) A new caBt'iroB pipe is 2000 ft. loDg and 3 ins. diameter. Itisbo
discharge 50 c. ft, of water per miniite. Find the lofis of bead in tndma
and the virtual elope.
(2) What is the head loet per mile in a pipe 2 ft. diameter* diflchitfiiiC
6,000,000 gallons in 24 hours ? /= 007.
(3) A pipe tig to r^upply 40,000 gallons in 24 hours. Head of witcf
above point of discharge = 86 ft. Length of pipe=2J miles* Find iti
diameter. Take C from Table XII.
(4) A pipe is 12 ms. in diameter and 3 rnile^ in length. It oohmscH
two FeaerroirB with a difference of level of 20 ft. Find the disehazga pflf
minute in a ft. Use Darcy's coefficient for corroded pipes.
{5) A water main has a Tirtual slope of 1 in 900 and di£chafge636C*(L
per second. Find the diameter of the main. Coefficient / is 0*007.
(0) A pipe 12 inB. diameter is suddenly enlarged to IB ins.» and tlian to
24 ins. diameter. Each section of pipe is 100 foet long* Find the h&k d
head in friction in each length, and the loss due to shock at eaeli ealMge
ment. The discharge is 10 c. ft. per second^ and the coefficient of fricti»
/=^'O106, Draw, to scale, tlie hydraulic gradient of the pipe.
(7) Find an oxpresaion for the relative discharge of a square^ aad *
circular pipe of the same section and slope.
(8) A pipe is 6 ins. diameter, and is laid for a quarter mile at a ^ops
of 1 in 60: for another quarter mile at a slope of 1 in 100; and for ftUM
quarter mile is level. The level of the water is 20 ft. above the inlet efli
and 9 ft. above the outlet end. Find the diacharge (neglecting all
except skin friction) and draw the hydrauMc gradient. Mark Uie
in the pipe at each quarter mile.
(9) A pipe 2000 ft. long discharges Q c. ft. per second. Find bybo*
much the discharge would be increased if to the last 1000 ft* a second ppl
of the same siase were laid alongside the first and the water allowed to
equally well along either pipe.
FLOW THBOUOH PIPES
171
.Ibei
I)
►) A naacrffoir , the level of wkicl) is 50 ft, above datum, diBcharges
neeood rmmvm 80 ft. ^bove datum, through a 12 in. pipe, 5000 ft.
find the disc^hiurge. AIbd, taking the levelB of tlie pipe at the
; Rud at each successive 1000 ft., to be 40, 25, 12, 12, 10, 15,
d»liiin, wriie down Uie pressure at each of tbeie poiats, and
posltieii of the line of hydraulic gradient,
1% m reqained to draw off the water of a reservoir Uirough a
lOed harixoatAllj. Diameter of pipe 6 ins. Length 40 ft* Ef
lead 20 ft Find the disch^^e per geoond.
Given the data of Ex. 11 find the diBcharge^ taking into atccount
oi h#ad if the pipe ifi not bell mouthed at either end*
A pipe 4 ins. diameter and 100 ft. long discharges ^ c. ft* per
Find the head expended in giving velocity of entry^ in overcoming
reatstasce, and in Mctdon,
14) BeqQired the diameter of a pipe having a fall of 10 ft. per mUe,
Mm ol delivering water at a velocity of 3 ft per second when dirty.
TaMsg the coefficient / as 0*01 (l + f^^)i ^^ ^ow much water
be diaehatged through a 12^inch pipe a mile long, connecting two
i with a differeoce ol level of 20 feet,
Watetr flows through a 12 inch pipe liaving a virtual slope of 8 feet
I feel at a velocity of 8 feet per second.
I tlie Action per sq. ft. of surface of pipe iu lbs,
iihc Talne of / in the ordinary formula for flow in pipes.
Find the relative discharge of a 6 inch main witli a slope of
40Ov and a 4 Inch main with a slope of 1 in 50.
A 6'inch main 7 mUes in length vrith a virtual slope of 1 in 100
hj 4 miles of Sinch main, and S miles of 4iiich main. Wowld
► be altered, and, if so* by how much ?
[ (i^) Find the velocity of flow in a water main 10 miles longi con
[ Iwo reaervoirs with a dificTence of level of 200 feet. Diameter of
> ifldiea. Goef&cient /=U^009.
' (10) Ftod the discharge, if the pipe of the last question is replaced for
I flxnt 5 BuLas by a pipe 20 inches diameter and the remainder by a pipe
idkmeter.
Ctl) Calculate the loss of head per mile in a 10 inch pipe (area of cross
1 0^54 fl(. ftj when the discliarge is 2^ c, ft per second.
A pipe canaiata 6f ^ a mile of 10 inch, and 4 a mile of 5 inch pipe,
I oomr^ya $ e* It per second. State from the answer to the previous
k the losB of head in each section and sketch a hydraulic gradient*
» head al the ontlet is 5 ft.
What is the head lost in friction in a pipe 8 feet diameter
6,000,000 galioDS in 12 hours?
' {Uj A pipe 2000 feet long and 8 inches diameter is to discharge 65 o. ft.
f ndnnte* What mufit be the head of water f
ITi
1(19)
^^whargel
J^^b
172
HTDRAULICS
(25) A pipe 6 mcbes dlanaeter, 50 feet long, is connected to the bcitkrnti
of a tank 50 feet long by 40 feet wide. The original head over the openl
end of the pipe is 15 feet. Find the time of emptying the tank^ assmning]
the entrance to the pipe in sharp edged.
If /i^the head ovet the exit of the pipe at any moment,
t?* St?" 4/t^5Q^
°2g^ %^2£rx05'
from 'which, v
In time dt^ the discharge is
* 1*5 +400/'
144 l'5 + 400/
In time ct the Borface falls an amoimt dh.
Therefore 0_m^^^^^^^^h
15 + 400/ ^4
Integratmg,
^_ 2QQ0 (15 + 400/) 2 ^j^^ 79QQ0(l'5 + 4OO/) ^^
0196^% V2g
(26) The internal diameter of the tubea of a condenser is 5*654 Inc
The tubes are 7 feet long and the n amber of tubes is 400* The number <
gallons per minute flowing through the condenaer is 400. Find the lo» ot
head due to friction as the water flowB through the tubes* /^ 0*006.
(27) Aasundng fluid friction to vary as the square of the velodty, J
an expression for the work done in rotating a disc of diameter d at i
angular velocity a in water.
(26) What horse power can be convej'ed through a 6 in, main if ihs I
working presBure of the water supplied from the hydraulic power station tt
700 lbs, per sq, in;? Assume that the velocity of the water Is limited l9
8 ft, per second,
(29) Ten horaepower is to be transmitted by hydranlic presanie i
distance of a mile. Find the diameter of pipe and pfdssme required for li
efficiency of ^ when the velocity is 5 ft. per sec
The frictional loss is given by equation
2g d
(BO) Find the inclination necessary to produce a velocity of 4 leei p* I
second in a steel water main 31 inches diameter, when roniiing full ifl4
digchargiug with free outlet, using the formula
■ ■0005 tJ^^
*" di» '
(Bl) The following values of the slope i and the velocity v w«re
determined from an experiment on flow in a pipe '1296 ft diam*
i 00022 '00182 '00650 ^02889 '04348 12815 '22409
V 206 *606 1*252 2*585 8'693 6310 8'521
di^ii
FLOW THROOGH PIPES
173
Delermme k and n m the formola
i—ki)'^.
AIbo determine T&Itiee of O for this pipe for velocitieB of *5, 1, 3^ 5 and
1 feel per sec
(S2^ The total length of the Coolgsrdie steel aqueduct is SOTf miles
ftttd the di&nieter 30 inches. The discharge per daj Z£iay be 5f600000
The water is lifted a total height of 1499 feet,
(a) the head lo^t dtie to friction,
{b} the total work done per minote In raising the water.
A pipe 2 feet diameter and 500 feet long without bende furnishee
wmlier to a tnrhine* The tnrbine works under a head of 25 feet and uses
ID e. fL ol water per second. What )ercentage of work of the fall is lost
^A MctioD in the pipe ?
Find
im
Coiifficiaitt
/'007
{'""m)^
C84> Sight thousand gallons an hour ha^e to be disoharged through
^^ch of mx nozzles, and the jet haM to reach a height of 80 ft.
If the water supply i& 1^ tuilee away, at what elevation above the
ooj^lea wcpnld you place the required reseryoir, and what would you
make the diameter of the supply main ?
QtTB the dimensions of the refiervoir you would provide to keep a
OODStant supply for six hours. Loud. Uu. 1903.
(85 ^ The pipes laid to connect tlie Vymwy dam with Liverpool are
4S inebea diameter. How much water will snch a pipe snpply in gallons
^m diiBBi if the slope of the pipe is 4^ feet per mile 9
At 0Q« point on the line of pipes the gradient m 6 feet per mile« and the
fipe diam^er ia reduced to 89 inches; is thJB a reasonable reduction in the
^BieBsioii of the croas section ? Loud. Un. 1905.
"*'ater under a head of 60 feet is diachajged through a pipe
I meter and 150 fe^t long, and then through a nozzle the area of
 tenth tlie area of the pipe. Neglecting all losaes except friction,
locity witli which tbe water leaves the nozzle*
^ Two rectangular tanks each 50 feet long and 50 feet broad are
ftmnected by a horiiEuntal pipe 4 inches diameter, 1000 feet long. The
Wd over the centre of tlie pipe at one tank is 12 feet, and over the other
i ke% when flow commences.
Determine tlie time taken for the water in the two tanks to come to the
^am level. Aianme the coefficient € to be constant and equal to 90.
(885 Two reaervoirs are cotinectcd by a pipe 1 mile long and 10''
Kuieter; Ih^e difference in the water surface levels being 25 ft.
B^terminic the flow through the pipe in gallons per hour and find by
modi tlia discharge would be increased if for the last 2000 ft. a second
ol lOT diameter is laid alongside the first. Loud. Un. 1905.
(99^ A pipe 18^ diameter leads from a reservoir, SflO ft. above the ;
od ia continued for a lengtli of 5000 ft. at the datum, the length'
iSjOOO ft For the Last 5000 ft. of its length water is drawn off by
174
HYDRAULICS
fiendoe pipes at the rate of 10 c. it, per nun* per 500 fL tmiformly, FmiM
the pressure at the end of the pipe. Loud. Un. 1906. I
(40) B50 horse power ia to be transmitted by hydraulic preasoxe m
distance of 1^ miles. I
Find the nmnber of 6 ins. diameter pipes and the preadoi^e required fDcl
an efficiency of 92 per cent, /—Ol* Take t* as 3 ft, per see I
(41) Find the loaa of head due to friction in a water maiD L feet Iohh
whicJi receives Q cubic feet per second at the inlet end and dahTMl
P cubic feet to branch mains for each foot of its length* I
What is the form of the hydraulic gradient ? I
(42) A reservoir A aupphes water to two other reservoirs B and CLl
The difference of level between the surfaces of A and B is 75 feet, anil
between A and C 975 feet, A common 8 inch cast iron main supplies foci
the hrst 850 feet to a point T>* A 6'inch main of length 1400 feet m tbeal
earned on in the same straight line to B^ ajid a 5 inch main of lengtlil
630 feet goes to C* The entrance to the 8«iiich main is belltnouthed« and!
losses at pipe exits to the reservoirs and at the junction may be neglaoleU
Find tlie quantity discharged per minnte into the reservoirs B and GJ
Take the coefficient of friction (/) as *01. Lond, 0n. 1907. 1
(43) Describe a method of finding tlie '* loss of head " in a pipe due tol
the hydraulic resistances^ and sta,te how you would proceed to find IbJ
loss as a function of the velocity. I
(44) A pipe* I feet long and D feet in diameter, leads water from ■
tank to a nozzle who^e diameter is d, and whose centre is h feet bekM
the level of water in the tank* The jet impinges on a fixed plaal
surface. Assuming that the loss of head due to hydraulic resistanoe m
given by M
show that the preasure os the Boriace in a maxiiiiniii wheD I
^ m 1
(45) Find the flow through a sewer consisting of a castiron ptpJ
12 inches diameter, and having a fall of 8 feet per mllet when dLschafgufl
full bore, c = 100. 1
(4d) A pipe 9 inches diameter and one mile long alop^ for the Snfl
half mile at 1 in 200 and for the aeoond half mite at 1 in 100. The pr€ii^
head at the higher end is found to be 40 feet of water and at the Iohm
20 feet. I
Find the velocity and flow through the pipe. I
Draw the hjrdrauhc gradient and find the pressure in feet at 500 ktm
and lOOO feet from the higher end. I
(47) A town of 250,000 inhabitants is to be suppUed with water. EmM
the daily supply of 32 gallons per head is to be delivered in 8 hours* I
The service reservoir is two miles from the town^ and a fall of 10 §&m
per mile can be allowed in the pipe. M
What must be the size of the pipe? C = 90, I
PLOW THROtJOH PIPES
175
(4$) A w&iar pipe is lo be laid in a street 800 jardfl long with houseB
Mbolli Biidea of ibe street of 24 feet frontage. The average number ol
iiilml)itsiits ai aacb honse is 6, and the aTemge eoneumption of water far
eacli peCBcm ia 80 gallon b in 8 brs. On the assumption that the pipe is laid
In loar equal lengths of 200 yards and has a uniform dlope of j^, and that
Hie wbo)e of tha water flows through the first lengthy threefourtlis through
Ibe Mooml* one half through the third and ona quarter through the fourth,
lad Uwl Ibe Talne of G is 90 lor the whole pipe^ lind the diameters of the
ter parte of the pipe.
(49) A pipe 3 miles long has a nniform slope of 20 feet per mile, and is
19 inclMM diameter for the first mile, 30 inches for the second and 21
faicbes for the third. The pressure heads at the liigher and lower ends of
fbe pipe are 100 feet and 40 feet respectively. Find the discharge through
th«^ [Hprn and determine the pressure heads at the comman cement of the
3ij inchem diameter pipe, and abo of the 21 inches diameter pipe,
(SOi The difference of lerel of two i^eserroirs ten miJes apart in 60 feet,
A pq^ Is Tnquired to connect tliem and to convey 45,000 gaUona of water
pm boor fr«>m the higher to the lower reservoir.
Find the necefisary diameter of the pipe, and sketch the hydraulic
gtafdienU aMsanusg/^OOl.
Tlw middle part of the pii>e is 120 feet below the surface of the upper
reaerrotr. Osiculate the pressure head in the pipe at a point midway
betw^iSl tiie two reservoirs.
f51) Sonne hydraulic machines are served with water under pressure
hf a pipe 1000 feet long, the pressure at the machines being 600 lbs. per
ai^oara indi. The horBe^power developed by the machine is 300 and ihe
fricdoa iionepowes in the pipes 120. Find the necessary diameter of the
ptp&^ taking the loss of head in feet as 0*03 li^^ and *i3 lb. per square
laeli aa ac^aiTiilefit lo 1 foot head. Also determine the pressure at which
Aa w«t6r ia deliirered by the pump.
Wliat ts the ma^mum horse ^power at which it would be possible to
work the machines, the pump pressure remaining the same ? Lond. Un.
1906*
(53} Discnsa Beynolds' work on the critical velocity and on a general
law oi resistance, describing tJie experiments appaiatus, and showing the
ooB&edioii with the experiments of Poisemlle and D'Arcy. Load. Un,
(58) In a condenser, the water entars through a pipe (section A) at the
ol the lower water head, passes through the lower neat of tubes,
thzerii^ the uppernestof tubes into the upper water head (section B).
I '9m teetional areas at sections A and B are 0196 and 0^95 sq. ft. respec
#i«ly; the total sectional area ol flow of the tubes forming the lower nest
Ji C^S14 81]* ft^ and of the npper nest 75 sq. ft., the number of tnbea being
feoipecltrely 35a and 826. The length of all the tubes is 6 feet 2 incheSp
IVIifiii the Tolume of the circulating water was 1^21 c, ft. per sec., the
Ered dtfferance ol preasure head (by gauges) at A and B was 6 5 feet.
Ihe total actual head necessary to overcome frictional resistancei and
176
BYDaATJLICa
the coelScieiit of hydraulic reciBtanoe referred to A. If i^3^ oodEdest d
Motion (4/) for the tubed is t&ken to be *015t find the coefficient of bj^ilttiiifi .
reflifitanee for the tubes alone, and compare with the actual l niiiiTipcit ]
Lend. Un. 1906. iCr = head lost dimded bp veL he^ at A.)
(54) An open stream, which 19 discharging 20 c. ft. of water M^
BeGood ifl paeied under a road by bl siphon of smooth stoneware pipev H
eectioii of the siphon being cylindrical, and 2 feet in diameter. When toVi
stream enters ttiis siphon, the siphon descends vertioally 12 feet, H
then has a horizontal length of 100 feet« and again rises 12 feet^ If all Hm I
bends are sharp rightangled bends, what is the total Iohb of head in ibi I
tunnel due to the bends and to the friction ? G = 117. Loud. Un. 190?. I
(50) It has been shown on page 159 that when the kinetic ene^y of jfl
jet issuing from a nozzle on a long pipe line is a maximum^ H
^ 8/L' ^
Hence find the minimum diameter of a pipe that will supply a ¥^im I
Wheel of 70 per cent, efficiency and 500 brake horsepower, the airaikUJ
head being 600 feet and tlie length of pipe 3 miles. ^M
(56) A fire engine supphes water at a pressure of 40 lbs, per sqo^H
inch by gauge, and at a velocity of 6 feet per second into a pipe 3 indi^l
diameter. The pipe is led a distance of 100 feet to a nozzle 25 feet aba^
the pump. If the coeJScient / (of friction) in the pipe be '01, and the ad^l
lift of the jet is  of that due to the velocity of effiux, find the actnal tiei^
to which the jet will rise, and the diameter of the nozzle to eati^ ^
conditions of the problem. ^H
(57) Obtain expressions (a) for the head lost by friction (e^kpre^ej^l
feet of gas) in a main of given diameter, when the main is horizontaL i^
when the variations of pressure are not great enough to cause any importtiH
change of volume, and (b) for the discharge in cubic feet per saoond. ^
Apply year resnlte to the following example:— I
The main is 16 Inches diameter, the length of the main is 300 jrwfc j
the density of the gas is 0'56 (that of air^ l),and the difference of presMiii]
at the tw^o ends of the pipe is  inch of water j find:^ I
(a) The head lost in feet of gas, "
(h) The discharge of gas per hour in cubic feet.
Weight of 1 cubic foot of air=0'08 lb,; weight of 1 cubic foot of waier
J^ba.; coefficient / (of friction) for the gas against the waUs of the pipe
J*^"^ Loud, Un. 1905. J
on f t '^^ ^^^ ' substitute for w the weight of cubic foot of gas,) I
Find thetee reservoirs A, B and C are connected by a pipe le«di4g
Draw the a junction box P situated 450' above datum,
and 1000 feet ^ of the pipes are respectively 10^000' ♦ 5000' and 6000" and ilm
(47) A tow*^^ water sorface in A, B and C are SOff, 600' and 200' &bot9
^j^^^^^^^magnitude and indicate the direction of mean Telocity itt ^
permilecanbeaD€^^=>10*>*^'*^*' ^^ pip^s being aU the same
WliatmustbethP"'^^'
FLOW THROUGH PIPES 177
(00) A pipe 8* fiT diameter bends through 45 degrees on a radius of
7 IS leek Deiennine the dieplaGing foroe in the direction of the radial line
\ MinnHnc the aii^ between the two limbs of the pipe, when the head of
P wiler in the pipe is 260 feet
^ Show alao that, if a nniformly distributed pressure be applied in the
/ plftBe of the oentre lines of the pipe, normally to the pipe on its outer
'^ ^w**^, and of intensity
f _ 49M« ,^
par unit length, the bend is in equilibrium*
B»Tadins of bend in feet
d— diameter of pipe.
h»head of water in the pipe.
12
CHAPTER VL
FLOW IK OPEN CHANNELS,
117, Variety of the forms of channels.
The study of the flow of water in open cliannelB is much more
complicated than in the case of closed pipeSj because of the
infinite variety of the forms of the channels and of tlie dii&reit
degrees of ro ugliness of the wetted BurfaceSj varying^ as they do,
from channels lined with smooth boarda or cementj to the irregnkr ,
beds of rivers and the rough, pebble or rock strewn, moiEidiift
stream.
Attempts have been made to find formulae wliich are applicalJft I
to any one of these very variable conditions, bat as in the case d I
pipes, the logarithmic formulae vary with the roughness of ttel
pipe, so in this case the formulae for smooth regular shaped chanBeU I
cannot with any degree of assurance be applied to the CBlculahon I
of the flow in the irregular natural streams.
118. Steady motion in imifomi channels.
The experimental study of the distribution of velocitiefi
water flowing in open channels reveals the fact that, as in tlil'l
case of pipesj the particles of water at different points in a crowl
section of the stream may have very difFerent velocities, and the]
direction of flow is not always actually in the direction of the flow [
of the stream.
The particles of water have a sinuous motion, and at any jkxuh
it is probable that the condition of flow is continually changinfJ
In a channel of uniform section and slope, and in which the totJ
flow remains constant for an appreciable time, since the lyiii
quantity of water passes each section, the mean velocity i? in
direction of the stream is constant, and is the same for all
sections, and is simply equal to the discharge divided by the ami
of the cross section. This mean velocity is purely an artificial]
quantity, and does not represe^nt, either in direction or magnitit<}«^
the velocity of the particles of water as they pass the section.
FLOW IK OPEN CHANNELS
179
?fits with current meters, to determine the distribution
iii diannels, show, however, that &t any point in the
tion, the corapOBent of velocity in a direction parallel to
fefction of flow renmins practically constant. The considers
pf the motion m consequently simplified by assuming that
iter moves in paralh*] fillets or stream lines, the velocities in
•ne different, but the velocity in each stieam line remains
This is the aesutnption that is made in investigating
rational formulae for the velocity of flo^v in channels,
[tt should not be overWked that the actual motion may be
more complicated.
Formula for the flow when the motion is uniform
Otiftiuael of onifonn section and slope.
I this sasumptionj the conditions of flow at similarly situated
i C and D in any two cross sections AA and BB^ Figs. 112
[118^ of a channel of uniform slop© and section are exactly the
I; the velocities are equal, and since C and D are at the same
lice below the free surface the pressures are also equal For
it CD, therefore,
«? 2^ w 2g '
B, HiBce the same is true for any other filament,
for the two sectii
:»us.
C
i""~" — '—^
mm
fi».m.
Fig, 113.
kH be the mean velocity of the stream, i the fall per foot
J ^ the surface of the water, or the slope, al the length
tn AA and BB, w the cross sectional area EF&H of the
fttiit P the wetted perimetor, i.e, the length EF + FG + GH,
1 *e the weight of a cubic foot of water.
Let ^ s HI be called the hydraulic mean depth*
I bet iff be the fall of the surface between AA and BB*
11 dsi. cL
Since
12—1
180
HTDBAUUCS
If Q cubic feet per second fall from AA to BB, the wort dow
upon it by gravity will be :
Then, since ^ £ "^ 2^)
is constant for the two sections, the work done by gravity
be equal to the work done by the frictional and other resistaiios
opposing the motion of the wa*****
As remarked above, all the
so that there is relative motion
and since water is not a perfect
done by gravity is utilised in ova
relative motion. Energy is also
eddy motions, which are neglect
and some resistance is ako offer
surface of the water.
The principal cause of loss
of the sides of the channel^ a;.»
work done by gravity is utilised in overcoming this resistance
Let F . v be the work done per unit area of the sides of tlie
channel, v beiug the mean velocity of flow, F is often called tk
frictional resistance per unit area, but this assumes that tlie relatire
velocity of the water and the sides of the channel is equal to the
mean velocity, which is not correct.
The area of the surface of the channel between AA and BB
isP.dZ.
Then, wi^idlFvPdl^
its have not the same velodtf^
bween consecutive likmeDti,
id some portion of the work
Euing the friction daetotliifi
due to the cross currents or
1 assuming stream line flow,
o the flow by the air on ih
ever^ the frictional reskt&a*
s assumed that the whole A
therefore
or
« ,_ F
. F
w
F is found by experiment to be a function of the velocity ani
also of the hydraulic mean depth, and may be written J
b being a numerical coefficient.
Since for water to is constant  may be replaced by k and
therefore, mi^k ./{v) f(m). ]
The form of /(i?) /(?n) must be determined by experiment
120. Formula of Chezy.
The first attempts to determine the flow of water in channels
FLOW IN OPEN CHANNELS 181
ith precision wrere probably those of Chezy made on an earthen
Jialy at Coax^alet in 1775, from which he concluded that
/(t;) = t;»and/(m) = l,
id therefore m% = kv' (1).
Writing C for ^
v = C vmi,
'hich is known as the Chezy formula^ and has already been given
1 the chapter on pipes.
lai. JPormulae of Prony and E3rtelwem.
Prony adopted the same formula for channels and for pipes^ and
Bsmned that F was a function of v and also of t?', and therefore,
mi = av + W.
By an examination of the experiments of Chezy and those of
)a Buat* made in 1782 on wooden channels, 20 inches wide and
988 than 1 foot deep, and others on the Jard canal and the river
layne, Prony gave to a and h the values
a = 000044,
5 = 000094.
This formula may be written
mx
■(:*')•■■
>r ^ = / J mi.
s/~v
+ b
The coefficient C of the Chezy formula is then, according to Prony,
a function of the velocity v.
If the first term containing v be neglected, the formula is the
same as that of Chezy, or
V = 103 Jmi.
Eytelwein by a reexamination of the same experiments
together with others on the flow in the rivers Rhine t and Weser J,
gave values to a and b of
a = 000024,
6 = 0001114.
Xeglecting the term containing a,
V = 95 y/mi,
• Prifieipei d'hydraulique,
t ExperimeDts by Funk, 180S6.
X Experimenis by Braoings, 179092
182
HTDRAULICB
Afi in the ease of pipes, Prony and Eytelwein mcorrectlf
assumed that the constants a and h were independent of the j
nature of the bed of the channel.
122. Formula of Barcy and Bazin,
After completing his classical experiments on flow in pipes I
M, Darcy commenced a series of experiments upon open channels
— afterwards completed by M. Bazin — to determine, how the ]
frictional resistances varied with the material mth which the
channels were lined and also with the form of the channel.
Experimental channels of semicircular and rectangular secdon I
were construct^ at Dijon, and lined mth different materiak.
Experiments were also made upon the flow in small earthen
channels (branches of the Burgoyne canal), earthen channels hned
with stonesj and similar channels the beds of which were cove^^
with mud and aquatic herbs* The results of these experimental
published in 1858 in the monumental work, Rechsi^he^ Hydrtnc*
liquss^ very clearly demonstrated the inaccuracy of the assutop
tions of the old writers, that the frictional resistances were
independent of the nature of the wetted surface.
From the results of these experiments M. Bazin proposed for J
the coefficient k^ section 120, the form used by Darcy for pipes^
i'^i).
a and bein^ coefficients both of which depend upon the nat
of the lining of the channel.
Thus, mi = {ar Av^
or
V
1
y
a +
mt
The coefficient C in the Chezy formula is thus made to
with the hydraulic mean depth m^ as well as with the roughne
of the surface*
It is convenient to write the coefficient k as
\ am/
Taking the unit m 1 foot, Bazin^s values for a and ft mi
values of k are shown in Table XVIII.
It will be seen that the influence of the second term inc
very considerably with the roughness of the surface,
123. Ganguillet and Kutter, from an elimination of Baim*j
FLOW IN OPEN CHANNEI^
183
periments, together with some of their own, found that the
efficient C in the Chezy formula could be written in the form
I which a is a constant for all channels, and b is a coefficient of
TABLE XVin.
Showing the values of a, fiy and k in Bazin's formula for
channels.
1
a
P
k
Planed boards and smooth
cement
Bon^ boards, bricks and
concrete
Aiihl^r masonry
;Earth
Gravel (Ganguillet and
Katter)
•0000467
•0000580
•0000780
•0000864
•0001219
•0000046
•0000188
•00006
•00085
•00070
•0000457 (l+^*^«)
•000078 (l + ^^)
•0000854(1+^1)
•0001219(1 + ^"^^)
The results of experiments by Humphreys and Abbott upon
the flow in the Mississippi* were, however, found to give results
inconsistent with this formula and also that of Bazin.
They then proposed to make the coefficient depend upon the
slope of the channel as well as upon the hydraulic mean depth.
From experiments which they conducted in Switzerland, upon
the flow in rough channels of considerable slope, and from an
examination of the experiments of Humphreys and Abbott on the
flow in the Mississippi, in which the slope is very small, and
a large number of experiments on channels of intermediate slopes,
they gave to the coefficient C, the unit being 1 foot, the value
416 +
000281 1811
C =
n
l.(416.«2281) »'
\ * / Vm
^0 which n is a coefficient of roughness of the channel and has the
values given in Tables XIX and XIX A.
. * Report on the Hydraulics of the Missunippl River, 1861 ; Flow of water in
^^ and canaU, Tnutwine mnd HeriDg, 1893.
184 HYDRAULICS
TABLE XIX.
Showing values of n m the formala of Gaugtiillet and Kuttjer.
Chumd n
Very smooth, cement and planed boards »„ 009 to 01
Smooth, boards, brickHi, concrete ^latoW
Smooth, covered wiUi slime or tnberculated • .,, 015
Roagh ashlar or rubble masonry ., «^« Ditto 019
Very firm gravel or pitched witli stone ,.. *., ... D2
Earth, in ordinary conditioii free fron^ m and weeds *.. D25
Earth, not free from atones and weed *. ., "030
Gravel in bad condition , ... H)65 to ^
Torrential streamH with rough stony » , *0S
TABLE Ca.
Showing values of n in the fa i of GangiiiUet and Kutter^
determined from recent experimc
K
Rectangular wooden flume, very mno ,», ..» „. .^, *0M
Wood pipe 6 ft. diameter ... ... OlSI
Brick, washed with cementf basket sharped sewer^ 6'?c6'8*\ nearly
new Olto
Brick, washed witli cement, basket shaped newer, 6'x6'B", one
year old ,.. ,., '0148
Brick, washed with cement, basket shaped sewer, Wx&&\ font
years old .,, •0153
Brick, washed with cement, circular sewer, 9 ft diameter, nearly
new „ „, ^116
Brick, washed with cement, circular sewer, 9 ft. diameter, four
years old Om
Old Croton aqueduct, lined with brick *0i5
New Croton aqueduct* .. , ... D12
Sudbury aqueduct ^ Dl
Glasgow aqueduct, lined with cement ... D124
Steel pipe, wetted, clean, 1897 (mean) "0144
Steel pipe, 1899 (mean) DUB
This formula has found favour with English, American and
German engineers, but French writers favour the simpler formula
of Bazin.
It is a peculiarity of the formulaj that when m equals unity
then C =  and is independent of the slope; and also when m^B
n
large, C increases as the slope decreases.
It is also of iitiportance to notice that later experiments apou
the Mississippi by a special commissionj and others on the flow of
the Irrawaddi and various European rivers, are inconsistent with
* Meport New York Aqueduct Committion^ ^
FLOW IK OPEN CHANNELS
185
early experiments of Humplireya and Abbott, to which
G&Dg^illet and Katter attached very considerable import^ice in
fraiiittig^ their formula, and the later experiments show, as described
later^ that the experimental determination of the How in, and the
dope of, large natural etreams is beset with anch great difBcidtieB, ■
' ' any formula deduced for channels of uniform section and
'*.* cannot with confidence be applied to natural streams, and
trier v^rsS.^
The application of this formula to the calculation of uniform
«5hannels gri%'^Sj however^ excellent results, and providing the value
of n in known, it can be used with confidence*
It is, however, very^ cumbersome, and does not appear to give
results mare accurate than a new and simpler formula suggested
I lecsently hy Bazin and which is given in the next section.
124 BI. Spin's later formula for the flow in chaxmela.
M. Baxin had recently (AiirJiali^ des Fonts ei ChausseeB^ 1897,
VbL IV p, 20), made a careful examination of practically all the
ble experinient'S upon channels, and has propa^ed for the
icieut C in the Chezy formula a form originally proposed by
Gan^uillet and Kufcter, which he writes
1
c=
«t +
c=
when the unit is one foot,
1+4=
m
4:
c =
1575
m
.(2),
s/i
in which a is ccmstant for all channels and ^9 is a coefficient of
roo^hnass of the channel
Taking 1 metre as the unit a  '0115, and writing y for ,
c = 2I (1).
tike ralne of y in (2) being I'Slly, in formula (1),
The values of y as found by Bazin for various kinds of channels
ar^ shown in Table XX, and in Table XXI are shown values of
186
HYDRAULICS
C, to the nearest whole number, as deduced from Bazin's
coefficients for values of m from "2 to 50.
For the channels in the first four columns only a very feir
experimental values for C have been obtained for values of m
greater than 3, and none for m greater than 7*3. For the earth
channels, experimental values for C are wanting for small valuei
of m, so that the values as given in the table when m is greater
than 7*3 for the first four columns, and those for the first ihiee
columns for m less than 1, are obtained on the assumption, tlurff
Bazin's formula is true for all values of m within the limits dt tiifr i
table.
TABLE XX.
Values of y in the formula,
C =
157.5
1 +
s/m
Very smooth surfaces of cement and planed boards ...
Smooth surfaces of boards, bricks, concrete
Ashlar or rubble masonry
Earthen channels, very regular or pitched with stones,
tunnels and canals in rock
Earthen channels in ordinary condition
Earthem channels presenting an exceptional resistance,
the wetted surface being covered with detritus,
stones or weed, or very irregular rocky surface
unit metre
nnit fool
•06
•16
•46
106S
•88
•86
180
154
286
17
817
125. Glazed earthenware pipes.
Vellut* from experiments on the flow in earthenware pii>e8 has
given to C the value
in which
or
C =
417 + i
n
1 +
75^'
71 = 0072,
181
C =
, 54 '
This gives values of C, not very different from those given by
Bazin's formula when y is 0*29.
In Table XXI, column 2, glazed earthenware pipes have been
included with the linings given by Bazin.
♦ Proe, I, C. E,, Vol. cli. p. 482.
FX^OIV IM^ OPEX CHANNELS
187
TAJBLiE XXI.
Values of C in the f ormiila v = C^/m^ calculated from Bazin's
{ormiila, the unit of len^i^lx l>eiii^ 1 foot,
1575
C =
1 +
y/m
ChannelR
126 Basin.*« xnetliod of determining a and JS.
The meftiod used by Bazin to determine the values of a and /?
' of sufficieiit interest and importance tx) be considered in detail.
1
TTe first calculated values of t= and
■^ vm
>/:
mi
from experimental
^jita, and plotted these values as shown in Fig.
 Jim
abBciBme,and ^
114, ^
vm
as
as ordinates.
188
HTDBADLICS
As will be seen on reference to the figure, points Iiave bea
plotted for four classee of channels^ and the points lie close to km
straight lines passing through a common point P on the aja
of y.
•i
The equation to each of these lines is
y = a + )8aj,
5
\\
I
■
s
i \
B FI^OW m OPEN CHANNELS 189
a bein^ the intercept on the axis of ^, or the ordinate when j^ is
ieto, a^d /3, ^irluch is variable, is the inclination of any one of
Hhcfwi linae to tKe ajds of oj ; for when i= is ^ero,  — = a, and
traBspoamg ihB ©qaation,
whicli IB clearly the tangient of the angle of inclination of the line
to the aiUB of ae.
It should he notedj that since  —  = n j ^^^ ordinatea give
acteikl eiqperimental values of ^ , or by inverting the Bcale, values
of C. Two scales for ordinates are thua shown.
In addition to the points shown on the diagram, Fig, 114,
Bazin plotted the results of some hundreds of experiments for all
londs of channels^ and found that the points lay abont a series of
lines, all passing through the point P, Fig. 1 14, for which a is '00635,
B
and the values of  , i.e. y, are as shown in Table XX.
c
BsLsin therefore concluded, that for all channels
^* = '00636 + ^.
V Sim
I
the iralae of p depending upon the roughness of the channel,
For very smooth channels in cement and planed boards, Bazdn
plotted a large number of points, not shown in Fig. 114, and the
luie for which y = '109 passes very nearly through the centre of
the »cirie occupied by these points.
TTie line for which y is 0"29 coincides well with the mean of
the plotted points for smooth channels, but for some of the points
y msky he as high as 0'4*
It is further of interest to noticej that where the surfaces and
0ectioiiH of the channels are as nearly as possible of the same
character, as for instance in the Boston and New York aqueductS
the values of the coefficient C differ by about 6 per cent,, the
differenix* being probably due to the pointing of the sides and
^rch of the New York aqueduct not being so carefully executed
aa for the Boston atiueduct. By simply washing the walls of the
latter with cement^ E*teley found that its discharge was increased
20 per cant.
190
HYDRAULICS
y is also greater for rectangularshaped channalB, or those
which approximate to the rectangular form, than for those rf
circular form^ as is seen by comparing the two channels in wood
W and P, and also the circular and basketshaped sewers,
M* Bazin also found that y was slightly greater for a veiy
smooth rectangular channel lined with cement than for one of
semicircular section.
In the figure the author has also plotted the results of some
recent experimentSj which show clearly the effect of slime aid
tuberculatiousj in increasing the resistance of very smooth channels.
The %"alue of y for tlie basketshaped sewer lined with brici,
washed with cement, rising from % to "642 during 4 years* service.
12 7« Variations in the coefficient G.
For channels lined ^dth rubble, or similar materials, some iif
the experimental points give values of C differing very consideTjlily
from those given by points on the line for which y is 0'B3, Fig. 1
but the values of C deduced from experiments on parti
channels show similar discrepancies among themselves.
On reference to Bazin^s original paper it will be seen that,
channels in earthy there is a atilj greater variation between
experimental values of C, and those given by the formula, but
experimental results in these cases, for any given channeli
even more inconsistent amongst themselves.
An apparently more serious difficulty arises with respect
Bazin's formula in that C cannot be greater than 157'5.
maximum value of the hydraulic mean depth m recorded ii
any series of experiments is 74"3, obtained by Humphreys
Abbott from measurements of the Mississippi at Carrollton in l
Taking y as 2*35 the maximum value for C would then be 121
Humphreys and Abbott deduced from their experiments valoeft
of C as large as 254. If, therefore, the experiments are reliable
the formula of Basin evidently gives inaccurate results for axoep*
tional values of m.
Tlie values of C obtained at Carrollton are, however, incon
sistent %vith those obtained by the same workers at Yicksbuig,
and they are not confirmed by later experiments carried out at
CatTollton by the Mississippi commission. Further the Telocitiil
at Carrollton were obtained by double floats, and, according to
Grordon*, the apparent velocities determined by such floats should
be at least increased, when the depth of the water is large, by ten
per cent,
Bazin has applied this correction to the velocities obtained by
* Gordon, Proeeedinff§ Intt Civil Eng,, 189S.
FLOW IN OPEH CHAKNELS
iphrejB and Abbott at Yickabm^ and also to those obtained
be MiflstHsippi Commissioti at Catrollton, and shows, that the
immB vala€> for C is theHj probably, only 122.
%st the talties of C as deduced from the early experiments on
Mwiesippi are unreliable, is more than probable, since the
iUefi( slope, as measured, was only '00000*34, which is less than
kch per mile* It is almost impossible to believe that such small
ervnces of level could be measured with certainty, as the
^Il<?«t ripple would mean a very large percentage error, and
■^rther probable that the local variations in level would be
HR than this measured difference for a mile length. Further,
itoiug the slope is cx)rrectj it seems probable that the velocity
gijch a fait won Id be less than some critical velocity similar
that obtained in pipesp and that the velocity instead of being
>portional t4> the square root of the slope t, is proportional
i. That either the measured slope was unreliable, or that the
ocity was less than the critical velocity, seema certain from the
:, that experiments at other part8 of the Mississippi, upon the
ftwaddi by Gordon, and upon the large rivers of Europe, in no
gtva iralaas of C greater than 124.
The experimental endence for these natural streams tends,
er, clearly to show, that the formulae, which can with
ifidtrnce be applied to the calculation of flow in channels of
form, cannot with assurance be used to determine the
of riirerH* The reason for tins is not far to seek, as
lit ions obtaining in a river bed are generally very far
ed from those assumed, in obtaining the formula. The
pttoti that the motion is uniform over a length sufficiently
lo be able to measure with precision the fall of the surface,
be ixLT from the truth in the case of riv^ers, as the irregu
ia the crosa section must cause a corresponding variation
n velocities in those sections,
deri%'ation of the formula, frictional resistances only
n into account, whereas a considerable amount of the
tie on the falling water by gravity is probably dissipated
eddy motions, set up as the stream encounters obstructions in
bed of the river, Tliese eddy motions must depend very
li OB local circunuitances and will be much more serious in
liar channels and those strewn with weeds, stones or other
tioiis, than in the regTilar channels. Another and probably
Otis difficulty is the assumption that the slope is uniform
oui the whole length over which it is measured, whereas
between two cross sections may vary considerably
bank and bank. It is also doubtful whether locally
192
HYDRAULICS
there is always eqailibriam between the resisting and accele
forces. In those eases, therefore, in which the beds are rock
covered Mrith weeds, or in which the stream has a very ir
shape, the hypotheses of uniform motion, slope, and sectmi} i
not even be approximately realised*
128. Logarithmic formula for the flow In channels.
In the formulae discussed, it has been assumed that the fricti
resistance of the channel varies as the square of the velocity,
in order to make the formulae fit the qp^eriments, the coefficient
has been made to vary with the velo
As early as 1816, Du Buat* p
increased at a less rate than the
half a century later St Venant pr
mi = W040'
To determine the discharge of 1
Crimp has suggested the formula
t^ = 124m* •
and experiments show that for sewers that have been in use
time it gives good results. The formula may be written
0*00006 1>*
ited out, that the slope
[uare of the velocity,
>i»0d the formula
cklined sew^ers, Mr
i
m
An examiTiatton of the results of experiments, by logarit
plotting, shows that in any uniform channel the slope
^ ' m^ '
Ic being a numerical c inefficient which depends upon the rouglme«]
of the surface of the channel, and n and p also vary with
nature of tht.^ surface.
Therefore, in the formula.
fMfirn)^
.P't'
From what follows it will be seen that n varies between IIS ]
and 2'1, while p varies between 1 and I'S,
Since m is constant, the formula t  — ^ may be written i  friT,
m"
b being equal to
Therefore
log i = log b + n log 17.
♦ Principcs d'Hydraulique, Vol. i. p. 29, 1816.
FXOTV IN OPEN CHANNELS
193
Pig. 115 are shoiwii plotted the logarithms of i and v
A from an eaqperiment by Bazia on the flow in a semi
r cem^ntlixied pipe. The x)oint8 lie about a straight line,
Qgent of the inclination of which to the axis of v is 1*96
le intercept on the axis of % through v = ly or log v = 0, is
08.
Fig. 115.
liOgarithmio plottings of t and v to determine the index n in
the formola for channels, %=—=,
For this experimental channel, therefore,
i = 00008085t;^*«.
In the same figure are shown the plottings of log i and log v for
18 siphonaqueduct* of St Elvo lined with brick and for which
I is 2*78 feet. In this case n is 2 and b is 000283. Therefore
t=000283i?^
If, therefore, values of v and i are determined for a channel,
rhile m is kept constant, n can be found.
• AnnaUi d£$ Fonts et Chatusiei, Vol. tv. 1897.
U B.
\z
194 HYDRAULICS
To determine the ratio  . The formula,
may be written in the form,
or log m = log (J +  log v.
By determining experimentally m and r, while the slope i is
kept constant, and plotting log m as ordinates and log v as
abscissae, the plottings lie about a straight line, the tangent of the
inclination of which to the axis of v is equal to , and the
P
intercept on the axis of m is equal to
JcV
(;)
In Fig. 116 are shown the logarithmic plottings of m and v for
a number of channels, of varying degrees of roughness.
The ratio  varies considerably, and for very regular channels
increases with the roughness of the channel, being about 1"40 tat
very smooth channels, lined with pure cement, planed wood or
cement mixed with very fine sand, 1*54 for channels in unplaned
wood, and r635 for channels lined with hard brick, smooth
concrete, or brick washed with cement. For channels of greater
roughness,  is very variable and appears to become nearly equal
to or even less than its value for smooth channels. Only in one
case does the ratio  become equal to 2, and the values of m and
P
V for that case are of very doubtful accuracy.
As shown above, from experiments in which m is kept constant^
n can be determined, and since by keeping i constant ~ can be
found, n and p can be deduced from two sets of experiments.
Unfortunately, there are wanting experiments in which m is
kept constant, so that, except for a very few cases, n cannot
directly be determined.
There is, however^ a considerable amount of experimental data
for ciannels similarly lined, and of different slopes, but here
FLOW IN OPBN CHANNELS
195
Fig. 116. Logarithmie plottings of m and v to determine the
ratio  in the formula t= —  .
TABLE XXII.
Particulars of channels, plottings for which are shown in Fig. 116.
1. Semieirealar ehannel, very smooth, lined with wood
3. •> »t tf ft ft }» cement mixed with
Tery fine sand
3. Bectangnlar channel, very smooth, lined with cement
4. ,. », „ „ „ „ wood, 1' 7" wide
5. f, tf smooth „ „ „ slope '00208
6 t» «f fi »» »» M ti '0043
#. »i »f »t ♦» »» »» It '004y
8. „ „ M „ M „ „ 00824
9. New Croton aqnednet, smooth, lined with bricks (Report New York
Water Sapply)
10. Glasgow aqueduct, smooth, lined with concrete IProc. I, C. E. 1896)
11. Sodbary „ „ ,, ,, brick well pointed (Tr, Am.
S.C.E. 1883)
12. Boston sewer, circular, smooth, lined with brick washed with cement
{Tr. Am.S, C. E, 1901)
13. Hectangular channel, smooth, lined with brick
14. 9f tf •> tf tt ^o<xi
15. t, t, „ „ „ small pebbles
15^1. Bectangnlar slniee channel lined with hammered ashlar
l«>w tt tt tf ff »f ff If
16. tt ehannel lined with large pebbles
17. Torlofiia ionnel, rock, partly lined
18. Ordinary ehannel lined with stones covered with mud and weeds ...
19. tf 9» tt ff
50. BiTerWeeer
51. «. ff
S3. «t tt
23. Eitrth dianneL Ores bois
S4« Catoot canal
S5. Bhrer Seiiifl
n
P
145
136
144
138
164
164
154
164
174
1636
1635
1635
1635
1655
149
136
136
129
149
118
94
1615
165
21
149
15
XZ—'l
196
HYDRAULICS
again, as will appear in the context, a difficulty is encountered,
even with similarly lined channels, the roughness is in no t'
cases exactly the same, and as shown by the plottings in Fig. 1.
no two channels of any class give exactly the same yali
for  , but for certain classes the ratio is fairly constant.
Taking, for example, the wooden channels of the group (Nos
to 8), the values of  are all nearly equal to 1*54.
The plottings for these channels are again shown in Fig. 11
The intercepts on the axis of m vary from 0*043 to 0'14.
Fig. 117. Logarithmio plottings to determine the ratio  for smooth ch&nne
Let the intercepts on the axis of m be denoted by y, then
JcV
(!).
FLOW IN OPEN CHANNELS
197
and
logy=log*logi.
If fc and 'p are constant for these channels, and logt and
logy are plotted as abscissae and ordinates, the plottings should lie
about a straig^lit line, the tangent of the inclination of which to the
axis of i is — • and when logy = 0, or y is unity, the abscissa %lc.
V
we. the intercept on the axis of % is k.
In Y\%. 118 are shown the plottings of log % and log y for these
channels, froni which p=l'14 approximately, and % = '00023.
Therefore, n is approximately 1*76, and taking  as 1*54
._ '00023t?^^ ^
'
N
~
'
N.
\ ^

i
\
1
1
\
\
<
^
s
s
S^i
s
\
\
w
4
\
trr
T
\
« 
J
N
*t—
_
.__
—
'""
~"
J
" ^^ ' '"
"" ~'^' '"
:^

^*
—
~
_. ^ ,
■'^—^—'—'
^
«IUi
1 ,
toLn.d}%i
i
\
U
COOXt.
S
b^
■oc
Jff
'^
br
oc
f Qi
n
m
IS
^i
Log, iy
Fi^. 118. Logmrithmio plottings to determine the value of p for smooth
channels, in the formula t=— ;:•
Since the ratio  is not exactly 1*54 for all these channels, the
valaes of n and p cannot be exactly correct for the four channels,
but, as will be seen on reference to Table XXIII, in which are
shown yaloes of t? as observed and as calculated by the formula,
the calculated and observed values of v agree very nearly.
198
HYDEAUtlCS
TABLE XXm.
Values of t?, for rectangular channels lined with wi
determined experimentally, and as calculated from the
formn
Slope O020e
Slope 0019
Slope 00824
e ob
V ©ftJcu
V ob*
t; ealea
V ob
r^ftleu
m m
served
lated
m in
served
Jated
m m
served
UUd
metres
metres
metres
metres
metres
meirea
metres
metres
metres
per B«o.
per iee.
per sec.
per Bee.
pel sec.
per »a«.
0*1881
0*962
0W2
0^042
1825
1814
0882
1594
1580
'1609
1'076
107
1224
1479
1459
'1041
1776
1764
'1833
1162
1*165
1882
1612
158
1197
1902
liSSS
1076
1259
1'228
1585
1711
1690
1313
2053
2051
'2146
1324
1290
1668
1818
1782
1420
2186
2158
'2318
1874
1*854
1789
1898
1S5S
^548
2268
2276
^2441
1440
1402
a9l3
1967
1947
■1649
2S57
287T
*2578
1487
1*452
*2018
2045
2014
*1744
2447
2460
^2681
1562
149
2129
2102
2089
1842
2518
2*j^
■2809
1'687
1552
2215
2179
2143
1919
2612
2*618
As a further example, which also shows how n and p increase
with the roughness of the channel, consider two channels built ia
hammered ashlar^ for which the logarithmic plottings of m and
are shown in Fig* 116j Nos. 15 a and 15 6^ and  is 1"36.
The slopes of these channels are "101 and *037, By plotting
log't and logi^, p is found to be r43 and k '000149. So thai for
these two channels
The calculated and observed velocities are shown in Table XXXI
and agree remarkably well.
Very smooth channels.
The ratio  for the four very smooth
p
channels, shown in Fig. 116, varies between V36 and 1'45, tha
average value being about 1'4. On plotting log^ and logt the
points did not appear to lie about any particular line, so that
conld not be determined, and indicates that k is different for tbo
four channels. Trial values of n ^ 175 and p ^ 1*25 were teken, oe
._^k.v'^
and values of ft calculated for each channeL
FUm IN OPEN CHAKKELS
Velocities b& determined experimentally and as calculated for
' ^ ehaimels are shown in Tablo XXIII from which it will
it fe vanes from '00006516 for the channel lined with
eetnent, to "0001072 for tlie rectangular shaped section lined
th cai^fuHy planed boards.
It will be seen, that although the range of velocities is con
iderabie, there is a remarkable agreement between the calculated
obfierved values of t% so that for very smooth channels the
Ities of n and p taken, can be tised with considerable confidence,
Channeb moderattly imwoik. Ite plottings of logm and logtj
chatHiels Uned with brick, concrete, and brick washed with
^izLetit are shown in Fig. 116, Mos. 9 to 13.
It will be seen that the value of  is not so constant as for the
P
ro classes prevjouBly considered, but the mean value is about
"635, which is exactly the value of  for the Sudbury aqueduct,
mi
Tut the New Croton aqueduct  is as Mgh as r74, and, as shown
^m Fi^* 114j this aqueduct is a little rougher than the Sudbuiy.
The variable values of  show that for any two of these
P
iel« either n, or p, or both, are different » On plotting log*
r a* was done in Fig* 115, the points, as in the last casei
lid not W said to lie about any particular straight line, and the
la^ of p IB therefore uncertain* It was assumed to be 115, and
n
pre, teJdng  as 1*635, n is I'88.
n
Since no two channels have the same value for , it is to be
P
ted that the coefficient k will not be constant.
lo the Tables XXIV to XXXHI the values of v as observed
as Gslciilated from the formula
aW the value of k are given.
It will be seen that k varies very considerably, but, for the
large aquedncts which were built with care, it is fairly
It.
The effect of the sides of the channel becoming dirty with
time^ i« revy well seen in the case of the circular and basket
sewers. In the one case the value of k^ during four years'
varied from 00006124 to 00007998 and in the other from
'•00006405 to 0001096 It is further of interest to note, that when
200
HYDRAULICS
m and v are both unity and k is equal to ^000067, the value of i a
the same as given by Bazin'a formula, when y is *29, and when k is ]
'0001096, as in the case of the dirty basketshaped sewer, th© value
of y is '642^ which agrees with that shown for this sewer on
Fig, 114 '
Channels in Tnasonry^ HamTnered ashlar and rfihble. AttentiaQ
has already been called^ page 198, to the results given in
Table XX XT for the two channels lined with hammered ashlar*
The values of n and p for these two chaonels were determined
directly from the logarithmic plottings, but the data is tnsuiBeietit
to give definite values, in general, to n, p, and k.
In addition to these two channels, the results for one id
Bazin's channels lined with small pebbles, and for other channels
lined with rabble masonry and large pebbles are given. The
n
ratio  is quoted at the head of the tables where possible.
In the,
other cases n and p were determined by trial.
The value of n, for these rough channels, approximates to 2,]
and appears to have a mean value of about r96, while p vari$lj
from 1*36 to 1"5,
Earthen channels, A very large number of experiments h*f
been made on the fiow in canals and rivers, but as it is generally J
impracticable to keep either i or m constant, the ratio  c&n only]
be determined in a very few cases, and in these, as wiJl be seen!
from the plottings in Fig, 116, the results are not satisfactory, and]
n
appear to be unreliable, as  varies between '94 and 2*18,
P
It
probable that p is between 1 and 1*5 and n from 1*96 to 2'15<
Logarithmic formulae for tmrimts classes of channels.
Very smooth channels, lined with cement, or planed boards^
„17S
1 ("000065 to '00011)^,
m*
Smooth channels, lined with brick well pointed, or concrete,
* = '000065 to *00011^,
Channels lined with ashlar masonry, or small pebbles,
i = '00015^.,
m}*
Channels lined with rubble masonry, large pebbles, rock^ and]
exceptionally smooth earth channels free from deposits,
i = *00023 ^,,^,;, .
\
FLOW IN OPEN CHANNELS 201
Earth channels,
h varies {rem '00033 to '00050 for channels in ordinary condition
and from '00050 to *00085 for channels of exceptional resistance.
120. Approximate formula for the flow in earth
nhamialii
The author has by trial found n and p for a number of
channels, and except for very rough channels, n is not very
different from 2, and p is nearly 1*5. The approximate formula
t? = C vm^i,
may, therefore, be taken for earth channels, in which C is about
50 for channels in ordinary condition.
In Table XXXTTT are shown values of t? as observed and
calculated from this formula.
The hydraulic mean depth varies from '958 to 14*1 and for all
valaes between these external limits, the calculated velocities
Wbgree with the observed, within 10 per cent., whereas the variation
of C in the ordinary Chezy formula is from 40 to 103, and
according to Bazin's formula, C would vary from about 60 to 115.
With this formula velocities can be readily calculated with the
ordinary slide rule.
TABLE XXIV.
Very smooth channels.
Planed wood, rectangular, 1*575 wide.
i
= •0001072^,
log* = 40300.
mfeet
V ft. per sec v
observed (
ft. per
»loalai
•2872
865
857
•2811
400
408
•8044
420
426
•8468
467
468
•8717
494
494
•8980
511
512
•4124
526
580
•4811
5*49
547
202
HTDRAULICS
TABLE XXIV (continued).
Pure cement, semicircular.
*=m>
•00006516^,
log* = 58141.
m
V observed v
oalonlated
•508
872
866
•682
469
455
•760
487
4^87
•916
557
562
1084
614
614
y fine
sand, semicircular.
^I'n
1 = 0000759^,
log* = 68802.
t; ft. per seo. v
ft per see.
mfeet
observed (
Bsloalated
•879
287
274
•629
844
8*49
•686
887
898
•706
480
480
•787
451
4*59
•889
480
484
•900
494
610
•941
520
526
•988
5^88
548
1^006
548
558
1^02
555
558
104
566
TABLE XXV.
566
Boston circular sewer, 9 ft. diameter.
Brick, washed with cement, t = 7xnnr (Horton).
i = 00006124
m'
log V = 6118 log m + 5319 log t + 22401.
V ft. per sec.
V ft. per seo.
mfeet
observed
caloalated
•928
221
2*84
1208
2*70
276
1^408
808
808
1880
8*48
8^56
1*999
878
875
2809
4^18
410
FLOW IN OPEN CHANNELS 208
TABLE XXV (continued).
The same sewer after 4 years' service.
t = 00007998^,
log t; = 6118 log m + SSIO log i + 21795.
m V ohsenred v oaloulated
1120 2*88 2*29
1606 2*82 276
1952 816 822
2180 880 889
TABLE XXVI.
New Croton aqueduct. Lined with concrete.
i = 000073^,
log V = 6118 log m + 5319 log i + 2'200.
V ft. per 860. V ft. per seo.
iifeet obsenred . ealcolated
1000 187 187
1260 159 167
1499 179 176
1748 196 198
2001 211 210
2260 227 226
2600 241 240
2749 252 265
2998 266 2*68
8261 278 282
8508 289 296
8750 800 808
8888 802 812
TABLE XXVII.
Sudbury aqueduct. Lined with well pointed brick.
i =
•00006427^.,
081ogt? =
•6118 log m +
•5319 log 1
V ft. per 860.
t; ft. per seo.
iifeet
obBenred
oaloulated
•4987
1186
1142
6004
1269
1279
8005
1616
1526
1000
1765
1762
1200
1948
1964
1400
2149
2147
1601
2882
2881
1801
2518
2511
2001
2651
2672
2201
2844
2882
2886
2929
2987
204 HYDRAULICS
TABLE XXVm.
Rectangular channel lined with brick (Baziii).
i=000107^.
V ft. per 860. V ft. per seo.
m feet observed calealated
•1922 276 290
2888 867 8*68
8654 418 480
•4285 472 471
•4812 5^10 509
•540 584 546
•5828 568 577
•6197 601 594
6682 615 622
6968 647 689
7888 660 662
•7788 672 688
Glasgow aqueduct. lined with concrete.
i = *0000696^„
log V = 6118 log m + 5319 log i + 22118.
V ft. per seo. t; ft. per eeo.
m feet observed calculated
1227
1478
1478
1^489
1499
1499
1548
1597
1607
1610
1620
1627
1788
1811
Charlestown basketshaped sewer 6' x 6' 8".
Brick, washed with cement, i = ^Tnnr (Horton),
t = 00008405 ^5,
log V = 6118 log m + 5319 log i + 21678.
V ft. per sec. t; ft. per see.
m feet observed calculated
•688 1^99 2^05
•958 246 252
1187 282 287
1589 844 8*86
187
189
207
211
2106
211
2214
218
218
214
215
214
218
222
221
228
228
228
222
224
224
224
225
227
226
288
247
240
3 XXTX,
FLOW IN OPEN CHANNELS
205
TABLE XXIX (continued).
rhe same se^irer after 4 years' service,
i=0001096
m'
log V = 6118 log m + 5319 log i + 2* 1065.
mfeet
1842
1506
1645
V ft. per aeo. v ft. per see.
obserred oalcnlated
266
286
804
268
288
804
TABLE XXX,
Left aqnednct of the Solani canal, rectangular in section, lined
ill rabble masonry (Cunningham),
i = 00026^.
i
mfeet
V ft. per sec.
observed
V ft. per seo.
calculated
000225
•000206
•000222
•000207
•000189?
648
681
721
7648
794
846
349
870
387
406
350
347
384
383
388
aeduct,
i = 0002213 ^,.
t
m
V observed
V calculated
•000195
000225
000205
000198
000198
000190
342
586
676
743
777
796
243
361
373
387
393
406
226
358
376
389
404
406
"orlonia tunnel, partly in hammered ashlar, partly in solid
i= 00104,
•00022
m'
m
V observed
V calculated
1932
3382
345
2172
3625
873
2552
4232
416
2696
4324
432
8251
5046
490
8488
4965
508
8581
4908
518
8718
5858
537
HYDRAUUCS
TABLE XXXI.
Channel lined with hammered ashlar,
 = 136,
m feet
•824
•467
•580
•662
t = 000149
log * = 41740.
m*
{=•101
V ft. per seo.
observed
1280
1618
1868
2109
t; ft. per sec.
calculated
1280
1618
1897
208
t = 087
m feet
424
620
746
862
V ft. per see.
observed
904
1146
1866
1608
V ft. per MflL
oal(»lattd
902
1186
1852
1498
Channel lined with small pebbles, i = '0049 (w=l'96, p = l'32
will give equally good results).
^ = 149,
■ P
i = 000152 ■
log ifc = 41913.
m'
V ft. per sec.
V ft. per sec.
nfeet
observed
calculated
250
216
284
867
295
297
450
840
847
•520
884
882
•588
414
416
644
448
448
•700
464
466
746
488
4^88
786
612
505
882
526
525
871
548
648
910
567
668
FU>W IN OPSN CHANNELS
207
TABLE XXXn.
Cluumel lined with large pebbles (Bazin),
t = 000229^,
log i = 43605.
V ft. per seo. v ft. per see.
m feet observed oalcalated
179 184
248 244
290 290
827 818
856 845
886 867
4^08 891
428 488
448 458
460 469
478 484
490 600
TABLE XXXm.
Velocities as obsery^and as calculated by the formula
c^X'
t;=Cvm*
i. C = 50.
f'""
GaTiges
Canal.
i
mfeet
V ft. per sec.
observed
V ft. per sec.
calcolated
•000155
•000229
•000174
•000227
•000291
540
869
782
984
450
24
871
296
402
282
284
880
808
400
2d8
River
Weser.
t
m
V obserred
t; calcolated
•0006608
•0005608
•0002494
•0002494
898
1886
141
106
629
790
669
475
60
818
670
478
Missouri.
i
m
V observed
V calculated
•0001188
•0001782
•0001714
•0002180
107
128
154
177
86
488
603
619
828
487
480
6*26
208
HYDRAULICS
Cavon/r Caned.
i
m
V obsenred
t; calculated
•00029
•00029
•00088
•00088
782
516
663
474
870
8^10
840
8^04
8*80
292
814
291
Earth channel (branch ofBurgoyne canal).
Some stones and a few herbs upon the swrfcLce.
C = 48.
17 ft. per sec. v ft. per sec
t m feet observed calculated
•000957 958 1^248 1^80
•000929
1181
1702
166
•000998
1405
1797
1^94
•000986
1588
1958
2^06
•000792
•958
1288
126
•000808
1210
1666
166
•000858
1486
1814
179
•000842
1558
1^998
208
130. Distribution of the velocity in the oroas aectioii
of open channels.
The mean velocity of flow in channels and pipes of small ctosb
sectional area can be determined by actually measuring the weight
or the volume of the water discharged, as shown in Chapter Vll,
and dividing the volume discharged per second by the ckw
section of the pipe. For large channels this is impossible, and
the mean velocity has to be determined by other means, usually
by observing the velocity at a large number of points in the same
transverse section by means of floats, current meters*, or Htot
tubes t. If the bed of the stream is carefully sounded, the cross
section can be plotted and divided into small areas, at tiie centres
of which the velocities have been observed. K then, the observed
velocity be assumed equal to the mean velocity over the smaD
area, the discharge is found by adding the products of the areas
and velocities.
Or Q = Sa . t;.
M. Bazint, with a thoroughness that has characterised lus
exi)eriments in other branches of hydraulics, has investigated ths
distribution of velocities in experimental channels and also in
natural streams.
In Figs. 119 and 120 respectively are shown the cross section
of an open and closed rectangular channel with curves of eaas
* See page 288. t See page 241.
X Bazin, Recherches HydraiUique^
FLOW IN OPEN CHANNELS
209
velocity drawn on the section. Curves showing the distribution
of velocities at different depths on vertical and horizontal sections
are alao shown.
^aj^j^sar ofequat VeU^cify
ion Rectcuiffular Chaitttd/,
Fig. 119.
VertLcaLS^cUjons.
Eori^ciUal S fictions.
1
Fig. 120.
L Xt will be seen that the maximum velocity does not occur in
L^ free surface of the water, but on the central vertical section
Sit some distance from the surface, and that the surface velocity
\ may be very different from the mean velocity. As the maximum
m velocity does not occur at the surface, it would appear t\iat \n
I U H. \4
210
MTDRAULIC^
assuming the wetted perimeter to be only the wetted sarbeej
the chaniii'l^ some error is iritroducefL That the air hsi aOll
same influcfna; tm if the water wer*? in ccmtact with a sur
similar to that of the sides of the channel, is very cle
shown by comparing the carves of equal velocity for the ch
rectangular channel as showii in Fig. 119 with thase of Fig. 1211
The air resistancej no doubt, accounts in some measure for tltl
surface velocity not being the nmxinmm velocity, but that it dosij
not wholly acc<mnt for it^ in shinm by the fact that^ whether to]
wind is blowing up oi
below the surface. J
why the maximum veio
the water is \em eonstraini
movements of all k:
utilised in giving mouio
translation.
Depth rm a7ty vsrtkal at
velocity. Later is discussed,
on the verticals of any crosa
the maximum velocity i&«t3t I
ggest^ as the principal reason [
ot ocrcur at the surface,
e surface, and that ir
op, and energy is thepel»
rater not in the direction dfl
i$mheity u equal to ihi mmX
3^ tiie dktrtbntion of reloci^
, and it will be seen^ thst it \
is the mean veUx*ity on any vertical st^ctioti of the channt'I, tte]
depth at which the velocity is equal to the mean velocity is about
0'6 of tht' total depth. This depth varies with the roughne«!i of
the stream, and u deeper the greater the ratio of the depth to
the width of the stream. It varies between '5 and *55 of the depth
for rivers of small depth, ha\ing beds of fine sand, and from '5S
to 'Q^ in large rivers from 1 to 3f feet deep and ha\ing stitffig
bedst.
As the banks of the stream are approached, the point at wliidi
the mean veh^city occurs falls nearer still to the bed of the stream,
but if it falls v^ery low there m generally a second point near tk
surface at which the velocity is also equal to the mean velocity.
When the river is covered with ice the maxim ima velocity of
the current \^ at a depth of '35 to '45 of the total depth, and tl«
mean velocity at two points at depths of '08 to '13 and '68 to'JV
of the total depth J. ■
If, therefore, on various verticals of the cross section of a strfsS^
the velocity is determined, by means of a current meter, or I*ik>t
tube, at a depth of about '6 of the total depth from the surface,
the velocity obtained may be taken as the mean velocity upon ibe
vertical.
* Hydraulique.
f I,f GSnie Civil, April, 1906, ''Analysis of a communication by Marphj to
the Hydrological section of the Institute of Geology of the United States."
"^ Cunningham, Experiments on the Ganges Canal.
FLOW IN OPEN CHANNELS
211
The total discliarge can then be found, approximately, by
ividing the cross section into a number of rectangles, such as
M, Fig. 120a, and multiplying the area of the rectangle by the
"velocity measured on the median line at 0*6 of its depth.
Fig. 120 a.
The flow of the Upper Nile has recently been determined in
Akway.
Gaptain Cunningham has given several formulae, for the mean
idocity u upon a vertical section, of which two are here quoted.
u=\(y^^{) (1),
tt = i(2t?jt;j + 2yj) (2),
V being the velocity at the surface, v^ the velocity at i of the depth,
V at one quarter of the depth, and so on.
Form of the curve of velocities on a vertical
13L
notion.
M. Bazin* and Cunningham have both taken the curve of
telocities upon a vertical section as a parabola, the maximum
velocity being at some distance hm below the free surface of the
water.
Let V be the velocity measured at the centre of a current and
as near the surface as possible. This point will really be at 1 inch
iff more below the surface, but it is supposed to be at the surface.
Let V be the velocity on the same vertical section at any depth
A, and H the depth of the stream.
Bazin found that, if the stream is wide compared to its depth,
the relationship between v, V, h, and i the slope, is expressed by
the formula,
Vt ;
»r i; = Vi(g)VHi (1),
; being a numerical coefficient, whicli has a nearly constant value
»f 36'2 when the unit of length is one foot.
ns.
Recherches Hydrauliquef p. 228 ; Annales des Fonts et Chansn^es, 2nd Vol.,
\4— 'i
tV
tV
;,\ocw;
• \ ...uve ^^V »:, v\vU«*V' , ,..t '>^'^;".„ and t^*' \w, "
\\p
^^^•'l >rg>' "a\iuc^
\V
ateY
lAOt
\n
.•^'",f;;
«^:^s:^??t;>cs;■:f>
3<'
ivu"
to
tlieP^;",
..euv<:::t'^*"«^*
to*?'
opoi
litits
atvV'^^^faeptVv:
t\tet«t^^...^.otv
ot t\^«
,Tio*!
\cal»
oi*«»T
vevt^'^,. rf^
a>y
^^''*;e\octty
vottv
,V.eBS^^!!lt:Ui>^
Cttii^
.tiit^»^*'
212
HYBRAULICS
lb detetmtne ihe depth tyn aiiy vertical at wM^M the velocity i
eqtml to the mean veI>ocity, Let u be the mean velocity on
vertical section, and hu the depth at which the velocity is equal I
the mean velocity.
The discharge through a vertical Btrip of width* dl is
rH
uEdl^dlj v.dh.
Therefore uR = j^ (y  ^s/Ri )dA,
and
u^y^Jm
(2).
Substitnting u and A, in (1) and equating to (2),
and
A« = '577H.
This depth, at which the velocity is equal to the mean velocityJ
is determined on the assumption that Jc is constant, which is
true for sections very near to the centre of strea^ms which si«
wide compared with their depth.
It will be seen from the curves of Fig. 120 that the deptli ai 1
which the maximum velocity occurs becomes greater as the sides J
of the channel are approached, and the law of variation of \'elocitf J
also becomes more complicated ♦ M, Baxin ako found that ,
depth at the centre of the stream, at winch the maximum vek
occurSj depends upon the ratio of the width to the depth, j
reason apparently being that, in a stream which is wide comf
to its depth J the flow at the centre is but slightly affected byl
resistance of the sides, but if the depth m large compared with 1
width, the effect of the sides is felt even at the centre of
stream* The farther the vertical section considered is remO
from the centre, the effect of the resistance of the aides*
increased, and the distribution of velocity is influenced to
greater degree. This effect of the sides, Bazin expressed
making the coefficient k to vary with the depth h^ at wl
the maximum velocity occurs.
The coefficient is then,
36"2
Jfc
('fer
Further, the equation to the parabola can be written in
of v„^ the maximum velocity, instead of V.
FLOW IN OPEN CHANNELS 213
Th», ,.,,. 882Vffiftt.) (3,
The mean velocity t*, upon the vertical section, is then,
1 f^
riJo
Therefore
'"*(r:)(5HH.);:(^<'w
When t7 = tA, fe = /i«,
« 1 fcm /l«' 2huhm
and therefore, 3 " H "^ H* W~ '
The depth ^ at which the velocity is a maximum is generally
len than *2H, except very near the sides, and hu is, therefore, not
very different from '6H, as stated above.
Ratio of maximum velocity to the mean velocity. From
equation (4),
.,  36'2>/Hi7 1 K^K\
In a wide stream in which the depth of a cross section is fairly
constant the hydraulic mean depth m does not differ very much
rfrom H, and since the mean velocity of flow through the section is
( C ^mi and is approximately equal to u, therefore,
v^ . 362 /I K , hj\
^ CflM^^^ H hV
Assuming A. to vary from to '2 and C to be 100, — varies
from 1*12 to r09. The ratio of maximum velocity to mean
"Vidocity is, therefore, probably not very different from Tl.
132. The slopes of channels and the velocities allowed
in them.
The discharge of a channel being the product of the area and
ihe velocity, a given discharge can be obtained by making the
rea small and the velocity great, or vice versa. And since the
riocity is equal to Cvmi, a given velocity can be obtam^id \>y
214
HYDRAULICS
varying' either vt or i. Since m will in general increase with the
area> the area will be a niininuiuj when i is as lar^e as possible.
^Butf as the coat of a channel, including land, excavation and
Icxmstructionj will^ in many cases, be almost proportional to ite
tcroes eectiona! area, for the first cost to be small it is desirable
rthat i should be large. It should be noted, however, that the
discharge ia generally increased in a greater proportion, by an
increase in A, than for the same proportional increase in L
Assume, for instance, the channel to be semicirctilar*
The area is proportional to ^, and the velocity t* to vd , i.
Therefore Q oc cp Jdl
If d is kept constant and i donfeled, the discharge i^ increased
to %^2Q, but if d 18 doubled, i being kept constant^ the diacbargi?
will be increased to 5"6Q, The maximum slope that can bt* given
will in many cases be determined by the difference in level of til
two points connected by the channel.
When water is to be conveyed long distances, it is often
necessary tci have several pumping stations en rout^^ as gutficient
fall cannot be obtained to admit of the aqueduct or pipe line being
laid in one continuous lengtb.
The mean velocity in large aqueducts is about 3 feet per
second, while the slopes vary from 1 in 2000 to 1 in 10,000, ITje
slope may be as high as 1 in 1000^ but should not^ only in eioep"
tional circumatancesj be less than 1 in 1 0,000,
In Table XXXIV are given the slopes and the maximoiB
velocities in them, of a number of brick and masonry liuai
aqueducts and earthen channels, from which it will be seen thai
the maximum velocities are between 2 and 5J feet per second,
and the slopes vary from 1 in 2000 to 1 in 7700 for the brick aad
masonry lined aciueducts, and from 1 in 300 to 1 in 20,000 for the
earth channels. The slopes of large natural streams are in iomt
Leases even less than 1 in 100^000, If the velocity is too small
P suspended matter is deposited and slimy growths adhere to the sidefi
It is desirable that the smallest velocity in the channel aliall
such, that the channel is " self cleansing," and as £ar as poiiibi
the growth of low forms of plant life prevented.
In sewers, or ctiannels conveying unfiltered wTaters^ it
especially desiiable that the velocity shall not be too stnall,
ehouldj if possible, not be less than 2 ft, per second.
^ TABLE XXXIV.
Showing the slopes of, and maximum velocities^ as deterauoe
experimental lyj in some existing channels.
FLOW IN OPEN CHANNELS
215
/
Smooth aqueducts.
Slope
Maximum velocity
New Croton aqueduct 0001326
3 ft. pel
• second
Sudbory aqoednct
•000189
294 „
))
Glasgow aqueduct
•000182
225 „
»»
Paris Dhuis
•000180
Avre, Istpart
0004
„ 2nd part
00038
Manchester Thirlmere 000815
Naples
00050
4^08
»>
Boston Sewer
0005
844 „
>»
»» >f
•000388
Earth chamiels.
4^18 „
*»
Slope Maximum velocity
Lining
ages canal
•000806 416 ft. per second
earth
±er „
008 408
>» »»
»»
ith „
•00037 563
» »»
gravel and
some. stones
FOOT „
•00083 342
)) fi
amen „
•0070 874
» »♦
earth
azilly cut
•00085 1^70
t )9
1 earth, stony,
\ few weeds
•00048 1^70
» »»
the bottom of the canal)
•00005 3
» >»
»» M
TABLE XXXV.
Showing for varying values of the hydraulic mean depth m, the
limum slopes, which brick channels and glazed earthenware
es should have, that the velocity may not be less than 2 ft.
second.
m feet
slope
•1 ]
L in 93
•2 3
L „ 275
•3 J
I „ 510
•4 ]
L „ 775
•5 ]
L „ 1058
•6 ]
L „ 1880
•8 ^
L „ 2040
io ]
L „ 2760
m feet
slope
125 1
in 8700
15 1
„ 4700
175 1
„ 5710
20 1
„ 6675
25 1
„ 9000
30 1
„ 11200
40 1
„ 15850
lie slopes are calculated from the formula
1575 ,.
1 +
n/w
m
he value of y is taken as 0*5 to allow for the channel becoming
For the minimum slope for any other velocity v, multiply
(2\*
 j . For example, the minimum slope
velocity of 3 feet per second when m is 1, is 1 in 1227.
216 HYDRAULICS
Velocity offimv in, and $lope of earth channels* If th© relodtf
is high in earth channels, the sides and bed of the ehaji&el sii 
eroded, while on the other hand if it is too sroall, the c:i>;u ify of
the channel will be rapidly diminished by the depositi.u *►! ;<n4
and other suspend til nmfcter, and the growth of aijuittic plant*.
Da Buat gives '*"> f(H>t per second as the minimum velocity that
mad shall not be dei)o«ite<l, while Belgrand ttlluws a miminma
of '8 foot per second.
TABLE XXX\T
Showing the velocities above ih^ according to Du Bttiit, '
and as quoted by Rankine, erofiioi hanuebof vanous matemli j
takes place.
Soft clay Om tL per sec^iid
Fine sand 0^^ „ „
Coarse sand and gravel m large a^ p OW „ n
Gravel 1 inch diameter 2*25 „ ^
Pebbles 1^ inchea diameteT 8*33 „ ,,
Heavy sbmgle 4'00 „ «.
Soft rock, brick, eartbeuware 4'50 „ ,
Rock, various kinds 600 „ ,, and upwurii
133. Sections of aqueducts and sewers.
The forms ^jf sections given to scjme aqueducts and sewers are
shown in Figs. 121 to 131, In dt^signing snch aqueducts ^d
sewers, con side nit ion has to be given to problems other than tlie
comparatively simple one of determining the size and slope to
be given to the channel to convey a certain quantity of water.
The nature of the strata through which the aqueduct is to be
cut, and whether the excavation can best be accomplished by
tunnelling, or by cut and cover, and also^ whether the aqnediict
is to be lined, or cut in eolid rock, must be considered. In many
cases it is desinible that the aqueduct or sewer should have 9QcI
a form that a man can conveniently walk along it, although its
sectional area is not required to be exceptionally large. la
such cases the section of the channel is made deep and narrow.
For sewers, the oval section. Figs. 126 and 127, is largely
adopted because of the facilities it gives in this respect, and it has '
the further advantage that, as the flow diminishes, the cross
section also diniinishes, and the velocity remains nearly constant
for all, except very small, discharges. ITiis is im]>ortant, as at
small velocities sediment tends to collect at the bottom of the
sewer.
134. Siphons forming part of aqueducts.
It is frequently necessary for some part of an aqueduct to be
constructed as a siphon, as when a valley has to be crossed or the
FLOW IN OPEN CHANNELS
217
aciaedact taken nnder a stream or other obstruction, and the
aqaednct must, therefore, be made capable of resisting con
siderable pressure. As an example the New Croton* aqueduct
from Croton Lake to Jerome Park reservoir, which is 33*1 miles
rrs
Fig. 121.
Fig. 122.
<— 7.^— >
I
_ IT
Fig. 123.
i< is:s^ >i
Fig. 126.
rr\
«
Fig. 127.
Fig. 128.
Fig. 129.
Fig. ISO.
Fig. 131.
218 HYDRAULICS
long, is made up of two pnrt^a. The first m a tnasonry conduil
the section shown in Fig. 121, 23'9 miles long and haipHng a
of '0001326, the second consists almo«t pntirely of a brick lio^
siphon 6*83 miles long, 12' 3' diameter, the maximum head in
which is 126 feet, and the differenci* in level of the two ends is
6*19 feet. In such casen, however, the siphon is fretiuentljr made
of steel, or castiron pij^es, as in the case of the new Edinburgh
aqueduct (see Fig, 131) which j where it crosses the vaUeja, m
made of castiron pipes S3 inches diameter.
135. The best form of cham
The best form of channel, or nnel of least resistance, is
that which, for a gii'en slope and ?a, will give the majdmonL
discharge.
Since the mean velocity in a cl el of given slope is propor
A
tional to p , and the discharge is A the best form of channel for
a given area, is that for which P linimum.
The form of the cliannel whicj i the minimura wetted
meter for a given area is a semicir for which p if r is the nuiina,
the hydraulic mean depth is ^.
More convenient forms, for channels to be excavated in roct
or eartli, are those of the rectangular or trapezoidal section,
Fig. 133. For a given dis^charge, the best forms for these I
channels, will be those for which both A and P are a minimmD; I
that is, when the ilifTerentialw (^A and oV are respectively equal to ^
zero.
Rectangular chanwL Lot L be the >\ndth and h the depttj
Fig. 132, of a rectangular channel ; it is required to find the ratio
^ that the area A and the wetted perimeter P may both be a
minimum, for a given discharge.
A = L/i,
therefore dA = h.dh+hdh = (1),
P = L + 2/i,
therefore dP = dh + 2Jh = (2> 
Substituting the value of dL from (2) in (1),
L = 2A.
Therefore m = rj = ^ .
4/1 2
Since L = 2h, the sides and bottom of the channel touch a cir^^^
having h as radius and the centre of which is in the free surf^*^^
of the water.
FLOW IN OPEN CHANNELS
219
Earth ch4innels of trapezoidal form. In Fig. 133 let
I be tlie bottom width,
h tbe depth,
A. the cross sectional area FBCD,
P tbe length of FBCD or the wetted perimeter,
i tbe slope,
and let tbe slopes of the sides be t horizontal to one vertical ; CG
is then equal to th and tan CDG = t.
thH
and
Fig. 132. Fig. 133.
Let Q be the discharge in cubic feet per second.
Then K^hl^th?
P=Z + 2/iV^^"+~l
h(l±th)_
"^'U^hjwvi
.(3),
.(4),
.(5).
For tbe channel to be of the best form ^P and dA both equal
zero
From (3) K = hl^th\
and tberefore dA = hdl + ldh + 2thdh =
From (4) P = Z + 2hs/FTl
^^a dP = dl'i2s/¥^ldh=^0
Substituting the value of dl from (7) in (6)
Therefore,
l=2hy/F^2th .
.(6).
.(7).
.(8).
m =
4h^/¥Tl  2ht
h
T
L«et O be the centre of the water surface AD, then since from (8)
I
therefore, in Fig. 133, CD = EG = OD.
^SO STDBATUCS 1
Draw 'j¥ mii •'•£ geryniifnifar ta CD and BC respectirely.
T^usi. 'i«£aaae zsb ansfe «>FD s a TiaAt an^le, the mn^ks CD6
inii JOD M* OTiaL: ami mce OF = ODcc«FOD, auid DG = OE, J
Ami I«5 = »:D ^:«CIfe iu»5t. », OE=OF; mod ance OEC and
•jFC ir» T:jnir ing^i**. a orcie wim tj as centre will touch the sides
if The 'hannt*!, a6 in die v:at«e of die rectangular channeL
in a ebaimd of given fbim thai,
a TBaiimi i m , (b; tiie diacharge majr
and tracispiDfin^.
For a grven slope and ron^faness of the channel v is, therefore,
propcrti'^cal to the h3fiiraiiUc mean depth and will be a mRYimnm
wl^n H is a maxiniiim.
That isj when the differential of p is zero, or
PJAArfP = (1).
For nummom discharge, Ar is a maximum, and therefore,
.(p)'iB:
f SL maxm[iQm.
Differentiating and equating to zero,
^?^PdA»A^P=0 (2).
Affixing values to n and p this differential equation can be
solved for special cases. It will generally be sufficiently accurate
to assume w is 2 and p = 1, as in the Chezy formula, then
n^'p _ 3
n ""2'
and the equation becomes
3PdAAdP = (3).
137. Depth of flow in a circnlar channel of given
radius and slope, when the velocity is a maxinoLum.
Lot r bo tlio radius of the channel, and 2<^ the angle subtended
by tho Hiirfaco of the water at the centre of the channel, Pig. 184,
FLOW IN OPEN CHANNELS
221
Then the i^etted perimeter
The area A = r»*r»8in*co8* = r^(^5^V
; and dA = r»<i^ 7^(508 2^(2^.
Snbstitating these values of dP and dA in equation (3),
■ectaon 136,
tan2<^ = 2<^.
The solution in this case is obtained
directly as follows,
A r /^ sin 2<^N
m
P~2V 24 J'
This will be a TnaTrimuTn when sin2<^
18 n^;ative, and
sin2<»
2*
18 a maximnni, or when
d /sin2^\_^
d^\ 24 )^ '
Fig. 184.
.'. 2*cos2<^8in2<^ = 0,
and tan2^ = 2^.
The solution to this equation, for which 24 is less than 360^ is
2^ = 257'' 27'.
Then A = 2738r»,
P = 4494r,
m = 608r,
and the depth of flow d = l626r.
138. Depth of flow hi a ch'cular channel for maximum
diflcharge.
Substituting for dP and dA in equation (3), section 136,
6r*^<^  6r»«^ co8 2iM«^  2r»^«^ + r»sin 2«^d</» = 0,
from which 4^66 cos 2<^ + sin 2<^ = 0,
and therefore <^ = 154'.
Then A = 3044r»,
P = 530r,
m = •573r,
and the depth of flow d = l*899r.
Similar solutions can be obtained for other forms of channels,
and may be taken hj the student as useful mathematical exercises
bat thejr ure not of much practical utility.
222
HYDRA trues
139. Curves of velocity and (iiscliarge for a giif
channel.
The depth of How for maximum vrfocity, or discharge, aai U*
determined very readily hy drawing curres of velocity atid dis
charge for different depths of flow in the channeL This metliud
is useful and instructive, especially to those students who are not
familiar with the differential calculus.
As an exiumple, velocities and discharge, for different depths
of flow, have l>een calculated for a ^ge aqueductt the profUi^ of
which is shown in Fig. 135, and the pe i of which is 000013^31
The velocities and discharges are m by the cnrves drawn in
the figure.
Fig. 135.
Values of A and P for different depths of flow were first deter
mined and m calculated from them.
Tlie velocities were calculated by the formula
using values of C from column Z, Table XXI.
It will be seen that the velocity does not vary very much for
all depths of rt<jvv greater than 3 feet^ and that neither the velocity
nor the discharge is a niaxinnini when the aqueduct is full; the
reason being that, as in the circular channel, as the surface of the
water approaches the top of the aqueduct the wetted perimeter \
increases muuh muic! i apidly than the area.
The maximum velocity is obtained when m is a maximum
and equal to 3*87, but the maximum discharge is given, when the
depth of flow is greater than that which gives the greatest
FLOW IX OPEN CHANNELS 223
Telocity. A. circle is shown on the figure which gives the same
maxinmm discliarg^e.
The student should draw similar curves for the eggshaped
aewer or otKer form of channel.
140. AjfpiiBmtiflM of the finrmiiU.
Problewi 1. To find the flow in a channel of given section and slope.
This is the simplest problem and can be solved by the application of either the
logftnllmiie fonnuls or by Bazin's formula.
The only difficoltj that presents itself, is to affix values to k, n, and p in the
logsnthmie fonnohi or to 7 in Bazin's formula.
(1) By the logdritkmie formula.
Wnwt
agn some value to k^ n, and p by comparing the lining of the channel
mfA those given in Tables XXIV to XXXin. Let w be the cross sectional area of
the water.
Then since i = ^^ ,
log V =  log t + — log ffl   log Ae,
n n ti
mnd Q=«.r,
^^ logQ = log« + logi+^logm log*.
n 11 n
(3) By the Chezy formula, using BazirC$ coefficient.
The coefficient for a given value of m must be first calculated from the formula
c=. "^^
or taken from Table XXI.
Then
1575
1 + ^^
Example'. Determine the flow in a circular culvert 9 ft. diameter, lined with
sooth brick, the slope being 1 in 2000, and the channel half full.
^^ =i=225'.
Wetted perimeter 4
(1) By the logarithmic formula
t= 00073^
m
116 •
Therefore, log ,=:^ log 0005+ J;^ log 225  ^ log 00007.
17=4*55 ft. per sec.,
«='2— =318 sq.ft.,
Q=145 cubic feet per sec.
i2) By the Chery formula, unng Baxin's coefficient,
^225
r = 182 V2~25T^^= 443 ft. per sec.
Q = 318 X 335 = 141 cubic ft. per sec.
224 HYDRAULICS
Problem 2. To find the diameter of a eiroular ohaxm^ of giTen slope, for whidi
the maximum discharge is Q cubic feet pear second.
The hydraulic mean depUi m for maximum discharge is *678r (oeetion 138) tad
A = 3044r».
Then the velocity is
t;=•767C^/^^
and
Q=2•37Cr^^/^.
therefore
*'=Ri3V OT'
and the diameter
»— v^
The coefficient is unknown, but by assuming a value for it, an approzunatioa
to D can be obtained ; a new value for can then be taken and a nearer appraxi
mation to D determined ; a third value for G will give a still nearer approximatiiii
to D.
Example. A circular aqueduct lined with concrete has a diameter of 6' 9" and
a slope of 1 foot per mile.
To find the diameter of two oastiron siphon pipes 5 miles long, to be put m
series with the aqueduct, and which shall have the same disd^arge ; the diffenoot
of level between the two ends of the siphon being 12*6 feet.
The value of m for the brick lined aqueduct of cirenlar section when tilt
discharge is a maximum is *573r=64 feet.
The area A=:8'044rS=25 sq. ft.
Taking C as 130 from Table XXI for the brick culvert and 110 for the caitinn
pipe from Table XII, then
64
5280
Therefore dl=__^__ ^_
226'
d=400 feet.
Problem 3. Having given the bottom width {, the slope t, and the side slopes I
of a trapezoidal earth channel, to calculate the discharge for a given depth. 
First calculate m from equation (5), section 135.
From Table XXI determine the corresponding value of C, or calculate C from
Bazin's formula,
then
and
A convenient formula to remember is the approximate formula for oidinHj
earth channels
t; = 50vm*i
For values of m greater than 2, v as calculated from this formula ia yen
equal to v obtained by using Bazin's formula.
rrv r , . •CH)037t;«'»
The formula i= ^^ —
may also be used.
FLOW IN OPEN CHANNELS
225
Examiple. An og6iDMXj euth duumel has a width 1= 10 feet, a depth, d=4 feet,
adailope< = v^T Sidealopes 1 to 1. To find Q
A=46 aq. ft.,
P=21212 ft.,
»=2*16ft,
1576
C=
1 +
Ta
=606,
From tbe formnla
^/?16
vs 1625 ft per sec.,
Q=r74*7 oabio fk. per see.
v=l*68 ft. per see.,
Q=75 cubic ft. per sec.,
From the logarithmio fbrmnla
. •00087t?«^
• W^^'
vs 1649 ft. per sec.,
Q=75*8 cubic feet per sec.
Prtiblew^ 4. Haying given the flow in a canal, the slope, and the side slopes, to
(hwl the dimensions of the profile and the mean velocity of flow,
(a) When the canal is of the best form.
(5) When the depth is given.
In the first ease m=^ * ^'^^ ^^ equations (8) and (4) respectively, section 18i>
T=l + 2h ,JF+T.
A
Therefore
Sabetitoting ^ for si
m= r=r
UHl
Bet
Therefore
uid
4hjt'»+l2th
2>j€^ + lt
A«=M(2N/«*+it)2.
2 hH2y/wnt)^
(1).
c«.t(2^A>+rt)«
A ▼alne for C should be chosen, say C=70, and h calculated, from which a mean
vmine for m=^aaihe obtained.
A neeJ'er approximation to h can then be determined by choosing a new valae of C,
fgffgo T»ble XXI corresponding to this approximate valae of m, and recalcalatiD^
h from equation (1).
jr^ampU, An earthen channel to be kept in very good condition, having a slope
ai 1 in 10,000, and side slopes 2 to 1, is required to discharge 100 cubic feet
«MMad : to find the dimensions of the channel; take C=70.
troni
the ad» iloMt C» »e> an4 Uie M
TV »Mui T^omKj
c^ m whkh is nni
tytr nf dJK Ke sol
ja»i
:^ r aoififlMnt C i* unknowfi, ainm k ^peads upon ih» ^
. ml «vm if 1 tmlm: fnvChv^ uemned the equAUaa cusot I
' * !< dcwftlkk, Uieretore, u> toltt bj ftppn)^iiuLtioQ>
ini. fisid from mlamn 4« T^e XXI, the con^sponjuf
ate
vhkli viU probfthlT not btttl^
Fv wmtm iDt ^ &oin the fbnnala
TW ycv^tMi cift:^ hw fodtwl In m
TW wiJ^M^M « loii^ I
A Moe^ Tilac for L 1
Igr ■■hillT^hmt ^titdm fonnok (2), it ^ in ^paai!^
. if eo. ^ »FF^ ,
d as 4iiiMimil bj luing thi i vtiM <£ '
mmiij Avail )ir«eii«»l ptir poMp
' ««3r by ^ kifaritbi&k fom lik
t«al^miI'$i^^EBttf^, mdllMflOflg
•
FLOW TS OPEN CHANNELS
227
£xmmtpir. The depth of &a ardinary earth ch&nn&t Ib 4 feet, the uide slopes
1 ta I4 tbe Alope 1 ID €0€0 and the diiohaife is to be 7000 oublc feet per minute.
Find the bottom width of the fhnnn^K
Ammaaxe m "emi&t for m, iaj 2 feet.
Frc»iii the lof^tilhinic formula
tl log r=log 1 + 1*6 log TO 46682 , (3),
V  1*122 feet per aeo*
7000
'^^^ ^"1022:00"^^ '^'l*^*^*'
, 104 IS „„,^
.. 1= — 3 ^32 feet.
4
SulvtitiiliiiK thi^ mine for I in eqtuitioo (2)
4x23 + 16 „ ,«
22 + 8^2
Bcwlcnlatinir p &om fonucLla (S^
Tlnflii A = 75 feet,
1=1475 feet,
id in =2*88 feet.
Tbe ^iTst ralue of i ia, therefore, too large, and this second valati U too small
Tbird irmloes were foiuijd to be 17 =£ 1*455,
A=eo2,
f= 16*05.
in = 29B5*
X^i* ^vmlna cf 1 19 again too lafge.
▲ tatrtti ^coiatioD gave 17 1*475,
A = 79 2,
i=15 8,
afifirDiimatioii has been carried auffidentlj far^ and ereQ further tlian ii
_ tor met chattnele the coeffiqient of roughness k eaiiuot b» trusted to
mm Acenimcjr correvpondlng to the stDall differeDoe between tli^ third and four^i
wm^lnes of ^
PfiM^tm 6. HftTiDg giTen the boltom width £, the slope t and the tilde alnpes of
tjm p e« Qidal channel, to hud the depth d for a given diBcharge*
Thlm problem is aolTsd exaetlj as the last, by first afisumlng a value for m, and
J an approiima(« value for p from the formula v = C Vm7.
bj BQbetitntion in ei^Qation (I) of the last i^oblem^ and solving the
Tbm^
/q I I
atmg this valae for d In equation (2), a new valne for m can be found »
beoeev a Moond approiimation to fi, and bo on.
Um«g t^ logadthjnio formula the procedure is exaotlj the same as for
^Pro^Uml*. HaTing a natural stieam BC, Fig. 13dri, of ^ven slope, it in required
iluiafTiiIpe the point C, at whiqh a canal, of trapezoidal s^ciioti, whkh is to
jf«r m ^finite qoantity of water to a given point A at a given level, shall be
I to jptn ibe stream so that the cost of the canal is a minimum.
Tbe aolnlion here given 10 praotioally the same as that given bj M. Flamant
t fiim «3L0Qlktit treatise Hydmuli^ue,
and tbKt^fore
from which wfi=
228
Let I be the ilope of the ntrmtn, i of the msud, k Ihfi height ftboTt ioa» j
of the Burfooe of the watef at A, and h^ of ib«
water in the tttr^tu at B^ at «om€ diatanoe X4
from C.
Let L be alfto the length and A the
MCtioiial area of the canal, aod t«t it ba
aaaamed that the ««ctLOD of the canal !» of the
moat eoonomieo] form, or m =  .
Th« nde elopes of Ihe canal wil) be tw^
aooording to the tiatai^ of the atrata through which the canal u eat, uid 1
8uppof»«<i to be known.
Then the level of the water at C ia
Therefore
Let f be the bottom width of the
■ectio& la then dl^t^^ and
i
SobttitDtiug f m for d,
4
The ooefficient C in the formula v=C iJnU may be assmned constant.
Then r«=C*iiM,
and r*=C*m«i«.
Q
For V snbetitnting ^ , and for m' the above value,
<y C^At<
A*'
and ^'•'= ^ <2 >/?+l  1).
Therefore
The cost of the canal will be approximately proportional to the product of the
length L and the eroes sectional area, or to the cubical content of the exctvitioii.
Let £ik be the price per cubic yard including buying of land, excavation etc Let Ir
be the total cost.
Then £x=£k,h.A
This will be a minimum when jt =0.
Differentiating therefore, and equating to zero,
i*=tl.i,
and t = fL
The most economical slope is therefore  of the slope of the natural streun.
If instead of taking the channel of the best form the depth is find, tli^
slope tss^.I.
FliOW IN OPEN CHANNELS
229
There hate heea l#o lusiiiDptions mftde m the calcuJAlioti, neitliei of whifih is
ri^dljr trite^ tho &i^t beiog that the coefficieQt C ia cod slant, And the second th&X
the price of the isftnaJ ia proportional to ita cross sectiontU aTCft.
It win not &]vvjs be po««ible to adopt the elope tbuH founds as the mean
^reloctt^ mafit be maiDlained within thif limits Kiven on page 216, and It ta not
adviaable llial the slope shonld be loss than 1 in 10,000.
EXAMPLES.
(1) The areft of flow in a sewer was found to be (h2d sq. feet; the
peruneter 1*60 feet; the mcliaation 1 in 38' 7. The mean Telocity
llo^r was S'12 ^t per second* Find the value of G in the formula
(2) Tbe dramage area of a certain difitrict was 19'32 aerefli the whole
being impermeahle to rain water. The maximum intensity of the
was 0*300 ins^ per hour and the maximum rate of discharge regis 
in the sewer was 96% of the total rainfall.
Find the size of a circular glazed earthenware culvert having a slope of
is 50 aoi table for carrying the storm water.
(0) Draw a curve of moan velocities and a curve of discharge for an
«ig^Hili&ped brick iewer, using Bazin's coefficient. Sewer, 6 feet high by
^ ieei groatesi width; dope 1 in 1200.
(4> The sewer of the previous question is required to join into a main
Odt^ftll «ewer. To cheapen the junction with the main outfall it is thought
ad^satile to make the last 100 feet of the sewer of a circular steel pipe
feet diameter, the junction between tbe oval sewer and the pipe being
Jy shaped so that tliere is no impediment to the flow*
Find what fall the circular pipe should have no that its maximum
^ shall be equal to the maxlmmn discharge of the sewer. Having
ihe slope* draw out a curve of velocity and discharge.
i$) A oftnal in earth has a slope of 1 foot in 20,000, side slopes of
faodflcmtal to 1 vertical > a depth of 22 feet, and a bottom width of
find the volume of discharge.
Baxin^ coefficient y^2'35.
^fi) Oire ihe diameter of a circular brick sewer to rnn half full for a
peipQlation of 80,000, the dinmal volume of sewage being 75 gallons per
head, the period of maximum flow 6 hours^ and tlie available fall 1 in lOOO.
Inst C. E. 1906,
(7) A channel is to be cat with aide slopes of 1 to 1 ; depth of water,
ak^iet 9 inches per mile: discharge, 6,000 cubic feet per minute,
f approximation dimensions of clianneL
0f An area of irrigated land requires 2 cubic yards of water per hoar
f acre. Find dimensions of a channel 3 feet deep and with a side slope
^1 lo L Fall. 1^ feet per mile. Area to be irrigated, 6000 acres, (Solve
*F apfiroadmation^) y = 2*35*
(9) A trapezoidal channel in earth of the most economical form has a
i 0f 10 feet uad side slopes of 1 to 1. Find the discharge when the
\ li IS inches per mile. y=2*35.
S30
HYDRAULtCS
(10) A river has tlie loUowiiig section :— top width, 800 feet ; depth
water^ 20 feet ; mde slopes 1 to 1 ; fail, 1 foot per mile* Find the disdiAig^
iismg Bazin'B coefficient for earth channels*
(U) A channel ib bo be ooiwtnicted for a, discharge of 2000 cubic feet
per second ; the fall ii 1 feet per mile ; side ^opes, 1 to I ; bottom widlhr
10 times the depth. Find dimensions of diannel. Use the approximate'
formula, v=50^ftrL
(12) Find the dimenstoim of a trapezoidal earth channel, of the
economical form, to convey 800 cubic feet per second, with a fall of
per mile, and side slopes, 1 to h (Approxiniate formula,)
(13) An irrigation channel, with side slopes of H to 1, receiver 600
cubic feet per second. Design a suitable channel of 3 feet depth lad
determine its dimensions and slope* The mean velocity is not to exceed
2^ feet per second, y = 2 ■ 3 5 .
(14) A eanal^ excavated in rock, has vertical sides, a bottom widtli d
160 feet, a depth of 22 feet, and the slope is 1 foot in 20^000 feet. Find tlm
discharge, y = 1 '64,
(15) A length of the canal referred to in question (14) is in earth. It
has side slopes of 2 horizontal to 1 vertical ; its width at the water ]kt
is 290 feet and its depth 22 feet.
Fmd the slope this x>rtion of the canal should have, taJdng y aa S"!^
(16) An aqueduct 95 iJ miles long is made op of a cud vert &0J miJn
long and two steel pipes 8 feet diameter and 45 miles long laid side byst^
The gradient of the culvert is 20 inches to tlie mile, and of the pipes 2 1«e*
to the mile. Find the dimensions of a rectangnlar culvert lined with ^
pointed bricks so that the deptli of flow shall be equal to the wid^ crf lb*
culvert^ when the pipes are giving their maximum dischaxge.
Take for the culvert the formula
and for the pipes the formula
. 000061 tJ^*88
i^
■00050. t?"
(17) The Ganges canal at Taoli waa found to ha^e a slope el
and its hydraulic mean deptli m was 7^0 feet ; tlie velocity 8
by vertical floats was 2*80 feet per second; find the value of C io3"'
value of y in Bazin's equation,
(18) The following data were obtainad from an aqueduct lined fi^
brick carefully pointed :
mmeties
ID me Ires per S4».
■229
00001326
'336
■S81
tt
484
•5S8
t*
■596
'680
It
'691
'888
II
*769
991
fi
'848
1148
If
'918
M70
n
'922
PLOW IN OPEN CHANNELS 231.
^^= is oirdin&tes, —  ba absciesae; Imd T&liieaof a and ^ in Barb's
1^ and thos dedaee a valne of y for this aqueduct.
Aa aqtiedoct l(fl\ miles long eonsbts of 13 miles of siphon, and
niodsr of a masonry cnivert 6 feet 10 inches diameter with a gradient
I iOPCl The niphone coniiit of two lines of caetirQii pipes 43 inches
Air tiaruig a slope of 1 in 500, Determine the maximum cUaaharge.
An sqaednet ocmststs partly of the section shown in Fig, 131,
iT» ft&d fMutly (ie, when crossing valleys) of 38 inches diameter cast
tbe minimnm dope of the siphons^ so that the aqueduct
liacisafge 15,000,000 giillons per day^ aiid the slope of the masonry
so that the water shall not be more than 4 feet 6 inches deep in
) Calealaie the qtiantity delivered by the water main in question (30),
Lper day of 34 hours,
^Hpiooiit, representing the water supply of a city, is discharged into
Hn at tha rate of one half the total daily volume in 6 hours, and m
lebled by rainfalL Find the diameter of the circular brick outfall
which wUi carry off the combined flow when running half full, the
hie Ml being 1 in 1500, Use Bamn's eoefiicient for brick channels.
\) Determine for a smooth eyhndrical castiron pipe the angle
at the centre by the wetted perimeter, when tlie velocity of flow
Determine the hydraulic mean depth of the pipe nnder
Land, Un, 1905.
% A 8*fliQh drain pipe is Laid at a slope of 1 in 150, and the valne of
17 {v^e^mi). Find a general expression for the angle subtended at
mtre by the water hue. and the velocity of flow ; and indicate how Uie
d e^uatioB^ may be solved when the discharge is given. Loud. Un.
Ir Sifter^ Qit^e^unt ef tf%« hUiarical ihi^^hpm^nt of the pip^ and ehannel formulae.
kui reniJirki^bte Uint, although tti^ pmctiise of conduGting water ulnng piped
bapek for doniaitio and other purpnieg haa been cfLrried on for many
^Hhio aerioiii ittempt bo discover the la^'fi regulating th^ fiow fteems
Wmmtk att^topted nntiJ Ih*^ eigblc^nth centurj. It soema diMcult to realise
It giffmnlie «i?b ernes of wnti^r ihaiiibiiiion of the ancstetit Gitiea cnuld hftve be^n
~ viibout E^ycb kouw ledge, but ci'rtain it i«, tb&t whatever infonnnticin they
I. it WM lost diiriuf; ibe middle ages,
■i if pMoiiar interest to note tbe trouble taken by the Boman engineers in
of their aqueducts. In order to keep the elope oonBtftnl they
thnjugb biUs and a&rried their fujuedocts on mnf^iSeant arehea. The^ j
a^nedtiet wai 38 milei loug and dad a coiii^tAnt dope of five feel per mile,
mtiy they were unaware of ihe »tmple fact that it is not necessary for a pipe
idaet aonneoting two r^aervoira to be laid perfectly straight t or sine they
the water tt all parla of the aqoedticts to b^ at atmospheric pressure,
Seliwet^er in his interesting treatise on bydroHtatiiiB and hydraulics
in 1729 quotes experiments by Marriott Ehowitig that, a pipe 1400 yards
i loobaA diameter, only gave  of the diiM^harge which a bole 1 inches diameter
idt of m tiitik wonld giv« under the saoie heudp and also eiplama that the
of the Hquid in the pipes is diminished by friction, but he is entirely
biws regulatinjj the flow of Huida through pipes. Even as late as
2^2 HYDRAULICS 1
17B6 Dq Bta^t* wrote, "We mm jet m absolute ignoranoe of the laws lo wblek ih«]
moTement of wat^r is enbjected.*^ I
Thei <»adii^»t recorded experimetils of any valu<} on long pipes lyne tboM oil
Couplet, in whiish he measoced the flow through the pipes wm^ aupplicHl tbal
Ibmoiia fountains of Versailles in 1732, In 1771 Abb^ Bossni made experimefite nal
iow in pipes and ohannals, these bein^ followed by tba expedments of Do BuaU wbal
eironeotiflly argued that the loss of head due p f notion iu a pip« was indsiiendeall
of the internal »urfa43e of the pipe, and gave a oomplieated formula for the Tdoei^l
of ^ow when the head and the leo^th of the pipe were known. I
In 1T75 M. Chezj from expenmeuts upon the flow in an open eatial, caine IdH
the conclusion that the 6uiJ friction was propoTtioual to the velooitj squared, and'
that the nlope of Ihe channel multiplied by the croB» sectional area of the stt«*in,
wag equal to the prodoct of the length of the wetted surface measured on tbe enw
aeetiont the relooitj iquared^ and aotue eotiatant* or
iA^Fav* :,, .,.41},
I being the slope of the bed of the channel, A the croe^ sectional area of the atieam,
P Ihe wetted perimeter^ and d a coefficient.
From thi^ ie dedtieed the wellknown Ghezj formula
^^^ ^iojmu
Pronyf , applying to the iow of water in pipes the results of the cl&aaical esp«ii*
ments of Coulotoli on fluid frietion, from whieh Coulomb had dednoed the lav thst
fluid frieiioB was proportionai to avkbty^t arriyed at the formula
«i(= at? + ^« = [ + p\ r/^.
This is elmilar to the Chesj formola* (  + /9 j being equal io ^ ,
By an eiatnination of the experimentii of Couplet, Boseut, atid Du Bnat, Proii
gave values to a and ^ whiah when trane formed into British units are,
0^00001733,
/3 = '00010614,
For TclooitieSi above S feet per seeond, Pronj negleotod the term containing tbej
first power of the velocity and deduced the formula
Ue continued the mistake uf Du Bunt and assumed that the Criotion wai in<
dependent of the eoodition of the internal ^orfaoe of the pipe and gave the foUowioir
eiplanation : *' When the fluid flow? in a pipe or upon a wette^l surface a Slnx''
fluid adhere;; to the surface, and tltis film may be regarded an enoloiing the tail
of fluid in raotiont*' That such a film encloses the moving water reoeiyes soppenl
from the eiperiments of ProfesHor Hele Shswf. The expeiiment!? were made npAl
sueh K KmaD seals that it is dilBcult to say how far the results obtained arft indie**'
tivti of the conditiona of flow in large pipes, and if the ilm exists it do^ not sera
to m\ in the way argued by Prony.
H
The value of t in Prony's formula was equal tn y , H inaluding, not onlj tt»
loaa of head doe to friction bat* as measured by Couplet, Bossut and Da I'
it ako included the head neoensary to give velocity to the water and to overc.
leflistance^ at the entrance to the pipe.
Eytelwein and also Aubisson, both niade aUowaoces for these loisea^ hs ^
tracting from H a quantity x— , and then determined new yalnes for a and I ia tb«
formnla
* Zf XhVcowTf priliminaire de m* PHnciptt d'k^draulique,
f See also Girard'a Movement dc9 fluids dans let tub^t eapittairew, 1817,
^ Traiti d*hydrauliqut, f Engineer, Aug, 1897 and Maj ISUB.
FLOW m OPEN CHANNELS
233
Th gy iAv« lo a and b the rollowing TaJaes.
Eytelwem (r  ^000023584,
h= ^000085434.
Aubiaaon' a == 'OOOO 1 8837,
6=1)00104392.
By iMg)«ctui^ the term oonlaiiiiiig v to the first power, and transform ing the
I Atibisflon'B formolA reduces to
•'=«\^JT
Hd
3S6er ■
ToiiJDfc* tQ tbd Efuryc^cpQ^dta Bnl^nf^t^u, gave a complicated formula for ? when
»cul ^ were known, bnt gave the fiimfilified formula, for veJocitiee unoh oa
Are geiQ^imllj met with m practice,
8t Teoant made ft decided depftrtEue by making  prdportlonal to v'r imtead of
I r* M to lite Ch«^ IbimiilA.
Wbioi eipiMKjd IP EngHflh feei aa unitSi hia formula becomes
v = 206(>?fi)T^^
I by an examination of the early experiments together with ten others bv
avH
and on* by M. GneyttArd gave to the ooeflScient a in the formula ft = — 
rftlne
, b« be made it to vaiy with the Telocity*
valiiia of a and ^ being ol^QOI U ,
Wram lins formnlA iablas were drawn up by Weiabach, and in England by
* J, which were considerably need for calciilations relating to flow of
in pipes.
l^iflcy, as explained in Chapter \\ made the coefticient a to vary with the
r, and Qag«n proposed to make it v&ry with both the velocity aod the
M^ formula then beoame
«■=&.")'"•
The fbrtnclae of Qangmllet and Etitter and of Biusin hare b#eii given in
I V and VI.
lAmpe &om axperimentB on the Dantzig maina and other pipes proposed
ElonBUlft
*"ffi* <
modifying St Tenant's formula «nd antioipattng the formulae of Beynoldif
a^d Unwrnn, Id which,
i ftfid p being variable coeMcienta,
* Traiti d'htfdfaulique.
CHAPTER VII.
GAUGING THE FLOW OF WATER,
142. MeaBuidng the ^ow of water by weigMng.
In the laboratory or workshop a flow of water can geiws
be tueRsiired by collecting the water in tanks ^ and either
direct weigh iiigi or by measuring the voltime from the \m
capacity of the tank, the discharge in a given time cau
determined, lliis is the most accurate method of measu
water and should be adopted where possible in ex per
work.
In pump trials or in measuring the supply of water to boilfl
determining the quantity by direct weighing has the di^ct
advantage that the results are not materially afTtctetl by
changes of temperature* It is generally necessiirj" tu have W
tanks, one of which is filling while the other is being weifbeJ
and emptied. For facility in weighing the tanks should etJ
on the tables of weighing machines.
143, Meters.
Lirwri meter. An ingenious direct weighing meter
gauging practically any kind of liquid* is constrnckHl .
Figs. 136 and 137.
It consists of two tanks A" and A*, each of which can
on knife edges BB. The liquid is allowed to fall into a she
wliich sl\^vels about the centre J, and fi'om which it falls
either A* or A' according to the position of the shoot. The 1
have weights D at one end, which are so adjusted that wi
** tain weight of water has run into a tank* it swings ovt*r i
formu.j^l^^ position, Fig. 136j and flow commences thrtvagt
H "oe C. When the level of the liquid in tlie tank
^^H , j^ Dhc^^^^' ^^^ i^t^igbts D cause the tank to come back]
^^H t See also csitioUj but the siphon continues in action until I
OAUOiya THE FLOW OF WATEB
235
lenly tilts over tlie shoot F^ and the liquid is discharged
other tank. An indicator H registers th© nuTiiber of
le tanks are tUledjatid as at each tippling a detinite weight
is etnptied from the tank, the indicator can be marked
>unds or in any other unit.
Fig. im.
Fig. VAl.
Liner t direct weighing meier.
Meaatirmg the flow by means of an orifice
c<:»t?fficient of discharge of sharpedged orifices can be
ith cousiderablo precisioUj from the tables of Chapter IVj
coefficient for any given orifice can be deterniined for
hm he^ids by direct measurement of the How in a given time^
tdeicnbed above* Tlien, knowing the coefficient of discharge at
■tidii a curve of rate of discharge for the orifice, as in
may be drawn, and the orifice can then b© uised to
* a continuous flow of water.
! orilice should be made in the side or bottom of a tank. If
I the side i>f the tank the lower edge sh<juld be at k^ast one and
If to twice its depth above the bottom of the tank, and the
of the orifice whether hori?.ontal or vertical should be at
0tie and a half to twice the width from the sides of the tank,
ink i»hould be provided with baffle plates, or some other
pnient, fur destroying the velocity of the incoming water
t cnFuring quiet water in the neighbourhood of th© orifice. The
pent of dischargu i^ otherwise indefinite. Th© head over the
jnhould be observed at stated intervals. A headtime curve
\ h^ftd aas ordinate^ and time as abscissae can then be plotted
, 189.
From the headdischarge curve of Fig. 138 the rate of discharge
found for any head h^ and the curve of Fig. 139 plotted
of this curve between any two ordinates AB and CD,
I
I
I
I
I
]
236
HYDRAULICS
which is the mean ordinate between AS and CD multiplied bytbej
time tj gives the discharge from the orifice in tinae t.
The head h can be measured by fixing a scale, hBrlng '• "
coinciding ^vith the centre of the orifice, behind a tube on :
of the tank.
Fig, im.
145. Meaauring tlie flow in open channels.
Large open channels : floats. The oldest and simplest me
of determining approximately the discharge in an open cha
by means of floats.
A part of the channel as straight as possible is selected, i
which i\m flow may be considered as uniform.
The readings should be taken on a calm day as a down^st
wind will accelei*ate the floats and an upstream wind retard 1
Two cords are stretched across the channel, as near to J
surface as possible, and perpendicular to the direction of flow,
distance apart of the cords should be as great as possible coi: 
with uniform fiow^ and should not be less than 150 feet, i
boat, anchored at a point not less than 50 to 70 feet aboTe alreafflt j
so that the float shall acquire before reaching the first Hneaj
imiform velocity, the float is allowed to fall into the streiim audi
GAUOIKG THE FLOW OF WATEH
237
m time carefuDy^ noted hy means of a clironometer at wliich it
KM oa l>ot]i the first aod aecond line. If the velocity ig slow, the
Issenrei* ma^y walk along the bank while the float is moving. from
me cortl to the other, but if it is greater than 200 feet per minute
M observers will generally be reqaired, one at each line,
^kA better method, and one which enables any deviation of the
Hi fmni a path perpendicular to the lines to be determined, is,
Sr two observers provided with box sextan tSj or theodoHtes, to be
ioned at the points A and B, which are in the planes of the
hneo. As the float passes the line AA at D, the observer
L fign&Is^ and the observer at B measures the angle ABD
if both ane ppivided with watches, each notes the time.
en the float passes the line BB at E, the observer at B signals,
the obsen^er at A measures the angle BAE, and both
again note the time. The distance DE can then be
arately determined by calculation of by a sc^le drawing, and
\ mean velocity of the float obtained, by dividing by the time.
I To ensure the mean velocities of the floats being nearly equal
be mean velocity of the particles of water in contact with
, their horizontal dimensions should be as small as possible,
to reduce friction, and the portion of the float above the
of the water should be very small to diminish the effect of
^.pointed out in section 130, the distribution of velocity in
averse section is not by any means uniform and it is
(T, therefore^ to obtain the mean velocity on a number of
planes, by finding not only the surface velocity, but also
JvekxHty at various depths on each vertical.
1 146. Surface floats,
5m^€e floats may consist of washers of cork, or wood, or
small fioating bodies, weighted so as to Just project above
surface The surface velocity is, however, so likely to
It?cted by wind, that it is better to obtain the velocity a
distance below the surface*
147. I>ouble floats.
To uiesi^ur*? the velocity at points below the aur&ce double
are employed* They consist of two bodies connected by
of a fine wire or cord, the upper one being made as small
lible so as to reduce its resistance,
1% on the Irrawaddi, used two wooden floats connected
\ fine fittkmg linej the lower float being a cylinder I foot longi
• Proe.lmL V. E„ IS98.
riib
238
HYDRAULICS
and 6 iBches diameter, hollow undemeatli and loaded with
gink it to any required depth } the upper floaty which swam od
surface, waa of light wcKMi 1 inch tliiuk, and carried a small Am
The surface velocity was obtained by sinking the lower I
to a depth of 3J feet, the velocity at this depth being not i
different from the surface velocity and the motion of the float i
independent of the effect of the wind*
Kg. 141. Gurby'f ourrent m©t«r<
Subsurface velocities were measured by increasing the
of the lower float by lengths of Si feet until the bof
reached.
OAnGINe THE FLOW OF WATER
H Gordon has cM>mpared the results ob tamed hy floats with those
^Btftizied hf mestis of a cuiTeiit meter (see section 149). For
HpaU depths and low velocities the results obtained by double
^■KtB are ^rly accurate^ but at high velocities and great depths,
^Ee Talocities obtained are too high. The error is from to 10
per cent.
Double floats are soinetiinea made with two eimilar floats, of
the same dimensioiiS one of which is ballasted so as to float at any
reiiuired depth and the other floats just below the surface. The
wwiodtf^ of the float is then the mean of the surface velocity
mad the velocity at the depth of the lower float.
148. Bod floats.
The mean velocity, on any verticalj may be obtained ap
pro jdmately by means of a rod float, which consists of a long rod
baring at the lower end a small hollow cylinder, which may be
fillad with lead or other ballast so a^ to keep the rod nearly
vertical.
The rc»d is made sufficiently long, and the ballast adjusted, so
thBt ita lower end is near to the bed of the stream, and its upper
end project* slightly above the water. Its velocity is approximately
tn^&n velocity in the vertical plane in wliich it floats.
149. The onrrent nieteT.
The discharge of large channels or rivers can be obtained most
ccitiv'oni*?ntly and accurately by determining the velocity of flow
number of pi>ints in a transverse section by means of a current
The arrangement shown in Fig. 141 is a meter of the anemo
ineter type, A wheel is mounted on a vertical spindle and has
fi%e conical buckets. The spindle revolves in bearings, from
^rkicfa all water is excluded, and which are carefully made so
tbat the fricrion shall remain constant. The upper end of the
«tpixidle extends above its bearingj into an airtight chamber, and
im shaped to form an eccentric. A light spring presses against
eecentric, and successively makes and breaks an electric
r ■  the wheel revolves. The number of revolutions of the
r njcorded by an electric register, which can be arranged
any convenient tlistance from the wheeL When the circuit is
taade, an electro^magnet in the register moves a lever, at the end
of which is a pawl carrying forward a mtchet wheel one tooth
I for each revolution of the spindle. The fi'ame of the meter, which
im made of bronze^ is pivoted to a hollow cyUnder which can be
clamped in any desired position to a vertical rod. At the rightr
240
HTDBAtTttCS
hand side is a rudder having four lig^ht metal wing^ which
balances the wheel and its frame. Wlien the meter is being miei
in deep waters it m suspended by means of a tine cable, and
the lower end of the rod is fijced a lead weight. The el
circuit wires are passed through the trunnion and so haw
tendency to pull the meter out of the line of current,
placed in a current the meter is free to move about the hoi
axis, and also about a vertical axis, ao that it adjosta il
the direction of the current.
The meters are rated by experiment and the makers
the following method. The meter should be attached to
al a boati as shown in Fig. 1^, and immersed in still wmter
less til an two feet deep* A thin rope should be attached
boatj and pa^ed round a pulley in lint* %vith the course in
the boat is to move. Two parallel lines about 200 feet
should be staked on shore and at right angles to the course
boat. The boat should be without a rudder, but in the huai
the observer should be a boatman to keep the boat from
Fig, 142,
into the shore. The boat should then be liauled between i
ranging lines at varying speeds, which during each passage i
be as unifonn as possible. With each meter a reduction
supplied from which the velocity of the stream in feet jm^^t b
can be at once determined from the niunber of revolutions j
per second of the wheel.
The Haftkell meter has a wheel of the screw ]
revolving upon a horizontal axis. Its mode of a.
eimilar to the one described.
Comparative tests of the discharges along a rectangoiarj
as measured by these two meters and by a sharpedged '
had been carefully calibrated, in no case differed by moit^
5 per cent, and the agreement was generally much closer*.
* Mnrphj on current Meter and Wdr lU^abargesv Pruceedinp i»
Vol TZYiu p. 779.
GAT7GINO THE FLOW OF WATEB
241
150. ntot tube.
Another apparatus which can be used for determining the
docity at a point in a flowing stream, even when the stream is of
ull dimensions, as for example a small pipe, is called a Pitot
die.
In its simplest form, as originally proposed by Pitot in 1732,
=*=,
Jll
E
Fig. 143. Pitot tube.
3k ooDBisEtB of a glass tube, with a
BmII orifice at one end which may
be tomed to receive the impact of
Ik stream as shown in Fig. 143.
Be water in the tube rises to a
iMght h above the free surface of
Me water, the value of h depending
i the velocity v at the orifice of
Btnbe. If a second tube is placed
I the first with an orifice parallel to the direction of flow,
tte water will rise in this tube nearly to the level of the free
■Bzface, the &11 h being due to a slight diminution in pressure
9X the mouth of the tube, caused probably by the stream lines
laving their directions changed at the mouth of the tube. A
&rther depression of the free surface in the tube takes place>
if the tube, as EF, is turned so that the orifice faces down stream.
Theory of the Pitot tube. Let v be the velocity of the stream
at the orifice of the tube in ft. per sec. and a the area of the
orifice in sq. ft.
The quantity of water striking the orifice per second is wav
pound;s.
to
The momentum is therefore  . a . v' pounds feet.
If the momentum of this water is entirely destroyed, the
pressure on the orifice which, according to Newton's second law of
motion is equal to the rate of change of momentum, is
P =
wav
find the pressure per unit area is
9
The equivalent head
h  V!^ = 
According to this theory, the head of water in the tube, due to
le impact, is therefore twice  , the head due to the velocity v, and
242
the water shotild rise in the tube bo a height above th©(
equal to h.
Experimoiit allows that the actual height the water ]
tube is more nearljr equal to the velocity head
and the head b ig thum generally taken as
c being a coefficient for any
is fairly conKtantv
Similarly for given tubes
tube, which escperunent i
and
The coefRcienta are detei
the velocities of which are
body which moves through
carefully meitj^uring h for diUereu^ velocities
t
7
by placing the tubes in i
I, or by attaching them \
iter with a known vclocit
.i.
Fig. 144.
Darcy* was the first to use the Pitot tube as an instrumei
precision. His improved apparatus as used in open channels
sisted of two tubes placed side by side as in Fig. 144, the or
in the tubes facing upstream and downstream respectively.
Reeherchef Hydrauliques, etc., 1857.
QAI7GIKO THE FLOW OF WATER
2m
bobes were connected at the top^ a cock C^ being placed in the
tnon tube to allow the tubes to be opened or closed to the
At the lower end both tubes could be closed at the
e time by uieans of cock C. Wlien the apparatus is put into
in^ water, the cc»cks C and C* being open, the free surface
is the tube B a height hi and is depressed in D an amount
Tbfi cock C^ i&> then closed, and the apparatus can be taken
i the wster and the difference in the leYel of the two columns,
hhi^ fhj
with cotusiderable accuracy*
deeiredi air can be aspirated from the tubes and the eolunms
to rise to convenient levels for observation, without moving
The difference of level will be the same, whatever
in the upper part of the tubeB,
i 145 shows one of the forms of Pitot tubes, as experimented
by Pnifeasor Gardner Williams*, and used to determine
ribution of velocities of the water flowing in circular pipes,
arrangement shown in Fig. 146, ia a modified form of the
tias used by Freeman t to determine the distribution of
in a jet of water issuing from a fire hose under con
ile pressure. As shown in the sketch, the small orifice
the impact of the stream and two small holes Q are drilled
be T in a direction perpendicular to the Bow, The lower
^paratns OV, as shown in the sectional plan, is made
so as to prevent the formation of eddies in the
hood of the orifices. The pressure at the orifice is
ed through the tube OS, and the pressure at Q through
QR. To measure the difference of pressure, or head,
two tubeS) OS and QR were connected to a differential
SLttiilar to that described in section 13 and very small
of head could thus be obtained with great accuracy,
lobe shown iii Pig* 145 has a cigarshaped bulb, the
orifice O being at one end and communicating with the
OS. There are four small openings in the side of the bulb,
M any Tariations of pressure outside are equalised in the
The pres^upes are transmitted through the tubes OS and
> m differential gauge as in the case above,
r I%» 147 is shown a special atuffingbojc used by IVofessor
mOBBtto allow the tube to be moved to the various positions in
'•the I
ipri
of Fit0t tiibea api atetl by ProfeSBor WiHiflma, E. S< Oele tLnd
of the Jm^\C,E,, VpL uli.
16^^
1
I
I
244
HYDRAULICS
the cross section of a pipe, at which it was desired to detenmne
the velocity of translation of the water*.
Mr E. S. Colet has nsed the Pitot tube as a continaoas meteri
the arrangement being shown in Fig. 148. The tabes were con
nected to a U tube containing a mixture of carbon tetrachloride
and gasoline of specific gravity 1'25. The difference of level ct
the two colunms was registered continuously by photography.
^s^szsi^^ssm
\^A^WA^JJ?MJJ^JM'fJJ^J>>JJJi^M>}^>. ■^■gry?g
Fig. 147.
Fig. 146.
Fig. 148.
The tubes shown in Figs. 149 — 150, were used by Bazin to
determine the distribution of velocity in the interior of jets issoiiig
* See page 144.
t Proc, A.M.S.C.E., Vol. xztu. See also experiments by Murphy and TomuMi
in B&me voiume.
OAUGIKG THE FLOW OF WATER
245
{rem orifices, and in the interior of tlie nappes of weirs. Each
tobe consisted of a copper plate 1*89 inches wide, by '1181 inch
thick, sharpened on the upjier edge and having two brass tubes
"0787 inch diameter, soldered along the other edge, and having
orifices '059 inch diameter, 0'394 inch apart. The opening in tube
A was arranged perpendicular to the stream, and in B on the face
of the plate parallel to the stream.
cai —
I£
2
Kg. 149.
Fig. 160.
15L Calibration of Pilot tubes.
Whatever the form of the Ktot tube, the head h can be
expressed as
h =
cv'
or
2g'
= ksf2gh,
Ic being called the coefficient of the tube.
This coefficient h must be determined by exi)eriment under
conditions as near as possible like those under which the tube ^vill
be used to determine velocities.
To calibrate the tubes used in the determination of the distri
bution of velocities in open channels, Darcy * and Bazin used three
distinct methods.
(a) The tube was placed in front of a boat which was drawn
through still water at different velocities. The coefficient was
1"034. This was considered too large as the bow of the boat
probably tilted a little, as it moved through the water, thus tilting
the tube so that the orifice was not exactly vertical.
(h) The tube was placed in a stream, the velocity of which
was determined by floats. The coefficient was TOOG.
(c) Readings were taken at different points in the cross
section of a channel, the total flow Q through which was carefully
measured by means of a weir. The water section was divided
* J^eherehes Hydrauliquet,
246
HTDIUUUC3
[ two metikoAB of caUbit
^nbes throQgli still wfttlflr;
a cmmmlerential
i 8 inches de^p wa& buflt
sntre line, which W8§ tbs
hes. The tube to b^ rafe
i a central shaft wkicli m
and oeilitigrf and whicb
observer. The gang^ ^as
into areas, and about the cemm of mdt a raaiiitto' of tW
waa taken. CaDtng m the araa of oa»of theaa Hctio&9, and ^
reading of the tube, the coefficient
i = — 9=
and was foimd to be *fl93
Darcy* and Ba^o also fotmd that by changing the positioti
the orifice in the pressure tube the coefficients changed
giderably.
Williama, Hnbbell and Fenkel
which gave very diffei^nt re^nlts
The first methi^d was to mo'v
known velocities. For this pc
rectangolar in section, 9 inches i
galvanised iron. The diameter t
the path of the tube, wag 1 1 feet
WB8 supported npon an arm atta^
free to revolve in bearings on i
supported the gauge and a sear n.
connected with the tabe by rubber hose. The arm carrying the
tube was revolved by a man walking behind it, at as unifonaa
rate as possible, the time of the revolution being taken by means
of a watch reading to i of a second. The velocity was mail*
tained as nearly constant as possible for at least a period of
5 minutes. The \'a!ue of A: as determined by this method was ■9'2(
for the tube sho^^n in Fig* 145,
In the second method adopted by these workers, the tabe iras
inserted into a brass pipe 2 inches in diameter, the discharge
through which was obtained by weighing. Readings were tak^a
at various positions on a diameter of the pipe, while the 6ow in th?
pipe was kept constant. The values of J2gk^ which may be called
the tube velocities, could then be calculated, and the mean value «
Vfl, of them obtained. It was found that, in the caaes in which the I
form of the tube was such that the volume occupied by it in the pipe
was not sufficient to modify the fiow, the velocity was a maxim am
at, or near, the centre of the pipe* Calling this maximum velocity
V
Vc, the ratio ^ for a given set of readings was found to be '8L J
Previous experiments on a castiron pipe line at Detroit having
shown that the ratio ^ was practically constant for all velocities,
a similar condition was assumed to obtain in the case of the brass
* Reeherchet Hydraulique$,
ailFOmG THE FLOW OF WATER
ii
ipe. The fcube wae then fixed at the centre of the pipe, anfl
lugs taken for varinaB rates of discharge, the mean velocity
determined by weight, varying from ] to 6 feet per second.
For the valnes of h thna determined, it was found that /—
sf2gh
i practically cgnstanti This ratio was *729 for the tube shown
Fig 145.
Then since for any reading h of the tuhe^ the velocity v is
u
^_ U V.
I acttml m^n velocity
fe.
Bui
Thi^rcffore
&
ratio of U to V.
814' ^^'
ratio of Ym to V*
For the tube ahowTi in Fig. 146, some of the values of h
'by the two methods differed very considerably*
i^'fm fff the values of k by the two methods. It will
ihat the value of k as determined by moving the tube throug]
iQ water differs very considerably from that obtained i
l^rotining water. In the latter case the pressure was considerably
higher than in the former, and it appears therefore, that k depends
only upon the form of the tube but upon the pressure under
ch it is workings It is, clearly, of considerable importance
the value of k shall be determined for conditions similar
those under which the tube is to be finally used. This
aty of the value of the coefficient under varying con
of prensure, and the difficulty in any caae of accurately
silling it, and the danger of its alteration by objects floating
Lflw ftream, makes the use of the Pitot tube as a velocity
somewhat uncertain, and it should be used with con*
able care. In the handg of Darcy and Bazin it proved an
client instrument in the measurement of small velocities in
canakj but for the determination of velocities in closed
^Is in which the pressure is greater^ it does not seem so
Hit.
U2. Ganglxig by a weir.
( When a stream is so small that a barrier or dam can be easily
ett?d acrc^s it, or when a large quantity of water is required
^ be gauged in the laboratory, the flow can be determined bjr
^*^*" of a notch or weir.
it
248
HYDRAtTHOS
The chaimel as it approachBe the weir should be as far aa
poBflible uniform in sectioTij aud it is desirable for accurate
gauging*, that the sides of the channel be made verticalj and fhe
width equal to the mdth of the weir. The sill should be sharp
edged, and perfectly horizontalj and as high as possible abore th&
bed of the stream, and the dowiisti^ani channel
should be vender than the weir to ensure atmospheric
pressure under the nappe* The difference in level
of the sill and the surface of the water^ before it
begins to slope towards the weir, should be ac
curately Tneasured. This is best done by a Boyden
hook gauge.
153. Tlie hook gauge.
A simple form of hook gauge as made by Gurley
is shown in Fig> 15L In a rectangular groove foruied
in a frame of wood, three or four feet long, slides
another piece of wood S to which is attached a scale
graduated in feet and hundredths, similar to a leTel
staff* To the lower end of the scale is connected a
hook Hj which has a sharp point. At the upper end
of the scale is a screw T which passes through a lug,
connected to a second sliding piece L. This sliding
piece can be clamped to the frame in any position
by means of a nut, not shown. The scale can then
be moved J either up or down, by means of the milled
nut. A vernier V is fixed to the frame by two small
screw^s passing through slot holes, which allow for a
slight adjustment of the zero. At some point a few
feet upstream from the weir*, the frame can be
fixed to a post, or better still to the side of a box
from which a pipe runs into the stream. The level
of the water in the box will thus be the same as the
level in the stream. The exact level of the crest of
the weir must be obtained by means of a level and a
line marked on the box at the same height as the
crest. The slider L can be moved, so that the hook
point is nearly coincident with the mark, and the
final adjustment made by means of the screw T.
The vernier can be adjusted so that its zero is
coincident with the i^ero of the scale, and the slider
again raised until the hook approaches the surface of
By means of the screw, the hook is raised slowly^ until,
* See Hootton 82.
GA^UaiNQ THE FLOW OF WATER
249
J^ 15S. SMgJn'B Hook Gaage.
250
HYDRAULICS
fche surface of the water, it causes a distortion of the light refleete
from the surface. On moving the hook downwards again vm
slightly, the exact surface will be indicated when the distortio
disappears^
A more elaborate hook gauge, as used by Bazin for his expef
mental work, is shown in Fig. 152.
For rough gauging® a po«t can be driven into the bed of tl
channel J a few feet above the weir, until the top of the post
level with the sill of the weir. Tlie height of the water sutCm
Fig. 154. Eecording App«ratms Kent Teaturi Meier.
GAUGING THE FLOW OF WATER 251
the top of the post can then be measured by any convenient
154. Ga^gSjig the flow In plpea; Venturi meter*
Sucb methods as already dei^ribed are inapplicable to the
?meiit of the flow in pipes, in which it is necessary that
ah&II be no discontinuity in the flow, and special meters have i
ingiy been devised*
For large pipes, the Ventnri meter, Fig. 153^ is largely used in
f and is coming into favonr in this country.
The theory of the meter hae already been discussed (p, 4i),
it waa shown that the discharge is proportional to the square
* of the difference H of the head at the throat and the bend in
^pipCi or
^* being a coefficient.
For measuring the pressure heads at the two ends of the conei
W* G. Kent nses the arrangement shown in Fig. 154.
Fig> 154. B«oording drum of the Kent TeDtmi Meter.
• Seepage 4a
i^^li
252
HYDBAULICS
The two pressure tubes from the meter are connected to a U tube
consisting of two iron cylinders containing mercury. Upon the
surface of the mercury in each cylinder is a float made of iron and
vulcanite; these floats rise or fall with the surfaces of the mercmy.
Fig. 156. Integrating dram of the Kent Yentori Meter.
When no water is passing through the meter, the mercury in tibl
two cylinders stands at the same level. When flow takes platt
the mercury in the left cylinder rises, and that in the xi^
cylinder is depressed until the difference of level of the sox&oh
OAUGINO THE FLOW OF WATER
253
of the mercoiy is equal to — , « being the specific gravity of the
mercury and H the difference of pressure head in the two
cylinders. The two tabes are equal in diameter, so that the rise
in the one is exactly equal to the fall in the other, and the move
ment of either rack is proportional to H. The discharge is
proportional to VH, and arrangements are made in the recording
apparatus to make the revolutions of the counter proportional to
^/H. To the floats, inside the cylinders, are connected racks, as
shown in Fig. 154, gearing with small pinions. Outside the
mercury cylinders are two other racks, to each of which vertical
motion is given by a pinion fixed to the same spindle as the pinion
gearing with the rack in the cylinder. The rack outside the left
cylinder has connected to it a light pen carriage, the pen of which
Fig. 157. Kent Vestori Meter. Development of Integrating drum.
makes a continuous record on the diagram drum shown in
Fig 155. This drum is rotated at a uniform rate by clockwork,
and on suitably prepared paper a curve showing the rate of
discharge at any instant is thus recorded. The rack outside the
right cylinder is connected to a carriage, the function of which is
to regulate the rotations of the counter which records tlie total
flow. Concentric with the diagram drum shown in Fig. 155, and
within it, is a second drum, shown in Fig. 156, wliich also rotates
at a uniform rate. Fig. 157 shows this internal drum developed.
Hie snrface of the drum below the parabolic curve FEG is recessed.
If the rightband carriage is touching the drum on ttie xece'g.^eA.
HTDRAUUGS
, the coonter gearmg is in sc^on^ but is pat out of a^tioB
th* c&rmge tottches the cylinderr on the nuaad portm
*boTe FB. Sn^pomB ihie mercury in the right cylindE^r to fill a
beight i » o port io a»l to H, then the carriage will be m KXiwm
with the dram^ aa the dmm rofeales, along the line CD, bm i»
wiU imly be in operBtion while the CArriage is ia
along the kfngth CE, Since FG ts a parabolic carve ik
frmctkm of the drnunlerence CE = m , ^fW^ vi being & constant;
r for M*w «iiftn]st^«*>«n^«ti Tf cif the Hoats the ct juater k
I action far a poricxl pToptff'
ae top of the right cytind^frt
im^ and in contact wiili i^
ferolntion and no flow i
s in itB lowest position tht
*inn» and flow is registW
111' recording apparati^cii
« ism than 1000 feet fnM
'jp^ larger as the diHtaon
of
boulto^. Wk
tfae carnage m at
nised portifii for u
Whm tl
is al the b
tianng the whole of a .«,
bo placed ai any oonTOi
tlie meter, tbe conaecaog
B tQcreaeed.
155. Deacon's wastewater meter.
An ingenious and very diuple meter designed by Mr G. K
Deacon prmcipally for detecting the leakage of water from pipes
is a^ shc'Wn in Fig, lad.
The body of the meter which is made of cast*iron, has fittefl 
into it a hollow cone C made of brass. A disc D of the same diameter 
3fc^ the upper end of the cone is suspended in this cone bymeao^cl {
a fine winv — er a pulley not shown ; the other end
of the >Wre carries a balance weight.
GAUOtKG THK FLOW OF WATER
255
no water paaaes throu^li the meter the disc is drawn to
{top of the cone, bat when water is drawn through, the disc is
downwards to a position depending upon the quantity of
pasmiig* A pencil is attached to the wire, and the motion
imc can then be recorded upon a drum mad© to revolve by
rork The posirion of the pencil indicates the rate of flow
through the meter at any instant.
used as a wastewater meter, it is placed in a bypass
^ from the main, as shown diagrammatically in Fig, 159,
(^
s.v:
Bi sy
3
s.\:
D
Fig, 16a
\ulv»ii A and B are closed and the valve C opened. The
€v*ii!naniption in the pipe AD at those hours of the night
the actual consumption is very amall, can thus be detei^
i^ and an estimate made as to the probable amount wasted.
* If wm^te is taking place, a caiefiil inspection of the district
Hiliod by the main AD may then be made to detect where the
Hte is occurring.
^i&B. Kennedy's meter.
This is a positive meter in which the volume of water passing
ragh the meter is measured by the displacement of a piston
ing in the measuring cylinder.
long hollow piston P^ Fig. 157, fits loosely in the cylinder
hot m made watertight bj* means of a cylindrical ring of
iter which rijUs between the piston and the inside of the
ider, the friction being thus reduced to a minimum. At each
of the cylinder is a rubber ring, which makes a watertight
ii when the pisrton is forced to either end of the cylinder, so
it the rtibber roller has only to make a joint while the piston is
to move,
water ©titers the meter at A, Fig. 161 i, and for the
ition shown of the regulating cock, it Hows down the passage
id under the piston* ^Flie piston rises, and as it does so the
R turns the pinion 8, and thug the pinion p which is keyed
he .same spindle as S* This spindle also carries loosely
ighted !ever W, which is moved as the spindle revolves by
r of two projecting fingers. As the piston continues to
id, the weighted lever is moved by one of the fingers until its
^ ^ — ^
250
HYDRAULICS
centre of gravity passes the vertical positioHi wlien it suddei
falls on to a buffer j and in its motion moves the lever L^ whi
tuma the cock. Fig. 161 fe, into a position at right angles to tl
r~7 rn ^
R^ibb^SeaHiiff
Fig. 160.
GArOlKG THE rLOW OF WATER
257
lown. The water now passes from A throngh the paasage C,
syUnder, and as the piston descends,
Fig. 161 6,
k
jMilk
HYBIUtrUCS
Ao wmter that is below it pas(s<^ to the outlet B The
the pinkm H is now re^ersedj and the weight W lilted
mgam x^eecbee the vertical podttion, wheti it faik, hxm%
opek C into Ibo position shown in the fignre^ and another i
Fig. 161 c,
oke i« commenced. The rise illations of the pinion p bt^\
to the counter niecVmnisui through the pinions p,
161 a^ in each of which i^ a ratchet and pawl. The eofl
thtta rotated in the same direction whichever way the p
IBl. dauging the flow of streams by chemical meauL
ilr Stromeyer* has very sacceesfully ganged the quantity
water supplied to boilers, and also
the flow of stretams by mixing
with the stream doring a deliuite
ime and at a uniform rate, a
auwn quantity of a concentrated
llation of some chemical, the
91100 of which in watar^ even
in very small quantities, can be
]y detected by some sensitive
&nt. Suppose for instance
iraler ia flowing along a small
Two stations at a known
i apart are taken^ and the
determined which it takes
the water to traverse the dis
"HDce between them. At a stated
*e, by means of a special ap
"ns— Mr Stromeyer uses the gLL
'^ment shown in Fig. 162 Fig, 169.
nc acid, say, of known
run into the stream at a known rate» at the up.
f Navai AnMifCti, 1896 ; Proctidingf inwL C,B*, Vol, CLI.
^
^^i"^
K
J
GAUaiKG THE FLOW OF WATER
259
While the acid is being pat into the stream, a small
k»ce opstream from where the acid is introduced samples of
rr are taken at definit€r ititervals* At the lower station
tg is commenced, at a time, after the insertion of the
the apper station is started, equal to that required by the
W to trarerse the distance between the stations, and samples
then taken, at the same intervalsj as at the upper station,
quantity of acid in a known volume of the samples taken
he tipper and lower station is then determined by analyaiB*
I Ttjlnme V^ of the samples, let the difference in the amount of
hfiric acid be equivalent to a volume tv of pure sulphuric
k If in a time ^, a volume V of water j has flowed down the
^ and there has been mixed with this a volume v of pure
ric acid, then, H the acid has mixed uniformly Anth the
^ the ratio of the quantity of water flowing down the stream
BqBanlity of acid put into the stream, is the same as the
^fe the volnme of the sample tested to the difference of the
me of the acid in the samples at the two stations, or
Pf Stromeyer considers that the flow in the largest rivers can
iterxnined by this method within one per cent, of its true value.
d large streams special precautions have to be taken in
the chemical solution into the water, to ensure a uniform
and also special precautions must be adopted in taking
[okliar important information upon this interesting method
lorin^ the flow of water the reader is referred to the two
uted. above.
iratos for accurately gauging the flow of the solution
in Fig, 162* The chemical solution is delivered into
lindrieal tank by means of a pipe L On the surface of the
1 floats a cork which carries a siphon pipe SS, and a balance
keep the cork horizontal. After the flow has been
the head h above the orifice is clearly maintained
irhaterer the level of the surface of the solution in the
\1—1
260
HYDBAtJLlCS
EXAMPLES,
(1) Same obaerrfttions aro made by towing a coireot meter, witli ifa4
following resnlta:—
Speed in ft per lec.
1
5
Find an eqii^tion far the m^ter^
(2) Deecribe two methods of gi
in vertical and horizontal planes; ,
obtained.
If the croBs section of a met is
discharge may bo oaUmated by ob
alone.
(8) The following observations dl
were made ill connection with a weir
Head m feet ... ... 01 01
Discharge in cubic feet per
sec per foot width ,., 017
Be?i. of meter per min,
80
560
bulge river, from observatioiii
^te the nature of the resultB
explain how the apptosiiii«be
n of the midanzface f^pi^
nd the oorresponding disdiai^H
et wide.
80 ! 35 I 4i)
■1 1982
25
18*08
1703 2154 126^4
12 1»,
Assuming the law connecting the hoad h with the discharge Q as
Q = mh . A",
find m and n. ^Plot log&ritlimH of Q and h,}
(4) The following values of Q and h were obtained for a shacped^
weir 6'58 feet longT without lateral contraction. Find the coefl&djent d
discharge at veltious heads.
lead A ...
•1 j4 6
8
1*0
1*5
2*0 1 25 30
35
40
€5
W
H
1 per foot
1
tength ...
•17 87 j 156
237
sas
ei
932 1508 1703
3154
264
31*62
3709
43^81
(5) The following values of the head over a weir 10 feet long were
obtained at 5 minutes Intervals,
Head m feet 35 36 37 37 38 80 40 *41 *42 40 *8& "41
Taking tlie coefficient of discharge C as 8 36, find the discharge m
one hour.
(6) A Pitot tube was calibrated by moving it through still water in &
tank, the tube being tixed to an arm wliidi was made to revolve at
constant speed about a fixed centre. The following were the velocities of
the tube and the heads measured in inches of water.
Velocities ft. per sec. 1*432
Head in inches
of water 448
1788
•663
2275 I 2718
102 1 169
8286
207
8878
288
4988 I 5*584
540 697
6143
851
Determine the coefficient of the tube.
For examples on Venturi meters see Chapter U.
CHAPTER VIIL
IMPACT OF WATER ON VANES.
168. Definition of a vector. A right line AS, considered as
haying not only length, bat also direction, and sense, is said to be
a vector*. The initial point A is said to be the origin.
It is important that the difference between sense and direction
should be clearly recognised.
Suppose for example, from any point A, a line AB of
definite length is drawn in a northerly direction, then the
direction of the line is either from south to north or north to
soath, but the sense of the vector is definite, and is from A to B,
that is from sooth to north.
The vector AB is equal in magnitude to the vector BA, but
they are of opposite sign or,
AB = BA.
The sense of the vector is indicated by an arrow, as on AB,
Fig. 163.
Any quantity which has magnitude, direction, and sense, may
be represented by a vector.
D
For example, a body is moving with a given velocity in a
given direction, sense being now implied. Then a line AB drawn
parallel to the direction of motion, and on some scale equal in
* Sir W. Hamilton, QuaUmioru,
262
HYDRAULICS
length to the velocity of the body is the velocity vector; the seise
is from A to B<
159. ♦ Stun of two vectora.
If a and ^, Fig. 163, are two vectors the euoi of these vecton
is found, by drawing the vectors, so that the beginniiig of j8 ii il j
the end of a, and joining the beginning of a to the end of ftj
Thus y is the vector sum of a and fi.
160. ReanltaJEit of two
When a body has
velocities^ the resultant
direction is the vector si
may be stated in a way
problems to be hereai
moving with a given i
velocity is inipressed op
vector sum 6f the initial
Teiocitias*
)n it at any instant two
16 body in magnitude nni
impressed velociti**^, TIlei
iefiaitely applicable to the
as follows. If a body ii
en direction, and a ieoomd
he resnltant Telocity is the
velocities.
Example* Bfippos^ ^ pAri
with a velocitj V^, rolative U
If the Tune is at rest, the »i ^u*^ ^" at A with this velocity.
If the ^ane iti timd« to move m the ^±44^iiiOEL EF with a velocity r, and l3ot
particle haa still a Vf^locitj V^ r«lalive to the vane, and remams in contact with tbs
vane until th« point A i» reach^, the vdocity of the water as it learea the ^n« it
A, will be the ^^otor (lum 7 of a and ^, i.e^ of V,. and V, or h equal to i^,
161. DtfiTerencQ of two vectors.
The difference of two vectors a and is f onnd by drawing bo^
vectors from a common origin A, and joining the end of ^ to thei
end of a. Thus, CB, Fig, 165, is the difference of the two vectora
a and A or y = aft and BC is equal to i^of, or^a^7.
Fig. 166.
162. Absolute yelooity.
By the terms " absolute velocity " or " velocity " without the
adjective, as used in this chapter, it should be clearly understood,
is meant the velocity of the moving water relative to the earth, or
to the fixed part of any machine in which the water is moving.
Henrioi and Tamer, Veeton tmd Baton,
IMPACT OF WATER ON VANES 263
To avoid repetition of the word absolute, the adjective is
■eqaently dropped and " velocity " only is used.
163. When a body is moving with a velocity U, Fig. 166, in
ny direction, and has its velocity changed to U' in any other
iirection, by an impressed force, the change in velocity, or the
'elocity that is impressed on the body, is the vector difference of
he final and the initial velocities. It AB is U, and AC, U', the
mpressed velocity is BC.
By Newton's second law of motion, the resultant impressed
Sorce is in the direction of the change of velocity, and if W is the
wmght of the body in pounds and t is the time taken to change
ihe velocity, the magnitude of the impressed force is
W
P = T (change of velocity) lbs.
gt
This may be stated more generally as follows.
The rate of change of momentum, in any direction, is equal to
ihe impressed force in that direction, or
P= — .;^lbs.
g at
In hydranlic machine problems, it is generally only necessary
o consider the change of momentum of the mass of water that
fccts open the machine per second. W in the above equation then
lecomes the weight of water per second, and t being one second,
W
P = — (change of velocity).
164. Impulse of water on vanes.
It follows that when water strikes a vane which is either
noving or at rest, and has its velocity changed, either in magni
nde or direction, pressure is exerted on the vane.
As an example, suppose in one second a mass of water, weighing
iV lbs, and moving with a velocity U feet per second, strikes a
ixed vane AD, and let it glide upon the vane at A, Fig. 167, and
eave at D in a direction at right angles to its original direction
>f motion. The velocity of the water is altered in direction but
lot in magnitude, the original velocity being changed to a velocity
kt right angles to it by the impressed force the vane exerts upon
he water.
The change of velocity in the direction AC is, therefore,
W
qual to U, and the change of momentum per second is — .U
oot lbs.
264
HYDRAULICS
Since W lbs. of water strike the vane per second, the pressi
P, acting in the direction C A, required to hold the vane in positi
is, therefore,
W
Pig. 167.
Again, the vane has impressed upon the water a velocity U
the direction DF which it originally did not possess.
The pressure Pi in the direction DF is, therefore,
W
Pi = P = — .U.
9
The resultant reaction of the vane in magnitude and directi
is, therefore, R the resultant of P and Pi.
This resultant force could have been
found at once by finding the resultant
change in velocity. Set out clc^ Fig. 168,
equal to the initial velocity in magnitude
and direction, and ad equal to the final
velocity. The change in velocity is the
vector difference cd, or cd is the velocity
that must be impressed on a particle of
water to change its velocity from ac to
od. ^« ^^
The impressed velocity cd is V = VU* + TP, and the to)
impressed force is
W
W
9 9
n/2W
9
IMPACT OF WATEB ON VANES
S0S
It at once folIow*s, that if a jnt of water strikes a fixed plan€3
[perpendicularly^ with a velocitjr U, and glides along the plane, the
[normal proas ure on the plane is^ ^ U.
A ttrtuD of water 1 eq, fcN?t in seetion ft&d having a Telocity of
^inft par Mooad glMea on lo a fixed vane ia a dUreolion makiDg an angle of
iifMft wUh a gifen direction AB.
Tbm vaae tatna tbe jet ihroogh an angle ot §C degrees,
Whid Iha Pfcwmm cm Ihe Tane io the direction parsiLlsl lo AB and the reittltant
wmom tibe taaa.
Fle» 167. A€ ia the oiiginal direction of the jet and DF the tnal direction.
lae ctntply ehangea the directioti of the water* the ^nal velocity being oqnal
triaagle it ar4, Fif?^ ISS^ ae and ad being equal,
of vel<>£i^ In magnitude and direction is cd^ the vector difference of
; roaolTing cd parallel to, and perp^ndienlar to AB^ ce Is the ohangie of
pafsOlel to aB.
eallog off C9 and oalling it v^ » the for^e to be applied along BA to keep the
•I rmt ia,
Bat c4=j2,l0
C4 = cd oo» 1 5^
Tli0 premsum aomml to AB ii^
= 2fi4lbfl.
■ 9
Tbe rvsoltaat ia
B=
m,m4 , 1007^2.62 4
32*2
3SS
274 Ibe.
IBS. RelaUye yeloclty.
Betfore going on to the consideration of moving vanes it
^^Bfi^tit that the student should have clear ideaa as to what iS
f/fmk' by relative i^elocity.
I A rrain is said to have a velocity of dxty miles an honr when
iti . ' Tied in a Btraight line at a constant velocity for one
^^j^ Ad travel sixty miles. What is meant is that the train
^ fir^ying «^ sixty miles an hour relative to the earth.
^^ t^wo trains run on parallel lines in the same direction, oi
.ty And the other at forty miles an hour, they have a
!• Other of 20 miles an hour. If they move
^^^^^^,_ [hey havo a relative velocity of 100 miles
boor. If one of the trains T is travelling in the direction AB,
^ 169^ and th^ other T, in the direction AC, and it be supposed
fnt the line© on which they are travelling cross each ottei at A^
Li n ^
n(^^
266
HYDRAULICS
and the trains are at any instant over each other at A, at the end
of one minute the two trains will be at B and C respectively, at
distances of one mile and twothirds of a
mile from A. Relatively to the train T
moving along AB, the train Ti moving
along AC has, therefore, a velocity equal
to BC, in magnitude and direction, and
relatively to the train Ti the train T has
a velocity equal to CB. But AB and AC
may be taken as the vectors of the two
velocities, and BC is the vector difference
%
Fig. 169.
of AC and AB, that is, the velocity of Ti relative to T is the
vector difference of AC and AB.
166. Definition of relative velocity as a vector.
If two bodies A and B are moving with given velocities v and
t?i in given directions, the relative velocity of A to B is the vector
difference of the velocities v and Vi.
Thus when a stream of water strikes a moving vane the
magnitude and direction of the relative velocity of the water and
the vane is the vector difference of the velocity of the water and
the edge of the vane where the water meets it.
167. To find the pressure on a moving vane, and the
rate of doing work.
A jet of water having a velocity U strikes a flat vane, tie
plane of which is perpendicular to the direction of the jet, and
which is moving in the same direction as the jet with a velocitv ft
I
Fig. 170.
U^
Fig. 171.
The relative velocity of the water and the vane is U— «, the
vector difference of U and v. Fig. 170. If the water as it strikei
the vane is supposed to glide along it as in Fig. 171, it will do
IMPACT OF WATER ON VANES 267
80 with a velocity equal to (U— v), and as it moves with the vane
it will still have a velocity v in the direction of motion of the
vazie. Instead of the water gliding along the vane, the velocity
U — V may be destroyed by eddy motions, but the water will still
liave a velocity v in the direction of the vane. The change in
nelocity in the direction of motion is, therefore, the relative
relocity Ur, Fig. 170.
For every pound of water striking the vane, the horizontal
XJ — t?
ihan^e in momentum is , and this equals the norm^/l pressure
^ on the vane, per pound of water striking the vane.
The work done per second per pound is
Pt? = . V foot lbs.
9
The original kinetic energy of the jet per pound of water
. XP
triking the vane is s— > and the efficiency of the vane is, therefore,
"liicli is a maximum when v is ^U, and e = J. An application of
icH vanes is illustrated in Fig. 185, page 292.
Nozzle and single vane. Let the water striking a vane issue
XMn a nozzle of area a, and suppose that there is only one vane.
"Let the vane at a given instant be supposed at A, Fig. 172. At
le end of one second the front of the jet, if perfectly free to
love^ would have arrived at B and the vane at C. Of the water
lat has issued from the jet, therefore, only the quantity BC will
ave hit the vane.
Fig. 172.
The discbarge from the nozzle is
W = 624.a.U,
id the 'Weight that hits the vane per second is
W.(U t>)
u
The chftn^r^ of momentum per second is
W(Ut))'
g U '
HTDKAUUCa
froiQ ilie posxkil
ttitd hsm no
^ wmmm^ If them wre m
€tiket, tlie wholB of tlie w^r
mnti Urn work done is
IP •
^ w»ter wheel, witb imdial bbdoi^l
m P% 19§^ cuiBoC IxftTe an effici^^cj^ of more than 50 per cent
IG6 Impact of wftter on & T&ne when the dlxectioni €
wrf^"^ of the Ymne and jet are not parallel.
L«« r be tke ^ckiciix erf » j^ ol wuier and AB its directkitf
J>
ne
fif. tm
Ltfi «bt» f<48« A,<ii Ae na^ AC be morbg witli a Tielodtrn
tbe i tti i iTT ^ velocity Vr of tbe water and the Taoe at A is DB
tfe tnu^le DAB it is aeeti tbAt.^ tbe rector sunt of tht
of the Tane and tb«^ r^latire f^kMriljr of the jet ai>d d4
«^W ii tqasl to tlie T^k^^it^ ol tbe jec; for clearl^ IT is tbe t&^
aaB vi r at»l Tr.
If tbe dbvetiiJii vi tbe tip ol tbe Tmne at A is made panUel td
DB tbe water will gbde on to tbe T%ne in exaetlj^ the same vsf
J
IMPACT OF WATER ON VANIS
2G9
it were mt rest, and the water were moving in tha direction
L This h the condition that no eDerg7 shall be lost by shock,
"Wli^n the water leaves the vane, the relative velocity of the
and the vBne must be parallel to the diroction of the
^mit to the vane at the point where it leaves, and it is equal to
vector difference of the absolute velocity of the water, and
Tane. Or the absolute velocity with which the water leaves
mape is the vector sum of the velocity of the tip of the vane
I the relative velocity of the water to the vane*
l^t CO be the direction of the tangent to the vane at C. Let
i be ri p the velocity of C in magnitude and direction, and let CF
ihe absoliite velocity Ui with which the water leaves the vane.
Draw EF paiaUel to C(t to meet the direction CF in Fj then
\ relative velocity of the water and the vane is EF, and the
ocity with which the water leaves the vane is equal to CF.
If Vi and the direction CG are given, and the direction in which
I water leaves the vane is given^ the triangle CEF can be
IWB, and CF determined.
If on the other hand Vj is given, and the relative velocity Vr is
in magnitude and direction, CF can be found by measuring
along EF the known relative velocity iv and joining CF.
If t?j and Ui are given, the direction of the tangent to the vane
then, a« at inlet, the vector difference of Ui and t^i.
It will be seen that when the water either strikes or leaves the
ne, the relative velocity of the water and the vane m the vector
erence of the velocity of the water and the vane, and the actual
Oi3ty of the water as it leaves the vane is the vector sum of the
ocity of the van© and the relative velocity of the water and
rane.
I
I
I
■
I
I
I
I
, The difectiott of tbe tip of the vivne at the out^r circumferrence of a
witli T&fiai, m*ke« aa Angle of 165 degiee:^ with the direction of motioa
_ kip of the TADe.
7li« Tclodty of the tip at the outer circtimf«rf tjc«; in 82 feet p&r F^cnzid^
«mier leaves the ^h&e\ m auch a direct ii>n and with such a Velocity that the '
■enl i^ 13 fmx per second
wbmAute velocLty of tbe water in direetion and magnitude and the
of the wat«r and the wheeL ^
m tbe tri*iigl« of velocities, set out A 15 uqual to B2 feet, and make tha^
^ eQiiaJI to IS degneefi. BQ Ih then parallei to the tip of the vane. V
irallel lo ABp and at a diittatjce from It eqn&l to 1^ fe«t &aA
in C
,\C t* tha vector Aum of AB and UC, and ia the absolate velocitj of the
directiaa and magnttTide,
Ihgoiiometrically
AC^  (82  13 oot isy + 133
= 33^5*HhlB* and AC = air? ft per seo.
i
270
HYDRAULICS
169. Conditions wMch the vanes of bydraalio naadiini
should satiflfy*
In all properly deBigned hydraulic machines, sneh as tnrbti
veater wheels, and centrifugal pumps, in which water flowing id
a definite direction impinges on mo^^ng vanes, the relative ^
of the water and the vanes should be parallel to the din
the vaoes at the point of contact. If not, the water breaks into
eddies aa it moves on to the vanes and energy is lost.
Again, if in such machines the water is required to leave the
Tanes with a given velocity in magnitude and direction, it is only
neceaaary to make the tip of the vane parallel to the vect<jr
difference of the given velocity with which the water is to leav^?
the vane and the velocity of the tip of the vane,
ExampU (1, A jet of water, Fig. 174, movea in a direction AB making as as . •
of 30 def^ees with the direct! od of motion AC of a vane moving in tlie atmosi/t
The jet has a velocity of 30 ft. per second and the vaue of lo ft. per si^ood. Tt» umiI
(ii} the direction ^f the vnoe at A eo tbal tlie water may cnttir wiChoQl aliook; {h} tk*
direction of the taugeni to tiie vane where the water leaves it, so that the abtolulf
velocity of th€ water when it teavei^ the vane is in a direction perpendiotiJAr to ^Cfi
(r) the pressure on the vane and the work done par second per poand of '
striking the vane, Frictioii U neglected.
Fig, 174.
The relative Telocity T^ of Ihe water and the vatje %i A ie CB, and for no i
the vane at A must be parallel to CB.
Sinee there is no friction, the relative velocity V,, of the watar ajid ibe hm]
cannot alter, and tharefore, the triangte of velocities at exit is ACD or FA*C*.
The point D is fouodt by taking G as centre and CB as radiua uid sitwoff \
are ED to cut the known dlreotion AD in D.
The total change of velocity of the jet i« the vector difference DB of the L^ltkl ]
and final velocities, and the change of velocity in the direction of motion u E£^ 1
Calling this velocity Y^ the presaare exerted npon ihe vane in the direolkc <i
motion In
V
— lbs, per lb. of water striking the vane^
Tlie work done p^ lb. is, therefore^ — ft. lbs, asd the efEdencj, sinc« Ibtl* ii 
no loss by friction, or shock, h
\v _ %Vv
•'25
IMPACT OP WATER ON VANES 271
The change in the kinetic energy of the Jet i$ equal to the work done by the jet.
The kinetio energy per lb. of the original jet is ^ and the final kinetic energy is
El!
The work done is, therefore, ^ — o^ ^^ ^^^* ^^^ ^^® efficiency is
'2g
It can at once be seen from the geometry of the figure that
Vr __ Ua Ui«
For
AB«=AC«+CB«+2AC.CG,
knd aince
CD=CB and CD^=AC« + AD»,
lierefore.
AB«AD*=2AC(AC + CG)
= 2i;V.
Bat
AB«AD«=U«Uj«,
herefore,
2g g '
If the water instead of leaving the vane in a direction perpendicular to t\ leaves
t with a velocity Uj having a component V^ parallel to v, the work done ou the
ane per pound of water is
9
If Ui be drawn on the figure it will be seen that the change of velocity in the
V V
Lirection of motion is now (V  V^), the impressed force per pound is ^ , and
/ V — V \
be work done is, therefore, ( j ^'i '^' ^^^' P^^ pound.
A!» before, the work done on the vane is the loss of kinetic energy of the jet, and
.herefore,
(VV,)v,^U»Ui«
9 'k
The work done on the vane per pound of water for any given value of Uj , is,
Jierefore, independent of the direction of U^ .
Example (2). A series of vanes such as AB, Fig. 175, are fixed to a (turbine)
whmel which revolves about a fixed centre C, with an angular velocity u.
The radius of B is B and of A, r. Within the wheel are a number of guide
na«rr(p"i through which water is directed with a velocity U, at a definite inchnatiou
f with the tangent to the wheel. The air is supposed to have free accesn to the
wbe«L
To draw the triangles of velocity, at iulet and outlet, and to find the directions
yi tlM tips of the vanes, so that the water moves on to the vanes without shock and
Learee the wheel with a given velocity Uj. Friction neglected.
Aa in the last example the velocity relative to the vane must remain constant,
^a%A therefore, V^ and v^ are equal, but v and v^ are unequal.
The tangent AH to the vane at A makes an angle with the tangent AD to the
irheel, so (hat CD makes an angle with AD. The triangle of velocities ACD at
nlgt 18, therefore, as shown in the figure and does not need explanation.
To draw the triangle of velocities at exit, set out BQ equal to t'j and perpen
lieolar to the radius BC, and with B and G as centres, describe circles with U, and
. which is equal to V^— as radii respectively, intersecting in E. Then G£ is
l^rmliel to the tangent to the vane at B.
272
HYDRAULICS
If there is a loss of head, /t/, by friction, as the water moves over the vane tl
Vf IB less than V^, if h/ is known, it oan be found from
29 2g "^^
(See Impulse turbines.)
Work done on the wheel. Neglecting Motion etc. the work done per poond
water passing through the wheel, since the pressure is constant, bong equal to '
atmospheric pressure, is the loss of kinetic energy of the water, and is
2g
The work done on the wheel can also be found from the consideration of 1
change of the angular momentum of the water passing through the whed. Befi
going on however to determine the work per pound by this method, tiie notati
that has been used is summarised and sevoral important principles oonsideredL
^  ^ ft. lbs.
Notation used in connection with vaneSy tturbines and centrifug
pwm/pa. Let U be the velocity with which the water approach
the vane, Fig. 175, and v the velocity, perpendicular to the radii
AC, of the edge A of the vane at which water enters the wheel.
Let V be the component of U in the direction of t?,
u the component of U perpendicular to t?,
Vr the relative velocity of the water and vane at A,
t?i the velocity, perpendicular to BC, of the edge B of the vw
at which water leaves the wheel, ^
Ui the velocity with which the water leaves the wheel,
Vi the component of Ui in the direction of t^i,
IMPACT OF WATER OV VANES
273
Ui the component of Ui perpendicular to r,, or along BC,
tv the relative velocity of the water and the vane at B.
ITelociiies of whirl. The component velocities V and Vi are
called the velocities of whirl at inlet and outlet respectively.
This temi will frequently be used in the following chapters.
170. Definition of angular momentum.
If a weight of W pounds is moving with a velocity U, Figs. 175
and 176, in a given direction, the perpendicular distance of which
is S feet from a fixed centre C, the angular momentum of W is
W
— . U . S pounds feet.
171. Change of angular momentiun.
If after a small time t the mass is moving with a velocity Ui in
a direction, which is at a perpendicular distance Si from C, the
W
angular momentum is now — UiSi; the change of angular
momentum in time t is
J (US U, SO;
and the rate of change of angular momentum is
^(USUaSO.
Fig. 176. Fig. 177.
172. Two important principles.
(1) Work done by a coti/pUy or turning moment. When a
body is turned through an angle a measured in radians, under the
action of a constant turning moment, or couple, of T pounds feet,
the work done is Ta foot pounds.
If the body is rotating with an angular velocity w radians
per second, the rate of doing work is Tw foot pounds per second,
and the horsepower is g^.
I^ H.
\^
274 HYDRAULICS
Suppose a body rotates about a fixed centre C, Fig. 177, and
a force P lbs. acts on the* body, the perpendicular distance from
C to the direction of P being S.
The moment of P about C is
T = P.S.
If the body turns through an angle ca in one second, the
distance moved through by the force P is « • S, and the work
done by P in foot pounds is
Pa>S=T«.
And since one horsepower is equivalent to 33,000 foot pounds
per minute or 550 foot pounds per second the horsepower is
(2) The rate of change of angular momentv/m of a bodf
rotating about a fixed centre is equal to the couple acting wpm
the body. Suppose a weight of W pounds is moving at any instant
with a velocity U, Fig. 176, the perpendicular distance of which
from a fixed centre C is S, and that a couple is exerted upon W
so as to change its velocity from U to Ui in magnitude and
direction.
The reader may be helped by assuming the velocity U is
changed to Ui by a wheel such as that shown in Fig. 175.
Suppose now at the point A the velocity Ui is destroyed in a
time dt^ then a force will be exerted at the point A equal to
PW U
g 'cV
and the moment of this force about C is P . S.
At the end of the time d^, let the weight W leave the wheel
with a velocity Ui. During this time dt the velocity Ui might
have been given to the moving body by a force
p_WU,
acting at the radius Si.
The moment of Pi is Pi Si ; and therefore if the body has been
acting on a wheel. Fig. 175, the reaction of the wheel thus exerting
the couple upon the body, or a couple has been exerted upon it in
any other way, the couple required to change the velocity of W
from U to Ui is
T = J^(USU.S,) (1).
Let the wheel of Fig. 175, or the couple which is acting npoA
the body, have an angular velocity w.
DCPAGT OF WATER ON VANES 275
In a time dt the angle moved through by the couple is cud^,
and therefore the work done in time dt is
T.a)3^ = — co(USUiSO (2).
Suppose now W is the weight of water in pounds per second
iprhich strikes the vanes of a moving wheel of any form, and this
water has its velocity changed from U to Ui, then by making dt
in either equation (1) or (2) equal to unity, the work done per
second is
Ta>= — co(USU,S,),
and the work done per second per pound of water entering the
wheel is
^(USUiSx).
This result, as will be seen later (page 337), is entirely inde
pendent of the change of pressure as the water passes through the
wheely or of the direction in which the water passes.
173. Work done on a aeries of vanes fixed to a wheel
expreased in terms of the velocities of whirl of the water
entering and leaving the wheel.
Outvard flaw ta/rbine. If water enters a whgelat the inner
circumference, as in Pig. 175, the flow is said to be outward.
On reference to the figure it is seen that since r is perpendicular
to V, and S to U, therefore
r_U
And for a similar reason
R_Ui
A^ain the angular velocity of the wheel
" = r = R'
therefore the work done per second is
And the work done per pound of flow is
g g
Inward fl^ow twrhine. If the water enters at the outer cir
cunrference of a wheel with a velocity of whirl V, and leaves at
the inner circumference with a velocity of whirl Vi, tYve veVoaVdfe^
276
HYDRAULICS
of the inlet and outlet tips of the vanes being v and ri respectively
the work done on the wheel is still
Vr ViTi
V 9
The flow in this case is said to be inward.
Parallel flour or axial fl^vc turbine. If vanes, such as those
shown in Fig. 174, are fixed to a wheel, the flow is parallel to tiie
axis of the wheel, and is said to be axial.
For any given radius of the wheel, Vi is equal to Vy and the
work done per pound is
which agrees with the result already found on page 271.
174. Curved vanes. Pelton wheel.
Let a series of cups, similar to Figs. 178 and 179, be moving
with a velocity t?, and a stream with a greater velocity U in the
same direction.
The relative velocity is
Vr=(Ur).
Neglecting friction, the relative velocity Vr will remain con
stant, and the water will, therefore, leave the cup at the point B
with a velocity, Vr, relative to the cup.
If the tip of the cup at B, Fig. 178, makes an angle * with the
direction of v, the absolute velocity with which the water leaves
the cup will be the vector sum of v and Vr, jftnd is therefore Ui.
The work done on the cups is then
IP U^«
2g 2g
IMPACT OF WATER ON VANES 277
per lb. of water, and the efficiency is
For TJi, the value
Ui = 'J{v  (U  v) cos e\' + (U  vy sin ^r*
can be substituted, and the efficiency thus determined in terms of
r, U and 0.
JPelton wheel (mps. If is zero, as in Fig. 178, and Uv is
equal to t?, or XJ is twice v, Ui clearly becomes zero, and the water
drops away from the cup, under the action of gravity, without
possessing velocity in the direction of motion.
The whole of the kinetic energy of the jet is thus absorbed
and the theoretical efficiency of the cups is unity.
TTie work done determined from consideration of the cltange of
mofnenium. The component" of Ui, Fig. 178, in the direction of
motion, is
17 — (U  V) COS d,
and the change of momentum per pound of water striking the
vanes is, therefore,
U~t?f (U  v)jDOS
9
The work done per lb. is
t? {U  1? + (U  1?) coa^}
9
and the eflSciency is
^ 2v{Uv + (Ut7)cos 0}
W
When is 0, cosO is unity, and
e= u— >
which is a maximum, and equal to unity, when v is ^^ .
175. Force tending to move a vessel firom which water
is iflsnintf through an orifice.
When water issues from a vertical orifice of area a sq. feet,
in the side of a vessel at rest, in which the surface of the water is
maintained at a height h feet above the centre of the OT\&ce^t\v^
278 HYDRAULICS
pressure on the orifice, or the force tending to move the vessel
in the opposite direction to the movement of the water, is
F=2w.a,h\hs.,
w being the weight of a cubic foot of water in pounds.
The vessel being at rest, the velocity with which the water
leaves the orifice, neglecting friction, is
t7= J2ghy
and the quantity discharged per second in cubic feet is
The momentum given to the water per second is
9
= 2w ,a.h.
But the momentum given to the water per second is equal to
the impressed force, and therefore the force tending to move the
vessel is
F = 2w.a.h,
or is equal to twice the pressure that would be exerted upon a
plate covering the orifice. When a fireman holds the nozzle of a
hosepipe through which water is issuing with a velocity r, there
is, therefore, a pressure on his hand equal to
2wav^ _ wav^
"W" 9 •
If the vessel has a velocity V backwards, the velocity U of the
water relative to the earth is
and the pressure exerted upon the vessel is
F=^^'^lbs.
9
The work done per second is
F . V = ^^ foot lbs.,
9
or = — ^ foot lbs.
9
per lb. of flow from the nozzle.
The efficiency is e = — ^—r — 
_ 2V(t?V)
which is a maximum, when
.i? = 2V
and e = J.
IMPACT OF WATER ON VANES 279
176. The proptdflion of Bhips by water jets.
A method of proi)eUing ships by means of jets of water issuing
from orifices at the back of the ship, has been used with some
sxiccess, and is still employed to a very limited extent, for the
propalsion of lifeboats.
Wat^r is taken by pumjw carried by the ship from that
surrounding the vessel, and is forced through the orifices. Let
tr be the velocity of the water issuing from the orifice relative
to the ship, and V the velocity of the ship. Then ^ is the
hea.d h forcing water from the ship, and the available energy
per pound of water leaving the ship is h foot pounds.
The whole of this energy need not, however, be given to the
water by the pumps.
Ima^ne the ship to be moving through the water and having
a pip^e with an open end at the front of the ship. The water in
front of the ship being at rest, water will enter the pipe with a
velocity V relative to the ship, and having a kinetic energy ^y
per pound. K friction and other losses are neglected, the work
that the pomps will have to do upon each pound of water to eject
it at the back with a velocity v is, clearly,
2g 2g'
As in the previous example, the velocity of the water issuing
from the nozzles relative to the water behind the ship is v  Y,
vY
and the change of momentum per pound is, therefore,  . If a
is the area of the nozzles the propelling force on the ship is
and the work done is
9
9
The efficiency is the work done on the ship divided by the
work done by the engines, which equals wav (5 — n " ) ^"*^'
therefore,
.2V(i;V)
e =
2V
■« + ¥'
280
HYDBAULICS
which can be made ae near unity as is desired hy makiiig r and
V approxiraate to equality.
Bat for a given area a of the orifices^ and velocity t\ the nearer
V approximates to V the less the propelling fore© F becomes, and
the size of ship that can be driven at a g:iven velocity V for ihn
given area a of the orifices diminishes.
If ris2Y, e.
EXAMPLES.
(1) Ten cubic feet of water per second ore diiicharged ttom & ^tationtfj
jet* llie eectional area of which is 1 square foot. The water irapiagea not*
mally on a fiat surface, moring in the directiDii of the jet w^tb a velodiy
of % feet per second. Find the pressure on the plane in lbB,« and the ^mak
done on tlie plane in horse iK>wer.
(2) A jet of water deUyering 100 gallons per Becond with a Telocity d
20 feet per second iiQpiuge& perpendicularly on a wall. Find the pre^itre
on the w^all*
(8) A jet delivers 160 cubic feet of water per minute at a velocity of
I feet per Heeond and strikes a plane perpendicularly. Find the preestiM
tlie plane^(l) when it is at rest; {%} when it is moviug at 5 feet per
second In tlie direction of the jet. En the latter case find the work c]oo«
per second in driving the plane.
(4) A fire*engine hose^ 8 inches bore, discharges water at a velocity of
100 feet per seccnd. Supposing the jet directed normally to tJie side d a
binding, find the presaure.
(6) Water iBsues horizontally from a fixed thin edged orifice^ indbai
sqnare, under a head of 25 feet. Tlie jet impinges normally on a pbii^
moving in tlie same direction at 10 feet per second. Find tlie pres^suns <A
the plane in lbs., and the work dome in horsepower. Take the coefficicBl
of discharge as *64 and the coefficient of velocity as '97.
(6 1 A jet and a plane surface move in directions inclined at 30% witb
velocities of 30 feet and 10 feet per second resjiectively. \Mm% i» tlw
relative velocity of the jet and surface ?
(7) Let AB and BC be two lines inclined at 80% A jet of water moffli
in the direction AB, w^th a velocity of 25 feet per second, and a »eric» o(
vanes move in the direction CB with a velocity of 15 feet per second. Fm^
the form of the vane so that the w^ater may come on to it tangentially, afiJ
leave it in the direction ED, perpendicular to CB.
Supposing that the jet is 1 foot wide and 1 inch thick before impinging
find the effort of the jet on the vanes.
(8) A enrved plate is mounted on a slide so that tlie plate i
move along tlie shde. It receives a jet of water at an angle of 3i •
normal to the direction of sliding, and the jet leaves tlie plate at m iagi»
IMPACT OF WATEE ON VANES
281
"With the «eiiia noimial. Find the force which must be applied to
be to the direction of slidiag to hold it at rest^ and also tii© nt^moL
!e on the slide. Quantity of water flawing is 500 Iha. per minute
■vekxaty of B5 feet per second.
A fixed Fane receives a jet of water at an angle of 120^ with a
a AB Find what angle the jet must be turned tlirough in order
I pffcasore on the vane in the direction AB may be 40 Ibs.^ when the
\ water m 45 Ibs^. per second at a velocity of 30 feet per second.
Water under a head of 60 feet m discharged through a pipe 6 indies
t and 150 feet long, and then tlirough a nois^le, the area of which
nth the area of the pipe,
all losses but the friciioti of the piper determine the preaaure
ed plate placed in front of the nozzle.
A jet of watier 4 tnehee diameter impinges on a fixed cone, the
Adding with that of the jet, and the apex angle being BO degEeeSi
llocity of 10 feet per second. Pind the pressure tending to move the
tJie direction of its asds.
A veeael containing water and having in one of it» vertical sides
orMoe 1 inch diameter^ which at first is phigged up, in
ided in such a way tliat any diiiplacing force can be accurately
^ed. On the removal of the plug, the horii^ontai force rec}uired to
& venal in place, applied opposite to the orifice^ is 3*6 lbs. By the
measimng tank the discharge m found to be 31 gallons (>er minute^
tol of the water in the vessel being maintained at a constant height
M wihove the orifica Determine the coefficients of velc>cityi con^
X and di»cliaige.
A train carrying a Eamsbottom's scoop for taking water into the
18 nmiiiiig at 24 miles an hour. What is the greatest height at
Ibe 8000k will deliver tlie water ?
A locomotive going at 40 miles an hour scoops up water from a
The tank is 8 feet above the mouth of the scoop* and tlie delivery
M ftJi area of 50 square inches. If half the available head is wasted
msce, find tlie velocity at wliich the water is delivered into the tank,
number of tons Ufted in a trench 500 yards long. What, utider
ions* is the increased resistance; and what is the minimum
at which the tank can be filled ? Lond. Un. 1906.
A stream deliyeiing 3000 gallons of water per minute with a
<d 40 feet per second, by impinging on vanes is caused freely to
Ihroogh an angle of 10'", the velc^city being dimmialied to M feet
ond. Delejmine the pressure on tlie vanes due to impact. If the
m moving in the direction of that pressure, find their velocity and
the luefni hor»epower* Lond. Un. 1906,
Wsttif flows from a 2inch pipe, without contraction, at 45 feet per
imtne the maadmuni work done on a machine carrying moving
I^AjoUowiDg emm and tlie respective efficiencieti;^
282 HYDRAULICS
(a) When the water impinges on a single flat plate at right angles and
leaves tangentiaUy.
(b) Similar to (a) but a large number of equidistant flat plates an
interposed in the path of the jet.
(c) When the water glides on and off a single semicylindrical cup.
{(i) When a large number of cups are used as in a Pelton wheeL
(17) In hydraulic mining, a jet 6 inches in diameter, discharged unte
a hesid of 400 feet, is delivered horizontally against a vertical cliff ho$B
Find the pressure on the face. What is the horsepower delivered by tin
jet?
(18) If the action on a Pelton wheel is equivalent to that ol a jet on a
series of hemispherical cups, find the efficiency when the speed of the
wheel is fiveeighths of the speed of the jet.
(19) If in the last question the jet velocity is 50 feet per second,
and the jet area 0*15 square foot, find the horsepower of the wheeL
(20) A ship has jet orifices 8 square feet in aggregate area, and dii
charges through the jets 100 cubic feet of water per second. The speed «(
the ship is 15 feet per second. Find the propelling force of the jets, te
efficiency of the propeller, and, neglecting friction, the horsepower of fli
engines.
CHAPTER IX.
WATER WHEELS AND TURBINES.
Water ^wKeels can be divided into two classes as follows.
(a) Wheels upon which the water does work partly by
mpolae but almost entirely by weight, the velocity of the water
rfien it strikes the wheel being small. There are two types of
bis class of wheel, Overshot Wheels, Figs. 180 and 181, and
Ireast Wheels, Figs. 182 and 184.
(6) Wheels on which the water acts by impulse as when
he wheel utilises the kinetic energy of a stream, or if a head h is
available the whole of the head is converted into velocity before
lie water comes in contact with the wheel. In most impulse
rheels the water is made to flow under the wheel and hence
hey are called Undershot Wheels.
It will be seen that in principle, there is no line of demarcation
letween impulse water wheels and impulse turbines, the latter
mly differing from the former in constructional detail.
177. Overshot water wheels.
This type of wheel is not suitable for very low or very high
leads as the diameter of the wheel cannot be made greater than
he head, neither can it conveniently be made much less.
Figs, 180 and 181 show two arrangements of the wheel, the
mly difference in the two cases being that in Fig. 181, the top of
he wheel is some distance below the surface of the water in the
ip0tream channel or penstock, so that the velocity v with which
he water reaches the wheel is larger than in Fig. 180. Tliis has
he advantage of allowing the periphery of the wheel to have a
i^her velocity, and the size and weight of the wheel is conse
uently diminished.
The buckets, which are generally of the form shown in the
gures, or are curved similar to those of Fig. 182, are coll
ected to a rim M coupled to the central hub of the >N\\ee\ b^
284
HYDRAULICS
suitable spokes or framework. This class of wheel has
considerably used for heads varying from 6 to 70 feet, but u
becoming obsolete, being replaced by the modem turbine, y
for the same head and power can be made much more com
and can be run at a much greater number of revolutions pei
time.
Fig. 180. Overshot Water Wheel.
Fig. 181. Overshot Water Wheel.
The direction of the tangent to the blade at inlet for no si
can be found by drawing the triangle of velocities as in Figs,
and 181. The velocity of the periphery of the wheel is t? and
velocity of the water U. The tip of the blade should be par
to Vr. The mean velocity U, of the water, as it enters the wl
WATER WHEELS 285
in Fig. 181, will be Vo •^kyj2gB,, v^ being the velocity of approach
t{ the water in the channel, H the fall of the free surface and k
a ooeflScient of velocity. The water is generally brought to the
wlieel along a wooden flume, and thus the velocity U and the
•opply to the wheel can be maintained fairly constant by a simple
dmce placed in the flume.
The best velocity v for the periphery is, as shown below,
equal to iU cos tf, but in practice the velocity v is frequently
much greater than this.
In order that XJ may be about 2v the water must enter the
wheel at a depth not less than
below the water in the penstock. When
r = 45 feet, H = 063 feet,
mnd when r = 8 feet, H = 1 foot.
If the total fall to the level of the water in the tail race is h,
the diameter of the wheel may, therefore, be between h and
9
Since IT is equal to >/2flrH, for given values of U and of /i, the
larger the wheel is made the greater must be the angular distance
from the top of the wheel at which the water enters.
With the type of wheel and penstock shown in Fig. 181, the
bead H is likely to vary and the velocity U will not, therefore, be
constant.
If, however, the wheel is designed for the required power at
minimnm flow, when the head increases, and there is a greater
quantity of water available, a loss in efficiency \vill not be
important.
The horse^fxncer of the wheel. Let D be the diameter of the
wheel in feet which in actual wheels is from 10 to 70 feet.
Let N be the number of buckets, which in actual wheels is
generally from 2 to 3D.
Let Q be the volume of water in cubic feet of water supplied
per second.
Let •• be the angular velocity of the wheel in radians, and n
the number of revolutions per sec.
Let b be the width of the wheel.
Let d, which equals rj — n, be the depth of the shroud, which
en actual wheels is from 10" to 20".
i
286
Wbmfii the form of tbe backete the capacity of each bucket j
S
Tfce BTiiiiber of bocketa whieh pa^ tbe str^m pet m:miii
U « fnctbii i of each bucket: is filled mth water
or
Tlie Iractidii 1 % from  to .
If k » die &L flie level of die tail race ^i
the efficiefic7 of ti lpower is
. 50 '
and the widtli b for a given horsepower, HP, 18
llOOHP ^j^,^ HP
^<p*ci o/ reiUn/wi^of /ot^^. As the wteel revolves, the siir&«
of the water in the buckets, due to centrifugal forces, takes up i
parabolic form.
It h shown on page 33S that when a mass of water having tt
inner radios r^ and outer radius ri revolves about a fijced cenlie
^^ith angular velocity «, the pressure headj due to centrifogi
forces, at any radius r, is
ir" 2g
To balance this prepare head the surface of the water in asj
bucket, at the point Cj of radius r, must be raised above tb
hori^>iitaI through A a distance
This is the equation to a parabola, and the surface of the water,
therefore, assumes the form of a parabolic curve.
Let To be the radius at the centre of the surface of the watoin
any cup and ^ the inclination of the radius r© to the horisontaL
Then since n is nearly equal to ri, ^ = n nearly.
WATER WHEELS 287
Then y=2^(n+r) (rrO
= ^r.(rn) nearly.
Therefore, y is approximately proportional to r— n, and the
irface AB is approximately a straight line inclined at an angle
, the tangent of which is
tan B = cos ^.
g
Losses of energy in overshot wheels.
V * .
(a) The whole of the velocity head ^ is lost in eddies in the
nekets.
In addition, as the water falls in the bucket through the
ertical distance EM, its velocity will be increased by gravity,
nd the velocity thus given will be practically all lost by eddies.
Again, if the direction of the tip of the bucket is not parallel to
Tr the water will enter with shock, and a further head will be
Dst. The total loss by eddies and shock may, therefore, be
rritten
'^^^^^
[)r Ai + *i2^,
fc and Tci being coefficients and hi the vertical distance EM.
(6) The water begins to leave the buckets before the level of
the tail race is reached. This is increased by the centrifugal
forces, as clearly, due to these forces, the water will leave the
buckets earlier than it otherwise would do. If h^ is the mean
Iieight above the tail level at which the water leaves the buckets,
» h«id equal to fc» is lost. By fitting an apron GH in front of the
wheel the water can be prevented from leaving the wheel until it
i very near the tail race.
(c) The water leaves the buckets with a velocity of whirl
equal to the velocity of the periphery of the wheel and a further
l^ead JT is lost.
(d) If the level of the tail water rises above the bottom of
Awheel there will be a further loss due to, (1) the head K equal to
•Ke height of the water above the bottom of the wheel, (2) the
^pact of the tail water stream on the buckets, and (3) the
■C^ndency for the buckets to lift the water on the ascending side of
^ wheel.
288 HTDRAUUCS
In times of flood there may be a considerable rise of Urn
downstream, and h^ may then be a large fraction of A. If on'
the other hand the wheel is raised to such a height above the td
water that the bottom of the wheel may be always clear^ Urn
head km will be considerable during dry weather flow, and d»
greatest possible amount of energy will not be obtained from tfa
water, just when it is desirable that no energy shall be wasted.
If h is the difF erence in level between the up and downstresa
surfaces, the maximum hydraulic efficiency possible is
J^'^t^^ ,.
and the actual hydraulic efficiency will be
^^ h
k, ki and h being coefficients.
The efficiency as calculated from equation (1), for any giToi
value of hm, is a maximum when
2g^2g
is a minimum.
From the triangles EKF and KDF, Fig. 180,
(U cos evy^ (U sin ey = va
Therefore, adding v^ to both sides of the equation,
Vr" + i;' = U*cos''^2Ut;cos^ + 2v» + U>sin*tf,
which is a minimum for a given value of U, when 2Uv cos^2f*
is a maximum. Differentiating and equating to zero this, and
therefore the efficiency, is seen to be a maximum, when
U /J
V = ^^ cos u.
The actual efficiencies obtained from overshot wheels vary
from 60 to 80 per cent.
178. Breast wheel.
This type of wheel, like the overshot wheel, is becoming
obsolete. Fig. 182 shows the form of the wheel, as designed by
Fairbairn.
The water is admitted to the wheel through a number of
passages, which may be opened or closed by a sluice as shown in
the figure. The directions of these passages may be made so thai
the water enters the wheel without shock. The water is retained
WATER WHEELS
* backet) by the breast, until tb© bucket reaches the tail race,
greater fraction of tlie head m therefore utilised than in
verahot wheel. In order that the air may enter and leave
i^keta freely, they are partly open at the inner vim. Since
iter in the tail iBce runs in the direction of the motion of
9ttom of the wheel there is no serious objection to the tail
level being 6 inches above the bottom of the wheel.
The losses of head will be the same as for the overshot wheel
S^t that k^ will be practically ^ero, and in addition, there will
I08B by friction in the guide passages, by friction of the water
t moves over the breast, and further loss due to leakage
&n the breast and the wheel*
ling to Rankine the velocity of the rim for ovei^hot and
wbeeb, shoald Le from 4^ to 8 feet per second, and the
j/dty XJ fihoaltl be about 2i^
depth o! the shroud which is equal to ntt is from 1 to
liet it be denoted by d. Let H be the total fall and let
aed that the efficiency of the wheel is 65 per eeiit. Then,
mmM
mi
290
HYDRAUUGS
the quantity of water required per second in cubic feet for a
given horsepower N is
^_ N.550
^"■624xHxO65
_ 13'5N
H •
From ^ to f of the volume of each bucket, or from ^ to  of the
total volume of the buckets on the
loaded part of the wheel is filled with
water.
Let 6 be the breadth of the buckets.
If now V is the velocity of the rim, and
an arc AB, Fig. 183, is set off on the
outer rim equal to v, and each bucket
is half full, the quantity of water
carried down per second is
iABCD.6.
Therefore
«»(^')
vdb.
Equating this value of Q to the above value, the width b is
^^ 27KD
(ti + Ti) vdR '
D being the outer diameter of the wheel.
Breast wheels are used for falls of from 5 to 15 feet and the
diameter should be from 12 to 25 feet. The width may be as
great as 10 feet.
Example, A breaat wheel 20 feet diameter and 6 feet wide, working on a fifl
of 14 feet and having a depth of shroud of 1' S", has its backets f falL The meta
velocity of the backets is 5 feet per second. Find the horsepower of the wh«d»
assaming the efficiency 70 per cent.
xjxy K ^oK a ^ 624 X 070x14'
HP = 5x 125x6x5 X =^77
o ooU
= 261.
The dimensions of this wheel should be compared with those calonlated for M
inward flow turbine working under the same h^ and developing the aame hofM*
power. See page 339.
179. Sagebien wheels.
These wheels, Fig. 184, have straight buckets inclined to the
radius at an angle of from 30 to 45 degrees.
The velocity of the periphery of the wheel is very small, nerer
exceeding 2i to 3 feet per second, so that the loss due to the witer
leaving the wheel with this velocity and due to leakage betweoi
the wheel and breast is small.
WATER WHEELS
291
An efficiency of over 80 per cent, has been obtained wi^
wheels*
The water enters the wheel in a horizontal direction wit
velocity V equal to that in the penstock, and the triangle of
reloeities is therefore ABC*
If the bucket is made parallel to Vr the water enteis without
ck, while at the same time there is no lofts of bead due to
icnon of guide passages, or to contraction as the water enters or
iv€3 them ; moreover the direction of the stream has not to be
aoged.
Fig. 1S4. Sftgfbien WUeel.
Tlie iucUned srraight bucket has one disadvantage ; when the
[lower part of the whee! is drowaedj the buckets as they ascend are
{more nearly perpendicular to the surface of the tail water than
rheti the blades are radial, but as the peripheral speed is very
flow the renistance due to this cause is not considerable,
ISO. Impulse wheels.
In OvLrshot ami Breast wheels the work is done principally
Iby the weight of the water. In the wheels now to be considered
ie whole of the head available is converted into velocity before
le wAter strikes the wheel, and the work is done on the wheel
"rig the niomeutum of the mass of moving water, or in
i , by changing the kinetic energy o£ tlae w&^«nc.
J
292
HYDRAULICS
Undershot wheel with fiat blades. The simplest case is wlien
a wheel with radial blades, similar to that shown in Fig*. 185, is
put into a nmning stream.
If 6 is the width of the wheel, d the depth of the stream under
the wheel, and U the velocity in feet per second, the weight of
water that will strike the wheel per second isb.d.w.TJ lbs,, and
the energy available per second is
b.d.WcT foot lbs.
Let V be the mean velocity of the blades.
The radios of the wheel being large the blades are similar to
a series of flat blades moving parallel to the stream and the water
leaves them with a velocity v in the direction of motion.
As shown on page 268, the best theoretical value for the
velocity v of such blades is iXJ and the maximum possible
efficiency of the wheel is 0'5.
Fig. 185. Impalse Wheel.
By placing a gate across the channel and making the bed near
the wheel circular as in Fig. 185, and the width of the wheel
equal to that of the channel, the supply is more under control, and
loss by leakage is reduced to a minimum.
The conditions are now somewhat different to those assmned
for the large number of flat vanes, and the maximum possiUe
efficiency is determined as follows.
Let Q be the number of cubic feet of water passing through
the wheel per second. The mean velocity with which the wrtir
leaves the penstock at ah is \5 = hJ2gh. Let the defyth of tht
WATER WHEELS 293
stream at a& be ^. The velocity with which the water leaves the
wheel at the section cd is Vy the velocity of the blades. K the
width of the stream at c(2 is the same as at a& and the depth
is A«, then,
^ X t? = ^ X XJ,
or ^ = — .
V
Since TT is greater than v, h^ is greater than ty as shown in
the figure.
The hydrostatic pressure on the section cd is ^ho^bw and on
the section a6 it is ifbw.
The change in momentum per second is
and this must be equal to the impressed forces acting on the mass
of water flowing per second through ab or cd.
These impressed forces are P the driving pressure on the wheel
blades, and the difference between the hydrostatic pressures acting
on cd and ab.
If, therefore, the driving force acting on the wheel is P lbs.,
then,
P + iWfcw;  ^^"fctr = Q^ (U  tj).
Substituting for hoy — , the work done per second is
Or, since Q = 6 . ^ . U,
w9=(n.)«j(2i).
The efficiency is then,
f (Uv) t /U v\
g 2\v U/
*= — — w '
2g
which is a maximum when
2v^  4u»U + gtU'' + gtv^ = 0.
The best velocity, v, for the mean velocity of the blades, has
been found in practice to be about 04U, the actual efficiency is
from 30 to 35 per cent., and the diameters of the wheel are
generally from 10 to 23 feet.
Floating wheels. To adapt the wheel to the rising and
lowering of the waters of a stream, the wheel may be mounted ou
294
HYDRAULICS
a frame which may be raised or lowered as the stream rises, Gr thd
axle carried upon pontoons so that the wheel rises automaticalljr
r'th the stream.
181, Poncelet wheel.
^rhe efficiency of the straight hlade impulse wheels is
small, due to the large amount of energj^ lost by shock, and to
velocity with which the water leaves the w^heel in the direction
motion,
Tlie efficiency of the wheel is doubled, if the blades are of such
a form, that the direction of the blade at entiance is parallel to
■the relative velocity of the water and the blade, as fiist suggestad
l)y Poncelet, and the water is made to leave the wheel with do
component in the direction of mution of the periphery of tha
wheel.
Fig, 186 shows a Poncelet wheel.
I
Fig. 186* Undershot WlieeL
Suppose the water to approach the edge A of a blade mil
velocity U making an angle ^ mth the tangent to the wheel at t
Then if the direction of motion of the water is in the directifl
AC the triangle of velocities for entrance is ABC*
The relative velocity of the water and the wheel is V^ and if
the blade is made sufficiently deep that the water does not overflow
the upper edge and there is no loss by shock and by fiictiun, i
particle of water will rise up the blade a vertical height
WATER WHEELS 295
a begins to fall and arrives at the tip of the blade with the
ty Vr relative to the blade in the inverse direction BE.
he triangle of velocities for exit is, therefore, ABE, BE being
1 to BC.
.'he velocity with which the water leaves the wheel is then
AE = U,.
It has been assomed that no energy is lost by friction or by
»ck, and therefore the work done on the wheel is
ad the theoretical hydraulic efficiency* is
= l§* (1).
This will be a maximum when Ui is a minimum.
Now since BE = BC, the perpendiculars EF and CD, on to
AB and AB produced, from the points E and C respectively, are
equaL And since AC and the angle are constant, CD is constant
for all values of Vy and therefore FE is constant. But AE, that is
Ui, is always greater than FE except when AE is perpendicular
to AD. The velocity Ui will have its minimum value, therefore,
when AE is equal to FE or Ui is perpendicular to v.
The triangles of velocities are then as in Fig. 187, the point B
bisects AD, and
V = JU cos 6,
For maximum efficiency, therefore,
V = fU cos 0.
* In what follows, the terms theoretical hydraulic efficiency and hydniulio
•flSeiency will be freqaently osed. The niazimom work per lb. that can be utilised
bj ftoy hydxmolio maohine supplied with water under a head H, and from which
the water ezluuuts with a velocity u is H  ^t . The ratio
H Hi
11 the theoratieel hydraulic effidenoy. If there are other hydraulic losses in th
m nifhiw^i eqairalent to a head h/ per lb. of flow, the hydraulic efficiency is
— H — •
The aetoal effioieiiey of the maohine is the ratio of the external work done per
of water hj the machine to H.
BTDftlDIiCS
' cmn »bo be found t>K oaoitdeiiiig Qm dtangf oT
TW totfti cb»oge of Telocity jmprewed <m the water is CE^ aoJ
ife cittiig^ in th^ directioii of tsMmx is
fwm^ FD, 1%. im AIL^C
A«d»aiBEi.ttloltoBC,FBiii L^^^^'h'^^
FD = 2(Uco6ir). Kg. m.
uid tke
r
9e)
4
DtteRaDtiMtmg wttli i ind eqoatiii^ to zero,
or r = yU cos d.
The velocity Ui with which the water leaves the wheel, is then
perpendicular to r and is
Ui=Usind.
Substitnting for r its value JU cos B in (2), the maximum efficient
is cos^ 0,
The same result is obtained from equation (1), by substitutini
for Ui, Usin^.
The maximum efficiency is then
^ , IPsin'^ ,.
E = 1 ™ — = cos'^.
A common ^'alue for is 15 degrees, and the theoretia
hydraulic efficiency is then 0"933.
This increases as diminishes, and would become unity if
could be made zero.
If, however, 6 is zero, U and r are parallel and the tip of tl
blade will be perpendicular to the radius of the wheel.
This is clearly the limiting case, which practically is n
realisable, Avithout modifying the construction of the wheel. Tl
necessarj' modification is shown in the Pelton wheel described <
page 377.
The actual efficiency of Poncelet wheels is from 55 to 65 p
nt.
B'ATER WHEELS
tirfthe bed. Water enters i\w wheel at all points between
untl R, and for no shock the bed of the channel PQ should be
it of such a fomi tlmt the diroction of the stream, where it,
aliTs fhe wheel at any point A between E and Q, should make
constant angle & ^vith the radius of the wheel at A,
Willi as centre, draw a circle touching the line AS which,
liriikef^ the given angle ^ with the radius AO. Take several
point* on the circumference of the wheel between R and
and draw tangents t4> the circle STV* If then a curve
' m drawn normal to thcfle several tangents, and the stream
are parallel to PQ, the ivater entering any part of the
fl between R and Qj will make a constant angle \rith the
iiijss and if it enters without shock at A, it will do so at all
rinti*. Tlje actual velocity of the water U, as it moves along the
PQj will be less than V2gH, due to friction, etc. The
efficient of velocity iv in most cases will probably be between
I and 0*95, so that taking a mean value for fey of 0*92*1,
U = 0925 s/%H.
The best value for tfte velocity v tahhig frictum mio ctcc&wni,
\ determining the best velocity for the periphery of the wheel no
Qowance has been made for the loss of energy due to friction in
* wheel.
If V, is the relative velocity of the water and wheel at entrance^
is Uy be expected that the velocity relative to the wheel at exit
! be le^ than Vr, due to friction and interference of the rising
3d filing particles of water.
Tlie case is somewhat analogous to that of a stone thrown
ertically up in the atmosphere with a velocity v. If there were
Mnesistance to its motion, it would rise to a certain height,
:
A.=
2ff'
, di^ii descend, and when it again reached the earth it would j
a velocity equal to its initial velocity t*. Due to resistances,^
height to which it rises will be less than h^ j and the velocity
ith which it reaches the gi^ound will be even less than that due
foiling freely through this diminished height.
Let the velocity relative to the wheel at exit be nVr, « being
, fraction less than unity.
Tlie triangle of velocities at exit will then be ABE, Fig, 188. j
he change of velocity in the direction of motion is GH, which
Itials
BHHGB = BH(l + n)
= {1 +n)(U cos er).
298 HTDRAUUCS
If the velocity at exit relative to the wheel is only nVn thai
must have been lost by friction etc., a head equal to
The work done on the wheel per lb. of water is, therefore,
{(l+n)(Ucosev)}v Yr\. „r,
g ^(l"^
Fig. 188.
Let (1  TJ?) be denoted by /, then since
V/ = BH" + CH» = (U cos ««)' + U» sin'tf,
the efficiency
{(l + n)(Ucos««)}^{(Ucostft>)» + U»8in»tf}
e = a^ .
Differentiating with respect to v and equating to zero,
2 (1 + w) Ucos^4 (1 + n) t; + 2U/cos 6  2t7/=0,
from which
_ {(l+n)+/}Ucos^
"" /+2(l + n)
^(2 + n~?i')Ucosg
3n*4.2n
If /is now supposed to be 0*5, i.e. the head lost by friction, et
is — ^.^ — , n IS 71 and
V = 5617 cos ^.
If /is taken as 075,
v = 0'6Ucos^.
Dimemsiotis of Ponceht wheels. The diameter of the wheel
should not be less than 10 feet when the bed is curved, and noi
less than 15 feet for a straight bed, otherwise there will be con
siderable loss by shock at entrance, due to the variation of the
angle which the stream lines make with the blades between B
and Q, Fig. 186. The water will rise on the buckets to a heigte
WATER WHEELS
%m
Bttrlj equal to ^ , and since the water first entera at a point R,
be blade depth d must, therefore, be greater than this, or the
f»t^* will overflow at the upper edge. The clearance between
lie bed and the bottom of the wheel should not be less than f\
Ibe peripheral distance between the consecutive blades is taken
torn S inches to 18 inches,
^ Morse^p&wer of Ponctlet wheels. If H is the height of the
Ir&oe of water in the penstock above the bottom of the wheel,
be velocity U will be about ^
0*92 v^^, ,
tid V m&Y be taken aa
0*55 X 0*92 n/%H = 0'5 V%H.
Let D be the diametei* of the wheel, and & the breadth, and let
Ibe the depth of the orifice EP* Then the number of revolutions
BT minute is
I 05 V2ffH
I IT, D
The coeflicient of contraction c for the orifice may be from 0*6,
I it is shaTpedged, to 1 if it is carefully rounded^ and may be
iketi as 0^8 if the orifice is formed by a flatedged sluice.
The quantity of water striking the wheel per second is, then^
Q = 0'92d6%/2^.
1 If the efficiency is taken as 60 per cent., the work done per
bond is 06 ^ 624QH ft. lbs,
I The horsepower N is then
^ ■ 5,50 • I
182« Turbines.
Although the water wheel has been developed to a considemble
kgre^ of perfection, efficiencies of over 80 per cent having been
plained, it is being ahuost entirely superseded by the turbine.
I The old water wheels were required to drive slow moving
hichiiiery, and the great disadvantage attaching to them ot
iring a small angular velocity was not felt. Such slow moving
WMth are however entirely unsuited to the driving of modern
lacKinery, and especially for the d^i^^Ilg of dynamos, and they
1^ further quite unsuited for the high heads which are now
Billed for the generation of power.
I Turbine wheels on the other hand can be made to run at either
pr or veiy high speeds, and to work under any head vaTrjm^
^1^
^dM
800
BYBRAOLICS
fn^jiii 1 foot to 2000 feet, and the speed can be regulated
much greater precision.
Due to the slow B}eeds, the old water wheels ecmld not deirelop
large power, the niaximmn being about 100 horsepower, wherea*!
at Niagara Falls, turbines of 10,000 horsepjower have recently 
been installed.
Types of Tm'hifM&.
Turbines are generally divided into two cl&saes^ impulse,^
free deviation turbiues, and reaction or pressure turbines.
In both kinds of turbines an attempt is made to shape
vanes so that the water enters the wheel without shock ; tliat is
the direction of the relative velocity of the water and the vi
parallel to the tip of the vane^ and the direction of the leai
edge of the vane is made so that the water leaves in a speciie^
direetion.
In the first class, the whole of the available head is com
into velocity before the water strikes the turbine wheel, am
pressure in the driving fluid as it moves over the vanes rei
constant^ and equal to the atmospheric pressure. The wheel
vanes, therefore, must be so formed that the air has free
between the vanes, and the space between two consecutive vaneij
must not be full of water. Work is done upon the vanes, or i
other wordsj upon the turbine wheel to which they are fixedti!
virtue of the change of momentum or kinetic energy of
moving water, as in examples on pages 270 — 2*
Suppose water supplied to a turbine, as in Fig. 258, under ic^
effective head H, which may be suppoiied equal to the total Ht^J
miiius losses of head in the supply pipe and at the noEsle* Tin
water issues from the nozzle with a velocity U = j2gKj and ihi
available energy per pound is
IP
H =
%^
Work is done on the wheel by the absorption of the who?e, c^r
part, of this kinetic energy.
If Uj is the velocity with which the water leaves the wb«r:u
the energy lost by the water per pound is
2g 2g *
and this is equal to the work done on the wheel together w'4
energy lost by friction etc* in the wheel.
In the second class, only part of the available head ib
verted into velocity before the water enters the wheel, and
Blocity and preesore both vary as the water pasaefi through the
rbeel It is therelore essential, that the wlieel shall always be
pe»t full of water* Work is done upon the wlieel, ae will be seen
B the sequence, partly by changing the kinetic energy the water
when it enters the wheel, and partly by changing itB
or potential energy,
Suppose water is supplied to the turbine of Fig. 191, under
[ effective hi?ad H ; the velocity U with which the water enters
wheel, is only Borae fraction of J2gilj and the pressure head
the inlet fco the wheel mil depend upon the magnitude of U
And upon the position of the wheel relative to the head and tail
■vmter siirface^i* The turbine wheel always being full of water,
jUiere is continuity of flow through the wheel^ and if the !iead
faBpreeaed upon the water by centrifugal action is determint^d, as
pa pAge 335, the equations of Bernouilli * can be used to determine
Ed aajr gi^en case the difference of pressure head at the inlet and
of the wheeL
tlie preasure head at inlet is — and at outlet — , and the
ity with which the water leavea the wheel is Ui, the v^ork
on the wheel (see page 338) is
^  — + ,ir  TT per pound of water,
ir work is done on the wheel, partly by changing the velocity
lead and partly by changing the presBure head. Such a turbine
i called a reaction turbine, and the amount of reaction is measured
^ the ratio
r p Pi
riy, if p is made equal to p^ the limiting case is reached,
f tho turbine becomes an impntsej or freede\^ation turbine.
Id be clearly understood that in a reaction turbine no
ue on the wheel merely by hydrostatic pressure, in the
III which work m done by the pressure on the piston of a
engine or the ram of a hydraulic Hft.
13, Eeactioii turbiBes.
tie oldest furm of turbine is the simple reaction, or Scotch
ine, which in its simplest form is illustrated in Fig. 189.
%*ertieal cnbe T has two horizontal tubes connected to it, the
ends of which are bent round at right angles to the direction
* See page SSC
302
HTDRAUUCS
of length of the tube, or two holes O and Oi are drilled as in tiie
figure.
Water is supplied to the central tnbe at such a rate as to keep
the level of the water in the tube
constant, and at a height h above
the horizontal tubes. Water escapes
through the orifices O and Oi and
the wheel rotates in a direction
opposite to the direction of flow of
the water from the orifices. Tur
bines of this class are frequently
used to act as sprinklers for distri
buting liquids, as for example for
distributing sewage on to bacteria
beds.
A better practical form, known as the Whitelaw turbine, is
shown in Fig. 190.
KRs
Fig. 189. Sootoh TuImim.
1
1
Fig. 190. Whitelaw Turbine.
To understand the action of the turbine it is first necessary t<
consider the effect of the whirling of the water in the arm upot
TURBINES 303
4ie discharge from the wheel. Let v be the velocity of rotation
^ the orifices, and h the head of water above the orifices.
Imagine the wheel to be held at rest and the orifices opened ;
the head causing velocity of flow relative to the arm is
iply A, and neglecting friction the water will leave the nozzle
with a velocity
Vo = 'J2gh.
Now suppose the wheel is filled with water and made to rotate
ttt an angular velocity f», the orifices being closed. There will
aofw be a xnressure head at the orifice equal to h plus the head
iminesBed on the water due to the whirling of each particle of
water in the arm.
Assume the arm to be a straight tube, Fig. 189, having a cross
sectional area a. At any radius r take an element of thickness dr.
The centrifugal force due to this element is
a/= —  — .
The pressure per unit area at the outer periphery is, therefore,
9
~ 2g '
and the head impressed on the water is
P __ Ctf V
Let V be the velocity of the orifice, then t? = cor, and therefore
w''2g'
If now the wheel be assumed frictionless and the orifices are
opened, and the wheel rotates with the angular velocity co, the
head causing velocity of flow relative to the wheel is
K=.h + ^ = h+^ (1).
w 2g
Let Vr be the velocity relative to the wheel >vith which the
water leaves the orifice.
^^ ^ = ^^1 (2).
The velocity relative to the ground, with which the water
leaves the wheel, is Vrt?, the vector sum of Vr and v.
_1 f^ wau^rdr
^"ajo g
304
The water leaves the wheel, therefore, with a velocity relttnf 
to the ground of M= Vr », »nd the kinetic energy lost ia
The theoretical hydraulic efficiency is theni
Since from (2) Tr becomes
increases, the energy lost per 
and the eflicieacy E, therefor
The efficifmty of the reuctiou .
As before,
6 nearly equal to t^ a& f
diminishes b& v im
a^a with !?•
C whenfnetion is cofwidertd.
^9
(3),
kY/
Assuming the head lost by friction to be ^h^, the total head
must be equal to
2ff
(4).
The work done on the wheel, per pound, is now
h
and the hydraulic efficiency is
A
2ff 2ff'
Substituting for h from (4) and for ^j Y^v^
2v(Yrv)
Let
then
e =
{l+k)Yr'v''
Yr = nv,
2(711)
(l+/L)w»r
Differentiating and equating to zero,
n2(l + A:)2n(l4.fr) + l = 0.
TURBINES
305
the efficiency is a maxiYnnni when
T
1 + k'
Fig. 191. Outward Flow Torbine.
U. B.
"l^
806
HTDRAUUC8
184. Outward flow turbines.
The outward ilow tarhine was invented in 1838 hy
neyron. A cylindrical wheel W, Figs. 191, 192, and 201, har
a number of suitably shaped vanes, is &xed to a vertical ami
The water enters a cylindrical chamber at the centre of d»J
turbine, and is directed to the wheel by auitable tiated
blades G, and flows through the wheel in a radial dir
outwards. Between the guide blades and the wheel is a ryhndiij
cal sluice R which is used Uj ' >1 the flow of water th
the wheel.
*v
f
Fjg. 191 <t.
This method of regnlating the flow is very imperfect, as wbm
the gate partially closes the pas^sageSj tbere must be a suddii^
enlargement as the water enters the wheel, and a loss of h
ensues, "llie efficiency at " part gate " is consequently reif
much less than when the flow in unchecked. This difficaltf
partly o%'ercome by dividing the wheel into sevei^l distinct
compartments by horizontal diaphragmsj as shown in Pig. 19^
so that when working at part load, only the efficiency of oa*
compartment is affected*
The wheels of outward flow turbines may have their asse^
either horizuntal or vertical, and raay be put either above, *
below, the tail water level.
The '' atuclion hihty If plact.Mi above the tail water, ti*
exhaust must take place down a '* suction pipe," as in Fig. 2!0l
page 317, the end of which must be kept drowned, and the pip*
airtight, so that at the outlet of the wheel a pressure less tbii
the atmospheric pressure raay be maintained* If A i is the he^
of the centre of the discharge periphery of the wheel al
tail water level, and pa is the atmospheric pressure in pow
square foot, the pressure head at the discharge circumferoiui?
w
TURBINES
SOT
I wiieel catmot be more than 34 feet above the level of the tail
iter, or the preasore at the outlet of the wheel will be negativei
\d practTcally, it cannot be greater than 25 feet
It is Ehowii later that the effective head, under which the
rbine works, whether it h drowned, or placed in a suction tnbe,
the total fall of the water to the level of the tail race.
Fif, 19a, FoQTfieyron Out word Flow Turbine.
ttm af the suction tube has the advantage of allowing the
' I be placed at some distance above the tail water J
r.j bearings can be readily got atj and repairB caili
r emmiy executed.
mokiDg the suction tube to enlarge as it deseendBt the
of exit can be dimimghed very gradually, aiii '\^ %mX.
308 HYDRAULICS
value kept small. If the exhaust takes place direct froi
wheel, as in Fig. 192, into the air, the mean head available
head of water above the centre of the wheel.
Triangles of velocities at inlet and outlet For the wai
enter the wheel without shock, the relative velocity of the
and the wheel at inlet must be parallel to the inner tips <
vanes. The triangles of velocities at inlet and outlet are i
in Figs. 193 and 194.
Fig. 193.
Let AC, Fig. 193, be the velocity XJ in direction and magr
of the water as it flows out of the guide passages, and let A
the velocity v of the receiving edge of the wheel. Then DC
the relative velocity of the water and vane, and the recc
edge of the vane must be parallel to DC. The radial comp
GC, of AC, determines the quantity of water entering the ^
per unit area of the inlet circumference. Let this radial vel
be denoted by u. Then if A is the peripheral area of the
face of the wheel, the number of cubic feet Q per #3cond ent
the wheel is
Q = A.i^,
or, if (2 is the diameter and b the depth of the wheel at inlet
t is the thickness of the vanes, and n the number of vanes,
Q = (vd n.t) .b.u.
Let D be the diameter, and Ai the area of the discharge
phery of the wheel.
The peripheral velocity Vi at the outlet circumference is
v.D
TtJRBINES
309
t Ui he tlie radial component of velocity of exit, then what
per the direction with which the water leayes the wheel the
tciiAl component of velocity for a given discharge is coustant
The trmngle of velocity can now be drawn as follows :
Set off BE equal to t^i. Fig. 194, and BX radial and equal
Let it BOW be supposed that the direction EF of the tip of the
Itne at discharge is knoiJ^Ti, Draw EF parallel to the tip of the
^ne at D, and tlirotigh K draw KF parallel to BE to meet EF
F,
Then BF is the velocity in direction and magnitude with which
lie ir^ter leaves the wheel j relative to the gi'ound, or to the fixed
of the turbine. Let this velocity be denoted by Uj. If,
istead of the direction EF being given, the velocity Uj is given
I direction and magnitude, the triangle of velocity at exit can be
ri&im by setting out BE and BF equal to Vi and Ui respectively,
ttd joining EF, Then the tip of the blade must be made parallel
EF,
Fof »uy given value of Ui the quantity of water flowing
iroQgh the wheel is
Q = AiUiCotj^=A,w,.
Work drjtw tm the wheel neglecting friction, etc. The kinetic
a©rgy of the wmter as it leaves the turbine wheel is
^ per pound,
if the discharge is into the air or into the tail water this
is of necessity lost. Neglecting friction and other losses,
' available energy per pound of water is then
H5i! foot lbs.,
. the theoretical hydraulic efficiency is
E =
H '
"ant for any given value of Ui, and independent of the
.„., f Uj This efficiency must not be confused with the
1 efficiency, which is much less than E»
be smaller Uj , the greater the theoretical hydraulic efficiency,
^nnce for a given flow through the wheel, Uj will be least
it 18 radial and equal to ttj, the greatest amount of work
be obtained for the given flow, or the efficiency will be a
imtuti, when the water leaves the wheel radially. M tToss
ML
310 HTDRAULICS
water leaves with a velocity Ui in any other direction, the
efficiency will be the same, but the power of the wheel wfll be
diminished. If the discharge takes place down a snction tnbe^
and there is no loss between the wheel and the outlet from the
tube, the velocity head lost then depends upon the velocity TTi
with which the water leaves the tube, and is independent of the
velocity or direction with which the water leaves the wheel.
The velocity of whirl at inlet and outlet The component of
XJ, Fig. 193, in the direction of v is the velocity of whirl at inlel^
and the component of Ui, Fig. 194, in the direction of ih, is the
velocity of whirl at exit.
Let y and Yi be the velocities of whirl at inlet and oatlek
respectively, then
V = Ucos^
and Vi = Uisini8 = t^itani8.
Work done on the wheel. It has already been shown,
section 173, page 275, that when water enters a wheel, rotating
about a fixed centre, with a velocity XJ, and leaves it with velodi^
Ui, the component Yi of which is in the same direction as Vi, the
work done on the wheel is
Yv Yii?i ,
per pound,
and therefore, neglecting friction,
yir^^ w
This is a general formula for all classes of turbines and shaoU
be carefully considered by the student.
Expressed trigonometrically,
t;U cos S __ ViUitoxiP _ TT _ Hl fc%\
9 9 ^ 2g <2).
If F is to the left of BK, Yi is negative.
Again, since the radial flow at inlet must equal the radial floir
at outlet, therefore
AUsintf = AiXJiCos/9 ,,..,, ^...(g).
When Ui is radial, Yi is zero, and th equals t^i tan a.
'^^'^ T=°* • ^^
from which ?l£2i« = H?^^ ®
g 2g
and from (3) ATTsin^^ AiVitana (f^
TURBINES
311
and
If the tip of the Tana is radial at inlet, i.e. Vr is radial,
V = r
2sr
V* tan' a
9 9 '
= H
29
..(7)
.(8).
In actual turbines P is from '02H to '07H.
Exaw^U. An oatwmrd flow turbine wheel. Fig. 195, has an internal diameter of
6'349 feet, and an external diameter of 6*25 feet, and it makes 250 revolutions per
minnte. The wheel has 32 yanes, which may be taken as  inch thick at inlet and
1} inches thick at outlet The head is 141*5 feet above the centre of the wheel and
the ezhaoat takes place into the atmosphere. The effective width of the wheel face
mt inlet and outlet is 10 inches. The quantity of water supplied per second is
915 cubic feet.
Neglecting all frictional losses, determine the angles of the tips of the vanes at
inlet and oaUiet to that the water shall leave radially.
The peripheral velocity at inlet is
v=« X 6249 X W=69 ft P«r 8«o.
ftodatootlet Vi=«'x625x V/=:82ft. „ „
Fig. 195.
The i»dial Tcloeity of flow at inlet is
215
w X 5249 x li  f I X J
= 18*35 ft. per sec.
The zadiAl Tekwty of flow at exit is
^ 215
*^"»x6*25xHHxf'
= 16*5 ft. per sec.
^=423 ft.
312
Then
and
^ = 14^5
= 13727 ft,
137'«7>cS22
= 64 ft. p^ Mc»
To dr&w the triangle of Tdocitks at mtel Bet out p and u at Hglit ai]ig]e&
Then mnee V is 64^ and ia b« tan^eotia] coEDpooeat of U> k\A u in tkniM
compon<?tit of U, tb« ^ir^tioD and magnitude of U is determined.
By joinmg B and C the relative velocity' \% is ohtained, and BC i« p&Ealkl lo Ik
tip of thi' \'une.
The triangle of v«lodtkfl at exit ii DBF, and the ti^ of the tane mmt be ptfiOK
toEF.
 ~V^*»"
^
tf,S^
! B
t/"''
"c"
E
Fig, m
Fig. 197,
aitW —
The angiefi i?» ^, and a can be caloulated; for
tanS== ^1^02867,
b'4
tan^=  — !^= 3670
and
and, therefore,
tanB^^0vt994,
^ = 105° ir,
a=iri7'.
It will be seen later how these aiiKles are modt^ed when friction ia oonsiderei
Fig. 1^8 rHows the form the guide blades and ?anea of the wheel woold
probably lake.
7*^ path of the water thrQugh thi wheel. The average l^adial vetoc^itj chroogfa
the wheel may be taken aa 17*35 feet.
The time taken for a i>article of water to get through the wheel i«t thereibrc^
R ^ r 0'5 ^ ^
The an^le turned through bj the wheel in thia time is 0^S9 radians.
Set off the arc AB, Fig. 198, ecual to 39 radian, and divide it into four eqntl
parts, and draw the radii fUffb, pc and Bit.
Divide AD atiM into four equal parts, Aud draw cireles through A^^ A,, &nd A,'
Suppose a paiticlo of ^ater to enk^r the vrhed &i A in contact wiib a vooe aM
Buppoef it to remam id contact with the vau<* during itB poBsage through the vhts^ '
Then, a&snmiug the radial velocity is coo&tant, while the wheel turns throTJcb tbe
ft the water will mote radially a dietanoe AA^ and a particle that came o^ ^
TURBINES
313
B TA&e ftt A will, therefore, be in oontaot with the Tane on the arc throngh A^ .
i« Tane initially passing throngh A will be now in the position el, al being
Qml to hJ and the partiele will therefore be at 1. When the particle arrives on
e are throng A, the vane will pass throngh/, and the particle will consequently
at S, 63 bdng eqoal to mn. The curve A4 drawn through Al 2 etc. gives the
^tli of the water relative to the fixed casing.
Fig. 198.
185. Losses of head due to Motional and other resistances
i outward flow turbines.
The losses of head may be enumerated as follows :
(a) Loss by friction at the sluice and in the penstock or
ipply pipe.
If Vo is the velocity, and ha the head lost by friction in
18 pipe,
(b) As the water enters and moves through the guide
assages there will be a loss due to friction and by sudden changes
1 the velocity of flow.
This head may be expressed as
being a coefficient.
* See page 119.
814
HYDRAULICS
(c) There is a loss of head at entrance due to shock as
the direction of the vane at entrance cannot be determined
with precision.
This may be written
V«
2a '
that is, it is made to depend upon Yr the relative velocity of the
water, and the tip of the vane.
(d) In the wheel there is a loss of head hd, due to fricticniy
which depends upon the relative velocity of the water and the
wheel. This relative velocity may be changing, and on any small
element of surface of the wheel the head lost will diminish, as the
relative velocity diminishes.
It will be seen on reference to Figs. 193 and 194, that as the
velocity of whirl Vi is diminished the relative velocity of flow tv at
exit increases, but the relative velocity Vr at inlet passes through
a minimum when V is equal to r, or the tip of the vane is radial
If Vo is the relative velocity of the water and the vane at any
radius, and b is the width of the vane, and dl an element of
length, then,
Jci being a third coefficient.
If there is any sudden change of velocity as the water poonrw
through the wheel there will be a further loss, and if the turbine
has a suction tube there may be also a small loss as the water
enters the tube from the wheel.
The whole loss of head in the penstock and guide passages may
be called H/ and the loss in the wheel h/. Then if U* is the
Fig. 199.
TUJEIBINES
315
city with which the water leaves the tufbme the effective
is
In well designed inward and outward flow turbines
2ff
+ A/+H/
^^' = eH
ies from O'lOH to *22H and the hydraulic efficiency is, therefore^
[jm 90 to 78 per cent
The efficiency of inward and outward flow tnrbines including
Dhanical losses is fruin 75 to 88 per cent.
Calling the hydraulic efficiency e, the general formula (1),
tion 184, may now be written
& 9
= 78to9H
Outward flow turbines were made by Boyden* about 1848 for
rhich he claimed an efficiency of 88 per cent. The workmanship
of the highest quality and great care was taken to reduce
losses by friction and shock. The section of the crowns of the
rhoel of the Boyden turbine is shown in Pig, 199, Outside of
turbine wheel was fitted a "diffuser" through which, after
Irving the wheel, the water moved radially with a continuously
imlni^iung velocity, and finally entered the tail race with a
!ity much less, than if it had done so direct from the wheeL
lo«s by velocity head was thus diminished, and Boyden
claimed that the diffuser increased the efficiency by 3 per cent.
186. Some actual outward flow turbines.
l^'uhh' oufward flaw turhin&f. The genera! arrangement of an
ontward flow turbine as installed at Ch^vres is shown in Fig. 200,
bere are four wheels fixed to a vertical shaft, two of which
ire the water from below, and two from above. The fall
i.ries from 27 feet in dry weather to 14 feet in time of flood.
Tlje uppt.*r wheels only work in time of flood, while at other
imes the full power is developed by the lower wheels alone, the
'cylindrical sluices which surround the upi>er wheels being set in
^«uch a position as to cover completely the exit to the wheel.
The water after leaving the wheels, diminishes gradually in
locity, in the concrete passages leading to the tail race, and the
of head due to the velocity with which the water enters the
4
Lamdl HffdrauUc E^eHmettttt J. B. Franoisi 1S56.
i
316
tail race is consequently small. These passagee eemre _, ™^,
purpose as Bo^den^s diifuser, and as tlie enlarging saction tabej
in that they allow the velocity of exit to diminish gradiiaUj^.
High
Fig. 200. Double Outward Flow Turbine, (E so her W3*sa and Co.)
Outward fioie furhine with horizontal tu^m Fig, 201 shows b
section through the wheel, and the supply and exhaust pipes, of Rn
outward flow turbine, having a horizontal axis and exhausting
dowm a " suction pipe." The water after leaving the wheel enters
a large chamber, and then passes down the exhaust pipe, the
lower end of which is below the tail race.
The supply of water to the wheel is regulated by a horizontal
cylindrical gate S, between the guide blades G and the wheel. The
gate is connected to the ring R, which slides on guides, outside
the supply pipe P, and is under the control of the governor.
The pressure of the water in the supply pipe is prevented from
causing end thrust on the shaft by the partition T, and between
T and the wheel the exhaust water has free access.
Outward flow turbines at Niagara Falls. The first turbines
installed at Niagara Falls for the generation of electric power,
TURBINES
817
^rere outward flow turbinee of the type shown in Figs* 202 and
203.
TTier© are two wheels on the same vertical ahaft^ the water
being brought to the chamber between the wheels by a Teiticai
penBtock 7' &* diameter. The water passes upwards to one wheel
And downwards to the other.
Fig, SOI. Ontw&rd Flow TnTbine with Hoetion Tube.
Am shown in Fig. 202 the water preBBiire in the chamber is
vented from acting on the lower wheel by the partition MN,
fa allowed to act on the lower side of the upper wlieel, the
{MrtitioD HK having holes in it to allow the watar free access
iderneath the wheoL The weight of the vertical shaft; and of
wheels, is thus balanced, by the water pressure itself.
The lower wheel is fixed to a solid shaft, which passes through
Iho centre of the upper wheel, and is connected to the hollow
of the upper wheel as shown diagram matically in Fig. 202,
i¥e this connection^ the vertical shaft is formed of a hollow
thD
318
tube 38 inclieB diameter, except where it passes through tb
bearii)gB, where it is solid, and 11 inches diameter.
A thrust block is also provided to carry the tmbaiftnced I
weight*
The regulatiug sluice is external to the wheel. To maintaias I
high efficiency at part gate, the wheel is divided into three sepaniti ]
compartments as in Fourneyron's wheeL
Fig. 20*2. Diagrammatic section of OQt\rard Flow Turbine it Niagtftft Fftlk
A vertical section through the lower wheel is shown in Fig.
203, ami a part sectional plan of the wheel and guide blades in
Fig. 1115.
(Further particulars of these turbines and a description of the
governor will be found in Cassier's Magazine, Vol. III., and in
Turbines Actuelle) Buchetti, Paris 1901.
187. Inward flow turbines.
In an inward flow turbine the water is directed to the wheel
through guide passages external to the wheel, and after flowing
radially finally leaves the wheel in a direction parallel to the axis.
Like the outward flow turbine it may work drowned or with a
suction tube.
The water only acts upon the blades during the radial
movement.
819
improvud by Francis •, in 1840, the wheel was of the form
in Pig. 204 and was called by its inventor a "central rent
I
ted
O
1
§
M
g
1^
1^
I'.'fi m t ill runi uii a vertical shaft., resting cm a footstep,
rted by a collar bearing placed above the gtaging S.
• Ltwetl Hjfdmtilii E^peHmenUf F, B. Ffioeii, 1856.
320
HTURAULICS
LboT0 tie wheel is a heavy castm^ C, supported by l>d
from the staging S, which acts as a guide for the cylindric
,,alaice F, and carries the bearing B for the shaft. There
\ wmnm in the wheel shown, and 40 fixed guide blades, the f ormf
aing made of iron one quarter of an inch thick and the lat
threeeixteenthB of an inch.
Fig. 204. Francis" Inward flow or Centrml v^ni TiubiB«.
The triangles of velocities at inlet and outlet, Fig. ^''
drawn, exactly as for the outward flow turbine, the only d/>
being that the velocities v, U, V, Vr and u refer to the owlet
TURBINES
321
eriplieryy and th, Ui, Vi, Vr and lii to the inner periphery of the
rheel.
The work done on the wheel is
YH_Yl5lft.ib8.perlb.,
9 9 *~ '
iiid neglecting friction,
9 9 ^
For maximum efficiency, for a given flow through the wheel,
Ji should be radial exactly as for the outward flow turbine.
Fig. 205.
The student should work the following example.
, The oater diameter of an inward flow tortfine wheel is 7*70 feet, and the inner
^bmeler 68 feet, the wheel makes 55 revolntions per minute. The head is
'^'8 feet, the Telocity at inlet is 25 feet per sec., and the radiid velocity may be
— iiimJ constant and equal to 7*5 feet. Neglecting friction, draw the triangles of
2^oeities at inlet and outlet, and find the directions of the tips of the vanes at
^Jet and oatlet so that there may be no shock and the water may leave radially.
Lass of head by friction. The losses of head by friction are
•*«u3ar to those for an outward flow turbine (see page 313) and
^e general formula becomes
9 9
■^^en the flow is radial at exit,
9
The value of e varying as before between 0*78 and 0*90.
^^uExaMj»£f (1). An inward flow turbine working under a head of 80 feet lias
2^ial blades at inlet, and discharges radially. The angle the tip of the guide
rJJ^le makes with the tangent at the inlet is 30 degrees and the radial velocity is
^^^tant. The ratio of the radii at inlet and outlet is 1*75. Find the velocity of
^ inlet drenmference of the wheel. Neglect friction.
L. H. ^\
^^^ = 6H.
322
HYDRAULICS
Sinoe tlM diach&rge it radial, tli« Tvlodtj' ftt wt h
Ui=:=Pjtan3(r
V
175
taoSr.
Then
7" 1T5«""^^*
and sinoe the Uiid^ A^m radial «i inlet V ia eqnaX Ui i^,
therefore
from whieh
/82 r an
E^ r^
Fig. 206.
Example (2). The outer diameter of the ^he«l of an inward flow tcrbiiii*
200 horsepower U 2*41^ fret, the inner diameter is l^^tfiS feet. The eDfeenT*iwhb
of the wheel at inlets ILS feet. The head b 39'5 feet and 59 eobic feet of
water per second ^iv supplied. The radial velocity with which the wat^r \^^^
the wheel may he taken us 10 feet per ftecond.
Determine the theori^tical hydraulic eMciencj £ aod the aeitial effioien^'i^
the turbine, and de^vign suitable vane>^.
300x550
^i"a95x5yxti35" '"*
Theoretical hydranUo efficiency
S9'5"
10*
mb
= 96%
The radial velocity of flow at in let,
59
2 46xrxM5
= fi*7 feet per sec*
TURBINES
323
ripbenJ yekmij
r=246. XX W=886 feet
loctty 0/ wkiH V. Aasamiiig a hydraulic effioienqj of B6%, from
^_ S9'5x32ax'85
886
=:S8*0 feet per lee.
gU e. Sinoe if=67 ft. per sec. and y=38*0 ft. per sec.
tan ^=^=0289,
igle ^. Since V is ]
^=180 27'.
\ than V, is greater than 90°.
0=162<>.
le water to discharge radially with a velocity of 10 feet per sec.
^ 10x60 ^^^^
*"'= l«68xxx800 =°"^'
a =18° nearly,
leoretical vanes are shown in Fig. 206.
9U (3). Find the valoes of ^ and a on the assumption that e is 0*80.
nsorCa inward flow twrhvne. In 1851 Professor James
n invented an inward flow turbine, the wheel of which
rounded by a large chamber set eccentrically to the wheel,
n in Figs. 207 to 210.
reen the wheel and the chamber is a parallel passage, in
ire four guide blades Gr, pivoted on fixed centres C and
:jan be moved about the centres C by bell crank levers,
1 to the casing, and connected together by levers as shown
207. The water is distributed to the wheel by these guide
and by turning the worm quadrant Q by means of the
the supply of water to the wheel, and thus the power of
bine, can be varied. The advantage of this method of
ing the flow, is that there is no sudden enlargement from
ide passages to the wheel, and the efficiency at part load
much less than at full load.
i, 209 and 210 show an enlarged section and part sectional
)n of the turbine wheel, and one of the guide blades G.
tails of the wheel and casing are made slightly different
bose shown in Figs. 207 and 208 to illustrate alternative
is.
t sides or crowns of the wheel are tapered, so that the
jral area of the wheel at the discharge is equal to the
3ral area at inlet. The radial velocities of flow at inlet
itlet are, therefore, equal.
41—2
TURBINES
325
iriea of velocities for the inlet and outlet are shown in
water leaving the wheel radially,
^of the water through the wheel, relative to the fixed
"^ shown and was obtained by the method described
•w turbines with adjustable guide blades, as made by
!ral makers, have a much greater number of guide
I'ig. 238, page 352).
tion through wheel and casing of Thomson Inward Flow Turbine.
»ine actual inward flow turbines.
jnn of the Francis inward flow turbine as designed by
/O., and having a horizontal shaft, is shown in Fig. 212.
id is doable and is surrounded by a large chamber
water flows through the guides G to the wheel W.
g tihe wheel, exhaust takes place down the two suction
38 allowing the turbine to be placed well above the
hile ntilifliiig the full head.
nhting dnioe F consists of a steel cylinder, which
parallel to the axis between the wheel and
3S6
HYDEAULICS
Fig. 209. Fig. 210*
Detail of wbeel and galde blade of Thomson Inward Flow Turbine.
Fig. 211,
mhlWi
TURBINES
327
Hie wheel is divided into five separata compartments, so that
iny time only one can be partially closedj and loss of head by
ittTaction and sudden enlargement of the streanij only takes
tee in this one compartment*
TUBBINES
329
ita^e or
mjh the
^■he tti
Ha h
Bessar
Ripened or clewed as required by the steel cylindrical sluice CC
^VDUtidtng the distributor*
^"WThen one of the stages is only partially closed by the eluice,
h loss of efficiency must take place, but the efficiency of this one
ita^ only is diminisbedj the stages that are still open working
their full efficiency. With this construction a high efficiency
turbine is maintained for partial How, With normal flowSj
head of about 625 feetj the three lower stages only are
iry to give full power, and the efficiency is then a
mnximum. In times of H<x>d there is a large volume of %vater
S reliable, but the tail watar rises so that the head is only about
\9 te0tf the two upper stages can then be brought into operation
to accommodate a larger floWj and thus the same power may be
obtained under a less head. The efficiency is less than when the
Oiree stages only are working, but as there is plenty of water
availablCj the loss of efficiency is not serious*
I The cylinder C is carried by four vertical spindles S, having
ks R fixed to their upper ends. Gearing with these racks, are
ions Pf Fig. 213, all of which are worked simultaneously by the
^lator, or by hand, A bevel wheel fixed Uj the vertical sliaft
f9 with a second bevel wheel on a horizontal shaft, the velocity
b being 3 to L
^89* The best peripheral velocity for inward and outward
tow turbines.
IjWheii the discharge is radial, the general formula, as showTi on
b 315, is
I — = eH = 078toO90H
L ^
K the blades are radial at inlet, for no shock, t^ should be equal
r , and
I ^^Yi^039 toO'45V2gH,
f t^ V0624to0 67v^2^,
This is aametimes called the best velocity for t;, but it should be
rly understood that it is only so when the blades are radial at
.(1).
W
190. Experimental determination of the best peripheral
Telocity for inward and outward flow turbines.
fw an outward flow turbine, working under a head of 14 feet,
mdial at inlet* Francis* found that when v was
626 V2^,
• Lowell, Hydmuth Exfifrlmentst
'
390
HYDRAULICS
From the formula —  ■
9
the efficiency was a maximum and equal to 79 "37 per cent. Tlie
efficiency however was over 78 per cent, for all \^lues of tf
between 0'545 J^E aud *671 J2gR. If 3 per cent, be allowt'd
for the mechanical losses the hydraulic efficiency may be taken
^B 82"4 per cent,
VtJ
 '824H, and taking V equal to i\
V ^ '64 %^yH,
so that the result of the experiment agrees well with the formnla*
For an inward flow turbine having vanes as shown in Fig, 205,
I the ttjtal efficiency was over 79 per cent, for values of r between
PO'624 V%B and 0'708 J2^, the greatest efficiency being W!
per cent* when v was 0*708 v2gH and again when t? WM
It will be seen from Fig. 205 that although the tip of the irwa^
at the convex side is nearly radial, the general direction of tha
vane at inlet is inclined at an angle greater than 90 degree \f>
the directioii of motion, and therefore for no shock Y ^should hi
less than v,
Wben V was 708 V2gH, V, Fig. 205^ was less than i\ Tbfi
value of V was deduced from the following data» which is abo
useful as being taken from a turbine of very high efficiency.
Diameter of wheel 9'338 feet.
Width between the crowns at inlet 0'999 foot.
There were 40 vanes in the wheel and an equal number of
fixed guides external to the wheel.
The minimum width of each guide passage was 0"1467 foot ini
the depth r0066 feet.
The quantity of water supplied to the wheel per second *i*
ni2'526 cubic feet, and the total fall of the water was 134 fe<*
Kthe radial velocity of flow u was^ therefore, 3*86 feet per secoii
The velocity through the minimtmi section of the guide
was 19 feet per second.
When the efficiency was a maximum, t' was 20'8 feet per
Then the radial velocity of flow at inlet to the wheel
3'S6 feet, and U being taken as 19 feet per second, the
of velocities at inlet is ABC, Pig. 205, and V is 18'4 feet per
K it is assumed that the water leaves the wheel radially, thfi^
L eH=— = 1185feet
r ^
11*85
The efficiency e should be j^rx ==88'5 per cent., which is 9
cent higher than the actual efficiency.
TURBINES
331
The actual efficiency howeirer mcladea not only the fluid losses
it also the mechanical losses, and these would probahly be from
to 8 per c^nt,, and the actual work done by the turbine on the
laft IB probably between 80 and S&5 per cent, of the work done
r the water. j
Vv I
19L Value of e to be used in the formula —  ^ eH* 1
i
In general, it may be said that, in using the formula — = eH,
e value of e to be used in any given case is doubtful, as even
oQ^b the efficiency of the class of turbiues may be known, it is
fficult to say exactly how much of the energy is lost mechanically
id how moch hydrauUcally,
A trial of a turbine without load, would be useless to deter
ine the mechanical efficiency, as the hydraulic losses in such a
tal would be very much larger than when the turbine is working
full load. By revolving the turbine without load by means of
i electric motor, or through the meditnn of a dynamometer, the
art to overcome friction of bearings and other mechanical losses
uld be found. At all loads, from no load to full load, the
iotional resistances of machines are fairly constant, and the
schanical losses for a given class of turbineSj at the normal load
r which the vane angles are calculated, could thus approximately
\ obtained. If, however, in making calculations the difference
tween the actual and the hydraulic efficiency be taken as, say,
per cent., the error cannot be very great, as a variation of 5 per
ntp in the value assumed for the hydraulic efficiency e, will only
kke a difference of a few degrees in the calculated value of
I angle <^.
The beat value for 0, for inward flow turbines, is probably 0*80,
1 ejcperience shows tliat this value may be used with confidence,
ffiwilpfr. Tftkiag thedjiU ikM u^^en in the exacnple of ae^tiou 184, and aasaming
MMtWQT t^' ^^ tutbioe of 75 per eeat.^ the horfiepower k
iKJK 215 y 624 >£Ul%5x 75x60 ■
==2600 horse power
If the hjdr&iUie efficiencj ij anppoaed to be SO per centt^ the velocitj of
d V sbonld be
^ eff.H ^ 08,32 Ul6
r " 60 I
=53 feet per a«e* I
riten tmm^ 18 35 18 85 
Cbcti *~5a6S~ 17 '
^182° 47',
Kippoae the torbiise to be itill ^nersting 2600 hor»e*power, and to h&ye
\j of 80 pet oenL, ^nd » hydrauito effioie^cy of 65 per eent.
332
HYDRAUUCS
Then the qaantity of water reqaired per second, is
^ 216x076 o^ u # *
Q = — zr^ — = 200 cubic feet per sec.
and the radial velocity of flow at inlet will be
1836x200
215
= 17*1 ft, per sec.
, •86.32.1416 __.^
V = nn =66*4 ft. per sec.
Then
tan0=:
69
171
66469
= 128°. 24'.
171
136
192. The ratio of the velocity of whirl V to the velodtj
of the inlet periphery v.
Experience shows that, consistent with Yv satisfying the gfenenl
formula, the ratio ^y may vary between very wide limits without
considerably altering the efficiency of the turbine.
Table XXXVII shows actual values of the ratio • , taken
from a number of existing turbines, and also corresponding valnai
Fig. 214.
TURBINES
333
=^ , V beinff calculated from — :
' 0'8H. The corresponding
.tion in the angle <t>y Fig. 214, is from 20 to 150 degrees,
or a given head, v may therefore vary within wide limits,
h allows a very large variation in the angular velocity of the
1 to suit particular circumstances.
TABLE XXXVII.
howing the heads, and the velocity of the receiving circum
ice V for some existing inward and outward, and mixed flow
nes.
Hfeet
vfeet
per see.
J29H
Ratio
V
sl2gR
H.P.
V
Ratio 7=
>/2pH
V being calculated
from— = 8H
9
.rdflaw:
gara Falls*
146
70
968
072
5000
0655
infelden
148
22
807
071
840
0666
Theodor)
and Co. j
284
89
426
091
044
604
822
628
052
077
et and Co.
1887
611
768
047
800
085
»»
1846
466
656
0505
800
079
625
166
20
088
048
ji
80
2575
44
058
700
069
♦»
885
508
077
200
052
Lz and Co.
112
648
846
054
074
»9
226
647
120
064
682
058
ber and Co.
1066
152
26
0685
80
069
^rdflow :
gara Falls
1416
69
952
0725)
5000
055
;et and Co.
1806
69
916
0750)
058
LZ and Co.
961
887
780
0495
290
081
»»
228
656
1200
046
1200
087
♦ Escher Wyss and Co.
'or example, if a turbine is required to drive alternators
rt, the number of revolutions will probably be fixed by the
nators, while, as shown later, the diameter of the wheel is
tically fixed by the quantity of water, which it is required to
through the wheel, consistent with the peripheral velocity of
v^heel, not being greater than 100 feet per second, unless, as
ie turbine described on page 373, special precautions are
1. This latter condition may necessitate the placing of two
3re wheels on one shaft.
334
BYDBAmJCa
Suppose then, the number of reTolutions of the wheel to
given and d is fixed, then v has a definite value, and V muii 1
made to satisfy the equation
s
Fig. 214 is drawn to illustrate three cas€«s for which Yri]
constant, Tht? angles of the vanes at outlet are the same fori
three, but the guide angle ^ and the vane angle ^ at inlet
considerably^.
193. The velocity witli
In a welldeaigned turbine
leaves tht* turbine should be as
keeping the turbine whrel an
dimensions.
In actual turbines the 1
varies from 2 to 8 per cei
suction pipe the water may t
with a fairly high velocity a
conical sv» ai^ to allow the actual
water leaves a turbme.
locity with which the '
as possible, condsteal i
>ftTitake within
h
^due to this velocity
I turbine is fitted with
jd to learc the wheel
iacharge pipe can be \
irge velocity to be as i
as desired. It should however be noted that if the w^ater leavtf*
the wheel with a high velocity it is more than probable that there
^^^ll be Jiome loss of head due to shock, as it is difficult to ensure
that wattT ?4* di^chargtKi shall have its velocity changed gradiiallT.
194. BemoniUi's equations applied to inward and out
ward flow tnrbmes neglecting firiction.
Ctntrifn^al head imprejtsed on the water by the wheei Tk
theory of rhe reaction turbines is best considered from the point
of Niew of Bemouilli's equations ; but before proceeding to discuss
them in detail, it is necessary to consider the " centrifugal head"
impivssed un the water by the w^heeh
This head has already been considered in connection with tk
Scotch turbim*, page 303,
Let r^ Kig. 216j be the internal radius of a wheel, and B tb
external radius.
At the internal circumference let the wheel be covered with i
cylinder e S4> that there can be no flow through the wheel, and 1h
it W suppixsed that the wheel is made to revolve at the angular
vehxnty f^ which it has as a turbine, rhe wheel being full of wat€f
and surrounded by water at rest, the pressure outside the wbed
being sufficient to prevent the water being whirled out of the
whet^l. Lei d be the depth of the wheel between the crowia
Consider any element of a ring of radius n and thickness dr, and
subtending a smaU angle 6* at the centre C, Fig. 210*
TURBINES
335
The weight of the element is
UDToO .dr,d,
and the centrifugal force acting on the element is
uTo^ ,dr,d, «Vo
9
lbs.
Iiet p be the pressure per unit area on the inner force of the
element and p^dp on the outer.
fjPToB ,dr .d . «Vo
op =
Then
g.nO.d
Fig. 215.
Fig. 216.
The increase in the pressure, due to centrifugal forces, between
r and R is, therefore,
r
T'*" I •'<"'"'■'
i^nd
For equilibrium, therefore, the pressure in the water surround
ing the wheel must be pe.
If now the cylinder c be removed and water is allowed to flow
tlnroiigh the wheel, either inwards or outwards, this centrifugal
lieftd will always be impressed upon the water, whether the wheel
u driven by the water as a turbine, or by some external agency,
mnA acts as a pump.
BernouilWa equations. The student on first reading these
equations will do well to confine his attention to the inward flow
terbine, Fig. 217, and then read them through again, confining his
attention to the outward flow turbine, Fig. 191.
336
HYDRAULICS
Let p be the pressure at A, the inlet to the wheel, or in the
clearance between the wheel and the guides, pi the pressure at
the outlet B, Fig. 217, and pa the atmospheric pressure, in pounds
per square foot. Let H be the total head, and H« the statical
head at the centre of the wheel. The triangles of velocities are
as shown in Figs. 218 and 219.
Then at A
Pa
TP
w w 2g
.(1).
Between B and A the wheel impresses upon the water the
centrifugal head
2g 2g'
V being greater than Vi for an inward flow turbine and less for the
outward flow.
Fig. 217.
Consider now the total head relative to the wheel at A and BL
The velocity head at A is ^ and the pressure head is ^i>ilH
at B the velocity and pressure heads are ^ and ^ respectivdy.
If no head were impressed on the water as it flows thtongli
the wheel, the pressure head plus the velocity head at A and B
would be equal to each other. But between A and B thero i«
impressed on the water the centrifugal head, and therefore,
Pi Vr^ v^ ^^' = P I ^^*
w 2^ 2g 2g w 2g
.(2).
TURBINES
.337
This equation can be used to deduce the fundamental equation,
^^ = fc (3).
9 9
From the triangles CDE and ADE, Fig. 218,
Vr»=(Vt7)« + ti'andV» + tt« = U',
ind from the triangle BFG, ¥ig 219,
vr' = (vi  YiY + u,' and Va» + V = Vi\
Therefore by substitution in (2),
i?"" 2^ 2g 2g^2g w^ 2g ^ 2g "'^^^'
From which
ti7 g 2g w 2g g '
ad
g g w w 2g 2g
Substituting f or *^ + ^5 from (1)
w
VV VlVl ^ ^ , Pa Pi Ul'
2sr
.(5).
(6).
Pig. 218.
Wheel in suction tube. K the centre of the wheel is ho feet
the snr&ce of the tail water, and Uo is the velocity witli
e water leaves the downpipe, then
^^m th
' 2g ^ w 2g
w
Snbstitntingr for ^ + §^' in (6),
w 2g
9 " 9
  WW 2g
= HU''
2ff
I. B.
^•1
338 HYDRAULICS
IfVisO, —^R^h.
9 2flr
The wheel can therefore take full advantag^e of the head H
even though it is placed at some distance above the level of the
tail water.
Drowned wheel. If the level of the tail water is CD, Fig. 217,
or the wheel is drowned, and hi is the depth of the centre of the
wheel below the tail race level,
W W
and the work done on the wheel per pound of water is again
vV Yiv, „ W ,
9 9 ^9
vY
IfViisO, — = h.
From equation (5),
vV _ ViYj ^ P _ Pi ^ U! _ W
9 9 w w 2g 2g ^
so that the work done on the wheel per pound is the difference
between the pressure head plus the velocity head at entrance and
the pressure head plus velocity head at exit.
In an impulse turbine p and pi are equal, and the work doi«
is then the change in the kinetic energy of the jet when it strikes
and when it leaves the wheel.
A special case arises when pi is equal to p. In this case a
considerable clearance may be allowed between the wheel and the
fixed guide without danger of leakage.
Equation (2), for this case, becomes
2g 2g 2g 2g'
and if at exit Vr is made equal to Vi, or the triangle BFG,
Fig. 219, is isosceles,
2g 2g^
and the triangle of velocities at entrance is also isosceles.
The pressure head at entrance is
w 2g'
and at exit is either — + fei, or —  ^.
w ^ w
TURBINES 339
lieref ore, since the pressures at entrance and exit are equal,
<r else Ho + /iD = H.
The water then enters the wheel with a velocity equal to that
Lue to the total head H, and the turbine becomes a freedeviation
NT impulse turbine.
195. Bemonilli's equations for the inward and outward
low turbines including friction.
If H/ is the loss of head in the penstock and guide passages,
^ the loss of head in the wheel, he the loss at exit from the wheel
ttid in the suction pipe, and Ui the velocity of exhaust,
£ + U!=H. + £eH, (1),
w^2g^2g2gw^2g^^ ^^^'
w w
w w
rom which — = Hf ^^ + A/+H/+fee) (4).
f the losses can be expressed as a fraction of H, or equal to KH,
hen
Vt?
— = (lK)H = eH
^ =078H to 090H*.
196. Turbine to develop a given horsepower.
Let H be the total head in feet under which the turbine works.
Let n be the number of revolutions of the wheel per minute.
Let Q be the number of cubic feet of water per second required
>y the turbine.
Let E be the theoretical hydraulic efficiency.
Let e be the hydraulic efficiency.
Let Sm be the mechanical efficiency.
Let fii be the actual efficiency including mechanical losses.
Let Ui be the radial velocity with which the water leaves the
irheel.
Let D be the diameter of the wheel in feet at the inlet circum
erence and d the diameter at the outlet circumference.
Let B be the width of the wheel in feet between the crowns
it the inlet circumference, and b be the width between the crowns
it the outlet circumference.
Let N be the horsepower of the turbine.
* See page 815.
340 HYDRAULICS
The number of cubic feet per second required is
^ e,H. 624.60 ^^'•
A reasonable value for ei is 75 per cent.
The velocity Uo with which the water leaves the turbine, since
is Uo=>/25f(lE)Hft.per8ec (2).
If it be assumed that this is equal to thy which would d
necessity be the case when the turbine works drowned, or
exhausts into the air, then, if Hs the peripheral thickness of the
vanes at outlet and m the number of vanes,
(irdmOUo6=Q.
If Uo is not equal to Ui, then
(^dmt)uib = Q (3).
The number of vanes m and the thickness t are somewhat
arbitrary, but in welldesigned turbines t is made as small as
As a first approximation mt may be taken as zero and (3)
becomes
7rdbUi=Q (4).
For an inward flow turbine the diameter d is fixed from
consideration of the velocity with which the water leaves the
wheel in an axial direction.
If the water leaves at both sides of the wheel as in Pig. 208,
and the diameter of the shaft is do, the axial velocity is
t^= y ft. per sec.
2j(d'd.')
The diameter do can generally be given an arbitrary value, or
for a first approximation to d it may be neglected, and u% may be
taken as equal to i^. Then
^^"^ <«•
From (4) and (5) b and d can now be determined.
A ratio for v having been decided upon, D can be calcvlatedy
and if the radial velocity at inlet is to be the same as at oatletk
and to is the thickness of the vanes at inlet,
(^DmMB = S = (irdm06 (6).
til
TURBINES 341
For rolled brass or wrought steel blades, to may be very small,
and for blades cast with the wheel, by shaping them as in Fig. 227,
to is practically zero. Then
If now the number of revolutions is fixed by any special
condition, such as having to drive an alternator direct, at some
definite speed, the peripheral velocity is
frDn . ,^,^.
t7 = ^ ft. per sec (/).
Yv
Then V"^^'
and if e is given a value, say 80 per cent.,
V = '^ ft. per sec (8).
V
Since u, V, and v are known, the triangle of velocities at inlet
can l>e drawn and the direction of flow and of the tip of vanes
at inlet determined. Or B and <^, Fig. 214, can be calculated from
tan^ = ^ (9)
and ' tan<^ = ;^^ (10).
Then U, the velocity of flow at inlet, is
U = V sec 0,
Trdn «,
At exit fi = ^ ft. per sec,
and taking Ux as radial and equal to Uy the triangle of velocities
can be drawn, or a calculated from
u
tan a =  .
If BU is the nead of water at the centre of the wheel and 11/ the
head lost by friction in the supply pipe and guide passages, the
pressure head at the inlet is
Example An inward flow turbine is required to develop 300 horsepower under
A bead 60 feet, and to run at 250 revolutions per minute.
To determine the leading dimensions of the turbine.
Aaeuming ^i to be 75 per cent.,
Q_ 300x33,000
^~ 76 x60 x~02Tx^60
= 58*7 cubic feet per sec.
342 HTDRAUUCS
Assmmiig E is 95 per cent., or five per cent, of the head iq lost by Telo^
of exit and u^^u,
=^=•05.60
2g
and tt= 13*8 feet per sec.
Then from (5), page 340,
^^^W^J^^^
= 165 feet,
say 20 inches to make allowance for shaft and to keep eyen dimenBion.
Then from (4), * = fH "= '^ ^^*
=9} inches say.
Taking  as 1*8, D=3*0 feet, and
a
V =rir . 8 . W = 39*3 feet per sec.,
and B=5} inches say.
Assuming « to be 80 per cent.,
„ 80x60x32 ^^^^
" "~
393 ~*>''""*P«^"«
"
* n 13*8
tan^= 39 '
and
^=19^80'.
*«'*=^'.  ^^
and
0=91'^ 15'.
13*8 X 1*8
**"*=39*3'
and
a = 32° 18'.
The velocity U at inlet is
U=>/39*0» + (13*8)«
=41*3 ft. per sec.
The absolute pressure head at the inlet to the wheel is
n n 41 'V
 = Ho + ^ — hf, the head lost by friction in the down pipe
=Ho+3426*6/i/.
The pressure head at the outlet of the wheel will depend upon the height of t
wheel above or below the tail water.
197. Parallel or axial flow turbines.
Fig. 220 shows a double compartment axial flow turbine, tl
guide blades bein g placed above the wheel and the flow throuj
the wheel being parallel to the axis. The circumferential secti
of the vanes at any radius when turned into the plane of t
paper is as shown in Fig. 221. A plan of the wheel is also shov
The triangles of velocities at inlet and outlet for any radi
are similar to those for inward and outward flow turbines, t
velocities v and t^i, Figs. 222 and 223, being equal.
TUUBINES
343
The general formula now becomes
For maximum efficiency, for a given flow, the water should
leave the wheel in a direction parallel to the axis, so that it has
no momentum in the direction of v.
Fig. 220. Doable Compartment Parallel Flow Turbine.
Figs. 221, 222, 223.
Then takii^ friction and other losses into account,
9
344
HrDRACILICS
The ineloeit^ r will be proportional to the radius, mi th^t if 1
w^ter is to enter and lettTO the wheel withoat shock, the anfltsl^l
^ and m tous^ faiy with the radius.
TW rariMion m the form of the r&ne with the imdios is skmr^]
by an exaai{iki^
A Jcfowml wheel has an internal diameter of 5 feet mud
extremal diameter of B"^. The depth of the wheel is 7 iiicli
TTie h^ttd is 15 feet and the wheel makes 55 tevolationf
minnre, The flow b 300 cubic feet per second.
Ti> find the h^iriMes power of th< — ^ id, and to design the wh
Let Ti be the immn radins, a:
ai the inner and oater ciirmnfer
r  i'o feet attd r = 2a
fi = ^^*> feet and rj = S
ff^= 4l25 ket and r^  &
The mean axial Telocity is
ind Ft the radii of the wh
e^iectiTely. Then
= 14"4 feet per sec,^
= 21*5 feet per sec,
=24^5 fe^ per sec.
. 1 jv =8'la !i, per sec*
*4*^^ —
^:,_H%^;AML ~I^^M,j , Jt C
F^. aa*.
TriAa^J^s f»l Tdodlies ai mki md aailet %t tbree dif ei«nt
imlit of m Pkimllel Fl^v Tarbine.
Taking ^ as 0^80 at each radius,
i4'4 ^ ^^ ®^''
14 4
385
Vt  ^YjT^ ==17^ ft per sec,,
\ , = ^ , i = la f ft, per sec
ImcJimatiim of ik^ ran^s at inlei. The triangles of velocities
for the ihrt*^^ T*dii n rj, r are siniwn in Fig. 224. For example,
at nftditt^ r, ADC i^ the triangle of Yelocities at inlet and ABC tie
TURBINES 345
ian^le of velocities at oatlet. The inclinations of the vanes at
let are found from
8*15
tan <^ = 2&f:^UA ' ^^^ ^*^^^ * " ^^ ^ '
^^"^^ 179 ^215 ^"^^ *! = 113^50', .
8'15
tan 4>i = 157^24^5 » ^^^ which «^ = 137** 6'.
2^he inclination of the guide blade at each of the three radii.
tan^ = 267'
om 'which ^ = 17',
tan^i = j^ and ^,=24^30',
tan^5 = ?~ and ^2 = 27'* 30'.
lo7
The inclination of the vanes at exit,
*''°" = 144 = 29'36',
tan a, = 5'^ = 20° 48',
Zl o
tan 03= 1^^=18^ 22'.
If now the lower tips of the guide blades and the upper tips
[)f the wheel vanes are made radial as in the plan, Fig. 221, the
inclination of the guide blade will have to vary from 17 to
27^ degrees or else there will be loss by shock. To get over this
diflBculty the upper edge only of each guide blade may be made
radial, the lower edge of the guide blade and the upper edge of
each vane, instead of being radial, being made parallel to the
npper edge of the guide. In Fig. 225 let r and R be the radii
of the inner and outer crowns of the wheel and also of the guide
hlades. Let MN be the plan of the upper edge of a guide blade
and let DG be the plan of the lower edge, DG being parallel to
HX. Then as the water runs along the guide at D, it will leave
"the guide in a direction perpendicular to OD. At G it will leave
in a direction HG perpendicular to OG. Now suppose the guide
«M; the edge DG to have an inclination fi to the plane of the paper.
H then a section of the guide is taken by a vertical plane XX
XKrpendicnlar to DG, the elevation of the tip of the vane on this
X^e will be AL, inclined at P to the horizontal line AB, awd AG
»46
HYDRAULICS
will be the intersection of the plane XX with the plane tangei
to the tip of the vane.
Now suppose DE and GH to be the projections on the plai
of the paper of two lines lying on the tangent plane AC ao
perpendicular to OD and OG respectively. Draw EF and HI
perpendicular to DE and GH respectively, and make each <
them equal to BC. Then the angle EDF is the inclination of tl
stream line at D to the plane of the paper, and the angle HGE :
the inclination of the stream line at G to the plane of the pape
These should be equal to and 0^,
JCZ/fl;
btcut&l. of upper edg& ofiwrvt
Fig. 225. Plan of guide blades and vanes of Parallel Flow Tarbines.
Let y be the perpendicular distance between MN and DG
Let the angles GOD and GrOH be denoted by <^ and a respectively
Since EF, BC and HK are equal,
ED tan^ = y tan)3 (1),
and GH tan ^2 = y tan ^ (2).
But
and
Therefore
and
Again,
^ = cos (a + «^),
JL
= cos a.
GH
tan B  cos (a + <^) tan fi.
tan ^3 = cos a tan j3
V
sina =
R
.(3),
.(4).
.(5).
There are thus three equations from which a, ^ and fi can ta
determined.
Let X and y be the coordinates of the point D, O being tta
intersection of the axes.
TURBINES
347
len
cos (a + ^) =
•om (5)
cosa
bstitating for cos (a + ^) and cos a and the known values of
»id tan ^s in the three equations (3 — 5), three equations are
led with X, y^ and fi as the unknowns.
Iving simultaneously
X = ri4 feet,
y = 223 feet,
tan /3 = 067,
«rhich i8 = 34\
Fip. 226.
Fig. 227.
Fig. 228.
e length of the guide blade is thus found, and the constant
at the edge DG so that the stream lines at D and G shall
he correct inclination.
now the upper edge of the vane is just below DG, and the
: the vane at D and G are made as in Figs. 226 — 228, ^ and
:U^
HTDKJLUIJCS
^ bein^ '^ '¥/ ami ITT ^ respectively, the wmter wiD more on to
the vTkEie wTtho'iit «hf xrk.
The plane •:•£ the lower edge of the Tame may now be taken ii
VG\ riz, 22^^ and che cErcnlar aectbos DIX, FQ, and GG' at th
three nuifi. r, r, and Tj are th^i as in FigSL 226 — ^228.
198. BegoIationofthellowtopanaidlloirtiiiliiniaL
To regulate the dow through a paralld flow tarbinesy FontuM
placed Juices in the guide passagesy as in Fig^. 229, connected to
a ring which could be raised or lowered by three Tertical lodi
having nuts at the upper ends fixed to toothed pinions. What
Fig. 32!^. Fontaine's Slniee*.
Fig. 230. Adjustable guide blades for Parallel Flow Turbine.
the sluices required adjustment, the nuts were revolved together
by a central toothed wheel gearing with the toothed piniotf
canying the nuts. Fontaine fixed the turbine wheel to a hoUoir
shaft which was carried on a footstep above the turbine. In son*
modem parallel flow turbines the guide blades are pivoted, as in
Fig. 230, so that the flow can be regulated. The wheel may ^
made v^ith the crowns opening outwards, in section, similar to
the Girard turbine shown in Fig. 254, so that the axial vdodtf
with which the water leaves the wheel may be small.
The axial flow turbine is well adapted to low falls with variable
head, and may be made in several compartments as in Fig. 220.
In this example, only the inner ring is provided with gates. In
dry weather flow the head is about 3 feet and the gates of the
inner ring can be almost closed as the outer ring will give the full
TURBINES 349
»r. During times of flood, and when there is plenty of water,
head faJls to 2 feet, and the sluices of the inner ring are
ed. A larger supply of water at less head can thus be
red to pass through the wheel, and although, due to the shock
le guide passages of the inner ring, the wheel is not so efficient,
ibundance of water renders this unimportant.
cample, A doable compartment Jonval turbine has an outer diameter of
and an inner diameter of 6 feet.
le radial width of the inner compartment is 1' 9" and of the outer compart
1' 6". Allowing a velocity of flow of 8*25 ft. per second and supposing the
Dum Call is V Q^, and the number of revolutions per minute 14, find the hone
: of the wheel when all the guide passages are open, and find what portion of
iner compartment must be shut off so that the horsepower shall be the same
> a head of 3 feet. Efficiency 70 per cent.
>gleoting the thickness of the blades,
the area of the outer compartments j (125'9'5^ = 52*6 sq. feet.
„ „ inner „ =^ (9'6»6«) = 428 sq. feet.
>tal area=954 sq. feet
16 weight of water passing through the wheel is
W=95'4 X 62*4 X 3*25 lbs. per sec.
= 19,800 lbs. per sec.
he horsepower is
19,800 X 166 X 07 .^.^
°^= 660 = *^®
(saming the velocity of flow constant the area required when the head
eet is
408x83,000
60x626x8x7
=556 sq. feet,
• outer wheel will nearly develop the horsepower required.
99. Bemouilli's equations for axial flow turbines.
Tie Bemouilli's equations for an axial flow turbine can be
ten down in exactly the same way as for the inward and
^rd flow turbines, page 335, except that for the axial flow
ine there is no centrifugal head impressed on the water
een inlet and outlet.
Tien, P^l^^.P^^'f^h,,
which, since v is equal to Vi,
p V*2Vt? + t?' u* pi r'~2ViV + V,^ t^» ,
p V Vi; t^'^p. V.'^t^' Y,v
tore — +0:: — Z"^ oZ^ ~'^ lyZ'^ oZ — 1~ + %>
tc 2g g 2g w 2g 2g g
g g w 2g 2g w ^'
330
BrDRAin.lC3
If U« ie udal and equal to u, as in Fig. 228,
200. Mix^d flow tturbinas*
By a taodtiicatioti of tht* sIuijh* uf the viineei of an jnwarl
turbiiu*, tht* mixed flow turbine in obtained. In tie inwwt
outward flow turbine the water only acti* upon the whfiel wl
is raaving in a radial direction, but in tho mix<?d flow tarbi;
viuies are so formed that the water aotfi u)Oii tliem aiais
flowing axially.
modern
Fig. 230,
made with
the (iirard tu
w^th which tbt
The axial fl
head, and may \
In this example,
dry weather flo>^^ ^ diagrammatic section through the whee
inner rinff can b^^^^^^> ^^® ^^® ^^ which is vertical. Tlie w
Fig. 231. Mixed Flow Turbine.
k
TU^KIS
351
tlie wheel in a horizontal direction and leaves it ¥@rtically^]
leaTes the di^hargiiig edge of the vanes in different
ions. At the upper part B it leaves the vanes nearly
lly, and at the lower part A, axially; The vanes are spoon
1, BM shown in Fig. 2*32, and should be so formed, or in other
fc, the inclination of the discharging edge gihould so vary,
i^herever the water leaves the vanes it should do so with no
ment in a direction perpendicular to the axis of the turbine,
ith no velocity of whirl. The regulation of the supply to
b] in the turbine of Fig, 231 is effected hy a cylindrical ]
' speed gate between tlie fixed guide blades and the wheel.
Fig. 232. Wht^l i^f Mixed Flow Turbme.
Sg 2*i*^ shows a section through the wheel and casing of a
le tnixrd flow turbine having adjustable guide blades to
ate the How, Fig, 234 shows a half longitndinal section of
rbine, and Fig. 23o an outside elevation of the guide blade
idng gean The guide blades are surrounded by a lar^e
852
HYDRAULICS
vortex chamber, and the outer tips of the guide blades are of
variable shapes, Fig. 283, so as to diminish shock at the entrance
to the guide passages. Each guide blade is really made in two
parts, one of which is made to revolve about the centre C, while
the outer tip is fixed. The moveable parts are made so that the
flow can be varied from zero to its maximum value. It will be
Fig. 233. Section through wheel and guide blades of Mixed Flow Turbine.
noticed that the mechanism for moving the guide blades is
entirely external to the turbine, and is consequently out of the
water. A further special feature is that between the ring B
and each of the guide blade cranks is interposed a spiral spring.
In the event of a solid body becoming wedged between two of
the guide blades, and thus locking one of them, the adjustment ol
the other guide blades is not interfered with, as the spring con
nected to the locked blade by its elongation will allow the linf
to rotate.
As with the inward and outward flow turbine, the mixec
flow turbine wheel may either work drowned, or exhaust into I
"suction tube."
^
TURBINES
353
For a given flow, and width of wheel, the axial velocity
vitli which the water finally flows away from the wheel being the
tame for the two cases, the diameter of a mixed flow turbine can
)e made less than an inward flow turbine. As shown on page 340,
;lie diameter of the inward flow turbine is in large measure flxed
{.
JL
=L
ZFegt
Fig. 234. HalfloDgitudinal section of Mixed Flow Tarbine.
\gy the diameter of the exhaust openings of the wheel. For the
■ame axial velocity, and the same total flow, whether the turbine
is an inward or mixed flow turbine, the diameter d of the exhaust
<^>eniiig8 must be about equal. The external diameter, therefore,
of the latter wiQ be much smaller than for the former, aivd ticva
L. H. 'i^
354
HTDRAUUGS
general dimensions of the turbine will be also diminished,
a given head H, the velocity v of the inlet edge being the san
the two cases, the mixed flow turbine can be run at a hi
angular velocity, which is sometimes an advantage in dri
dynamos.
o
a
!2
'a
TURBINES
355
J'Wm of ths varies. At the receiving edge, the direction of the
is found in the same way as for an inward flow turbine*
ABC, Fig, 236, is the triangle of velocities^ and BC is parallel
the tip of the blade* This triangle has been drawn for the data
he turbine shown in Figs. 23S— 235 ; v is 46*5 feet per secondj
from
9
Y = 33*5 feet per second*
The angle ^ is 189 degrees.
Triangles of V^ctHjeG
at raomrmff fdge^
Fig. 236,
The best form for the vane at the discharge is somewhat
icult to determine, as the exact direction of flow at any point
the discharging edge of the vane is not easily found. The
aditioii to be satisfied is that the water must leave the wheel
; any component in the direction of motion*
The following constmction gives approximately the form of
vane.
Hake a section through the wheel as in Fig. 237. The outline
the discharge edge FGH is shown. This edge of the vane is
til be on a radial plane, and the plan of it is, therefore,
f Tsdins of the wheels and upon this radius the section is taken*
It ifi now neoeasary to draw the form of the stream lines, as
would be approximatelyj if tlie water entered the wheel
ly and flowed out axially, the vanes being removed*
Ihvide 04, Fig. 237, at the inlet, into any number of equal
say four, and subdivide by the points a, i, /i, e.
Tike any point A, not far from Cj as centre, and describe
circle MMi touching the crowns of the wheel at M and M,.
AM and AMi,
]>i^w a flat curve Mi Mi touching the hues AM and AM, in M
Mt respectivety, and k% near as can be estimated, perpendicular j
856
HYDRAULICS
to the probable stream lines through a, 6, d, e, which can
sketched in approximately for a short distance from 04.
Taking this curve MMi as approximately perpendicular to
stream lines, two points/ and g near the centres of AM and A
are taken.
Fig. 287.
Let the radius of the points g and / be r and n respectively
If any point Ci on MMi is now taken not far from A, tfe
peripheral area of Mci is nearly 2irrMci, and the peripheral art*
of MiCi is nearly 2wriMiCi.
On the assumption that the mean velocity through MiM ^
constant, the flow through Mci will be equal to that throng*
MiCi, when,
Mci.r = MiCi.ri.
TURBINES
357
If, therefore, MM ib divided at the point Cj ao that
Mj_Ci _ r
thB point rj will approximateljr be on the stream line through c.
If now when the stream line cci is carefully drawn in^ it is
perpendicular to MMi, the point c, cannot be much in error.
A near€*r approximation to d can be found by taking new values
for r and n, obtained by moving the points / and g bo that they
more tiearly coincide with the centres of CjM and cjMi. If the
two corves are not perpend icularj the cui've MM^ and the point Ci
ajre not quite correct, and new vsiluea of r and Ti wiU have to be
obtained by moving the points / and g. By approximation Cj can
"be thus found ^vith considerable accuracy.
By diawing other circles to touch the crown of the wheelSj the
curv«i MrM«, M^Ma etc. normal to the stream lines, and the points
45i, Cs, etc. on the centre stream line, can be obtained.
The curve 22, thereforej divides the stream lines into equal
parf^
Proceeding in a similar manner, the curves 11 and 33 can be
'I'htaitied^ dividing the stream lines into four equal parts, and
iii^^ again subdivided by the curves oa, hh, rfd, and ee, which
inttfrst»ct the outlet edge of the vane at the points F, G, H and e
Tie*pectively.
To dei ermine the direction of the tip of the varw at points mi the
<^i^chirg{ng edge. At the pcdiUs F, Gr, H, the directions of the
<to?am lines are known, and the velocities tip, t%, uu can be found,
«mce the flows through 01, 12, etc, are equal, and therefore
uvR^qt = Uii'RiTtui = tiuRiW't'
Q
8tr
Draw a tangent FK to the stream line at F, This is the inter
*^*cti«ui, with the plane of the paper, of a plane perpendicular to
^ ^U' p*»>er and tangent to the stream line at F.
The piiint F in the plane of FK m moving perpendiculai* to the
Wane of the paper with a velocity equal to *w,Rm, «»* being the
^n^kr velocity of the wheel, and H, the radius of the point F.
If a circle be struck on this plane with K as centre, this circle
^ay be taken as an imaginary discharge circumference of an
*^^ward Row rnrbinej the velocity v of which is mR^^ and the tip of
^^ = '* Made i& w have such an inclination, that the water shall
'**>chiirgo radially* i.*;. along FK, with a velocity nr . Turning this
'ff'cle >nto the plane of the paper and drawing the triangle of
^>l^K.nties FST, the inclination a^ of the tip of the blade at F in
^hm pliiTie FK is obtained,
358
HYDRAULICS
At G the stream line is nearly vertical, but «R» can be set ont
in the plane of the paper, as before, perpendicular to uq and the
inclination oeq, on this plane, is found.
At H, an is found in the same way, and the direction of the
vane, in definite planes, at other points on its outlet edge* can be
similarly found.
Fig. 288.
Fig. 239.
Sections of the vane by planes OGby and OiHd. These are
shown in Figs. 238 and 239, and are determined as follows.
Imagine a vertical plane tangent to the tip of the vane at
inlet. The angle this plane makes with the tangent to the whed
at b is the angle <^, Fig. 236. Let BC of the same figure be the
TURBINES
359
plan of a liomontal line lying in this plane, and BD the plan of
tbe radius of the wheel at 6. The angle between these lines is y.
Liet 0be the inclination of the plane Oiyh to the horizontal.
From D, Fig, 236, set ont DE, inclined to BD at an angle ft
aiul intersecting AB produced in E, and draw BF perpendicular
to CB.
Make BF equal i<.\ BE and join CF.
Now set out a triangle BGDj ha^nng BG equal to CF, DiG
eqtial to DE, and the angle BGDj a right angle* In the figure
Di and D happen to coincide.
The angle BGD is the angle yi, which the line of intersection of
tte plane OG^, Fig, 237, witli the plane tangent to the inlet tip of
tlie vane, makeB irith the radius O^.
Ill Fig 238 the inclination of the inlet tip of the blade is yi as
To detamiine the angle a at the outlet edge, resolve tt©, Fig,
*2S7\ along and perpendicular to OS, ti^ being the component
along (Xt,
Draw the triangle of velocities DEF, Fig. 238,
Tlte tangent to the vane at D is parallel to FE,
In the same way^ the section on the plane Hti, Fig. 237, may be
letentiinetl ; the inclination at the inlet is y^. Fig, 239.
MU*d Jl^tt hirhim worki^ig m open stream, A double turbine
rnrking in open stream and discharging tlirough a suction tube
shown in Fig. 21^^. This is a convenient arrangement for
^^oderately low falls. Turbines, of this class, of 1500 horne
p(n«rer, having four wheels on the same shaft and working under
m head of 25 feet, and making 150 revolutions per minute, have
rt^'einly tieen installed by Messrs Escher Wyss at Wangen an der
A a re in SwitxcTland,
201. Cone ttirbltie.
Another type of inward flow turbine, which is partly axial and
ly radial, \b shown in Fig. 241, and is known aa the cone
Mne, It has been designed by Messrs Escher Wyss to meet
\ie demand for a turbine that can be atlapted t« variable flow^s.
l*he example shown has been erected at Cusset near Lyons and
ikes 120 revolutions jx?r minute.
Tlie wheel is divided into three distinct compartments, the
applj' of Waaler being regulated by three cylindrical sluices S, Si
S*. The sluices S and 8i are each moved by three vertical
?^uch as A and Aj which carry racks at their upper ends.
_ wo aluic^es move in opposite directions and thus balance
ch oilier* The sluice Sa is normally out of actioni the upper
360
HYDRAULICS
compartment being closed. At low heads this upper compartrxne
is allowed to come into operation. The sluice Sj carries a "ma
which engages with a pinion P, connected to the vertical sha. ^
Fig. 240.
The shaft T is turned by hand by means of a worm and
wheel W. When it is desired to raise the sluice Sg, it is revolved
by means of the pinion P until the arms F come between collar*
D and E on the spindles carrying the sluice Si, and the sluice S»
then rises and falls with Si . The pinion, gearing with racks on A
and Ai , is fixed to the shaft M, which is rotated by the rack B
gearing with the bevel pinion Q. The rack R is rotated by two
connecting rods, one of which C is shown, and which are under
the control of the hydraulic governor as described on page 378.
The wheel shaft can be adjusted by nuts working on the
squarethreaded screw shown, and is carried on a special collar
bearing supported by the bracket B. The weight of the shaft is
partly balanced by the waterpressure piston which has acting
underneath it a pressure per unit area equal to that in the supply
chamber. The dimensions shown are in millimetres.
TDRBINES
36t
362
HTDEAPLICS
202. Effect of changing the direction of the guide bladi
when altering the flow of inward flow and mixed floi
turbine 3«
As long as the velocity of a wheel remains constant, fhj
backward head impreased on the water by the wheel is the sam^
and the pressure head, at the inlet to the wheel, will remm
practically constant as the guides are moved^ The velocity d
flow U» through the guides, will, thereforej remain constajit
but as the angle ^, which the guide makes with the tangent to the
wheel, diminishes the radial component u^ of U, diminishes^
Fig. 242.
Let ABC, Fig. 242, he the triangle of velocities for full opariafi .
and suppose the inclination of the tip of the blade i.^ made paraM
to BC. On turning the guides into the dotted position* tlie incti*
nation being «^'), the triangle of %^elocities is ABCi, and the relati«
velocity of the water and the periphery of the wheel is now BC^
which is inclined to the vane, and there is, consequently, loss <te
to shock.
It vnW be seen that in the dotted position the tips of the gtt>«i*
blades are some distance from the periphery of the wheel and it ii
probable that the stream Hues on leaving the guide blades folio*
the dotted curves SS, and if so, the inclination of these ^tre^B
lines to the tangent to the wheel will be actually grealer thiiD^']»
and BCi will then be more nearly parallel to BC. The lt»ss owtf
be approximated to as follows :
As the water enters the wheel its radial component will remwfi
nnaltereilj but its direction will be suddenly changed from flCita
BCj and ita magnitude to BCj; CiCt is drawn parallel to Ai
A velocity equal to dCs has therefore to be suddenly impr
the water*
On page 68 it has been shown that on oertain assumptiuui ^
TURBINES 863
t ^when the velocity of a stream is suddenly changed
to 179 is
2g '
it is equal to the head due to the relative velocity of
?«.
CiC is the relative velocity of BCi and BCs, and therefore
1 lost at inlet may be taken as
a coefficient which may be taken as approximately unity.
. Effect of diTniniiihiTig the flow through turbines on
Locity of exit.
pater leaves a wheel radially when the flow is a maximum,
lot do so for any other flow.
angle of the tip of the blade at exit is unalterable, and if
Uo are the radial velocities of flow, at full and part load
ively, the triangles of velocity are DBF and DBFi, Fig. 243.
part flow, the velocity with which the water leaves the
s tt,. K this is greater than u, and the wheel is drowned,
3xhaust takes place into the air, the theoretical hydraulic
3y is less than for full load, but if the discharge is down a
tube the velocity with which the water leaves the tube is
iu for full flow and the theoretical hydraulic efficiency is
for the part flow. The loss of head, by friction in the
iue to the relative velocity of the water and the vane,
8 less than at full load, should also be diminished, as also,
3 of head by friction in the supply and exhaust pipes,
chanical losses remain practically constant at all loads.
Br^T ! i^A
Fig. 243. Fig. 244.
fact that the efficiency of turbines diminishes at part loads
lerefore, in large measure be due to the losses by shock
icreased more than the friction losses are diminished,
suitably designing the vanes, the greatest efficiency of
flow and mixed flow turbines can be obtained at some
of full load.
304
HTDBAULU^
204. Begulaiion of the £ow by cylindriGal gata&.
\\ lieu tlie &]>eed of the turbine is adjusted by a gate betw«i
the guidefi and the wheel, and the fio\r m less than the nomtftJ, tk
velfjtcity XJ with which the water leav^es the gmde is alteral ic
mBgnitade bat not in direction*
Let ABC be the triangle of velocitiaSj Fig* 244> whem the iiowii
nonimL
IM tht* How bo diminished until the velocity with wMchlit
water leRves the guides is Uoj equal to AD,
llien BD is the
rsdial velocity of flfi
Draw DK pan
the VMIB a sodden reiociiyi
the water, and there i» a ]
To keep the velocity
intrmluoed the gate shown
connected to the guide bla
bladee as well as tlie gat« &
^r of Uo and r, an*i «s ii lil
for the water to mmv aii
o KD must be im
k(KDY
equal to
%
early constant Mr Swain ha
245. The gate g i* npif
to adjust the flow tW p\k
L The effe<;tive width of i
guides is thereby inarle approximately pmpirtional to thetpiantitf
of flow, and the velocity D remains more nearly constant. If the
gate is raised, tlie width h of the wheel opening will be greater
than t, the widtli of the gate opening, and the radial velocif)'*!
Fig. 2iS. SwalQ Ga«.
Fig, 246,
TURBINES
365
wheel will consequently be less than the radial velocity u
guides. If n is assumed constant the relative velocity of
)r and the vane will suddenly change from BC to BCi,
Or it may be supposed that in the space between the
id the wheel the velocity U changes from AC to ACi.
k (CC,)^
OS8 of head will now be
2flr
The form of the wheel vanes between the inlet and
f turbines.
form of the vanes between inlet and outlet of turbines
»e such, that there is no sudden change in the relative
of the water and the wheel.
ider the case of an inward flow turbine. Having given
3 the vane and fixed the width between the crowns of the
le velocity relative to the wheel at any radius r can be
follows.
any circumferential section ef at radius r, Pig. 247. Let
? effective width between the crowns, and d the effective
" between the vanes, and let q be the flow in cubic feet
id between the vanes Ae and B/.
RelatiTe Telocity of the water and the Tanes.
Fig. 348.
366 HTDRAUUCS
The radial velocity through e/is
Find by trial a point O near the centre of ef such that a circle
drawn with as centre touches the vanes at M and Mi.
Suppose the vanes near 6 and / to be struck with arcs of cirdcB.
Join to the centres of these circles and draw a curve MCMi
touching the radii OM and OMi at M and Mi respectively.
Then MCMi will be practically normal to the stream lines
through the wheel. The centre of MCMi may not exactly
coincide with the centre of e/, but a second trial will probably
make it do so.
If then, h is the effective width between the crowns at C,
h . MMi . ^v = q,
MMi can be scaled off the drawing and Vr calculated.
The curve of relative velocities for varying radii can then be
plotted as shown in the figure.
Fig. 249.
It will be seen that in this case the curve of relative velocities
changes fairly suddenly between c and A. By trial, the vanes
should be made so that the variation of velocity is as unifonn
as possible.
If the vanes could be made involutes of a circle of radius &»
(
TURBINES 367
in Fig. 240, and tlie crowns of the wheel parallel, the relative
locity of the wheel and the water would remain constant,
lis form of vane is however entirely unsuitable for inward
>w turbines and could only be used in very special cases for
itward flow turbines, as the angles ^ and which the involute
akes with the circumferences at A and B are not independent,
IT from the figure it is seen that,
8ind = ?5
r
nd sin ^ = g^ ,
ainO R
r T— T =  .
8m9 r
The angle must clearly always be greater than <^.
206. The llmitiTig head for a single stage reaction
arbine.
Reaction turbines have not yet been made to work under heads
ligher than 430 feet, impulse turbines of the types to be presently
Lescribed being used for heads greater than this value.
Prom the triangle of velocities at inlet of a reaction turbine,
.g. Fig. 226, it IB seen that the whirling velocity V cannot be
greater than
V + u cot ^.
Assuming the smallest value for ^ to be 30 degrees, and the
oaximum value for u to be 0*25 v 2grH, the general formula
9
becomes, for the limiting case,
viv^2j3s/R)=e.g.R.
K t? is assumed to have a limiting value of 100 feet per second,
rhich is higher than generally allowed in practice, and e to
ye 0*8, then the maximum head H which can be utilised in a one
(tage reaction turbine, is given by the equation
256H 346 n/H = 10,000,
from which H = 530 feet.
207. Series or multiple stage reaction turbines.
Professor Osborne Reynolds has suggested the use of two
or more turbines in series, the same water passing through them
successively, and a portion of the head being utilised in each.
For parallel flow turbines, Reynolds proposed that the wheels
S6S
HYDRAUUCS
and fixed blades be arranged alternately as shown in Fig S
This arrangeinentj although not used in water turbines, is
largely caed in 3*eactioii steam turhinos.
^^^^^^^
^
^^*==^w
J J E^'t^ ^ ^
J\
.^^^si<^^^
Fig. 260.
ImihjtdL
Figfl, 251, 252. AiIaI Flow Impulse TorlJim.
* Taken from Prof, R^yuolda* Scimtijis Papyri, VoL i.
TURBINES
369
Fig, 353.
L impulse turbines.
md turbine. To overcome the difficulty of diimnution of
hf with diroinatioii of flow,
(intToduced, about 1850, the
krrmtion or partial admission
flead of the water beitig
to the wheel throughout
jle circumference as iu the
turbines, in the Girard
it 18 only allowed to enter
^1 through guide passages
I diametrically oppcis^ite
( as shown in Figs, 2*52—
tn the first two, the flow is
knd in the last radiah
[Fig, 252 abo%'e the guide crown are two quadrantshaped
l^nted 2 and 4^ which are made to rotate about a vertical
^Qmns of a toothed wheel. When the gates are over the
tJB 2 and 4, all the guide passages are open, and by turning
in the direction of the arrow, any desired number of the
can be clost^d. In Fig. 2*S4 the variation of flow i^
by means of a cylindrical quadrantshaped sluice, which,
c previous case^ can be made to close any desired number
gnide paseages* Several other types of regulators for
turbines were introduced by Girard and others,
. 25^^ shows a regulator employed by Fontaine. Above the
l>lade6, and fixed at the opposite ends of a diameter DD,
ro indianibber bands, the other ends of the bands being
Sled to two conical rollers. The conical rollers can rotate
artialB, formed on the end of the arms which are connected
coocbed wheel TVV, A pinion P gears with TW, and by
Ig the spindle carrying the pinion P, the rollers can be made
irrap, or wTap up, the indiarubber band, thus opening or
m the guide passages.
I the Girard turbine is not kept full of water, the whole of
reliable head is converted into velocity before the water
the wheel, and the turbine is a pure impulse turbine,
prevent loss of head by broken water in the wheel, the air
. be freely admitted to the buckets as shown in Figs» 252
A.
small heads the wheel must be horizontal but for large
it may be verticaL
b clase of turbine has the disadvantage that it. c^^^'&.c^'^
1
370
HYDBAUUCa
rati drowned, and tence must alwajB b© plaeod aboTe the
water. For low aod variable heads the full head cannot thef©^
bii utilisod, for if the wheel is to be clear of the tail water,
amount of hmd equal to half the width of the wheel mast
necessity be lost.
Fig» 26t. Girard Hftdial flow Impulse TurWoe,
To overGome this difficalty Girard placed tJie wheel
tight tube, Fig. 254, the lower end of which is below ihc
level, and into which air is pumped hy a small auxiliary
the pressure being maintained at the necessary value t*
surface of the water in the tube below the wheel.
TURBINES
an
t H be the total head above the tail watar level of the snppl^
~ the pressure head dae to the atmospheric preasare^ H,
lee of the centre of the wheel below the sarface of the
i?atery and h^ the distance of the surface of the water in
Inbe below the tail water level. Then the airpressure in
tttbe must be
W '
the head causing velocity of flow into the wheel is^ therefore,
w Xw 7
b that wherever the wheel is placed in the tube below the tail
the full fall H is utilised.
Ihis system, however, has not foand favour in practice, owing
difficulty of preserving the pressure in the tube*
The form of the vanee for impulfle turbines, neg
ng fkictloiL
Jhe receiving tip of the vane should be parallel to the relative
Atf Vr of the water and the edge of the vane, Fig< 255.
t exit the relative velocity iv, Fig. 256, neglecting friction,
be equal to the relative velocity V,. at inlet,
' the angle a which the tip of the vane at exit makes with
lirectiou of Vi is known the triangle of velocities can be drawn,
fffeting out DE equal to Vj and EF at an angle a with it and
1 to Vr. Then DF is the velocity with which the water leaves
irbeeh
W the aidal flow turbine Vi equals t*, and the triangle of
ntiee at e3dt is AGB, Fig. 255.
t the velocity with which the water leaves the wheel is Ui,
iieorotjcal hydraulic eflSciency is
E
H "^ IP
idependent of the direction of Uj .
»iild be observed, however, that in the radial flow turbine
of the section of the stream by the circuniference of the
for a given flow, will depend upon the radial component of
\xd in the axial flow turbine the area of the section of the
by a plane perpendicular to the axis will depend upon the
I component of Uj, That is, in each case the area ^vill depend
the component of Ui perpendicular to Vi.
372
HYDRAULICS
Now the section of tlie stream most not fill the outlet area''o{
the wheels and the minininTn area of this outlet so that it is just
not filled will clearly be obtained for a given value of XJi when Ui
is perpendicular to Vi*, or is radial in the outward flow and axial in
the parallel flow turbine.
For the parallel flow turbine since BC and B6, Fig. 255, are
equal, Ui is clearly perpendicular to Vi when
v = ^ = ^'J2gRcosO,
and the inclinations a and ^ of the tips of the vanes are equaL
Figs. 255, 256.
Fig. 257.
If R and r are the outer and inner radii of the radial flow
turbine respectively,
R
r
* It is often stated that this is the condition for mazimam effieieiiOT bat it ctif
is 80, as stated above, for mazimam flow for the given maohine. Tat
only depends upon the magnitude of Ui and not apon its direotion.
Far Ui ta be radial
TURBINES
Yr~Vi sec a
373
sec a.
Y Y
and if u is made equal to ^, Yr from Fig, 255 is equal to ^ sec <^a
and therefore,
sec ot = ^ «©c 9.
210. Txian^es of velocity for an axial flow Impulsa tur
bine confiidering friction*
The* velocity witb which the water leave© the guide passages
may be takeu as from 0'94r to 0'97 \^2^H, and the hydraulic losses
ia tlie wheel are from 5 to 10 per ceut*
If the angle between the jet and the direction of motion of the
vane is taken as 30 degrees, and U ia assumed as 0"95 n/2^H, and v
ms 0*45 v^2gH, the triangle of veloeitiL^s is ABC, Fig. 257.
lUdng 10 per cent, of the head as being lost in the wheel^ the
relative velocity tv at ©sit can be obtained from the expression
% ^
K now the velocity of exit Ui be taken as 0'22s/2^H, and
circles with A and B as centres, and Ui and iv as radii be
described, intersecting in D, ABD the triangle of velocities at exit
is obtaijiedj and U] is practically axial as shown in the figure.
On these assumptions the best velocity for the rim of the wheel is
iber<?fore "45 •J2gB. instead of *5 *J2gR.
The head lost due to the water leaving the wheel with velocity
u i» *M8H^ and the theoretical hydraulic efficiency is therefore
^"2 per cent.
The velocity head at entrance is 0*9025H and, therefore, *097H
lm& been lost when the water enters the wheel.
The efficiency^ neglecting axle friction, will be
H  O'lH  0O48H  O'OQTH
e = g
= 76 per centt nearly.
21L Imptdse turbine for bigli beads.
For high heads Girard introduced a form of impulse turbine,
of which the turbine shown in Figs. 258 and 259, is the modem
deirelopment.
J The water instead of being delivered through guides over an
mrc of a circle, is delivered through one or more adjustable nozzles.
^ib
TURBINES
876
pie shown, the wheel has a mean diameter of 6*9 feet
revolations per minute; it develops 1600 horse
lead of 1935 feet.
pipe is of steel and is 1'312 feet diameter.
the orifices has been developed hy experience, and
)re is no sadden change in the form of the liquid
juently no loss due to shock.
of water to the wheel is regulated by the sluices
258, which, as also the axles carrying the same,
the orifices, and can consequently be lubricated
ae is at work. The sluices are under the control
►vemor and special form of regulator.
d of the turbine tends to increase the regulator
11 crank lever and partially closes both the orifices.
I speed of the turbine causes the reverse action to
igh peripheral speed of the wheel, 205 feet per
?s a high stress in the wheel due to centrifugal
ng the weight of a bar of the metal of which the
le square inch in section and one foot long as
tress per sq. inch in the hoop surrounding the
/=
336. t;^
9
= 4400 lbs. per sq. inch.
iger of fracture, steel laminated hoops are shrunk
ery of the wheel.
arrying the blades is made independent of the disc
that it may be replaced when the blades become
ri entirely new wheel being provided.
of the vanes at the inner periphery is 171 feet per
herefore, 0*484 v2gH.
ity U wth which the water leaves the orifice is
5(7 H, and the angle the jet makes with the tangent
K) degrees, the triangle of velocities at entrance is
and the angle <f> is 53'5 degrees.
1*1 of the outer edges of the vanes is 205 feet per
iming there is a loss of head in the wheel, equal to
'^9
2^
006H,
= 123*5 ft. per second.
876
HTDRAULIGS
If then the angle a is 30 degrees the triangle of velocities !
exit is DBF, Fig. 261. .
The velocity with which the water leaves the wheel is th<
Ui = 95 feet per sec., and the head lost hy this velocity is 140 fe
or OTSH.
Fig. 260.
Fig. 261.
The head lost in the pipe and nozzle is, on the assumpti(
made above,
H(097)»H=006H,
and the total percentage loss of head is, therefore,
6 + 73 + 6 = 193,
and the hydraulic efficiency is 80*7 per cent.
Fig. 262. Pelton Wheel.
TURBINES
377
The actual efficiency of a similar turbine at full load was found
by experiment to be 78 per cent., which allows a mechanical loss
of 2*7 per cent.
212. Pelton wheel.
A form of impulse turbine now very largely used for high heads
is known as the Pelton wheel.
A number of cups, as shown in Figs. 262 and 266, is fixed to a
wheel which is generally mounted on a horizontal axis. The
water is delivered to the wheel through a rectangular shaped
nozzle, the opening of which is generally made adjustable, either
by means of a hand wheel as in Fig. 262, or automatically by a
regulator as in Fig. 266.
As shown on page 276, the theoretical efficiency of the wheel is
unity and the best velocity for the cups is onehalf the velocity of
the jet. This is also the velocity generally given to the cups
in actual examples. The width of the cups is from 2J to
4 times the thickness of the jet, and the width of the jet is about
twice its thickness.
The actual efficiency is between 70 and 82 per cent.
Table XXXVIII gives the numbers of revolutions per minute,
the diameters of the wheels and the nett head at the nozzle in
a number of examples.
TABLE XXXVIII.
Particulars of some actual Pelton wheels.
Head
infect
Diameter
of wheel
(two wheels)
Bevolations
per minute
r
U
H. p.
262
894"
875
645
129
500
*233
7"
2100
64
125
5
*197
20"
650
665
112
10
722
89"
650
111
215
167
882
60"
800
79
156
144
♦289
54"
810
73
186
400
508
90"
200
79
180
300
• Pieard Pictet and Co., the remainder by Eschar Wyss and Co.
213. Oil pressure governor or regulator.
The modem applications of turbines to the driving of electrical
^Xiachinery, has made it necessary for particular attention to be
X^^d to the regulation of the speed of the turbines.
The methods of regulating the flow by cylindrical speed gates
^»^:id moveable guide blades have been described in connection with
378
HTBRAULICS
various turbines but the means adopted for moving the gatae i
gtiidt?8 have not been discussed, ^M
Until recent years some form of differential goven^H
almoat entirely used, but these have been almost corople
superseded by hydranlic and oil governors.
Figs. 26^3 and 264 show an oil governor, aa construct
Messrs Escher Wym of Zurich. ~
FigB. 263. Mi. Oil PresBure Begalator for TarbineB.
A piston P having a larger diameter at one end than \
other^ and fitted with leathers I and ^i, fits into a double cylbS
Ci. Oil under pressure m continuously supplied through app*
into the annulus A between the pistons, while at the back of \
iarge piston the pressure of the oil is determined by the refuU
TURBINES
Fig, 265.
^ppo^ the regulator to be in a definite poBitioiij the spacej
»Iiinti the krge piston being full of oil, and the
rbine rtinning at its normal speed. The valve Y
enlarged diagrammatic section is ehowii in
1 265) will be in such a position that oil cannot
or e^icape from the large cylinder, and the
sure in the annular ring betfcveen the pistons
ill keep the regulator mechanism locked.
If the wheel increases in speed, due to a
inntioD of load, the balls of the spring loaded
remor G move outwards and the sleeve M
For the moment, the point D on the lever
is fixed, and the lever turns about D as a
ftilerum, and tljus raises the valve rod KV, This
Howe oil under pressure to enter the large
blinder and the piston in consequence moves to
ft '  1 moves the turbine gates in the manner descnbed later,
Hi moves to the right, the rod R, which rests on the
^edge W connected to the piston, falls, and the point D of the
Bver MD consequently falls and brings the valve V back to its
iginal ptjsition. The piston P thus takes np a new position
:irresponding to the required gate opening. The speed of the
irbine and of the governor is a little higher than before, the
> in speed depending upon the sonsitiverjess of the governor*
the other hand, if the speed of the wheel diminialies, the
M and also the valve V falls and the oil from behind the
irge piston escapes through the exhaust E, the piston moving
the left. The wedge W then lifts the fulcrum D, the valve V
I automatically brought to its central position^ and the piston P
ikes up a new position, consistent with the gate opening being
ifficient to supply the necessary water required by the wheeh
A hand wheel and screw. Fig* 264, are also pro\^ded, so that
gates can be moved by hand when necessary.
Tlie piston P is connected by the connecting rod BE to a crank
^f which rotates the vertical shaft T. A double crank KK is
OBnected by the two coupling rods shown to a rotating toothed
wheel R, Fig. 241^ turning about the vertical shaft of the turbine,
^and the movement, as described on page 360, causes the adjusts
&nt of the speed gates.
214. Water pressure regulators for impulse turbines.
Fig. 266 shows a water pressure regulator as applied to regulate
fche flow to a Pelton wheel,
Tlie ar^a of the supply noazle is adjusted by a beak B which
TURBINSS
381
: the centre O. The pressare of the water m the
icting on this beak tends to lift it and thos to <ipei]
The piston P, working in a cylinder C, is also acted
mder side, by the pressare of the water in the supply
connected to the beak by the connecting rod D£L
:he piston is made sofficiently large so that when the
{ton is relieved of pressare the pmll on the coDnecting
nt to close the orifice.
p conveys water nnder the same presBore. Uj the
;h maybe similar to that described in ocmnectKiD with
re governor, Fig. 265.
rod passes throngh the top of the cylinder.and carries
screws on to the square thread cut on the r>d. A
268, which is carried on the fixed f alcnuii <:, iut made
1 the piston. A link /A oonnect£ ef with the lerer
M of which moves with the governor d*jei'e and the
is connected to the valve rod XV. Tlje vaJie V m
neutral position.
■o;
"Sleteve
(a
now the speed
of til*: turf^;jj*r Vj iu*:r*ii»*i^. TLe
the lever MS tumfc i^'yjvi ti**: f ulcmin
The va;v*: V laii*s aiid o^iens the
Til*: ;;r<i*«ur»r Oli th* pist^.iij
the U'/ZZ^l^: \VU^ di»iM;iKnTr,p
he puRogu nw^ :t Lft^^ a^aiii tht
A/f aod C:io«* th*: va}Tf T. J^
vveacbefd. If the igteed cf the
382
HTBHAULICS
governor decreases the governor sleeve falls, th© valve T
and wfLter pressure ia admitted to the top of the piaton, which i
then in equihbriuin, and the pressure on the beak B causes it i
move upwards and thus open the noaszle*
Hydraulic ^^alve for water regulatfjr* Instead of the simple
piston valve controlled mechanically, Messi*s Escher Wyss use, fo
hi^h heads, a hydraulic doublepiston valve Pp, Fig* 260.
This piston valve has a small bore through its centre by meansi
of which high pressure water which is admitted below^ the valve j
can pass to the top of the large piston P, Above the piston is \
small plug valve Y which is opened and closed by the governor.
Fig. S69* Hjdranlio valve for autornKtle regalfttbn.
If the speed of the governor decreases, the valve V is open^
thus allowing water to escape from above the piston valve, and tif
pressure on the lower piston p raises the valve. Pressure water if
thus admitted above the regulator piston^ and the prefisiire on tk
beak opens the nozzle* Aa the governor falls the vslre V c]n(S«»
the exhaust is throttledj and the pressure above the piston P riitf^
"WTien the exhaust through V is throttled to such a degree dm
the pressure on P balances the pressure on the under face of tkt
piston pf the valve is in equilibrium and the regulator pfetoft tf
Joeked*
rittHy
TUHBINKS
383
the speed of the ^Fernor increases, the valve V is closed^
and tlie excess pressure on th^ tipper face of the pieton valve
eaojM^s it to descend, thus connecting tlie regulator cylinder to
ejEhatist. The pressure on tlie under face of the regulator piston
tbem closes the uozzle.
Filter. Between the conduit pipe and the governor valve V,
is placed a filter, Figs. 270 and 271, to remove any sand or grit
contained in the water.
Wit 111 n the cylinder, on a hexagonal fianje, is stretched a
pn'eee of canvas. Tlie water enters the cylinder by the pipe E, and
after passing through the canvas^ enters the central perforated
pipe and leaves by the pipe S.
FiffK 370, 27L Water Filter for Impuke Turbine Reguktor.
To clean the filter while at work, the canvas frame is revolved
by me^ns nf the handle shown, and the cock R is opened. Each
mde of tlie hexagonal frame is brought in turn opposite the
chamber A, and water tlowB outwards through the canvas and
ibroa^h the cock R^ carrying away any dirt that may have
collected outside the canvas.
Af^iliary valve to prei^eni hammer action. When the pipe line
^ long an auxiliary valve is frequently fitted on the pipe near to
nozzlet which is anfcomatically opened by means of a catai^ct
motion* as the nozzle closes, and when the movement of the nozxle
beak is finished^ the valve slow^ly closes again.
If no such provision is made a rapid closing of the nozzle
ixieans that a large mass of water must have its momentum
'^ly changed and very large pressures may be set up, or in
: words hammer action is produced, which may cause fracture
of the pipe.
WTien there is an abundant supply of water, the auxiliary
valve is connected to the piston rod of the regulator and opened
and elo3*4*d as the piston rod moves, the valve being adjusted so
tliat the opening increases by the same amount that the area of
tbe orifice dimiiiishes*
* See Enffin^ttt VoU ic, p. 265,
384
HYDRAULICS
If the load an the wheel does not vmiy through a large m
the quantity of water wasted is not large.
215. Hammer blow in a long turbine supply pipe.
Let L be the length of the pipe and d its diameter.
The weight of water in the pipe is
4
Let the velocity change by an
rate of change of momentum is
the lower end of the column of w
be applied equal to this.
Therefore
"}•
^nnt dv in time cL Then &9
J and on a cross sectioii rf
' in the pipe a force P tmat
ft'
lepth of the orifice and di it«
1 the centre about which UitS
Referring to Fig. 266, let b be
width.
Then, if r is the distance of 1
beak turns, and r, h the distance ot the elf ising edge oi the ^►ejik
from this centre, and if at any moment the velocity of the piston
is r» feet per sei^ond, the velocity of closing of the beak ^nll be
r
In any small element of time ct the amount by which the
nozzle will close is
BJ
r^Fi
Bt
Let it 1^ ass^umed that U* the velocity of flow through tis
nozzle, remains constant. It will actually vary, due to tk
ivs^istances varying with the vel^xnty, hut unless the pipe is very
long the errv^r is n*>t gr^^t in neglecting the variation. If then r
is the velocity in the pipe at the commencement of this element of
time and r  c r at the end of it, and A the area of the pipe,
rA=fr.d,.U ......a)
and (rcr>A(t^d(yd^.U
Subtracting \2i tnmi (I),
r
{il
ct ''
r A
.(3).
TURBINES 385
If W is the weight of water in the pipe, the force P in pounds
it will have to be applied to change the velocity of this water
cv in time dt is
g of
Therefore p^Wr.^o
id the pressure per sq. inch produced in the pipe near the
ozzle is
W r, diUi^o
^ = 7r"A» •
Suppose the nozzle to be completely closed in a time t seconds,
nd during tlie closing the piston P moves with simple harmonic
notion.
Then the distance moved by the piston to close the nozzle is
br
md the time taken to move this distance is t seconds.
The maximum velocity of the piston is then
vbr
ad substituting in (3), the maximum value of r is, therefore,
dv_ vbrridiU
ot" 2trirA '
ad the maximum pressure per square inch is
^* 2gtA' 2g.t.A' 2t' gA'
here Q is the flow in cubic feet per second before the orifice
e^an to close, and v is the velocity in the pipe.
ExamvU, A 500 horsepower Pel ton Wheel of 75 per cent, efficiency, and worki ng
Bder a head of 260 feet, is sapplied with water by a pipe 1000 feet long and
8" diameter. The load is suddenly taken off, and the time taken by the
igtilator to close the nozzle completely is 6 seconds.
On the assumption that the nozzle is completely cloned (1) at a uniform rate,
id (2) with simple harmonic motion, and that no relief valve is provided,
dtermine the pressure produced at the nozzle.
The quantity of water delivered to the wheel per second when working at fall
Dwer is
^ 500x33,000 „ _ , . , ,
Q= 260x624x'76x60 =^^^ <="'»* '*«'•
The weight of water in the pipe is
W = 624 X J. (225)5x1000
= 250,000 lbs.
L. H. ^"^
386
The Telodty is ^^ = 5^6 ft. per seo.
In case (1) the total pT«sffii]:« acting on the hyvet en^ of thm 4»lixmii nf «il»
the pipe is
= 8200 lbs.
The presson per sq. inch ia
y =  ^14 5 Iba. per sq, iiioh^
w W I?
In case (2) p^=^ ^ ^^ =221 u per wq.
exam: s.
(I) Find tteHMMWiic^l horne^pow*
diameter, usingt 9XCXXM)00 gailons of m
of 25 feet.
! an oversliot watetwbed S3 &et 
• per 24 hours under a IxM had
(2) An over^ot waterwheel has i imetez^ of 24 Ceet, and ma^aif^
revolutions per minute. The vekx'ity oi trie water a.^ it enters the bucketi
is to be twice that of the wheel" a periphery.
If the angle wliicli the water make« with the periphery is to be U
degrees, find the fhrection of the tip of the bucket, and the relkfeive velodty
of the water and the bucket.
(8) The sluiee of an overshot waterwheel 12 feet diameter \& vertioBj
above the centre of the wheel. The surface of the water in the shm*
channel is 2 feet 6 ineheH above the top of tlje wheel and the centre of Uae
sluice o{)ening is B iuclies above the top of thc^ wheel. The velocity of ^
wheel periphery is to be onehalf that of the water aa it enters the backets
Determine the nuuibt^r of rotationw of the whoeit the point at wludi the
water enters the buckets, and tlie direction of the edge of tiie bucket.
(4) An overt^iu^t wheel 25 feet diaujeter having a widtJi of 5 feeUMui
depth of crowns 12 inches, receiveit 450 cubic feet of water per minute^ ao(i
makes 6 revolutions jier minute. There are 64 buckets.
The water enters the wheel at 15 degreca from the cro^Ti of the wlieel
witli a velocity e^^iial to twice that of the periphery, and at an angle of D
degrees with the tan gent to the wheel.
Assuming the buckets to be of the form shown in Fig, 180, the lengtb
of the radial i>ortiou being one half tlie length of the outer face of the
bucket, tind how much water enters each bucket, and, allowing for oeatn
fugal forces, the point at wliich the water l)egins to leave the buckets^
(5) An overi^hot wheel S2 feet diameter has Rhrouds 14 inches deep^
and is required to give 9 horse ■i>ower when making 5 revolutions per micafe
Assuming tlie buckets to be one third filled with water and of the same
form as in the last question, find the width of the wheel, when the tot^l
fall is 32 feet and Uie efficiency 60 per cent
TUBBINIS
387
I Assodmiig the Telocity of tlie water la the penstock to Ve 1} times that
K tl]« wliael's periphciry, and the bottotD of the penstock level with the top
m. tba ^wbci&l, find the pomt at which the water entem the wheel. Fmd alao
vhere water begins to discharge frona the buckets.
■ (S> A ra^dial blade impnlie wheel of the name width as the channel in
vliicb it mud, is 15 feet diameter. The depth of tlie sluice opening i^
■^ focbaa and the head above the centre of the sluice is H feet. Assuming
K coefficient ol vielocity of 0*8 and ttiAt tlie edge ef the sluice is rounded so
iliAt there m no contraction, and tlie velocity of the rim of the wlicel is 0'4
pile velocity of flow through the aluicet iind the theoreticid efficiency of
lllie 'vrbeeL
I (7 1 An overshot wheel has a supply of 80 cubic feet per second on a fall
Lf Mfeet
I DetienDine tlie probable horsepower of the wheel, and a suitable
l^riillii for the wheeL
I (8) The water impinges on a Poncelet float at 15° with the tangent to
m^UB iHideL, and the velocity of the water is double that of the wheel. Find,
Ihj' oorastmciioD, the proper toclination of the tip of the float.
I <]d) In a Ponoelet wheel, the direction of tlie jet impinging on tlie floats
kwkflfi an angle of W^ with the tangent to the circumference and the tip of
nllB floats inakies an angle of W>° with the aame tangent. Supposing tlte
ftr^locitj of the jet to be 20 feet per second. And, graphically or otherwise»
■ (1) the proper velocity of ttie edge of the wheel, (2) the height to which the
nrater will rise on tlie float above the point of admission, (8) the velocity
B ttod direction of motion of the water leaving the float.
I 00) Show that the efficiency of a simple reaction wheel increases
■ niyi the speed when frictional resistances are neglected^ but is greatest
BMI a fiaiite speed when they are taken into account.
m U the speed of the orifices be that due to the head (1) And the efficiency,
I 0egl€<:ting friction ; f'2) assuming it to be the speed at maximum efficiency,
m ibow that j of the head is lost by friction, and } by final velocity of water,
I (ll.> Explain why^ in a vortex turbine, the inner ends of the vanes are
I lodined backwards instead of being radial.
I ii2< An inward flow turbine wheel has radial blades at tlie outer
I F^iphtfry. and at the inner periphery tlje blade makes an angle of 30"" with
K Ft
m iSbb tangeBt. The total bead is TO feet and ^"^ * Find the velocity of the
■ tUb of (lie wheel if the water discharges radially. Friction neglected.
■ III I The inner and outer diameters of an inward flow turbine wheel
■*& I loot and % feet resfjectively. The water enters the outer circumference
^P^ It with the tangent, and leaves the inner circumference radially. The
Br^^ velocity of flow is 6 feet at both circumferences. The wheel makes
^pJ rerqhitions per seconds Determine the angles of the vanes at both
V^tnsoferenceaf and the theoretical hydrauhc efficiency of the turbine.
V_ (14) Water ia supplied to an inward flow turbine at 44 feet per second,
r^ al 10 d^rees to the tangent to the wheel The wheel makes 200
88B
HYDRAULICS
nmsltitiQtts per miniite^ The inlet radius is 1 foot &iid the ontei nism I
% feet. The radiaJ velocity of flow Uirough the wheel is oom&tttut I
Find the iudlnation of the vanes at iniet and outlet ol the wheel I
Determine the ratio of the kinetic Gmstg^ of the water eiitanflg tbd I
wheel per pound to the work done on the whed per pound. I
(15) The suppler of water far an inward flow reaction turbine s S(K^ 1
cable feet per minnte and the available head is 40 feet. The vASea sn 1
radial at the inlet, Qxe outer radius is twice the inner, tlie couitisl I
velocity of fiow is 4 feet per aecond, and the revolutions are 3S0 pif I
minute. Find the velocity of the wheeU the guide and vane angles 1
tlie inner and outer diameters^ and the width of the bucket at inlet ill
outlet. Lomd. Un. 1906. I
(16) An inward tiow turbine on 15 feet fall has aji inlet radius of 1 Ml
and an outlet radius of 6 inches. Water enters at 15* with the tangent to j
the circumference and is discharged radially with a velocity ol S feet per J
Be<»ncL The actual velocity of water at inlet is 22 feet per secofid fhem
G^^^nnifereutial velocity of the Inlet surface of the wheel m ld feet pttl
second. I
Construct the inlet and outlet angles of the turbine vanes* I
Determine the theoretical hydrauhc efficiency of the turbine. I
If the hydraulic efficiency of the tmbine is assumed 80 per cent ind tli« I
vane angles. I
(17) A quantity of water Q cubic feet per second flows throi^ti A j
tnrbin&t a^^ ^^^ initial and final directions and velocities are kuavSiJ
Apply the principle ol equality of angular impulse and momani di
momentum to find tlie couple exerted on the turbine, I
(18) The wheel of an inward flow turbine has a peripheral vebcttftf 1
50 feet par second. The velocity^ of whirl of the incoming water ig 4«) tet I
per second^ and the radial velocity of flow 5 feet per second. DetenniBB I
the vane angle at inlet. I
Taking the flow as 20 cubic feet per second and the total losees •* I
20 per cent, ol the available energy, determine the horse *iKjwer of tJnJ
turbinoi and the head H. ^^1
If 5 per cent, of the head is lost in friction in the supply pipe, an^H
centare of the turbine is 15 feet above the tail race level, find the preaia]i
head at the inlet circumference of the wheel. I
(10) An inward flow turbine is required to give 200 harse*pewer is^y
a head of 100 feet when running at 500 revolutions per minnte. WM
velocity with which the water leaves tlie wheel axiaUy ruay be tall^H
10 feet per secoud^ aod tho wheel is to have a double outlet The diftd^H
of the outer circumference may be taken as 1 times tlie inner. Defeei^^l
tlio dimensionB of the tiubine and the angles of the guide bMay^
vanes of the turbine wheel. The actual efficiency is to be taken as I^H
cent, and the hydraulic efficiency as 80 per cent. ^H
(20) An outward flow turbine wheel has an internal diameter of fi^Sl
feet and an external diameter of 0*25 feet. The head above the turlitflUU
14V5 feet The width of the wheel at inlet is 10 inches, and the iifii^
I
TURBIKES
389
plied per HeconcT m 215 cnbic feei. Asanmmg the hydraulic
«?« nrc 211 ior oent., determine tlie angles of tips o! the Tanes ^a that
^Ixg w«i4^r shall leave the wheel radially, Determiiie the horse power o£
tiirbme and verify the work done per poand from the triangleB of
121) The total head arailable for an inward flow ttubitie is 100 feet
The tnrbine wheel is pl&oed 15 feet above the tail water ley eh
When the flow is somud, tliere is a loss of head in the supply pipe of
• jmi t i'ut of the head I in tlie goide passages a loss of 5 per cent. ; in the
'^'"lieel 9 pi?r cent ; in the down pipe I per cent, i and the velocity of flow
^«^m the wheel and in the supply pipe, and also from the down pipe ib
^ feet iMjr decond
Tljr diameter of the inner circnmference of the wheel is 9^ inches and
tht» outer 19 inches, and the water leaves the wheel vanes radially*
TtTie wheel haa nnUal vanes at inlet
f >et«^nnine tlie tiamber of revolutiona of the wheeL the pretssure head in
\ #ye of the wheel* the pressure head at the circumference to the wheel,
^ 3n%amire head at the eu trance tti the guide chamber, and the velocity
vliicb thr water Ijaa when it enters tlie wheeL From the data given
j^
2) A horizontal inward flow tnrbine has an internal dXameter of
4 iuchef* and an external diameter of 7 feet The crowns of the
I aro parallel and are 8 inches apart The diHerence in level of the
i and tail water in 6 feet, and the upper crown of the wheel is jost below
I latt water level. Find the angle the guide blade niakea witti the tangent
wbnel, when the wheel makea 32 revolutious per minute, and the
■15 cnbic foet per second. Neglecting friction, determine the vane
, lite horse power of the wheel and the theoretical hydranhc efficiency.
9) A parallel flow tnrbine has a mean diameter of 11 feet.
The number of revolutions per minute is 15, and the a^dal velocity of
is 3'5 feet per second. The velocity of the water along the tips of the
. is 15 feet per second.
LN^termine tlie inclination of tlie gnide blades and the vane angles that
li« water shall enter withont shock and leave the wheel axially,
Dc?ieriiiine the work done per pound of water passing thi'ough the wheeL
i*M) The diameter of Uie inner crown of a parallel flow pressure turbine
f 5 feet atid the diameter of tlie outer crown is B feet. The head over the
whie^l iit 11* feett The number of revolutions per minute is 52* The radial
Jocitv of flow tlirough the wheel is 4 feet per second.
Aaaiiining a hydraulic efficiency of 0*8, determine the guide blade angles
1 rani^ angles at inlet for the three radij 2 feet 6 inches^ 8 feet 8 inches
i4Ceet
Assrunlng the depth of the wheel is S inches, draw am table sections of
» Tanes at the three radii*
Find al»o ihe width of the guide blaile in plan, if the up])er and lower
are paraUel* and the lower edge makes a constant angle with the
HTDEArXICS
Huee mt the miidr And the
head of 64 iBtLl
Uml speed ol ibe whed^l
tmliiike has an inner diajneter of 5 leiti]
d iftdiBA, ftud makes 4S0 reTolatioiiA^ ]
at H lewes tlie nozzles is dotibla the Telodt*l
I Hm wlifiel, and tlie dtreetion of the water iDaJi«« m\
wUli Hie csmttnference ol the wheel.
at miei, and the angle of tlie Tane at outlet i
mier AmXt learo Ibe idieel nidiaUy.
llie tbeocHml bydr&idie efficiency. U 8 per cent, of ^e head!
at the mamM^ ia lost in the wheel, find tJie vane angle at exit thoft]
Uft leanie imdiaUj.
■0V the l^dnnfo didbncy of the turbine ?
P
In aA asdal flow Oinrd turbine, kt V be the velocity doe to Die I
head. Siqppose the water issiies from the guide blades with Qm 1
O^T, and It dbehazged tnillj with a velocity ^12 V* Lei ^ 
iJotily ctf the teoeinigMddiafahaigiiig edges be 055 V,
^1^ of the guide liladeB, receiving and dischajrging angles d
■d Iry^snbe effdepey cl the torbine.
m^
imi
Waier k sn^fiied to an axial flow impulse turbine, having a mma
r ef t fibit^ and niakmg 144 revolntioiis per minate, tmder a hmd d \
. Tin aag^ of t^ guide blade at entrance is 30% and the angja tbe
le direction of motion at exit is 80"". Eight per eeoi of
I the supply pipe and guide. Determine ihe ralAtlvt
wtkicHy «i mier end wtieil at entrance, and on the assumption Utat 10 f«f
eesi. ci fiie kpt^ hesd is lost in frictfon and shock in tlie wheel, detemunt
Hie veitoeily wHh which the water leav^ the wheel. Find the Uydnolk
ofthei
W Hke golde bladea ef an inward dow turbine are inclined M SD
iliyeui» «Bd the Telocity U tlong tl je tip of the blade is 60 feet pet scooni
The i^odtyoC Hie wheel periphery is 55 feet per second. The guide blidei
wm trailed an thnt tbey ate inclined at an angle of 15 degrees, the ve
C T^mftimng eoasluit. Find the loss of head due to »hock at entranc
If the nidtQt of the iimer peripheiy ie onehalf that of the outer aad^
tidkU velocity thrangb the wheel ie constant for any flow^ and the inl^
kfl Ibe wheel twlbliy in the first case, &id the direction in which it leaf««
JM liie eeeond om& The inlet radius is twice the outlet radius.
03f) Tile suppler of water to a turbbie is controlled by a speed \
between tlie guides and the wheel. If when tlie gate ia fully opea ^
i^ocity witli which the wmter approaches the wheel is 70 foot pe; seooA^
^y^
I
t
f,
! TURBINES 391
m and it TiMkkes an angle of 15 degrees with the tangent to the wheel, find
the loss of Ikead by shock when the gate is half closed. The velocity of
the inlet periphery of the wheel is 75 feet per second.
(81) A Pelton wheel, which may he assumed to have semicylindricaJ
buckets, is 2 feet diameter. The available pressure at the nozzle when it
is doeed is 200 lbs. per sqnare inch, and the supply when the nozzle is
open is lOO cuIhc feet per minute. If the revolutions are 600 per minute,
estJTnate the horsepower of the wheel and its efficiency.
(82) Show that the efficiency of a Pelton wheel is a maximum —
neig^ecting fric!tional and other losses — when the velocity of the cups equals
half the velocity of the jet.
25 cnbic feet of water are supplied per second to a Pelton wheel through
m nozzle, the area of which is 44 square inches. The velocity of the cups
is 41 feet per second. Determine the horsepower of the wheel assuming
an efficiency of 75 per cent.
CHAPTER X.
PUMPS.
Pumps are machines driven by some prime mover, and used
for raising fluids from a lower to a higher level, or for impartmg
energy to fluids. For example, when a mine has to be drained
the water may be simply raised from the mine to the surface, and
work done upon it against gravity. Instead of simply raising the
water through a height h, the same pumps might be used to
deliver water into pipes, the pressure in which is wh pounds per
square foot.
A pump can either be a suction pump, a pressure pump, or
both. If the pump is placed above the surface of the water in
the well or sump, the water has to be first raised by suction;
the maximum height through which a pump can draw water,
or in other words the maximum vertical distance the pump can
be placed above the water in the well, is theoretically 34 feet, but
practically the maximum is from 25 to 30 feet. If the pump
delivers the water to a height h above the pump, or against a
pressurehead A, it is called a force pump.
216. Centriftigal and turbine pumps.
Theoretically any reaction turbine could be made to work as
a pump by rotating the wheel in the opposite direction to that in
which it rotates as a turbine, and supplying it with water at the
circumference, with the same velocity, but in the inverse direction
to that at which it was discharged when acting as a turbine. Up
to the present, only outward flow pumps have been constructed,
and, as will be showTi later, difficulty would be experienced in
starting parallel flow or inward flow pumps.
Several types of centrifugal pumps (outward flow) are shown
in Figs. 272 to 276.
The principal difference between the several types is in the
form of the casing surrounding the wheel, and this form has con
siderable influence upon the efficiency of the pump. The reason
PtJMPS
393
this can be easily seen in a general way from the following
cnnsideration. The water approaches a turbine wheel with a
lii^li velocity and in a direction making a small angle with the
directioti of motion of the inlet circumference of the wheel, and
Fig, 1^3. Diagrum of Centrifui^al Pump.
[thus it has a large velocity of whirl When the water leaves the
] wheel its velocity is small and the velocity of whirl should be zero.
jIn the centrifugal pump these conditions are entirely reversed;
[the water enters the wheel with a small velocity, and leaves
394
HYDBAUUCS
it with a high velocity. If the case surronnding the whfid
admits of this velocity being diminished gradually, the Mneiac
energy of the water is converted into useful work, but if not, itia
destroyed by eddy motions in the casing, and the efficiency of tlid
pump is accordingly low.
In Fig. 272 a circular casing surrounds the wheel, and prac
tically the whole of the kinetic energy of the water when it leaves
the wheel is destroyed ; the efficiency of such pumps is generally
much less than 50 per cent.
Fig. 273. Centrifugal Pump with spiral casing.
The casing of Pig. 273 is made of spiral form, the sectional
area increasing uniformly towards the discharge pipe, and thus
being proportional to the quantity of water flowing through the
section. It may therefore be supposed that the mean velocity of
flow through any section is nearly constant, and that the stream
lines are continuous.
The wheel of Fig. 274 is surrounded by a large whirlpool
chamber in which, as shown later, the velocity with which the
water rotates round the wheel gradually diminishes, and tie
velocity head with which the water leaves the wheel is partly
converted into pressure head.
The same result is achieved in the pump of Figs. 275 and 276
PUMPS
395
'>; allowing the water as it leaves the wheel to enter giiide
p8«sag^, similar to those used in a turbine to direct the water
to the wheeL The area of these passages giadually increaaeB
md a eoiieiderahle portion of the velocity head is thus converted
into preeenre head and is available for lifting water.
Tliis cliiaB of centrifugal punip is known as the tarbine pump.
Pig, 274. EHa^ram of Centrifagftl Pump with Whirlpool Chamber.
21f . Starting centrifugal or turbine pmnps,
A centrifngal pump cannot commence delivery unless the wheel,
casings and suction pipe are full of water.
K the pump is below the water in the well there is no difficulty
in starting as the casing will be maintained full of water »
WTien the pump is above the water in the well, as in Fig, 272,
Isonretum valve V must be fitted in the suction pipe, to prevent
kj^tup when stopped from being drained. If the pump becomes
ff or when the pump is first set to work, special meuna have
to Ije provided for filling the pump case. In large pumps the air
may be expelled by means of steam, which becomes condensed and
ihe water rises from the well, or they should be provided with
HTBRAUI,ICS
i ftirpiimp or ejector as an auxiliaTy to the patdp* StiiaU pmnps
geiie?mll>r be easily filled by hand throug^h a pipe sach m \
Aomn at P, Fig, 276.
With some classes of pomps, if the pump ha& to commence
delivery against full head, a stop valv^e on the rising maiii, ,
Fig. 296, is closed until the painp has attained the speed necessary
to commence delivery*, after which the stop valve is dowlir 
opened.
It will be seen later that, under special circumstances, other
pronsions will have to be made to enable the piimp to commeDOJ
dt^liwry.
818. Form of the Tanea of centrlftigal pumps,
Tii^^^ conditions to be satisfied by the vanesi of a centrifuia!
pump are exactly the aame as for a turbine. At inlet the directioa
of the vane shoukl be parallel to the direction of the relativE
n^loeity ivf tlie water and the tip of the vane, and the velocity
witli which the Mater leaves the wheel, relative to the pump o»ft
\H tht^ vtvtor ffiini of the velocity of the tip of the vane and tl#
veUK*ity n*lative to the vane,
* Set ptge 4m,
PDlfPS
397
Suppoae the wheel and casing of Fig. 272 is full of water^ and
the wheel is rotated in the dii'ection of the arrow mth such a
velocity that water enters the wheel in a known direction with a
Telocity XT, Fig. 277 j not of necessity radial.
t^t r he the velocity of the receiving edge of the vane or inlet
eirciimferenee of the wheel; Vi the velocity of the discharging
circnmfereiice of the wheel ; Ui the absolute velocity of the water
&6 it leaves the wheel ; V and Vj the velocities of wlurl at inlet
and ontlet respectively; Vr and iv the relative velocities of the
water and the vane at inlet and outlet respectively; u and iti the
indial velocities at inlet and outlet respectively.
The triangle of velocities at inlet is ACD, Fig. 277, and if the
vsae at A, Fig* 272, ia made parallel to CD the water will enter
the wheel without shock.
wangle of vdifoiies
Fig. 277.
at e*ixt .
Fig. 278.
The wheel being full of watePj there is continuity of How, and
if A and Aj arc5 the circumferential areas of the inner and outer
cipDumferences, the radial component of the velocity of exit at the
Doter circumference ia
Au
«i =
A,"
If the direction of the tip of the vane at the outer circnm
ference is known the triangle of velocities at exit, Fig. 278, can be
drawn as follows.
Set out BG radially and equally to %, and BE equal to v^
Draw GF parallel to BE at a distance from BE equal to ft,,
ad EF parallel to the tip of the vane to meet GF in F.
Then BF is the vector sum of BE atid EF and is the velocity
ith which the water leaves the wheel relative to the fixed casing,
219. Work done on the water by the wheel,
Lt*t R and r be the radii of the discharging and receiving
enmferences respectively.
The change in angular momentum of the water as it passes
Ihrgugh the wheel is V,R + Yr/g per pound of flow, the plus sign
becing used when V is in the opposite direction to Vj, as in
Kgs, 277 and 278,
398 HYDRAULICS
Neglecting frictional and other losses, the work done by the
wheel on the water i)er pound (see page 275) is
g " g '
If n is radial, as in Fig. 272, Y is zero, and the work done on
the water by the wheel is
^ foot lbs. per lb. flow.
If then H«, Fig. 272, is the total height through which the water
is lifted from the sump or well, and Ud is the velocity with which
the water is delivered from the delivery pipe, the work done on
each pound of water is
and therefore,
g ^ 2g
Let (180*  <^) be the angle which the direction of the vane at
exit makes with the direction of motion, and (180"  &) the angle
which the vane makes with the direction of motion at inlet. Then
ACD is e and BEF is *.
In the triangle HEF, HE = HF cot <^, and therefore,
Vi " t'l — tti cot ^.
The theoretical lift, therefore, is
2g g
If Q is the discharge and Ai the peripheral area of the dis
charging circumference,
Q
Q .^
Vi  Vi ^ cot 4>
and H = ^ (1).
g
If, therefore, the water enters the wheel without shock and all
R
resistances are neglected, the lift is independent of the ratio — , and
dex^ends only on the velocity and inclination of the vane at the
discharging circumference.
220. 5B»tioofVitoVi.
As in the ca^ of the turbine, for any given head H, Vi and I'l
can theoretically' have any values consistent with the product
\
PUMPS
399
Vit?i heing equal to grH, the ratio of Vi to Vi simply depending upon
the magnitude of the angle ^.
The greater the angle ^ is made the less the velocity Vi of the
periphery must be for a given lift.
Fig. 279.
This is shown at once by equation (1), section 219, and is
illustrated in Fig. 279. The angle 4^ is given three values,
30 degrees, 90 degrees and 150 degrees, and the product Yv and
also the radial velocity of flow tti are kept constant. The theo
retical head and also the discharge for the three cases are there
fore the same. The diagrams are drawn to a common scale, and it
can therefore be seen that as ^ increases Vi diminishes, and Ui
the velocity with which the water leaves the wheel increases.
221. The kinetic energy of the water at exit from the
wheeL
Part of the head H impressed upon the water by the wheel
increases the pressure head between the inlet and outlet, and the
remainder appears as the kinetic energy of the water as it leaves
400 HTD&AULICS
the wheel. This kinetic energy is equal to ^ , and can only be
utilised to lift the water if the velocity can be gradually diminished
so as to convert velocity head into pressure head. This however
is not very easily accomplished, without h^ng accomi)anied by a
considerable loss by eddy motions. If it be assumed that the same
proportion of the head ^ in all cases is converted into useful
work, it is clear that the greater Ui, the greater the loss by eddy
motions, and the less efficient will be the pump. It is to be ex
pected, therefore, that the less the angle <^, the greater will be
the efficiency, and experiment shows that for a given form of
casing, the efficiency does increase as <^ is diminished.
222. Gross lift of a centriftigal pomp.
Let ha be the actual height through which water is lifted;
hg the head lost in the suction pipe ; ha the head lost in the delivery
pipe ; and it^ the velocity of flow along the delivery pipe.
Any other losses of head in the wheel and casing are incident
to the pump, but fe„ ha, and the head 5 should be considered as
external losses.
The gross lift of a pump is then
h=^ha + A, + fcd + o~ >
and this is always less than H.
223. Efficiencies of a centrifugal pump.
Manometric efficiency. The ratio =^ , or
e= ^
vi  vi ? cot 4>
Ai
is the manometric efficiency of the pump at normal discharge.
The reason for specifically defining e as the manometric
efficiency at normal discharge is simply that the theoretical lift H
has been deduced from consideration of a definite discharge Q,
and only for this one discharge can the conditions at the inlet edge
be as assumed.
A more general definition is, however, generally given to e, and
for any discharge Q, therefore, the manometric efficiency maj
be taken as the ratio of the gross lift at that discharge to the
theoretical head
Vi'  Vi A cot 4^
Ai
CENTRIFUGAL PUMPS 401
This manometric efficiency of the pump must not be confused
with the efficiency obtained by dividing the work done by the
pump, by the energy required to do that work, as the latter in
many pumps is zero, when the former has its maximum value.
Hydraulic efficienSy, The hydraulic efficiency of a pump is
the ratio of the work done on the pump wheel to the gross work
done by the pump.
Let W = the weight of water lifted per second.
Let h = the gross head
Let E = the work done on the pump wheel in foot pounds
per second.
Let ek = the hydraulic efficiency. Then
W.h
The work done on the pump wheel is less than the work done
on the pump shaft by the belt or motor which drives the pump,
by an amount equal to the energy lost by friction at the bearings
of the machine. This generally, in actual machines, can be
approximately determined by running the machine without load.
jkchuil efficiency. From a commercial point of view, what is
generally required is the ratio of the useful work done by the
pump, taking it as a whole, to the work done on the pump shaft.
Let E, be the energy given to the pump shaft per sec. and
em the mechanical efficiency of the pump, then
E = E«.em,
and the actual efficiency
_ W.fea
ea ^ .
Gross efficiency of the pump. The gross efficiency of the pump
itself, including mechanical as well as fluid losses, is
W.h
ea =
K
224. Experimental determination of the efficiency of a
eentriftigal pomp.
The actual and gross efficiencies of a pump can be determined
directly by experiment, but the hydraulic efficiency can only be
determined when at all loads the mechanical efficiency of the
pump is known.
To find the actual efficiency, it is only necessary to measure
the height through which water is lifted, the quantity of water
L. H. ^^
40S HfDRAITLICS
dischiirged, and tlie energy E, given to the pump shaft in oj
tim©,
A. very convenient method of determiniiig £« with a hk
degree of accuracy is to drive the pump shaft direct by an electrie
motor, the efficiency cun""e* for which at varying loads is kn
A better method ia to use some form of transmission dynjuui^
meter t.
225. Design of pump to give a discbarge Q.
If a pump is required to give a discliarge Q under a grom
lift hf and from pre\^ous experience the probable manom^tnc
efliciency e at this discharge is known, the problem of determinTtig
suitable dimensions for the wheel of the pump is not diificulL
The difficulty really arises in gi\nng a correct value to e and ia
making proper allowance for leakage.
Tins difficulty w^tl be better appreciated after the losses m
various kinds of pumps have been considered. It will then be
seen that e depends upon the angle ^, the velocity of the wbeeJ*
the dimensions of the wheeU the form of the vanes of the wheet
the discharge through the wheel, and upon the form of the casmg
surrounding the wheel ; the form of the casing being Just as
important, or more important, than the form of the wheel in
determining the probable value of e.
Design of the tvheel of a ptiffip fftr a given discharge und^fr o
given Jiead. If a pump is required to give a discharge Q under an
effective head h^^ the gross head h can only be determined if A»i
hit and ,j^ f are known.
Any suitable value can be given to the velocity tu^ If the
pipes are long it should not be nmch greater tlian 5 feet per sec<mi
for reasons explained in the chapter on pipes, and the velocity' %
in the suction pipe should be equal to or less tlian lij. Tlie
velocities ta, and n^t having been settled, the losses h* and h^ can k
approximated to and the gross head h found. In the suction pip^^
as explained on page 395, a foot valve is generally fitted, at ubicK
at high velocities, a loss of head of several feet may itcenr.
The angle <^ is generally made from 10 to 90 degrees. Theoreti
cally, as already stated^ it can be made much greater tim
90 degrees, but the efficiency of ordinary centrifugal pumps Mlglii
be very considerably diminished as <f> is increased*
The manometric eificiency e varies very considerably ; with
radial blades and a circular casing, the efficiency is not generally
' See Etfciriml Engineerinff, ThomilenHowe, p< 195*
t Bee >apef by Stanlon, Pmc. ItnL Meek Eufft^^ 1909,
CENTRIFUGAL PUMPS 403
more than 0*3 to 0*4. With a vortex chamber, or a spiral casing,
and the vanes at inlet inclined so that the tip is parallel to the
relative velocity of the water and the vane, and i> not greater than
90 degrees, the manometric efficiency e is from 0*5 to 0*75, being
greater the less the angle <^, and with properly designed guide
blades external to the wheel, e is from 0*6 to '85.
The ratio of the diameter of the discharging circumference to
the inlet circumference is somewhat arbitrary and is generally
made from 2 to 3. Except for the difficulty of starting (see
section 226), the ratio might with advantage be made much
smidler, as by so doing the frictional losses might be considerably
reduced. The radial velocity t^ may be taken from 2 to 10 feet
per second.
Having given suitable values to u, and to any two of the three
quantities, e, v, and <^, the third can be found from the equation
7 e W ~ Vitti cot 0)
n = .
9
The internal diameter d of the wheel will generally be settled from
consideration of the velocity of flow tt, into the wheel. This may
be taken as equal to or about equal to u, but in special cases
it may be larger than u.
Then if the water is admitted to the wheel at both sides, as in
Fig. 273,
from which d can be calculated when u^ and Q are known.
Let b be the width of the vane at inlet and B at outlet, and D
the diameter of the outlet circumference.
and B =
Then * = ^ ,
If the water moves toward the vanes at inlet radially, the
inclination of the vane that there shall be no shock is sucli that
tan ^ =  ,
and if guide blades are to be provided external to the wheel, as in
Fig. 275, the inclination a of the tip of the guide blade with the
direction of Vi is found from
tan a=  ,
The guide passages should be so proportioned that the velocity
Ui is gradually diminished to the velocity in the delivery pipe.
404
HYDRAULICS
Limitifig velocity of the rim of the whe&L Quite apart from
lead lost by friction in tlie wheel due to the relative motion of
he water and the wheel, there is also conBiderable loss of energy
external to the wheel due to the relative motdon ol the water mi
the wheel. Betiyeen the wheel and the casing there is in most
pumps a film of water, and between this film and the wheel,
frictional forces are set up which are practically proportional to
tlie square of the velocity of the wheel periphery and to the ane*
of the wheel crowns. An attempt is frequently made to dimitiisli
tlris loss by fixing tlit* vanes to a central diaphragm only, the
wheel thus being without crowns, the outer casing beiog so
fonned that there is but a small clearance between it and ih
outer edges of the vanes. At high velocities these frictional rdd/ft
anoes may be considerable. To keep them small the surface d
the wheel crowiis and vanes must be made smooth^ and to this
eiid many high speed wheels are carefully finished.
Until a few years ago the periphei'al velocity of pump wheals
twas generally less than 50 feet per second, and the best velocity
ms supposed to be about 30 feet per second. They are now, how
ever, run at much higher speeds, and the limiting velocities ar©
fixed from consideration of the stresses in the wheel due to centri
fugal forces. Peripheral velocities of nearly 200 feet per second
are now frequently used, and Bateau has constructed small pump*
with a peripheral velocity of 250 feet per second*.
Exatn^U. To hnd the proportions) of a pump with radial bl&d^ at oullH
(i.e« ^=±&0°) lo Uft 10 cubio fe«l uf water per »e<]oti(l figainat n head of 50 feet
Ai»mnie there are two euotion pipes and that the water ei^tera the vheelfrto
both itde«, a^ ti) Fig, S7S, ako that Uie velocity in the auction arid delii^^ ftip!«
ftnd the radial velocitj through the wheel are 6 feet per eecondt s^d the manooKtik
efficiei^f^y ia 75 per cent.
Fir&t to find lu. .
■7fi'^ = 50,
Sinoe ihe bladei aire mdi&l,
from which t*! := 46 feet per l^ee,
To find Che diameter of the suction pipes.
The diseharse is 10 oubk feet per second, therefore
4
from which ii = l*03'^12".
If the radiiiB R of the ^xteroal circ^mferenoe be taken as 2r and r is taken <
to the radiua of the auction pipes, then R = 12", and the namher of
p«T second win be
Th« felooit; of the inner edge of the vane ia
p=2S feel per see.
Engineer, 1903*
CENTRIFUGAL PUMPS 405
The inolination of the vane at inlet that the water may move on to the vane
^tlioQt abock U
tan<?=A,
lad the water when it leaves the wheel makes an angle a with v^ such that
tana=^.
If there are guide Uadea sarronnding the wheel, a gives the inclination of these
= •268'
The width of the wheel at discharge is
Q 10
».D.6'"». 206x6
=ft inches about.
The width of the n^ieel at inlet =6J inches.
226. The centrifligal head impressed on the water by
thm wheeL
Head against which a pump vrill commence to discharge. As
shown on page 335, the centrifugal head impressed on the water as
it passes through the wheel is
^^2g 2g^
1>at this is not the lift of the pump. Theoretically it is the head
which will be impressed on the water when there is no flow
through the wheel, and is accordingly the difference between the
pr oso nre at inlet and outlet when the pump is first set in motion ;
or it is the statical head which the pump will maintain when
ronning at its normal speed. If this is less than — ^^ , the pump
theoretically cannot start lifting against its full head without
being speeded up above its normal velocity.
The centrifugal head is, however, always greater than
2g 2g'
as the water in the eye of the wheel and in the casing surrounding
the wheel is made to rotate by friction.
For a pump having a wheel seven inches diameter surrounded
by a circular casing 20 inches diameter, Stanton* found that, when
the discharge was zero and the vanes were radial at exit, he was
— s— , and with curved vanes, i> being 30 degrees, he was —^ — .
For a pump with a spiral case surrounding the wheel, the
centrifugal head he when there is no discharge, cannot be much
greater than ^ , as the water surrounding the wheel is prevented
from rotating by the casing being brought near to the wheel at
one point.
♦ Proceedings Inst, M. £., 1903.
406
HY>RAriJCS
Parsons foimd for a pomp having a wheel 14 inches diameter
with radial vanes at outlet, and ninning at 300 reTolatians per
minute, that the head maintained without discharge was ^ ,
and with an Appold* wheel running at 320 revolutions per minitte
the statical head was ^ —  . For a pump, with spiral casing,
having a rotor 1*54 feet diameter, the least velocity at which
it commeTiced to discharge against a head of 14"67 feet wn&
392 revolutions per minute^ and thus he was ^ "^ > ^^^ ^^^ 1^*^
velocity against a head of 17'4 feet was 424 revolutions per
minute or K was again ^^—^ . For a pump with circular casing
1 '05t' *
larger than the wheel, he was ^ — . For a pump having guide
passages surrounding the wheel, and outside the guide passages
a circular chamber as in Fig. 275j the centrifugal head may alaa
he
larger than ^; the mean actual value for tJiis pump
found to be 1067^.
Stanton found, when the seven inches diameter wheels mentioned
above discharged into guide passages surrounded by a circuljix
chamber *20 inches diameter, that he was ^ when the vanes ot
the wheel were radial, and ,^ ^ when <^ was 30 degrees.
That the centrifugal head when the wheel has radial vanes is
likely to be greater than when the vanes of the wheel are set hnck
is to be seen by a consideration of the manner in which the water
in the chamber outside the guide passages is probably set m
motion. Fig, 2B0. Since there is no discharge, this rotation cauaot
be caused by the water passing through the pump, but mu^ he
due to internal motions set up in the wheel and casing, Th^
water in the guide chamber cannot obviously n.*tate about die
axis 0, but there is a tendency for it to do so, and consequeuily
stream line motions, as shown in the figure, are prt)bably «i
up. The layer of water nearest the outer circumference of the
wheel will no doubt be dragged along by friction in the directs
shown by the arrow, and water will flow from the outer casing to
take its place ; the stream lines will give motion to the wai/er in ^
the outer casing.
* See pftge 4l€.
CTNTEIFUOAL PUMPS
407
Wlien the vanes m the wheel are radial and as long as a vane is
'mo^'iTig between any two guide vanes, the straight vane prevents
the frictioii between the water outside the wheel and that insidsj
from dragging the wat^r backward:^ along the vanej but when the
vane is set back and the angle <^ is greater than 90 degrees, there j
will bt^ a tendency for the water in the wheel to tnove backwardi
while that in the guide chamber moves forward, and conseciuently
the velocity of the stream liTiea in the casing will be less in the
latter caee than in the former. In either case, the general
direction of fiqw of the stream liiieSj in the guide chamber, will
b© in the direction of rotation of the wheelj but due to frtction
and eddy raotions, even w4th radial vaneSj the velocity of the stream
Fig. 2m.
^ will he less than the velocity I'l of the periphery of the wheel.
outride the guide chambers the velocity of rotation will b©
than i\. In the outer chamber it is to he expected that the
rater will rotate as in a free vortex, or itB velocity of whirl will
be inversely proportional to the distance from the centre of the
TOtor, or will rotate in some manner approximating to this.
The liead which a pumpj with a vortes^ chambeTy tcill th^oreti^
llif rnaifUain when the discharge is zero. In this case it is
[)bable that as the discharge approaches zero, in addition to the
ktcfT in the wheel rotating, the water in the vortei chamber will
also rotate because of friction.
408 HYDRAUUCS
The centrifagal head dne to the water in the wheel is
If R= 2r, this becomes j ^ .
The centrifugal head due to the water in the chamber is,
Fig. 281,
f^wv^dr
Jr^ gn '
To and Vo being the radius and tangential velocity respectively of
any ring of water of thickness dr.
Fig. 281.
If it be assumed that t;oro is a constant, the centrifugal head
due to the vortex chamber is
g k n' 2g\Tj njJ'
The total centrifugal head is then
^'2g 2g^2g\rJ R«,V '
If rto is 2r and R^ is 2r«„
2g
The conditions here assumed, however, give K too high. In
Stanton's experiments he was only — ^ —  . Decouer from experi
^g
CENTRIFUGAL PUMPS 409
ments on a small pomp with a vortex chamber, the diameter being
l'3t' *
aboat twice the diameter of the wheel, found he to be rr^ .
Let it be assomed that K is ^ in any pump, and that the lift
of the pump when working normally is
7 e Vi Vi e W  Vit^ cot 4^)
/i, = — = .
Then if fe is greater than v^ \ the pump will not commence to
discharge unless speeded up to some velocity Va such that
mvj efa*t;ittiCot<^)
^g ^ g
After the discharge has been commenced, however, the speed
may be diminished, and the pump will continue to deliver against
the given head*.
For any given values of m and e the velocity fa at which delivery
commences decreases with the angle ^. If ^ is 90 or greater than
©0 degrees, and m is unity, the pump ynW only commence to
discharge against the normal head when the velocity is t'l, if the
manometric efficiency e is less than 0*5. If <^ is 30 degrees and m
is unity, v^ is equal to Vi when e is 0*6, but if <^ is 150 degrees rj
is equal to t?i when e is 0*428.
Nearly all actual pumps are run at such a speed that the
centrifugal head at that speed is greater than the actual head
against which the pump works, so that there is never any
difficulty in starting the pump. This is accounted for (1) by the
low manometric efficiencies of actual pumps, (2) by the angle <t>
never being greater than 90 degrees, and (3) by the wheels being
surrounded by casings which allow the centrifugal head to be
greater than ^.
It should be observed that it does not follow, because in many
cases the manometric efficiency is small, the actual efficiency of
the pump is of necessity low. (See Fig. 286.)
227. Headvelocity curve of a centrifugal pump at zero
discharge.
For any centrifugal pump a curve showing the head against
which it will start pumping at any given speed can easily be
determined as follows.
On the delivery pipe fit a pressure gauge, and at the top
* See pages 411 aud 410.
410
HYDRAULICS
of the suction pipe a vacuum gauge. Start the pomp with
the delivery valve closed, and observe the pressure on the two
gauges for various speeds of the pump. Let p be the absolute
pressure per sq. foot in the delivery pipe and pi the absolute
pressure per sq. foot at the top of the suction pii)ey then ^  ^
is the total centrifugal head K
teoo leoo 200Q 2200
RevoUttums per Minute.
Fig. 282.
240?
A curve may now be plotted similar to that shown in Fig.{282
which has been drawn from data obtained from the pump shown
in Fig. 275.
When the head is 44 feet, the speed at which delivery would
just start is 2000 revolutions per minute.
On reference to Fig. 293, which shows the discharge under
different heads at various speeds, the discharge at 2000 revolutions
per minute when the head is 44 feet is seen to be 12 cubic feet
per minute. This means, that if the pump is to discharge againrt
this head at this speed it cannot deliver less than 12 cubic to
per minute.
228. Variation of the discharge of a oentriftigal puiiV
with the head when the speed is kept constant*.
Headdischarge curve at constant velocity. If the speed of*
centrifugal pump is kept constant and the head varied, the dis
charge varies as shown in Figs. 283, 285, 289, and 292.
* See also page 418.
CENTRIFUGAL PUMPS
411
The curve No. 2, of Fig. 283, shows the variation of the head
with discharge for the pump shown in Fig. 275 when running at
1950 revolutions per minute; and that of Fig. 285 was plotted
from experimental data obtained by M. Bateau on a pump ha\nng
a wheel ll'S inches diameter.
The data for plotting the curve shown in Fig. 289* was
obtained from a large centrifugal pump having a spiral chamber.
In the case of the dotted curve the head is always less than the
centrifugal head when the flow is zero, and the discharge against
a given head has only one value.
RadilMod^ of How fronhWheA.
Fig. 283. Headdischarge curve for Centrifugal Pump. Velocity Constant.
Fig. 284. Velocitydischarge curve for Centrifugal Pump. Head Constant.
In Fig. 285 the discharge when the head is 80 feet may be
either '9 or 3*5 cubic feet per minute. The work required to drive
Che pamp will be however very different at the two discharges,
teid, as shown by the curves of efficiency, the actual efficiencies
for the two discharges are very different. At the given velocity
tlierefore and at 80 feet head, the flow is ambiguous and is
Unstable, and may suddenly change from one value to the other,
^>r it may actually cease, in which case the pump would not start
^gain without the velocity Vi being increased to 70*7 feet per
^lecond. This value is calculated from the equation
• Proeeedingi InsL Mech. Engs., 1903.
412
HYDRAULICS
the coefficient m for this pump being 1"02. For the flow to b
stable when delivering against a head of 80 feet, the pnmp shool
be run with a rim velocity greater than 70*7 feet x)er second, i
which case the discharge cannot be less than 4^ cubic feet pe
minate, as shown by the velocitydischarge curve of Fig. 2K
The method of determining this curve is discussed later.
Pump Wheel flScUanv.
Rew, per minute 1290.
Fig. 2S5.
1 Jl 3
DisduMT^ in, cfL per mJav.
Fig. 286.
Fig. 287.
Example, A oentrifngal pump, when disoharging normally, has a peripher
velocity of 50 feet per second.
Tbe angle ^ at exit is 30 degrees and the manometric effioienoy is 60 per ceo
The radial velocity of flow at exit is '^Jh.
Determine the lift h and the velocity of the wheel at which it will start delivei
nnder fall head.
S _
V=60(2VA)coBl30
501nJh.
CENTRIFUGAL PUMPS 413
9
from whieh A =87 feet.
Ijei Oj be the Telocity of the rim of the wheel at which pumping commences.
Then Mraming the centrifogal head, when there is no discharge, is
r,=48'6 ft. per sec.
229. Bemonilli's equations applied to centrifagal pumps.
Consider the motion of the water in any passage between two
consecutive vanes of a wheel. Let p be the pressure head at
inlet, pi at outlet and pa the atmospheric pressure per sq. foot.
If the wheel is at rest and the water passes through it in
the same way as it does when the wheel is in motion, and all
losses are neglected, and the wheel is supposed to be horizontal, by
Bemouilli's equations (see Figs. 277 and 278),
w 2g w 2g ^^^•
But since, due to the rotation, a centrifugal head
'•=1:1 <^)
is impressed on the water between inlet and outlet, therefore,
w 2g w 2g 2g 2g ^'*'''
**' w w 2g 2g* 2g 2g ^*^
From (3) by substitution as on page 337,
w^2g w^2g* g  g •••^^^.
and when U is radial and therefore equal to Uy
E. + U!«=£^^+Y..^i (6).
w 2g w 2g g ^^
If now the velocity Ui is diminished gradually and without
shock, so that the water leaves the delivery pipe with a velocity
U4, and if frictional losses be neglected, the height to which the
water can be lifted above the centre of the pump is, by Bernouilli's
equation,
w 2g w 2g ^'''•
If the centre of the wheel is K feet above the level of the water
in the sump or well, and the water in the well is at rest,
P^ = K^P^f (8).
w w 2g
414 HTDRAUUCS
Substituting from (7) and (8) in (6)
9 ^^
= H.+  = H (9).
This result verifies the fundamental equation given on page 888.
^ Further from equation (6)
Example. The centre of a centrifngal pmnp is 15 feei above Um level of tlit
water in the sump. The total lift U 60 feet and the velocity of dieehafge from flu
delivenr pipe is 5 feet per second. The angle at diachaige is 135 degrees, and
the radial velocity of flow through the wheel is 6 feet per second. AMnffiipg tfien
are no losses, find the pressure head at the inlet and oatlet cironmfiereiioes.
At inlet ^=34'16'^
tr 64
= 18*6 feet.
The radial velocity at outlet is
iii=5 feet per second,
and y^^.,'..u,.,cot 450^3^^
9 V 64' .
and therefore, ©i' + 5i7j = 1940 (1),
from which 17^ = 41*6 feet per second,
and V,=46r, „ „
Then l' = LLtii>^34feet.
The pressure head at outlet is then
tr IT 2g
= 45 feet.
To find the velocity v^ when <p is made 80 degrees.
cot ip=»JSt
therefore (1) becomes rj'  6 /^S . Vj = 1940,
from which V]=48*6 ft. per sec.
and V,=:40
Then 5l = 254 feet, and ?^=636 feet.
2*7 w
230. Losses in oentrifogal pumps.
The losses of head in a centrifugal pump are due to the same
causes as the losses in a turbine.
L088 of head at exit. The velocity Ui with which the water
leaves the wheel is, however, usually much larger than in the
case of the turbine, and as it is not an easy matter to diminisli
this velocity gradually, there is generally a much larger loss of
velocity head at exit from the wheel in the pump than in the
turbine.
CENTRIFUGAL PUMPS 415
In many of the earlier pumps, which had radial vanes at exit,
the whole of the velocity head ^ was lost, no special precautions
being taken to diminish it gradually and the efficiency was
constantly very low, being less than 40 per cent.
The effect of the angle 4^ on the efficiency of the pump. To
increase the efficiency Appold suggested that the blade should be
set back, the angle 4^ being thus less than 90 degrees. Fig. 272.
Theoretically, the effect on the efficiency can be seen by
considering the t^iree cases considered in section 220 and illustrated
in Fig. 279. When <^ is 90 degrees ^ is '543!, and when <^ is
30 degrees ^ is 'SdH. K, therefore, in these two cases this head
is lost, while the other losses remain constant, the efficiency in
the second case is 18 per cent, greater than in the first, and the
efficiencies cannot be greater than 46 per cent, and 64 per cent.
re8i)ectively.
In general when there is no precaution taken to utilise the
energy of motion at the outlet of the wheel, the theoretical lift is
^'"■7""2^ ^^^'
and the maximum possible manometric efficiency is
Sabstituting for Vi, i^i  tti cot <^, and for Uj', V,' + tt,',
TT V "l" »J.
^' = 2^2^^^'^^*'
, ^ . (vi u, cot <^)' + th^
2 W — ^\th cot <^)
_ V i^  Vg^ cosec' 4>
2Vi (Vi  Va cot <^) '
When r, is 30 feet per second, Ux 5 feet per second and <t>
150 degrees, e is 56 per cent, and when <^ is 90 degrees e is
48*5 per cent.
Experiments also show that in ordinary pumps for a given lift
and discharge the efficiency is greater the smaller the angle <t>.
Parsons* found that when i> was 90 degrees the efficiency of a
pamp in which the wheel was surrounded by a circular casing
was nearly 10 per cent, less than when the angle <t> was made
about 165 degrees.
• Proceedings Inst, C, E,, Vol. xLvn. p. 272.
416
HYBRA0LICS
Stanton found that a pump 7 inches diameter having niiial
vanes at discharge had an efficiency of S per cent, less than wto
the iiTigle ^ at delivery was 150 degrees. In the first case tht
maximum actual efficiency was only 39"6 per cent,, and in thi
second vnse 50 i>er cent*
It. han been suggested by Dr Stanton that a second reason fo
the greater efficiency of the pump having vanee curved hack ^
outlet is to be found in the fact that with these vanes the variatitm
of the relative velocity of the water and the wheel is less than
L when the vanes are radial at outlet. It has been shown erperi
l mentally that when the section of a stream is diverging, that is
the velocity is diminishing and the pressure increasing, there is
a tendency for the stream lines to flow backwards towards die
sections of least pressure. Tliese return stream lines cause a losi
of eiierg^'^ by eddy motions. Now in a pump, when the vanes am
radial, there is a greater difference between the relative velcunty
of the water and the vane at inlet and outlet than when the angb
L ^ is \em than fK) degrees (see Fig. 279), and it is prt^bahle tliere
i fore that there is more loss by eddy motions in the wheel in the
former case.
Liiss of head at entry. To avoid loss of head at entry the vanu
must bo pamnel to the relative velocity of the water and the
vane,
I Unless guide blades are provided the exact direction in whicl
pthe water approaches the edge of the vane is not knoivn. If there
were no friction between the water and the eye of the wheel it
would be expected that the stream lines, which in the suction piK
are parallel to tlie sides of the pipe, would be simply turned to
approach the vanes radially.
It has already been seen that when there is no flow tbe water
in the eye of the wheel is made to rotate by friction, and il ii
probable that at all flows the water has some rotation in the ep
of the wlieel, but as the delivery increases the velocity of rc»tati«i
probably diminishes. If the water has rotation in the swb©
direction as the wheel, the angle of the vane at inlet will c\mx}f
have to be larger for no shock than if the flow is radial. Thi*
the water has rotation before it strikes the vanes seems te he
indicated by the experiments of Mr Livens on a pumpj the Tanes
of which were nearly radial at the inlet edge. (See section 33^.)
Tlie efficiencies claimed for this pnmp are so high, that thm
conld have been very little loss at inlet*
If the pump has to work under variable conditions and the
water be assumed to enter the wheel at all discharges in the same
direction, the relative velocity of the water and the edge of the
CENTRIFUGAL PUMPS 417
»n only be parallel to the tip of the vane for one discharge,
; other discharges in order to make the water move along
me a sadden velocity must be impressed upon it, which
a loss of energy.
b t«s, Pig. 288, be the velocity with which the water enters a
and and v the inclination
jlocity of the tip of the vane i*  u^ >l
ft respectively.
e relative velocity of th and v
, the vector difference of u%
e radial component of flow
fh the opening of the wheel
be equal to the radial com ' j,. ^ •
t of Uij and therefore the
e velocity of the water along the tip of the vane is Vr.
Ui is assumed to be radial, a sudden velocity
u,  t?  tta cot
us to be given to the water.
ih has a component in the direction of rotation u» will be
Lshed.
has been shown (page 67), on certain assumptions, that if
Y of water changes its velocity from Va to Vd suddenly, the
ost is ^^^ — > or is the head due to the change of velocity.
this case the change of velocity is u«, and the head lost may
lably be taken as ^. K A; is assumed to be unity, the
ve work done on the water by the wheel is diminished by
u^_ (r  tta cot Oy
now this loss takes place in addition to the velocity head
lost outside the wheel, and friction losses are neglected,
jL_Vi^i tTi' (t?^,cot^)'
= L',L'eo8ec'» ^^"^"^^^)'
2g 2g ^ 2g
= —  ^ a cosec' 4^ 
(.gcot^y
2g 2^^^^^^^ ^ 2g
v^ u* u,'  . 2t?«, c<
= o — oT " o~ cosec' 4> + — s~"
2g 2g 2g 2g
H. Vk
r,' u* u,' ,..2t?«,cot^ a .a>, ,.v
418
HTDRJl0LtCS
Es^Mmpte. The rftdi&l Tdo«itj of flow thfoogh b pump w S f«« per
The u^a ^ ii SO degriec and the ADgle i if 15 de^ltem.. Tbe reiodt?
oqut cireiunfiertoae U 50 f««t per »ee. mnd the mdliu is twic^ ihas of tae
Find Ikw tbtonticy lift on the u«&m>tiofi ih&l the wtM»l« of Ihe kjnme mag
U lo«t &t ezii*
= S7*5 feet
The theorvtiefti lid &«glecticg aJl Io«
^f&cienej U Iberefore 68 ier cent.
231. Variation of the head
sp«ed of a centriftigal pump.
It is of intereet to study by »
the variation of the dischargic
wbeo fc ta constant, and th(
discharge when the relocitj
coiui>are the results with
experiment.
The full curve of Fig. 2
with the discharge when the
Tim data for which the cm.
the figure.
Ml ia 64*9 foet, u^ the naammm
with discharge and witb iM
^ of equation (])^ sed^j
th the velocity of the
ition of the head wiUi
e \mmp {» conetant, hid ^
tofll results obtained kam
i the ranatioBs of the hai
of a wheel is kept fionstot
been plotted is indicated to
Normal rajdtuML^tlacify^fSlaiviS'
\t \2 _ 3 ^ \5 ^'^% b
Fi^. 289, Head'diflcharge carve at constant Telocitj.
WHion the discharge is zero
2g 2g
10^5 feet.
Tlie velocity of flow f at outlet haa been assumed equal h^
2 at inlet.
Values of 1, 2, etc. were given to ^ and the correspondiji^
valuiiis of /* found from equation (1).
CENTRIFUGAL PUMPS
419
WTieii the discharge is normal, that is, the water enters the
thoat shock, ^ is 4 feet and ^ is 14 feet. The theoretical
ing no losses is then 28 feet and the manometrio
cy is thus SO per cent. For leas or greater values of ^
^e^atl diminishes and also the efficiency*
riie ciir%*e of Fig. 290 shows how the flow varies with the
city for a constant value of h^ which is taken as 12 feet.
if 2W1.
Rjadinl Vttod^ thrmtph WfieeL
VdocitjHiiicharg© curve at cans taut head for Ceatrifugal Pump.
It wU be seen that when the velocity Vi is 31*9 feet per second
& velocity of discharge may be either zero or 8*2 feet per second,
1^ meatm that if the head is 12 feet, the pnmp^ theoretically,
ill only start when the velocity is Zl'Q feet per second and the
Aocity of discharge will suddenly become 8"2 feet per second,
' iMJw the velocity v^ is diminished the pump still continues to
bdmrge^ and mil do so as long as i?i is greater than 26*4 feet per
Nx»tid. The flow is however tmstable, as at any velocity Vc it may
llddenly change from CE to CD, or it may suddenly cease, and it
BU not start again until Vj is increased to Sl'O feet per second.
232, The effect of the variation of the centrifugal besul
Hd the loss by firiction on the discharge of a piunp*
If then the lo.s.ses at. inlet and outlet were as above and were
be only lusses^ and the centrifugal head in an actual pump was
^ual to the theoretical centrifugal head, the pump could not be
tede to deliver water against the normal head at a small velocity
I discharge. In the case of the pump considered in section 281,
^■btd not safely be nm with a rim velocity less than 31*9 ft,
PPKec., and at any greater velocity the radial velocity of flow
[>iild not be less than 8 feet per second.
490 HTDRAULIGS
In mctoal pumpfi, however, it lias been seen that the oentrifagal
head at oommenoement is greater than
2^ 2sr
lliere is also loss of head, which at high velocities and in smsH
pumps is considerable, dne to friction. These two causes consider
ably modify the headdischarge curve at constant velocity and the
Tidocitydischarge curve at constant head, and the centrifugal
head at the normal speed of the pump when the discharge is zn%
is generally greater than any head under which the pump works,
and many actual pumps can deliver variable quantities of wator
against the head for which they are designed.
The centrifugal head when the flow is zero is
m being generally equal to, or greater than unity. As the flow
increases, the velocity of whirl in the eye of the wheel and in
the casing will climinish and the centrifugal head will there&nre
Let it be assumed that when the velocity of flow is u (supposed
constant) the centrifugal head is
k and n being constants which must be determined by experiment
When II is zero
and if «• is known Iq can at once be found.
Let it further be assumed that the loss by friction* and ed
C*tt*
melons, apart from the loss at inlet and outlet is ^ .
* The loss of besd by frieiioii will no doubt depend not only upon « bat slM
apon the Tek>eity r^ of the wheel, and should be written as
^ + ^+etc..
or, as ^"^^'*' ••®*^
If it be snppoeed it ean be expressed by the latter, then the oonvetion
if proper Tallies are giTen to &, n^ and Jk, takes into aoooont the Yariatioo of tki
OsntrifUgal head and also the firietion head «,.
Lz:
\
CENTRIFUGAL PUMPS
421
The gross bead h is then,
Vi
v' t^
'' = 2^2^2^°°^'*^
2vu cot S
w'cof^
2ff
(kvi — wit)* (^V?
2ff 23
.(2).
^(«*'>
If BOW the head Ai and flow Q be detepmined experimentally,
the difference between h as determined from equation (1), page 4 J 7,
atiid the experimental value of A, muat be equal to
2g 2^~W^ 'W
~ 2g 2g 2^ '
Iti bein^ equal to (c* — n*).
The coefficient fe being known from an experiment when u is
sero, two other exx)eriment9 giving corresponding values of k and
i» will determine the coefficients n and fcj*
The headdischarge curve at constant velocity, for a pump such
as the one already considered, would approximate to the dotted
Corve of Fig, 289. This curve has been plotted from equation (2),
by taking k as 0*5, n as 7*64 and h% as — 38.
Substituting \m)ue8 for fc, n, fti, cosec ^ and cot ^j equation (2)
beoomea
. * = ^*%H^H.C«' (3),
2ff ■ 2(7
■ and Ct being new coefficients ; or it may be written
».5^.^.C.Q. W,
Q being the flow in any desired units, the coefficients C^ and Ct
'varying with the units. If equation (4) is of the correct form,
three experiments will determine the constants m, Ca and Ci
din^ctly, and having given values to any two of the three
^variables h, \\ and Q the third can be found,
233. The effect of the diminution of the centrifugal head
and the inoreaa^ of the friction head as the flow increaaes, on
%he velocity discharge curve at constant head*
Using the corrected equation (2), section 232, and the given
*ir&laes of ^5 W and h the dotted curve of Fig. 200 has been plotted.
Frtjra the dotted curve of Fig> 289 it is seen that u cannot
Im.* greati^r than 5 feet when the head is 12 feet, and therefor© the
Hew curve of Fig. 290 is only drawn to the point where tt is 5.
ITie panip starts delivering when v is 27'7 feet per second and
the discharge increases gradually as the velocity increases.
MUM
422 HTDRAUUGS
The pump wfll deliver, therefore, water under a head of
12 feet at any velocity of flow from zero to 5 feet per secoBd.
In sach a pomp the manometric efficiency must have its
mayiTnnTn valae when the discharge is zero and it cannot be
greater than
w
fi* — txUx cot '
9
This is the case with many existing pomps and it explains wliy,
when running at constant speed, they can be made to give any
discharge varying from zero to a maximum, as the head is
diminished.
234. Special arrangements for converting the velocity
TP
head ^ with which the water leaves the wheel into preasoro
head.
The methods for converting the velocity head with which tlie
water leaves the wheel into pressure head have been indicated on
page 3W. They are now discussed in greater detail.
TfcowwowV vortex or tthirlpool chamber. Professor James
Thomson first suggested that the wheel should be surroimded by
a chamber in which the velocity of the water should gradoftlly
change from Ui to u^ the velocity of flow in the pipe. Such a
chamber is shown in Fig. 274. In this chamber the water forms
a frve vortex, so called because no impulse is given to the water
while moving in the chamber.
Any fluid particle ai, Fig. 281, may be considered as moving
in a circle of radius r# with a velocity ro and to have also a
radial velocity u outwards.
Let it be supposed the chamber is horizontal.
If W is the weight of the element in pounds, its momentum
Wr
perpendicular to the radius is ^ and the moment of mo
Wr r
meutum or angular momentum about the centre C is — —,
9
For the momentum of a body to change, a force must act upon
it> and for the moment of momentum to change, a couple must act
up^^n the body.
But since no turning effort, or couple, acts upon the element
after leaWng the wheel its moment of momentum must be
constant.
Therefore,
CENTRIFUGAL PUMPS 423
WVqTo
9
is constant or v^Vo = constant.
If the sides of the chamber are parallel the peripheral area of
the concentric rings is proportional to r©, and the radial velocity of
flow t* for any ring will be inversely proportional to Tq, and there
fore, the ratio — is constant, or the direction of motion of any
Vq
element with its radias Vq is constant, and the stream lines are
eqaiAngnlar spirals.
If no energy is lost, by friction and eddies, Bernouilli's theorem
iwill hold, and, therefore, when the chamber is horizontal
2g 2g w
is constant for the stream lines.
This is a general property of the free vortex.
If 1^ is constant
PT" + = constant.
2g w
Let the outer radius of the whirlpool chamber be Ru, and
the inner radius r^. Let Vr^ and Vr^ be the whirling velocities
at the inner and outer radii respectively.
Then since VqTo is a constant,
and ^' "^ 5!; = constant.
w 2g
w w 2g 2g
w 2g V njJ'
WTien R«, = 2r«.,
w w 4' 2<7 *
If the velocity head which the water possesses when it leaves
the vortex chamber is supposed to be lost, and i^i is the head of
crater ahove the pump and pa the atmospheric pressure, then
neglecting friction
or
w
h,+
Va
W '
hr
w
23
w
If duoL i, » die be^s of die psnp above the well, the total
Tlttrcfore
'^ w w 2g 2gV B.V 2g •
fiat from eqamdaa (8) paige 419,
Tberrfore
***2g g 2g"2i^V K.V
This mi^t hare been wntten down at once from equation (I),
section 230. For clearly if there is a gain of pressore head
in the Tortex chamber of ^ ( 1  ^"i), the velocity head to
be lost win be leas by this amount than when there is no Yortex
chamber.
Snbetitnting for Vi and Hi the theoretical lift h is now
, r,*r,ttiCot* tfi' (rit*,cot*)' R* r^^. m
*= 9 ^ ^— R^' ^^^ J
When the discharge or rim velocity is not normal, there is »
farther loss of head at entrance equal to
9
and
g 2gA^* 2g R.«
When there is no discharge Vr» is equal to Vi and
"' g 2gR«» 2g
.(2).
\
CENTRIFUGAL PUMPS 425
If R = iR«, and i? = ivi,
Correcting equation (1) in order to allow for the variation of
centrifugal head with the discharge, and the friction losses,
»i'  ViUi cot ^ tti' (t'l  Ml cot ^)*R*
h =
2
g 2g 2gBJ
(v — u cot Oy Vv^ ^ 2nkwvi _ kiu^
2^ ^ 2g 2g 2g '
^hich reduces to h=^^'%^^^.
2g 2g 2g
The experimental data on the value of the vortex chamber
^09r «c, in increasing the efficiency is very limited.
Stanton* showed that for a pump having a rotor 7 inches
^diameter surrounded by a parallel sided vortex chamber 18 inches
^Uameter, the efficiency of the chamber in converting velocity head
th> pressure head was about 40 per cent. It is however questionable
"whetiher the design of the pump was such as to give the best results
IKWsible.
So far as the author is aware, centrifugal pumps with vortex
chambers are not now being manufactured, but it seems very
probable that by the addition of a welldesigned chamber small
oentrifngal pumps might have their efficiencies considerably in
creased.
235. Turbine pomps.
Another method, first suggested by Professor Reynolds, and
now largely used, for diminishing the velocity of discharge Ui
gradually, is to discharge the water from the wheel into guide
passages the sectional area of which should gradually increase
from the wheel outwards. Figs. 275 and 276, and the tangents to the
tips of the guide blades should be made parallel to the direction
ofU,.
The number of guide passages in small pumps is generally four
or five.
If the guide blades are fixed as in Fig. 275, the direction of
the tips can only be correct for one discharge of the pump,
but except for large pumps, the very large increase in initial cost
of the pump, if adjustable guide blades were used, as well as
the mechaniical difficulties, would militate against their adoption.
Single wheel pumps of this type can be used up to a head of
100 feet wit i excellent results, efficiencies as high as 85 per cent.
* Proceedingi Irut. C.E., 1903.
/
4dl HnMULULiCS
mvaiff l«e& du^cd. They mre now bem^ used to deliver water
MKoiiS kaBd§ cf o'VBr dSO feet, mud M. Bateaa lias used a single
^ntf^ ^1^ mdtoK diameter Tmming at 18,000 revelations per
snuffle i£> detivvr against a Iiead of 936 feet.
iMm ^ Mai mi tke emiramee to ike guide passages. If the
i» aie fixed, tiie directicHi of the tips can only be correct
c£ xhe pomp. For any other discharge than the
^.^rsBhL i^ dxnKCiGQ cf the water as it leaves the wheel is not
piKraljei <o ik^ fixed grnde and there is a loss of head due to
Ijb% a lie tk^ indiBatioci of the gnide blade and ^ the vane
aaiic^ as ^xii.
tiS Ik fce Ae radsal velocity of ^ — 3P^r j
*:w. T^KL RE. Fig. 291, is the ^^^^^^^^^'"^ ^N. '^
Twxisy wA which the watm^ leaves o^ ^ ^ ^^ ^
Xii^ radial i^efcciiy with which
t^ wastnr e&ters^ the guide passages most be «i and the velocity
adco^ ^le jirside isv therefore, BF.
TVre is a swiden change of velocity from BE to BF, and on
die iBSSit^spdoii that the Ices of head is eqnal to the head due to the
TwaitTv Tvicvity FE. the head lost is
t T:  Ml cot4~ Hi cot g)^
A: ^rijec tfee kxsjs^ of head is
irncot^)*
*r5d rise rbev^r^^xal Hft is
^9
r^*  r\» t c\>t ^ _ (^r ~ « cot BY _ (ri  Ui cot ^ ~ tii cot a)'
r^* r* ir^nicota 2nico t^
^~2g
Ux^ (cot * ^ cot o)' tt* cot*
ij? 2g 2g
2g 2^ (1)
Tv> ov^r^^t for the dimination of the centrifugal head and to
allow fi.>r friciion^
iV_2tr,H.ti, , u^
2g 2g ^^ 2g'
mu$t be addeiU and the lift is then
* ^ «\* _ r* ^ 2tM*» cot a 2ri« cotf _ V (cot ^ ■^ cot a)*
ig ' 2g^ 2g ^ 2g 2g
t^cot^^ tV 2fatriUi fc«i'
2g '^ 2g 2g 2g '
CENTRIFUGAL PUMPS
427
which, since u can always be written as a multiple of Ui, reduces
to the form
2gh = mvi^ ^ CuiVii^ CiUi'' (2).
Equations for the turbine pum/p shown in Fig. 275. Character
istic cwrvea. Taking the data
tf = 5 degrees, cot tf = 1143
<^ = 30 „ cot*= 1732
o» 3 „ cot a =196
D = 25d
equation (2) above becomes
2^A = '84»i» + 483i^t7i587t^* (3).
piat^UMrge^ igv Cubic F^ perlHnu^
O 12 8 4.^
Sndial f VAoeUy aJbBadJb ffxnv Ovt WheA. Te9t par SmcorvdU
Fig. 292. Headdisoharge oarres at constant speed for Turbine Pump.
From equation (3) taking r, as 50 feet per second, the head
discharge curve No. 1, of Fig. 283, has been drawn, and taking h
as 35 feet, the velocitydischarge curve No. 1, of Fig. 284, has been
plotted.
In Figs. 292 — 4 are shown a as^'ies of headdischarge curves at
428
HTDRAUUGS
constant speed, velocitydischarge curves at constant head, and
headvelocity curves at constant discharge, respectively.
The points shown near to the curves were determined experi
mentally, and the curves, it will be seen, are practically the mean
curves drawn through the experimental points. They were how
ever plotted in all cases from the equation
2gh = r087t?i» + 2'26uiVi  62lu,*,
obtained by substituting for m, C and Ci in equation (2) the values
r087, 2*26 and 62*1 respectively. The value of m was obtuned
by determining the head A, when the stop valve was closed, for
speeds between 1500 and 2500 revolutions per minute. Fig. 282.
The values of C and Ci were first obtained, approximately, by
taking two values of v^ and Vi respectively from one of the
actual velocitydischarge curves near the middle of the series, for
which h was known, and from the two quadratic equations thus
obtained C and Ci were calculated. By trial C and Ci were then
corrected to make the equation more nearly fit the remaining
curves.
SpmecL —KeyolBUions per liUnuU^.
Fig. 293. VelocityDisoharge corYes at Constant Head.
No attempt has been made to draw the actual mean curves in
the figures, as in most cases the difPerence between them and the
calculated curves drawn, could hardly be distinguished. The
reader can observe for himself what discrepancies there are between
the mean curves through the points and the calculated curves. It
CENTRIFUGAL PUMPS
429
will be seen that for a very wide range of speed, head, and
discharge, the agreement between the curves and the observed
points is very close, and the equation can therefore be used with
confidence for this particular pump to determine its performance
under stated conditions.
It is interesting to note, that the experiments clearly indicated
the unstable condition of the discharge when the head was kept
constant and the velocity was diminished below that at which the
discharge commenced.
Fig. 294. Headvelocity carves at Constant Discharge.
236. Losses in the spiral casings of centrifugal pumps.
The spiral case allows the mean velocity of flow toward the
discharge pipe to be fairly constant and the results of experiment
seem to show that a large percentage of the velocity of the water
at the outlet of the wheel is converted into pressure head.
Mr Livens* obtained, for a pump having a wheel 19^ inches
diameter running at 550 revolutions per minute, an efficiency of
71 per cent, when delivering 1600 gallons per minute against a
head of 25 feet. The angle ^ was about 13 degrees and the mean
of the angle for the two sides of the vane 81 degrees.
For a similar pump 21f inches diameter an efficiency of 82 per
cent, was claimed.
* Proeeedingi Inst. Mech, Engs.^ 1903.
480
HTDRAULICS
Tho author finds th© equation to the headdischarge curreiar
the 19i inches diameter purop from Mr Livens' data to be
and for thiB 2U inckm diameter pump
l'lSvi*iMiVi = 2gh _. ,_®.
The velocitf of rotation of the water round the wheel will h
less than the velocity with wliich the i^ater leaves the wheel sni
there will be a losa of head due to the j§udden change in velodty*
/!='
Let this losa of head be v*
radial velocity of flow at eri
wheel nidially, is then
trj't^iUiCot^
9
Taking friction and the
account,
n 5= ~ ^ — 
9 2g
which again may hr written
I. TTt
^. Tlie head, when u, Utb
aasmniBg the water eaten tti^
tjoo of oentrifagal bed its
2g 2^'
h =
^g
Cuin Cifii'
% ^9
The vHluei> of ;«, C and Ci are given for two pumps in eqva&di
(1) uml VI).
237. General equation for a centrifugal pump.
The CH^ nations for the gross head ft at discharge Q as detennmfti
for the seveiTil classes of pumps have been shown to beof thefonn
'^ 2ff ^ 2g ^ 2g '
or, if a is the velocity of How from the wheel,
, mv^ Quv Ci?i*
'^9
2^
2g
in which m varies t jet ween 1 and To. The coefficients Ciandu
for any pump will depend upon the unit of discharge.
As a further example and illuj^trating the case in whieli ^
certain speeds the How may lie unstable, the curves of Flf^
285 — 2J^7 may be now considered. When Vi is 60 feet per saw
the equation to the head discharge curve is
Q being in cubic feet per minute.
CENTRIFUGAL PUMPS
431
The velocitydischarge cnrve for a constant head of 80 feet as
colated fmiu tluM equation is shown in Fig* 287.
Tu stiirt the pump against a head of 80 feet the periphf?ral
jdciiy has to be 707 feet per second, at which velocity the
^har^* Q suddenly rises to 4 "3 cubic feet per minute.
The curves of actual and nianonietric efficiency are shown in
b. 286, the maximum for the two cases occurring at different
leharges.
338 Tbe Limiting height to which a single wheel centri
Bil pump can he used to raise water.
The maximum height to which a centrifugal pump can raise
Iter, depends theoretically upon the niaxiinam %'elocity at which
* lim of thc^ wheel can be run.
It has already been stated that rim velocities up to 250 feet
•rsetoiiil IjavL* been used. Assuming radial vanes and a mano
Btric efficiency of 50 per cent., a pump running at this velocity
lift against a head of 9HQ feet,
t t}w^' very high velocities, however, the wheel must be of
mnterial such as brauKe or cast steel, having considerable
nee t^ tensile Rtresses, and ftpecial precautions must be
to balance the wheeL The hydraulic losses are also
derable, and manometric efficiencies greater than 50 per
hardly to be expected.
ding to M. Rateai!*, the limiting head against which it is
We to raise water by means of a single wheel is about
feet, and the maximum demrable velocity of the rim of the
i is about 100 feet per second.
ngle wheel pumps to lift up to 3oO feet are however being
At tliis velocity the stress in a hoop due to centrifugal forces
mt 7250 lbs. per sq. incht.
139. The sxictioii of a centrifugal pump.
The greatest height through which a centrifugal or other class
amp will draw water is about 27 feet. Special precaution has
I taken to ensure that all joints on the suction pipe are perfectly
^htj and especially is tliis so when the suction head is greater
15 feet; iJTify under hipecial circumstances is it therefore de*
for the suction head to be greater than this amount^ and it
^ advisable to keep the suction head as small as possible^
Centriftiaes,*' ©tc., Itnttetin d* la SociiU de I'Tjidtittrie minfmtfj
r. p. aafi, Mnrch, 1902.
witijis Sirrnfrih of SJatermlt; WooJ's Strength o/ Strttffural Membert;
Turbine^ Stodola.
CENTBIFUOAI. PUMPS
433
lO. Series or multiBtage turbine pomps,
has l>eeii stated that the limiting economical head for a single
L purap is about 100 feet, and for high heads series pumps
low generally used.
^f^^nss^
^i3i» 296. General Arrangement of Worthington Multistage Turbine Pump.
By putting several wheels or rotors in series on one shaft, each
>tor giving a head varying from 100 to 200 feet, water can be
^ed to practically any height, and such pumps have been
U H. "1^
HTOEAt?t.ICS
coDJ^tTuct^ to work
head of 2O0O feet. The
I
to work against a beail or W^M3 feet. iHe ntiiDD
of rocorsi oo one eimft* m^y be &QfEi one to twelve acconliisi
lo II10 total liead* For a given hefid, tlia greater the number 1
rtilors used, the leas the peripheral velocity^, aod within
hitiitB the greater the e&CMmej^
FlgB^ 295 and 296 show a longitndixial section and i
re gpe ct i%'e ly, of a seried, or muldstagt^ pump,
hf the Worthkigtoii Pnnip Conipaiij On the mi*
shaft are fixed three pboephor^brcmze rotors^ alternating with I
giudeSi which are Tigidl oonnected to the outer casings iind I
the bearings. The water is drawn in thrL>ugb the pipe at tb )
of the pisp and enters the fir«t wheel axialty. The watar lea^
the first wheel at the outer circmnlerence and paj^ises aloti^ 1
expanding pa^ssagB in which the velocttj' is gradually dtndtiisb
at^ enters the second wheel axially. The vane« in the
are of hard phosphorbronze made very smooth to rtnluce fricti^
losses to a minimum. The water parses through the remaintl
rotors and guides in a ^miJar manner and is finally discharf
into the caaiiig and thence into the delivery pipe.
Fig. f97. Siilx«r MaltisUge Turbine Ptiiiip.
The difference in pressure head at the entrances to any tw*>
consecutive wheels is the head impressed on the water by *J»«?
wheel. If the head is h feetj and there are n wheels tl^
lift is nearly nh feet. The vanes of each wheel and the dirt
the guide vanes are detennine<l as ejc plained for the mg\s
*Sf> that losses by shock are redxiced 10 a miniuiiitiif mi
's and gnide passages are made smooth so as to reduce
the back of each wheel, just above the boes^ *^
' holes which allow water to get behind part of tlw
*he pressure at which it enters the wheel, to balauct?
which would otherwise be set up.
CENTRIFUGAL PUMPS 435
The pomps can be arranged to work either vertically or
rizontally, and to be driven by belt, or directly by any form
motor.
Fig. 297 shows a multistage pump as made by Messrs Sulzer.
e rotors are arranged so that the water enters alternately
m the left and right and the end thrust is thus balanced.
Sciencies as high as 84 per cent, have been claimed for multi
•ge pumps lifting against heads of 1200 feet and upwards.
The Worthington Pump Company state that the efficiency
DiBishes as the ratio of the head to the quantity increases, the
it results being obtained when the number of gallons raised
r minute is about equal to the total head.
ExampU. A pamp is to be driven by a motor at 1450 revolutions per minute, and
•quired to lift 45 oubio feet of water per minute against a head of 320 feet.
i&iad the diameter of the suction, and delivery pipes, and the diameter and
r of the rotors, assuming a velocity of 5*5 feet per second in the suction and
^ I^pes, and a manometrio efficiency at the given delivery of 50 per cent.
one provisionally that tbe diameter of the boss of the wheel is 8 inches.
Lei d be the external diameter of the annular opening. Fig. 295.
144 "" 60 X 55 *
B which d=:6 inches nearly.
Fakiog the external diameter D of the wheel as 2d, D is 1 foot.
^ 1450 _^ ,
Phen, Vi = g^ X r = 76 feet per sec.
(kflBoming radial blades at outlet the head lifted by each wheel is
76^
A=0.5.ifeet
= 90 feet.
Poor wheels would therefore be required.
24L Advantages of centrifugal pumps.
There are several advantages possessed by centrifugal pumps.
In the first place, as there are no sliding parts, such as occur in
iprocating pumps, dirty water and even water containing coni
•atively large floating bodies can be pumped without greatlj
iangering the pump.
Another advantage is that as delivery from the wheel is
istant, there is no fluctuation of speed of the water in tlu»
tkni or delivery pipes, and consequently there is no necessity
air vessels such as are required on the suction and delivery
es of reciprocating pumps. There is also considerably less
iger of large stress being engendered in the pipe lines by
ater hammer*."
Another advantage is the impossibility of the pressure in tlu^
* See page 384.
436
HYDRAUMCS
in the pipe line it will gfatt
[n th€ second case a simikr
^heii the water falls below a
hough convenient 15 unooo
Oiing no effective work, ihs
%y be more than 50 per ceol
ing masimttm discharge.
ip may be made to deUwr
which water may be takai
pump casing riBing above that of the majdmnm head wiiici xh
rotor is capable of impressing npon the w^ter. If the dt'liirry
is closed the wheel vn\l rotate witliout any danger of the pfeasuf
in the casing becoraing greater than the centrifugal hmd (pa^
335). This may be of use in those cases where a pump is de
livering into a reservoir or pumping from a reaervoin hi tie iit
case a float valve may be fitted, which, when the Tnnter ri^ to
a particular height in the reservoir, closei^ the dehverr. Tk
pump wheel will continue to rotate but without delivering wat^r,
and if the wheel is running at s ^ a velocity that the ct'iitn
fugal head is greater than the hei
delivery when the valve is opened
valve may be used to stop the flo
certain level. This arrangement
nomical, as although the pump j
power required to drive the pumj.
of that required when the pump is
It follows that a centrifugal
water into a closed pipe system fr
regularly, or at intervals, while the punip continues to rotate at a
constant velocity.
Pump delivering into a long pipe lin€. When a centrifagiil
pump or air fun is delivering into a lung pipe line the resistances
will vary approximately as the square of the quantity of water
delivered by the pump*
Let p2 be the absolute pressure per square inch which h^
to be maintaimni at the end of the pipe line, and let the
resistances vary ns the square of the velocity r along the pip?.
Tlien if the re?iij*tances are equivalent to a head h/^ frr", tk
pressure head ] at the pump end tif the delivery pipe must be
w w
A being the sectional area of the pipe.
Let  be the pressure head at the top of the suction pipe, then
the gross lift of the pump is
u* w w A w
If, therefore, a curve, Fig, 298, be plotted having
CENTRIFUGAL PUMPS
437
a.s ordinates, and Q as abscissae, it will be a parabola. If on
"fclie same figure a cnrve having h as ordinates and Q as abscissae
'be drawn for any given speed, the intersection of these two
enures at the point P will give the maximum discharge the pump
*will deliver along the pipe at the given speed.
JKdfeharffe uv C.Ft,per SecondU
Fig. 29S.
242. Parallel flow turbine pump.
By reversing the parallel flow turbine a pump is obtained
which is similar in some respects to the centrifugal pump, but
differs from it in an essential feature, that no head is impressed on
the water by centrifugal forces between inlet and outlet. It
therefore cannot be called a centrifugal pump.
The vanes of such a pump might be arranged as in Fig. 299,
the triangles of velocities for inlet and outlet being as shown.
The discharge may be allowed to take place into guide
passages above or below the wheel, where the velocity can be
gradually reduced.
Since there is no centrifugal head impressed on the water
between inlet and outlet, Bemouilli's equation is
tr 2gr K? 2gr *
From which, as in the centrifugal pump,
XT Vifi pi J) . U* u^
g w w 2g 2g
If the wheel has parallel sides as in Fig. 299, the axial velocity
of flow will be constant and if the angles <^ and B are properly
chosen, Vr and Vr may be equal, in which case the pressure at
inlet and outlet of the wheel will be equal. This would have
the advantage of stopping the tendency for leakage through the
clearance between the wheel and casing.
438
UYDRAULICS
Such a pump is airnilar to a reversed impulBe tarbine, tk
guide passages tif which are kept full. The velocity witii wM
tlie water leavtB the wheel wuuld however be great and theB
above the pamp would depend upon the percentage of the Vt4tjciiy
liead that could be converted into preissure head.
Since there is no centrifugal d impresgsed upon tie watw,
the parallelflow pump cannot ci vence discharging unless ik
water in the pinup is first set in tion by some external ra^i^s
but as soon as the flow ia coroine«*.,d through the wheel, tbeM
discliarge under full head can be obtained.
Fij?. 300.
To commence the discharge, the pump would generally have to
be placed below the level of the water to be lifted, an auxilian*
discliarge pipe being fitted with a discharging valve, and a non
return valve in the discliarge pipe, arranged as in Fig. 300.
CENTRIFUGAL PUMPS 439
3 pamp could be started when placed at a height ho above
,ter in the sump, by using an ejector or air pump to exhaust
r from the discharge chamber, and thus start the flow
:h the wheel.
3. Inward flow turbine pump.
:e the parallel flow pump, an inward flow pump if constructed
not start pumping unless the water in the wheel were first
motion. If the wheel is started with the water at rest
ntrifugal head will tend to cause the flow to take place
•ds, but if flow can be commenced and the vanes are
ly designed, the wheel can be made to deliver water at its
periphery. As in the centrifugal and parallel flow pumps,
water enters the wheel radially, the total lift is
H = Yi^'=2>_£+U'_^ (1).
g w w 2g 2g ^ ^
>m the equation
£. Xl 'Pi Vl. a^ ^Vl
w'^ 2g~w '^2g'^2g 2g'
be seen that unless Vr* is greater than
2g 2sr 2g^
less than p, and ^ will then be greater than the total
ry special precautions must therefore be made to diminish
locity U gradually, or otherwise the efficiency of the pump
) very low.
3 centrifugal head can be made small by making the
nee of the inner and outer radii small.
2g^2g 2g
le equal to ^ , the pressure at inlet and outlet will be the
and if the wheel passages are carefully designed, the
re throughout the wheel may be kept constant, and the
becomes practically an impulse pump.
3re seems no advantage to be obtained by using either
llel flow pump or inward flow pump in place of the centri
pump, and as already suggested there are distinct dis
.ages.
I. Reoiprooating pumps.
simple form of reciprocating force pump is shown dia
latically in Fig. 301. It consists of a plunger P working in
440
Fig. sola. Vertical Singleacting Reciprocating Pump.
RECIPROCATING PUMPS 441
I cylinder C and has two valves Vs and Vd, known as the suction
ind delivery valves respectively. A section of an actual pump
8 shown in Pig. 301a.
Assume for simplicity the pump to be horizontal, with the
jentre of the barrel at a distance h from the level of the water
n the well; h may be negative or positive according as the
>ump is above or below the surface of the water in the well.
Let B be the height of the barometer in inches of mercury,
rhe equivalent head H, in feet of water, is
H.l?:f^ = 1133B,
which may be called the barometric height in feet of water.
When B is 30 inches H is 34 feet.
When the plunger is at rest, the valve V© is closed by the head
)f water above it, and the water in the suction pipe is sustained by
.he atmospheric pressure.
Let ha be the pressure head in the cylinder, then
At) = H — fe,
)r the pressure in pounds per square inch in the cylinder is
p = 43(Hfe),
cannot become less than the vapour tension of the water. At
)rdinary temperatures this is nearly zero, and hn cannot be greater
han 34 feet.
If now the plunger is moved outwards, very slowly, and there
s no air leakage the valve Vg opens, and the atmospheric pressure
^uses water to rise up the suction pipe and into the cylinder,
lo remaining practically constant.
On the motion of the plunger being reversed, the valve Vs
rloses, and the water is forced through Vd into the delivery
ripe.
In actual pumps if ho is less than from 4 to 9 feet the
lissolved gases that are in the water are liberated, and it is there
ore practically impossible to raise water more than from 25 to
W) feet.
Let A be the area of the plunger in square inches and L the
troke in feet. The pressure on the end of the plunger outside the
cylinder is equal to the atmospheric pressure, and neglecting
he friction between the plunger and the cylinder, the force neces
sary to move the plunger is
P = 43 {H  (H  A) } A = 43^ . A lbs.,
ind the work done by the plunger per stroke is
E = 43^ . A . L ft. lbs.
pet ffmte ti Ik )fain
• "ttAL Hm^ ud tb
■ft i — ,^^" , i.^cAiiL i fci^
^ vliefc dip waMir larii tk d^mf
K bf ilui pliiflftr dnna^tk
■ Ai ■ ■» «iiiv« tki«fc«t * * JS * ^ fool poooflipfrfKmid.
iv « ^firi bkmm m Urn pn^ md Am bmiitd Md
triwi. mi ikr vehae ^ vuer UM per ai^tioii ftn^b rH
WmmbM It ligWr ha tin ^nshuw moml tbi^ b; ^
1^ ' tr tk viifk U wuer lifted per minate, and K tiF
[ viki tkr wilpr is lifted,
^bjtlirpQiiipk W.^ foot poondipff
HP=
siooo*
i
j^T  _^.* t  "icbufe of thfi pomp. Slip.
^ ^p i & p)tingier pump is the rite I
' ^u. 4 It ^ l^^i^ P^ £tn)ke multiplied by the nmahsifi 
^ *^m «f ^ fcibigr pir stroke to ihe rolnme digpli<^lffl
:,xS'jct hiS^Km d^^ qvBiinei ti called the SKf.
^^i^' tM ««»I &cfcirgr H l» tbi tie tb^retical th0 (
^ la a ««dr ^^^rkiiig pm
4fai;«]a be le» than five r
Tbe c»a«* «>f »««*»"^
^^kg^pbw will be disc
Tnltai luii
RECIPROCATING PUMPS
443
246. Diagram of work done by the pomp.
Theoretical Diagram. Let a diagram be drawn, Fig. 302, the
"dinates representing the pressure in the cylinder and the abscissae
le corresponding volume displacements of the plunger. The
flumes will clearly be proportional to the displacement of the
lunger from the end of its stroke. During the suction stroke,
n the assumption made above that the plunger moves very
owly and that therefore all frictional resistances, and also the
lertia forces, may be neglected, the absolute pressure behind the
lunger is constant and efqnsA to H  fe feet of water, or 62*4 (H  h)
onnds per square foot, and on the delivery stroke the pressure is
62'4 f Z + H + o ) pounds per square foot.
he effective work done per suction stroke is ABCD which equals
I'i.h, V, and during the delivery stroke is EADF which equals
624 (Z.^),
nd EBCP is the work done per cycle, that is, during one suction
ad one delivery stroke.
Pressure^
Fig. 302. Theoretical diagram of pressare in a Reciprocating Pump.
l/V^
S$
Strokes per nunju/yo
^
Fig. 303.
Actual diagram. Fig. 303 shows an actual diagram taken by
eans of an indicator from a single acting pump, when running
'j a slow speed.
The diagram approximates to the rectangular form and only
444
HYDRAULICS
differs from the above in that at any point p in the soction strda^
pq in feet of water is equal to h plus the losses in the suction
pipe, including loss at the valve, pins th© head required to
accelerate the wat^r in the suction pipe, and qr is the lied
required to lift the water and overcome all losses, and to accelerate
the water in the deliveiy pipe* The velocity of the plunger beii^
small, these correcting quantities are practically inappreciable.
The area of this diagrani represents the actual work done cm
the water per cyck^ and is equal to W (Z ^ h}^ together with ^
head due to velocity of discharge i ^, all losses of energy in tiie
suction and delivery pipes.
It will be seen later that althot at any instant the pressnn
in the cylinder i« effected by the ^rtia forces, the total work
done in accelerating the water is *o.
247. The accelerationB of t pump piunger and of ^
water in the suction pipe.
The theoretical diagramj Fig. 2, has h^u dra^^Ti on die
assumption that the velc»city of ' plunger is very small and
without reference to the variatioTi f the velocity and of the
acceleration of the plunger, but it is now necessary to consider
this variation and its effect on the motion of the water in the suction
and (lch"vciy pipes. To realise how the velocity and acceleration
of tlic ])hnigcr varj", suppose it to be driven by a crank and
connecting rod, as in Fig. 304, and suppose the crank rotates with
a uniform angular velocitj' of ^ radians per second.
Fig. 304.
If r is tlie radius of the crank in feet, the velocity of the crank
pin is V = o>r feet per second. For any crank position OC, it is
proved in l)ooks on mechanism, that the velocity of the point B is
V ()f\
()\) • ^y luaking BD equal to CK a diagram of velocities
EDF is found.
When CB is very long compared with CO, OK is equal to
OC sin 6^, and the velocity v of the plunger is then Vsin^, and
RECIPROCATING PUMPS
445
EDF is a semicircle. The plunger then moves with simple
harmonic motion.
If now the suction pipe is as in Fig. 300, and there is to be
eontiniiity in the column of water in the pipe and cylinder, the
velocity of the water in the pipe must vary with the velocity of
the plunger.
Let V bo the velocity of the plunger at any instant, A and
a the crosssectional areas of the plunger and of the pipe respect
V A
ively. Then the velocity in the pipe must be — — .
Fig. 305.
As the velocity of the plunger is continuously changing, it is
continuously being accelerated, either positively or negatively.
Let I be the length of the connecting rod in feet. The
acceleration* F of the point B in Fig. 305, for any crank angle
0, is approximately
F = caV (cos ^ + ^ cos 2^ V
Plotting F as BG, Fig. 305, a curve of accelerations MNQ is
obtained.
When the connecting rod is very long compared with the
length of the crank, the motion is simple harmonic, and the
acceleration becomes
F = wV cos ^,
and the diagram of accelerations is then a straight line.
Velocity and acceleration of the water in the suction pipe. The
velocity and acceleration of the plunger being v and F respectively,
for continuity, the velocity of the water in the pipe must be
A
V — and the acceleration
a
/.=
F.A
* 8«e BaUmeing of Enginet, W. E. D«lby.
44*;
HYDRAULICS
248. TIM e£Eect of acccleratlofzi of the plnxi^er on Xt%
pressore in tiie cylinder during the suction stroke.
^^'!;en the Trfix iiy of ibe plmiig<?r is increaairig, F is poeitiT^,
ani :. mcoeleTmlki' Hie wmter in tte suction pipe m foroe P ii
T>e«::i:rvi. Tfce •tTi3>?pheric preasisiie has, thereforej not only U)
.'jr, thr v:»ter an^i oien^ouie the remtance in the snetion pipe,
Vu: :: h.As mbo to provide the necee^iry force to accelerate tk
'w^ter, ind the prr^'* in tW cylinder is ccinseqnently dimimshed
. &5 the Telocity of the plunger decreases,
piston has ** exert a reaction upon the
relocttyj or pre6sure on the plangvr is
:rie ocher
rr«tive, an
F
w^Tc^r : > diminisl.
L«r: L be the * :'h of the m
5^n „;il arm in t* feet, /» tl
:':: I ::>e at any in^iant in feet per
^c^xti: f a cohic fuiot c4 water,
Trvn the masfr of water in the i
;• ^T. 1, and since by Newton's sec
acce ermdng force = ai
:.  aCvv'eratinsr f vv rt^juired is
p fr . a, L
g
T:. :\><urv ivr unit arva is
:i' 1 :. .^ ;u:vLi\::: head .f water is
:7 ■
. F . A
tl pipe in feet, a its cross
Dceleration of the i^^ater iu
and per second, and u tke
to he accelerated is ii?.a«X
law of morion
« acceleration,
K lbs.
./.lbs.,
^.
(1
^i.=
LA
(7.a '
F
arce if any one of the three quantities, L, — , or
and other losses the pressure in the
v'v'..:..;r r :> ::• w
a:;d :V.v h^^ad r>:>:sring the UK^rion of the piston is h + ha,
249. Pressure in the cylinder during the suction stroke
when the plunger moves with simple harmonic motion.
It ::\o p'ur.c^^r Iv supp^^sed driven by a crank and very long
RECIPROCATING PUMPS
447
innecting rod, the crank rotating uniformly with angular velocity
radians per second, for any crank displacement ^,
F = <»i"rcos^,
J , L.A.wV /,
nd ha = . cos (f.
g.a
The pressure in the cylinder is
LAui'rcos^
HA
ga
When B is zero, cos B is unity, and when ^ is 90 degrees, cos B
\ zero. For values of B between 90 and 180 degrees, cos^ is
eB^tive.
The variation of the pressure in the cylinder is seen in
'ig. 306, which has been drawn for the following data.
Fig. 306.
Diameter of suction pipe 3 inches, length 12 feet 6 inches.
Oiameter of plunger 4 inches, length of stroke 7^ inches.
Number of strokes per minute 136. Height of the centre of
:he pump above the water in the sump, 8 feet. The plunger is
issumed to have simple harmonic motion.
The plunger, since its motion is simple harmonic, may be
supposed to be driven by a crank 3J inches long, making 68 revo
lutions per minute, and a very long connecting rod.
The angular velocity of the crank is
u>= ' = 7*1 radians per second.
The acceleration at the ends of the stroke is
F = a,>.r = 7rx 0312
= 15'7 feet per sec. per sec,
A / 4 V .
a \3
125/
and
K =
125. 157. 163
32
163,
= 10 feet.
448
HY0EAULICS
The preesnre in the cylinder neglecting the wn^tev m xh
cylinder at the beginning of the stroke is, therefore,
and at the end it is 348 + 10 = 36 feet That is, it is gnatrt
than the atmospheric pressure.
WTien ^ is 90 degrees, cos is aero, and A^ is therefore jscro,
and when is greater than 4K) degrees, eos ^ is negative.
The area AEDF is clearly equal to GADH, and the work iam
per suction stroke is, therefore, not altjered by the accielemtiiig
forces; but the rate at which thr ^^tnger ia working at varii'«i
points in the stroke is affected by m, and the force requiivi] t^>
move the plunger may be very m^
In the above example, for t
move the piston at the commei
more than doubled by the ace
remaining constant and equal t4
varies from
P= 4iJ(8
to P = '43(8
Air i^esseh, Tu quick running t^timpe, or when the leitgtk
of the pipe is long, the effects of these accelerating forces tend to
become serious, not only in causing a very large increase in the
stresses in the parts of the pump, but as will be shown later, under
certain circumstances they may cause separation of the water in
the pipe, and violi^nt hammer actions may be set up. To reduce
the effects of the accelerating forces, air vessels are put on the
suction ami delivery pipes. Figs. 310 and 311.
250. Accelerating forces in the delivery pipe of a plunger
pump when there is no air vessel.
When the plunger commences its return stroke it has not only
to lift the water against the head in the delivery pii^e, but, if no
air vessel is i)rovided, it has also to accelerate the water in the
cylinder and the delivery pipe. Let D be the diameter, a^ the area,
and Li the length of the pipe. Neglecting the water in the
cylinder, the acceleration head when the acceleration of the piston
is F, is
L,.A.F
nereaiied.
ice, the force necessary ki
ent of the stroke has be«n
ting force, and instead di
,8. A during the stroke, it]
)A
ha =
gcii
and neglecting head lost by friction etc., and the water in the
cylinder, the head resisting motion is
7j + ha +
If F is negative, ha is also negative.
'^9'
RECIPBOCATINO PUBCPS
449
When the plunger moves with simple harmonic motion the
iagram is as shown in Fig. 307, which is drawn for the same
ats as for Fig. 306, taking Z as 20 feet, In as 30 feet, and the
iameter D as S^ inches.
Fig. 307.
The total work done on the water in the cylinder is NJKM,
rhich is clearly equal to HJKL. If the atmospheric pressure is
cting on the outer end of the plunger, as in Fig. 301, the nett
rork done on the plunger will be SNRMT, which equals HSTL.
251. Variation of pressure in the cylinder due to firiction
irhen there is no air vessel.
Head lost by friction in the auction and delivery pipes. If t? is
he velocity of the plunger at any instant during the suction
troke, d the diameter, and a the area of the suction pipe, the
elocity of the water in the pipe, when there is no air vessel, is
A
— , and the head lost by friction at that velocity is
, _4fifJf^L
^^~ 2gda' '
Similarly, if ai, D, and Li are the area, diameter and length
espectively of the delivery pipe, the head lost by friction, when
he plunger is making the delivery stroke and has a velocity v, is
^^ 2gDa,' '
When the plunger moves with simple harmonic motion,
V = <i>r sin ^,
nd
, _ 4/A'<oVsin'gL
^•^■" 2gda'
L. H.
29
430
HVDRAUTJCS
If the pump mftkeA n strokes per aeoond, or tli@ nitmber d
revoloticiiis el tha cmnk is ^ per second, and 4 is tlia lengtK d
the strokei
and /, = 2r.
Snbvtttatiiig for i>» and r,
Plotting values of A/ at vai ' points along the stntke,
parabolic cnrv© E>tP, Fig. 808,
When ^ i» 90 degree®, ain^
The mean ordinate of the pan
head, is then
2^
3
.amed
QJty^ and ^ is a maTiinmru
which ia the mean fricti<nial
and since the mean frictional head is equal to the energy lost per
pound of water, the work done per stroke by friction is
2gda'
all dimensions being in feet.
foot lbs.,
r;^
G
H
'^"^ i
N
A
Sf
if
n
i !
•J I
R
£
B
1 [
N^^^
F i
S B
T
i
<3i
Fig. 309.
Let Do be the diameter of the plunger in feet. Then
and
RECIPROCATING PUMPS 451
Therefore, work done by friction per suction stroke, when
here is no air vessel on the suction pipe, is
The pressure in the cylinder for any position of the plunger
.uring the suction stroke is now. Fig. 309,
Ao = H — fe — fea~ fe/.
At the ends of the stroke h/ is zero, and for simple harmonic
lotion hm, is zero at the middle of the stroke.
The work done per suction stroke is equal to the area
LEMFD, which equals
ARSD + EMF = 624feV + ^^'P/^^'* .
Similarly, during the delivery stroke the work done is
624ZV + §^^^g^^*.
The friction diagram is HKG, Fig. 309, and the resultant
iagram of total work done during the two strokes is EMFGrKH.
252. Air vessel on the suction pipe.
Afi remarked above, in quick running pumps, or when the
sngths of the pipes are long, the effects of the accelerating forces
leeome serious, and air vessels are put on the suction and delivery
apes, as shown in Figs. 310 and 311. By this means the velocity
Q the part of the suction pipe between the well and the air
essel is practically kept constant, the water, which has its
elocity continually changing as the velocity of the piston
hanges, being practically confined to the water in the pipe
between the air vessel and the cylinder. The head required to
accelerate the water at any instant is consequently diminished,
>nd the friction head also remains nearly constant.
Let l^ be the length of the pipe between the air vessel and
he cylinder, I the length from the well to the air vessel, a the
rosasectional area of each of the pipes and d the diameter of the
ripe.
Let h^ be the pressure head in the air vessel and let the air
'essel be of such a size that the variation of the pressure may for
implicity be assumed negligible.
Suppose now that water flows from the well up the pipe AB
ontinuously and at a uniform velocity. The pump being single
icting, while the crank makes one revolution, the quantity of
rater which flows along AB must be equal to the volume the
)lunger displaces per stroke.
29—2
*:t
>^J^\
F^ 31 '
::.:i//.:'> :he;rv:i:, including friction and the velocity
c> r^rir.a: r.e^'.rvtcd. the atmospheric head
 A' «V 4f'A<uV/
.(1).
.1 r.uTth qaannnes of the righthand part of the
::^l'y be very small and hr is practically equal
u:: :: >rr>:ke is raking place, the water in the pip^
:l J ':v :r:r ;rvs5j^;irv head at the point B, when the velocity
u'^jTt r i> r :\v: ivr sev.vnd, and the acceleration F feet per
><x\ • .1 ;vr ><Ov
RECIPROCATING PUMPS 453
Let hf be the loss of head by friction in AB, and h/ the loss in
iC. The velocity of flow along BC is — , and the velocity of
ow from the air vessel is, therefore,
t?.A Aft>r
a ira
Then considering the pipe AB,
.nd from consideratioii of the pressures above B,
<vA A«i»r\'
ls=K*h{2—^^
2g /•
Neglecting losses at the valve, the pressure in the cylinder is
hen approximately
,, AkF
Tx J. A*o»V , ,, Ali¥
2gnra' ^ ^ ag
A*a)V
Neglecting the small quantity ^ 9 « ,
For a plunger moving with simple harmonic motion
I. XT 7, 4/wVAVZ^, . ,.\ AZiiuVcosd
ho = lB.h ^ — rj(3 + *i8m*^) .
2ga^d \ir Jag
By putting the air vessel near to the cylinder, thus making
I small, the acceleration head becomes very small and
Ao = H  fe  fe/ nearly,
nd for simple harmonic motion
The mean velocity in the suction pipe can very readily be
etermined as follows.
Let Q be the quantity of water lifted per second in cubic feet.
Then since the velocity along the suction pipe is practically
onstant Vm = — and the friction head is
BECIPROOATIKG PUMPS 455
Let k. Fig. 311, be the length of the pipe between the pump and
the air. vessel, Id be the length of the whole pipe, and ai and D the
area and diameter re8i)ectively of the pipe.
Let At be the height of the surface of the water in the air vessel
above the centre of the pipe at B, and let Ho be the pressure head
in the air vessel. On the assumption that Ho remains constant,
the velocity in the part BC of the pipe is practically constant.
Let Q be the quantity of water delivered per second.
The mean velocity in the part BC of the delivery pipe will be
Q
:=■
The friction head in this part of the pipe is constant and equal to
Considering then the part BC of the delivery pipe, the total
head at B required to force the water along the pipe will be
But the head at B must be equal to H» + /la nearly, therefore,
H. + A, = Z + ^^ga.W + H (1).
In the part AB of the pipe the velocity of the water will vary
with the velocity of the plunger.
Let V and F be the velocity and acceleration of the plunger
respectively.
Neglecting the water in the cylinder, the head Hr resisting the
motion of the plunger will be the head at B, plus the head
necessary to overcome friction in AB, and to accelerate the water
in AB.
Therefore, Hr = Ht, + fea + il V^ a + 
For the same total length of the delivery pipe the acceleration
head is clearly much smaller than when there is no air vessel.
Substituting for H» + /la from (1),
If the pump is single acting and the plunger moves with simple
harmonic motion and makes n strokes per second,
Q = A2rJ,
J Arn
and tt= .
be la. .4^ FA rr^'
RECIPBOCATINQ PUMPS
467
X)me below the zero line BC as in Pig. 312 •. The pressure in the
blinder then becomes negative; but it is impossible for a fluid
k> be in tension and therefore discontinuity in the flow must
CKxmrt,
In actual pumps the discontinuity will occur, if the curve EFG
aJls below the pressure at which the dissolved gases are liberated,
>r the pressure head becomes less than from 4 to 10 feet.
Fig. 812.
At the dead centre the pressure in the cylinder just becomes
sero when A + A^ = H, and will become negative when fe + fea > H.
Theoretically for no separation at the dead centre, therefore,
ha^Rh or ^^^Rh.
ga
If separation takes place when the pressure head is less than
iome head h^y for no separation,
/ia ^ H — hm "" fe,
md
a *^ I
Neglecting the water in the cylinder, at any other point in the
itroke, the pressure is negative when
v^ A *
^ 2g a'
h + + fe/+ ^ — a > H.
a g ^ 2g a:
That is, when
And the condition for no separation, therefore, is
FA
,(H/^/.;^.fe,)
.(2).
* See also Fig. 816, pafi^e 459.
t Sor&oe tension of flaids at rest is not alladed to.
458
HYDRAULICS
255. Separation during the suction etroke when tlu
plunger moves with simple harmonie motion.
When the plungt^r is driven by a crank and very long om
necting r<>d, the acceleration for any crank angle is
F = w'rcos^,
or if the pnmp makea n amgle strokes per second,
•1= wn.
and
I, being the length of the stroke,
F is k\ maximiim when ^ is ze
place at the end of the stroke if
9 a
a
nd separab'on will not tata
and ^^^ll just not take place whei
A , ^ A
— <tt'r or ^ .—n
a 2 a
The minimam area of the si
therefore,
^ Aoi^rL
""giUh^h)
«ind the maximum number of single strokes per second is
I pipe for no separation tSj
(3)
^_1 /2g{R}i^h)j
''"ttV " A./,.L
.(4).
Separation actually takes place at the dead centre at a less
number of strokes than given by formula (4), due to causes
which could not very well be considered in deducing the fornmla.
Example. A single acting pump has a 8troke of Ih inches and the plnngeris
4 inches diameter. The diameter of the suction pipe is 3^^ inches, the Ieni?th
125 feet, and the height of the centre of the pump above the water in the well is
8 feet.
To find the number of strokes per second at wliich separation ^^ill take plact,
assuming it to do so when the pressure head falls below 10 feet.
H/i = 26 feet,
:l63.
and, therefore.
_1 / 64x26
""ir V 163 X 75
xl2
xl25
= 11 = 36
= 210 strokes per minute.
Nearly all actual diagrams taken from pumps, Figs. 313—315,
have the corner at the commencement of the suction stroke
RECIPROCATING PUMPS
459
Toonded off, so that even at very slow speeds slight separation
occurs. The two principal causes of this are probably to be found
first, in the failure of the valves to open instantaneously, and
second, in the elastic jrielding of the air compressed in the water
at the end of the delivery stroke.
DMA/'cry
Zero JPre^suLre/
Fig. 314.
.Atnu
Line
Fig. 815.
The diagrams Figs. 303 and 313 — 315, taken from a singleacting
pomp, having a stroke of 7^ inches, and a ram 4 inches diameter,
illustrate the effect of the rounding of the comer in producing
separation at a less speed than that given by equation (4).
Even at 69 strokes per minute, Fig. 303, at the dead centre a
momentary separation appears to have taken place, and the water
has then overtaken the plunger, the hammer action producing
vibration of the indicator. In Figs. 313 — 315, the ordinates to the
line rs give the theoretical pressures during the suction stroke.
The actual pressures are shown by the diagram. At 136 strokes
.. .11 If resting to endeavour t
sh«Hild overtake the plunger ;
\NTiile the piston moves fi
70 degrees, in yts • Tr9 second
points the preesure in the c
therefore the head available
resistances and to accelerate tl
The height of the centre o
in the samp. The total len,
12*5 feet, and its diameter is
Assuming the loss of head a
to have a mean valne of 2*5 f et
rating the water in the pipe is
is, therefore,
205x82
125
When the piston is at g th
behind the piston. Let this c
velocity of the water be u f e<
takes the crank to torn throng
through a distance
= 0101tt + i5
= l2tt + 32i
The horizontal distance a& ii
should be equal tn i*^*
RECIPROGATINO PUMPS 461
The frictional resistances, which vary with the velocity, will be
small. Assmning the mean frictional head to be '25 foot, the head
canning acceleration is 12*55 feet and the mean acceleration of the
water in the pipe while the piston moves from rest to g is,
therefore,
. 1255x32 ^. ^
fm = — Toic — = ^^ *6®* P©r 86C. per sec.
The velocity in the pipe at the end of 0*058 second, should
therefore be
t? = 32 X 058 = 1*86 feet per sec.
and the velocity in the cylinder
u = y:^ = ri2 feet per sec.
Since the water in the pipe starts from rest the distance it
should move in 0'058 second is
12.i32.C058)« = 65in.,
and the distance it should advance in the cylinder is
065 . . , .
j;g3ms.= 4m.;
so that z is 0*4 in.
Then z + 4*2 ins. = 4*6,
and V2u + 3'2 ins. = 4'57 ins.
The agreement is, therefore, very close, and the assumptions
made are apparently justified.
256. Negative slip in a plunger pump.
Fig. 315 shows very clearly the momentary increase in the
pressure due to the blow, when the water overtakes the plunger,
the pressure rising above the delivery pressure, and causing
discharge before the end of the stroke is reached. If no separa
tion had taken place, the suction pressure diagram would have
approximated to the line rs and the delivery valve would still
have opened before the end of the stroke was reached.
The coeflBcient of discharge is 1*025, whereas at 59 strokes
per minute it is only 0*975.
257. Separation at points in the suction stroke other than
at the end of the stroke.
The acceleration of the plunger for a crank displacement B
mV A
is — ' — COS ^, and therefore for no separation at any crank
angle B
^costf^^Hfc^^fc^^^, h,) .(1).
•L=A.(l*4^)c«».
f 1 ^ ^~) » eqiml to
of tJbe cylisdM^
take i^ftae as tbe ikiad
. ~:a :*i_zi^>w rL :^^  * lie.T to be less than Ar, and
 : ' * > c 7 ^•T*jv<^skry Tv onsider the condition for nv
.^«5. S«c&rs=Dcir vt:^ & ^r^c ^^r Tessel on the snctioii pipe.
T_:i vir^itrr ?trcitr^rj:c: ^«^H lAke plaoe with a large air
^'>t^ .: "^♦r sijT'.c Ti^r^ ' > 'iiy neo^ssarr to substitute in
r»Mi*~ :  r^.'o c .v^ jkz>£ o . 4 , section 2o6, A» of Fig. 310
^ r r. ~ r _. i.:«i i r r >, Iz. FjC olO, k_ is negative.
y r :« ?^oij~ic:» c ^iie^ i^je ri.:Lz^>rr :> ai the end of the strc'ke
::it T..i^m Lzi 4r*f>i . z Z2sT ZLztz r^er^errr. ihe air vessel and the
:^r fr ^=lf^^
.^ . A  :
*. 1 ', section 253, and
.^A' 4"L*VA *
:•! ^> :t:t
frticiD.c ':>^fcas^ in the denominator, W
.^:c:irikrv\i w.ii^ < H  n L then.
RECIPROCATING PUMPS
463
The maxmmm number of strokes is
^1 / 2g(RhK)a
A pump can therefore be run at a much greater speed, without
of separation, with an air vessel on the suction pipe, than
*ithoat one.
259. Separation in the delivery pipe.
Consider a pipe as shown in Fig. 316, the centre of CD being at
height Z above the centre of AB.
Let the pressure head at D be Ho, which, when the pipe
ischarges into the atmosphere, becomes H.
Let ly 1% and k be the lengths of AB, EC and CD respectively,
f^ hf, and A^, the losses of head by friction in these pipes when the
lunger has a velocity v, and hm the pressure at which separation
ctoally takes place.
' — i/O
rw^
^3• N
(
> I M
fij^y.
; I
/
Fig. 316.
Suppose now the velocity of the plunger is diminishing, and its
"etardation is F feet per second per second. If there is to be
F A
jontinuity, the water in the pipe must be also retarded by — ' —
:eet per second per second, and the pressure must always be
positive and greater than fe.,.
Let Ho be the pressure at C ; then the head due to acceleration
in the pipe DC is
FAZ.
9
and if the pipe CD is full of water
H« = Ho :: — A/,
which becomes negative when
FAZ,
9
g
> Ho ~ h/.
alid
ar
tor
for
Qo $^t
andf,
or n.
b*^ deihvrv
P'aiv at B.
RECIPROCATING PUMPS 405
260. Diagram of pressure in the cylinder and work done
curing the suction stroke, considering the variable quantity of
rater in the cylinder.
It is instructive to consider the suction stroke a little more in
etail.
Let V and F be the velocity and acceleration respectively of
:xe piston at any point in the stroke.
As the piston moves forward, water will enter the pipe from the
""dl and its velocity will therefore be increased from zero to
• — ; the head required to give this velocity is
^'2^ (1)
On the other hand water that enters the cylinder from the pipe
3 diminished in velocity from — to v, and neglecting any loss due
o shock or due to contraction at the valve there is a gain of
iiressure head in the cylinder equal to
^2^^2^ ^2).
The friction head in the pipe is
, 4/Lv'A'
^^=2^^ (3).
The head required to accelerate the water in the pipe is
^' = ^ • W
The mass of water to be accelerated in the cyUnder is a
variable quantity and will depend upon the plunger displacement.
Let the displacement be x feet from the end of the stroke.
The mass of water in the cylinder is lbs. and the force
required to accelerate it is
and the equivalent head is
P ^x.F
wA g
The total acceleration head is therefore
9 \ a /
9
L. H. 30
I
466 I I BTDRACLICS
Now hit J be tli£ pressure kead in thte cylinder^ thm
„ , «» 4/LAV F/ LAV
When the plunger moves with simple harroonic motioii^ and a
driven by a crank of radium r rotating uniformly with angnlir
velocity % the displacement of the plmnger from the end of tta
stroke ife r(l cos ^), the reiocitv vr ein and ita aceelemtioa ie
w'r cos 0.
Therefor© i
wr — c ^  + .,M%
9 ^ g 9
Worh dime during iks swc> ok^* Assuming atmospli^c
pressure on the face of the pi the pressnr© per square fool
resisting itii motiDn ia
(E p.
For any small plunger displacement tj, the work done is,
therefore,
A (HHo)w'.aa',
and the total work done during the stroke is
E= r A(HHo)7r.caj.
The displacement from the end of the stroke is
a* = r (1  cos ^),
and therefore d.r = r sin OdO,
and E  I'u' . A (H  Ho) r sin OdO.
Jo
Substituting for Ho its value from equation (6)
, f' 4/LAVr\sin'^ a>Vsin^
2gda' 2g
a>VcOS^ (oVcOS*^ LA 2 A  n.^
+ + tuVcos 0} sm OdS.
9 g 9 (^ i
Tlie sum of the integration of the last four quantities of this
expression is equal to zero, so that the work done by the
accelerating forces is zero, and
E«'Ar r(h+hf)amOde
Jo
i/LAV
29 . da
RECIPROCATING PUMPS
467
Or the work done is that required to lift the water through
% height h together with the work done in overcoming the
resistance in the pipe.
Diagrams of pressn/re in the cylinder and of work done per
iCroJke. The resultant pressure in the cylinder, and the head
ransting the motion of the piston can be represented diagram
Btttically, by plotting curves the ordinates of which are equal to
B^ and H~Ho as calculated from equations (2) and (3). For
jdeamess the diagrams corresponding to each of the parts of
eqiiation (2) are drawn in Figs. 318 — 321 and in Fig. 317 is shown
Ihe combined diagram, any ordinate of which equals
H  A  (feZ + cd + ef gh).
t^^^—^rdS*
$h.
^
Figs. 318, 819, 820.
w
Figs. 821, 822.
In Fig. 318 the ordinate cd is equal to
and the carve H JK is a parabola, the area of which is
2 4/LA» , ,,
3 W"'^^
30—2
468
In Fig. 819, the ordinate efis
f".
and the ordinate gk of Fig. 320 is
+ COS* B,
9
The areas of the curves are respectively
1 wV
and are therefore equal ; and nin the orditiatea are atwav d
opposite sign the sum of the twc ^eas m K0ro,
In Fig. 822, km is equal to
and AZ to
— COSE'l JC J.
g \ a /
Since cos B is negative betweei )* and 180* the area ^TT is
equal to YZU.
Fig. 821 has for its ordinate at any p^int of the stroke, tke
head H  H,, resisting the niution of the piston.
This equals /* + kl i cd + efghf
and the curve NPS is clearly the cnrve GFE, inverted.
The area \'XST measured on the proper scale, is the work done
per stroke, and is equal to VMHT + HJK.
Tlie scale of the diagram can be determined as follows.
Since h feet of water = 62"4/i lbs. per square foot, the pressure
in pounds resisting the motion of the piston at any point in the
stroke is
624 . A . /i lbs.
If therefiue, VXST be measured in square feet the work done
per stroke in ft. lbs.
= 624 A . YNST.
261. Head lost at the suction valve.
In determining the pressure head Ho in the cylinder, no account
has been taken of the head lost due to the sudden enlargement
from the pij)e into the cylinder, or of the more serious loss of head
due to the water passing through the valve. It is probable that the
r*A^
whole of the velocity head, ^^ — g, of the water entering the cylinder
from the pipe is lost at the valve, in which case the available head
H will not only have to give this velocity to the water, but ynW
RECIPROCATING PUMPS 469
dso have to give a velocity head ^ to any water entering the
sylinder from the pipe.
The pressure head H© in the cylinder then becomes
„ „ , i;« A" t;* 4/Lt;'A' F/ IA\
262. yariation of the pressure in hydraulic motors due
lo inertia forces.
The description of hydraulic motors is reserved for the next
chapter, but as these motors are similar to reversed reciprocating
pumps, it is convenient here to refer to the effect of the inertia
forces in varying the effective pressure on the motor piston.
If L is the length of the supply pipe of a hydraulic motor, a
the crosssectional area of the supply, A the crosssectional area
of the piston of the motor, and F the acceleration, the acceleration
.FA
of the water in the pipe is —  — and the head required to accelerate
the water in the pipe is
, FAL
ga
If p is the pressure per square foot at the inlet end of the
supply pipe, and hf is equal to the losses of head by friction in the
pipe, and at the valve etc., when the velocity of the piston is v, the
pressure on the piston per square foot is
p« = p  wha  whf.
When the velocity of the piston is diminishing, F is negative,
ind the inertia of the water in the pipe increases the pressure on
ihe piston.
Example (1). The stroke of a double acting pump is 15 inches and the number of
strokes per minute is 80. The diameter of the plunger is 12 inches and it moves
vith simple harmonic motion. The centre of the pump is 13 feet above the water
n the well and the length of the suction pipe is 25 feet.
To find the diameter of the suction pipe that no separation shall take place,
Lssuming it to take place when the pressure head becomes less than 7 feet.
As the plunger moves with simple harmonic motion, it may be supposed driven
)y a crank of 7^ inches radius and a very long connecting rod, the angular
relooity of the crank being 2ir40 radians per minute.
The acceleration at the end of the stroke is then
4ir» . 40a . r
60
Therefore, Jx40»x^=84' 20',
rom which  = 1*64.
a
470
Therefore
and d^^r.
Ar is clearly lees th&n fjT^ tberefora a^pamtion o&iuiol take pkci mi u^olluo
point in the stroke.
Example (2). The pump of example (I) deli^rs water into a nsiAg mm
1225 feet long and 5 incbefi diameter, which U fitted with on aii T^asBi
The water is lifte^i thr^anh a total height of 920 feet.
Neglecting all lonieft escept frictic*a in the deliTery pipe* deleimine the h«ifr
power required to ^ork the pomp. /=0l0i5.
Since there is an air vesHel in the delivezj pipe the Teloei^ of flcrw « viD W
practically aniform.
Let A and a be tlie cross »eeUo&al areas
ively.
_ip 10 d
" 25 ' a ' 6i
The head h lost due to friction is
= 1764 fei
The total lift is therefore
220 + l764=l
The weight of water lift^ per minale is
A,2r,80
he pnmp (^Under and pipe respot
2r. 80
■6 ft, per WG,
feafc,.
Therefore,
. — . 80 X 625 lbs. = 4900 lbs.
12
«^= 33,000 ="«■«•
Exmuple (3). If in example (2) the air vessel is near the pump and the mean
level of the water in the vessel is to be kept at 2 feet above the centre of tlie
pump, find the pressure per sq. inch in the air vessel.
The head at the junction of the air vessel and the supply pipe is the head
necessary to lift the water 207 feet and overcome the friction of the pipe.
Th.refore, H, + 2' = 207 + 1764,
H„= 3814 feet,
3814 X 625
^=^—144
= 165 lbs. per sq. inch.
Kxdtnple (4). A single acting hydraulic motor making 50 strokes per minute
has a cylinder 8 inches diameter and the length of the stroke is 12 inches. The
diameter of the supply pipe is 3 inches and it is 500 feet long. The motor i
supplied with wat^r from an accumulator, see Fig. 339, at a constant pressure of
300 lbs. per sq. inch.
Neglecting the mass of water in the cylinder, and assuming the piston moves
with simple harmonic motion, find the pressure on the piston at the beginning and
the centre of its stroke. The student should draw a diagram of pressure for one
stroke.
There are 25 useful strokes per minute and the volume of water supplied
per minute is, therefore.
2,). rf'^ = 8725 cubic feet.
4
,502
At the commencement of the stroke the acceleration is ir^ — r, and the velocitT
oO^
in the supply pipe is zero.
RECIPROCATING PUMPS 471
The head required to accelerate the water in the pipe is, therefore,
ir».50a.l.8<.500
«~ 60». 2.3^32
^ tiou xeeiif
which is equivalent to 165 lbs. per sq. inch.
The effective pressure on the piston is therefore 135 lbs. per sq. inch.
At the end of the stroke the effective pressure on the piston is 465 lbs.
per sq. inch.
At the middle of the stroke the acceleration is zero and the velocity of the
piston is
II «T= 1 '31 feet per second.
The friction head is then
•04^1^12^8^ 600'
= 152 feet.
The pressure on the plunger at the middle of the stroke is
300 lbs.  1^^^^^= 2934 lbs. per sq. inch.
144
The mean friction head during the stroke is 1. 15*2=10*1 feet, and the mean
loss of pressure is 4*4 lbs. per sq. inch.
The work lost by friction in the supply pipe per stroke is 4*4 . j . 8^ . 2^
=222 ft. lbs.
The work lost per minute = 5500 ft. lbs.
The net work done per minute neglecting other losses is
(3001b8.4*4).^.Z..83.25
= 370,317 ft. lbs.,
and therefore the work lost by friction is comparatively small, being less than
2 per cent.
Other causes of loss in this case, are the loss of head due to shock where the
water enters the cylinder, and losses due to bends and contraction at the valves.
It can safely be asserted, that at any instant a head equal to the velocity head,
of the water in the pipe, will be lost by shock at the valves, and a similar quantity
at the entrance to the cylinder. These quantities are however always small, and
even if there are bends along the pipe, which cause a further loss of head equal to
the velocity head, or even some multiple of it, the percentage loss of head will still
be small, and the total hydraulic efficiency will be high.
This example shows clearly that power can be transmitted hydraulically very
efficiently over comparatively long distances.
263. High pressure plunger pump.
Fig. 323 shows a section through a high pressure pump
suitable for pressures of 700 or 800 lbs. per sq. inch.
Suction takes place on the outward stroke of the plunger, and
delivery on both strokes.
A brass liner is fitted in the cylinder and the plunger which,
as shown, is larger in diameter at the right end than at the left,
is also made of brass ; the piston rod is of steel. Hemp packing
is used to prevent leakage past the piston and also in the gland
box.
The plunger may have leather packing as in Fig. 324.
On the outward stroke neglecting slip the volume of water
RECIPEOCATINO PUMP8
473
into th© oylinder is j D/ » L cubic feet, D^ being the dia
leter of the piston and L the length of the stroke. Tlie quantity
t water forced into the delivery pipe through the valve V^,> is
■ J (Do'  <f ) L cubic feet,
being the diameter of the small part of the
"lunger.
On the instroke, the suction valve is
loeed and water is forced through the
elivery valve; part of this water enters
l» delivery pipe and part flows behind the
^^1 tJirough the port P>
^TThe amount that flowa into the delivery pipe is
Fig. 'i2A.
^T^ therefore* {D/^ef) is made equal to (^^ or D^ is V2(i, the
Lelivsry^ during each stroke, is ^ Da% cubic f eetj and if there are
ces per minute, the delivery is 4245D„%k gallons per
J^M^Sl^riUM^ii
^b Fig. 33S. TftDgye Duplex Pump.
^^■L Duplex feed pump.
^^P^ 32> show^s a section through one pump and steam cylinder
rf a Taiigye doubleacting pump.
474
HYDRAITUCS
There are mro stmm cylinders side hy aide^ one of
is afaowD, aad two pump cylinderB in Kne with the srteam cjHm
In iho pump the two lower valves an* sucticm valveF and lb
two upper delivery valvea. As the pnmp piston P moves to i^
right, the leflhaad lower valve opens and wat^^r is drawn iski^
pump from the auction chamber C. Daring this stroke the r^
upper valve is open, and vvater is delivered into the delivi^ Ci
^^^len the piston moves to the left, the water m drawn in thrmigii
the lower right valve and delivered through the upper left ^*alTe,
The steun engine has douhle j— * at each end. As the piston
appn ka«he« the end at ita stroke t^ earn valve, Fig. 82d, m at tm
and e*?veTs the steam port 1 whi
to eschauHl. VVTien the piston p
enck*!iied in the c>*linder acta as
and plunger gradually to rest.
inner stej^m port 2 m vspm
dia steatn ]:M>rt 2, rlie ^mm
ihkm and brings the friisou
F\^. 'A'lM.
Fig, S27
lji*r tilt* tnw t'ligino anil pump F^hown in ,'^'^ctiuTi be called A aini
the otlipr engine nud pmup, not j?huwn, Le called B,
As the piston of A moves from right to left, the lever L, Figs.
32o luiil 827, nitates a spindle to the othur end of whicli i^ fixed a
cnink ^!, wbiih moves tlie valve of the cylinder B from left to
right :ind ojx^ns the left j>irt of the cylinder B. Just befure tlie
piston of A reaches the left end of its strcfko, the piston of B,
then*fore, commences its stroke from left t<i rights and bv a lever
Li and cniiik Mj lnnvl^s the valve of cyli ruler A also from left lo
right, an si the piston of A can then commence its !^*turn i^troke.
It should be noted that while the piston of A is moving, that c^f
B is practically at rest, and ricr verm.
265. The hydraulic ram.
Tilt* bycininlic ratn is a machine which utilises the momentum
of a stream of water falling a small height to raise a part of the
watt^r tti a grt^att^r hciglit.
In the iirrruigcment sho\\ni in Fig. 328 water is supplied from a
tank, or stream, througli a pipe A into a chamber B, which ha^ rw^o
PUMPS
+75
V and Yi Wlien no flow is taking place the valve Y falls
seating' and the valve Vj rests on its seating. If water is
fed tci flow along the pipe B it vnl\ escape through the open
V. The contraction of the jet through the valve opening,
3r as in the case of the plate obstructing the flow in a pipe,
168, causes the pressure to b© ^eater on the under face of
ftlve, and when the pressure is sufficiently large the valve
x^mmence to close. As it closes the pressure will increase
le rate of closing will be continually accelerated. Tlie rapid
\g of the valve arrests the motion of the water in the pipe,
bere m a sudden riaa in pressare in B, which causes the
fVt to open, and a portion of the water passes into the air
C. Tlie water in the supply pipe and in the vessel B, after
\ brought to rest, recoils, like a ball thrown against a wall,
ikB pres.snre in the vessel is again diminished, allowing the
to once more escape through the valve V. The cycle of
l&cmB is then repeated, more water being forced into the air
ber C, in wliich the air is compressed, and water is forced up
rfivery pipe to any desired height.
Fig, 328,
it A be the height the water falls to the ram, H the height to
the water is lifted,
f W lbs. of water descend the pipe per second, the work
ble per second is Wh foot lbs., and if e is the efficiency of the
e weight of water lifted through a height H will be
H ■
iciency & diminishes as H increases and may be taken as
^ eent* at high heads.
j'ig. 329 shows a section through the De Cours hydraulic
[the valves of which Brre controlled by sprhigs. The springs
w =
nOetSt*
PUBfPS
471
the fitphon action of the water escaping from the discharge^
; sir to be drawn in past the spindle of the valve. A cushion
air is thus formed in the box B every stroke, and some of thil^H
is carritKl into C when the ^^Ive Vj opens. ^M
The extreme sirapHcity of the hydraulic ram, together with
with which it can be adjusted to work mth vaiying
im of water, render it particularly suitable for pumping
outoftheway places, and for supplying water, for fountains
nd domestic purposes j to country houses situated near a stream.*
260. Lifting water by compressed air.
A very simple method of raising water from deep wells is b;
fis of compressed air. A delivery pipe is sunk into a well^
[the open end of the pip© being placed at a considerable distance
[low the surface of the water in the well.
J
/O — Iw^
WaJLet*
f AirTuj^e
Fig. 030.
Fig. mi.
In the arrangement shown in Fig, 330, there is surrounding ths
delivery tube a pipe of larger diameter into which air is pumped
by a compressor.
The air rises up the delivery pipe carrying with it a quantity of
water. An alternative arrangement is shown in Fig. dSl.
Wbether the air acts as a piston and pushes the water in front
it, or forms a mixture with the water, according to Kelly**,
lepends very largely upon the rate at which air is supplied to the
[pump.
In the pump experimented upon by Kelly, at certain rates of
I
478
HYDEAUUCS
working the discharge was continuoue, the air and the water being
mixed together, while at low discharges the action was intermittent
and the pump worked in a definite? cycle; the discharge commeneed
slowly; the velocity then gradually increased imtil the pipi
di.sdiarged full bore; this was followed by a rush of air, after
whicli the flow gradually diminished and finally stopped ; after a
period of no flow the cycle commenced again. When the rate at
which air was supplied w^as further diminished, the water tme
up the delivery tube, but not sufficiently high to overflow, and tbe
air escaped without doing useful work.
Tlie efficiency of these pumps is very low and only in exeeptioial
cnsm does it reach 50 per cent. The volume v of air, in cubic feet^
at atuiospheric pressure, required to lift one cubic foot of ifnt^r
through a height k depends upon the efficiency. With an ef*
ticiency of 30 per cent, it is approximately ^' = 95'
and with an
efficiency of 40 per cent. *« = He approximately.
It is necessary that the lower end of the delivery be at a gresier
distance below the surface of the water in the well, than the hriglit
of the lift above the free surface, and the well has consequently ^
be made very deep.
On the other hand the well is much smaller in diameter than
would be required for reciprocating or centrifugal pumps, aiid the
initial cost of constructing the well per foot length is considerayj'
EXAMPLES,
(1) Pind tlie horsepower required to raise 100 cable feet of water p^f
minute to s. height of 125 fciet, by a pump whose efficiency is h
(2) A centrifugal pump has an inner radius of 4 Laches and an outer
radius of 12 inches* The angle the blade makes with the direction ^
motion at exit is 153 degrees. The wheel naakes 545 revolutions per miDnt^*
The discharge of the pump is 3 cubic feet per second. The sidej d tlio
wheel are jmrallel and 2 inches apart.
Determine the inch nation of the tip of the blades at inlet so that tb«^
shall be no shock, the velocity with which the water leayes the wheel ^
the theoretical lift. If the head due to tbe velocity with which the w»t0
leaves tlie wheel ia lost, find the theoretical lift.
(3) A centrifugal pump wheel has a diameter of 7 inches and ma^^
1358 revolutions jier minute.
The blades are formed so that the water enters and leaves the wheel
witliont shock and t!ie blades are ra^lial at exit The water is lifted bj tlw
pump 29 "4 feet* Find the manometric efficiency of the pnmp.
PUMPS
479
{41 A cefltrifng&l pump wheel 11 mchee diameter whicli runs at 1308
roToldtiotts per minute ib suiroimded by a Yortex chambet 22 iuchofi
di&tiieUT, autl has radi&l blades at exit The prensure head at the circum
lonuice of thti wheel is 23 feot. The water is hfted to a height of 43'5
ImI abov€ the centre of the pomp. Find the efficiency of the whirlpool
duutiber,
(5> The radial velocity of flow through a pump is 5 feet per second, i
Uie felocity of the outer periphery is 60 feet per aeeond.
The angle tlie tangent to the blade at outlet makes with the direction
of Diotion in 120 degrees. Determine the pressure bead and velocity head
wbi?re the water leaves the wheel* assuming the pressure head in tie eye
of tlie wheel in atmospheric, and thus determine the theoretical lift.
(6) A ctrutrifugal pump with vanes curved back has an outer radius of
10 indieH and an inlet rsdius of 4 inches, the tangents to the vanes at outlet
1>eing Lnelined at 40^ to the tangent at tlie outer periphery. The section of
the wheel is ^^ach that the radial velocity of 0ow is constant, 6 feet per
second ; and it runs at 700 revolutions per minute.
Deteruiipc :—
^1) the angle of the vane at inlet so that tliere shall be no shock,
^S tlie theoretical lift of tlie pump,
(3) Die Yelocity bead of the water as it leaves the wheel* Loud.
L1906.
(7) A centrifugal pump 4 feet diameter running at 200 revolotions per
pumps 5O0O tons of water from a dock in 45 minutes, the meaii
tliellig 00 feet. The area through the wheel periphery is 1200 sqoaroJ
\ and the angle of the vanes at outlet is 26^. Determine the hydraulic f
Dcy and estimate the average horse power. B'ind also the lowest i
I to start pumping against the head of 20 feet, the inner radius being
[ thti outer. Loud. Un. 1906.
(8) A centrifugal pump^ delivery 1500 gallons per minute with a lift of
I fckot^ has an outer diameter of 16 inches, and the vane angle i» 80^ All
\ kinetic energy at dischai*ge is tost, and is equivalent to 50 per cent, of
actual Uft^ Find the revolutions per minute and the breadth at the
Jet, tlie velocity of whirl being half the velocity of the wheel, Lond.
1906.
(9) A centrifugal pump has a rotor 19 inches diameter ; the width of
the outer periphery is 3^^^ inches. Using formula {1), section 236, deter
tnine the discbarge of the putnp when the hea<l is 30 feet and t^i is 50.
(10) The angle tp at the outlet of the pump of question (9) m IX.
Find the velocity witli which the water leaves the wheel, and the
;muin proportion of the velocity head that must be converted into work,
tlie other losses are 15 per cent, and the total efficiency 70 per cent*
fill The inner diameter of a centrifugal pump is 12 j^ inches^ the outer
diameter 21 1 inches. The width of the wheel at outlet is 8g i^chen. Using
uation 2k section 236, find the discharge of the pump when the head is
*5 f^ot^ and ^e number of revolutions per minute is 440.
tofOw
BT1ML4ITUCS
ntmiiiig at 550 leTola^
msi^B Ihe tip of ih^ vane makes
wdgp of tbe vane is 99 de^tes.
m& ti^ ^TOCtiOn of motion of the
of iiie wheel ui 11 tnehtft and tbe
idien a leaves the wheel.
is losi by frictioii, aitd that ooe
A& exit s Vsl^ Ibad tlie bead lost at inlet trhd^
I Ibe pvobabfe veldei;^ impcefieod on tbe ^^bsu
(U) Deaeoba a foeed TottaL, aad sketch tiia footn of tlici &^ sashcb
wbea the aaffolar w^locaty ia conatapt,
In a oiBlrifiiiial pmi^ A«fwuliiug bonsoBtaDj imder water, tbe dLimiet^r
tie CBBda frf Ibe paMka ^ 1 tac^ aad of tbe oatiide 2 feet, ajid tbe
9 tuiuliua at 400 ivivilotiaiES per aiinute. Find approximittclj bow
L tfie «m«ar won^ lie liftad abotv the tail waler level.
(14) KTp^aiii llie ac^oB of a oantrifogal pump, and deduce an exprmon
r iia ellcieBej. If sacii a paaii were requiicd to deliver ICXK^ gailosB iB
■r lo a liei^% cl ^ feet, liow wtwld jua. design it? Loud. Un. 190S.
(li) Fuid fte i^etd of rotasjoa of a wheel of a centrifu^ pomp wMxt
[ lo tin 30O toBB of water 5 feet high in one mmtite ; haviiig ^fiii
' w Ml The velocity of flow through ibe wheel is 4*5 feet pv
, aad Ihie vanes are carved backward so that the angle between ilm
and a tangent to the cirerunferenoe ts 20 degtees* Loml V^
(16) A oentrifogal ptunp is leqaired to lift 2000 galloiia ef water pf^
minnte ^iroogh 90 feetv The velocity ot liow throogh the wheel hi
per second and the efficiency 06, The angle the iipof tbe vane At ootU*
makes witli the direction of motion ib 150 degrees. The outer radius d tho
wheel is twioe tbe inner. Determine tbe dimensions of the wbeeL
(17) A donbleacting phinger pump hss a piston 6 inches cliametor
and tbe length of the strokes Is 12 inches. The gross heaxl i& 500 ttsei^
and the pomp makes BO strokes per minute. Assuming no alip, Md tbts
discharge and horsepower of the pump. Find also the necessary' diiuaeter
for the steam cylinder of an engine driving the pomp diret^ ajssumiiig
steam pressure is 100 lbs. per square inch, and the mechanical dficklK
of tbe combination is 85 per cent.
(18) A plunger pump is plaoed above a tank containing water tt &
temperature of 200' F, Tbe weight of the sncticni valve is 2 lbs. aud iUi
diameter 1 j inches. Find the maxim am height abc^ve the ^i^V at
the pump may be placed so that it will draw water, tJie barometer standlog
at 30 inches and the pump being assumed perfect and ^^itliout clearMCtt
(The vapour tension of water at 200" F. is about 11 "6 lbs, per aq. inch/
(19) A pump cylinder is 8 inches diameter and the stroke of the plung*
is one foot. Calculate ^e maximum velocity, and the acceleration of
1.
PUMPS
481
mtet in the eucUon and delivery pipes, aBBximing tlieir respective diametera
> be 7 inchee and 5 inches, the motion of the piston to be simple harroonic,
ad the pistcm to make 86 strokes per minute.
(20 1 Taking the data of question (19) ealculate the work done on the
iciion stroke of the pnmpi
(1) neglecting the friction in the suction pipet
• (2) indndiog the friction in the suctiou pipe and assuming that the
suction pipe is 25 feet long and that/— '01.
The height of the centre of the pump above the water in the aump is
(21) If the pump in queBtion (20) delivers into a rising main against
head of 120 feet, and if the length of the main itself is 250 feet,
nd the total mork done per revolution, Assuming the pnmp to be douhl©
efeing, find the i.h. p, required to drive the pumpj the efl&ciency being '72
Ii4 no slip in the pump. Find the delivery of the pumpt assuming a slip
f 5 per cent
(22) The piston of a pump moves with simple harmonic motion, and it
^fives at 40 itrokes per minute. The stroke is one foot. The suction
Wk ia 25 feet long, and the suction valve is 19 feet above the surface of the
mmt in the sump. Find the ratio between the diameter of the suction
lape and the pump cylinder, so that no separation may take place at the
bad pointe. Water barometer 84 feet.
(23) Two douhleactiug pumps deliver water into a main without an
lir TeflseL Each is driven by an engine with a flywheel heavy enough to
^ecp tbe speed of rotation uidf orm, and the connecting rods are very long.
Let Q be the mean dehvery of tlie pumps per second, Qi the quantity of
Witeif in the main. Find the pressure due to acceleration (a) at the begin 
aing of a stroke when one pump h delivering water ^ (&) at the beginning
fi the stroke of one of two doubleacting pumps driven by cranks at right
n^m when both are delivering* When is the acceleration ^ero ?
(S4) A double*acting horizontal pump has a piston 6 inches diameter
<tlie diameter of the piston rod is neglected^ and tlie stroke is one foot.
The water is pumped to a height of 250 feet along a delivery pipe 450 feet
kiiig and 4 incheH diameter. An air vessel is put on the dehvery pipe
10 feet from tlie delivery valve.
Find the press tire on the pump piston at the two ends of the stroke
when the pomp is making 40 strokes per minute^ assuming the piston
movea with simple harmonic motion and compare these l^esBiires with the
1 when there is no air vessel. /= *0075.
(36) A single acting hydraulic motor makes 160 strokes per minute and
E vrith simple harmonic motion.
The motor is supplied witli water from an accumulator iu which the
presanre is maintained at WO lbs. per square inch.
The cylinder is 6 inches diameter and 12 inches stroke. The delivery
iipG IS 200 feet long, and the coefficient, which includes bss at bends, etc,
may be taken as/^0'2.
^ UH.
^\
d ifae wiabfe
met p^ap m SSiBoi long and 4 infihes
r B 6 Bciw mud the stroke 1 foot
mmtaEK. 90lBet long, and the Iwad ai
t m mo wit wmml en the pomp. The
m above the levdd the water in the
2i Bflipie liamionic tno tJon and niskBi
cctical dfagrain far the pnnupi
qaaattjcl water in the cylinder and
'iT •oL «nzasKiL laks CKiee anywhere in the deliyery pipe of the
janm* itas saSA si wsndc. s f^vcn in qaestkm i26), if the pipe first rose
3iiE:»2icaJ.7 ±T yiil ieec aad •hea TcrticaDy for 40t» or rises 40 feet im
Tiwffnapij ^nxL 2Kf TozzLp aal ^mb mns horiaontally for 50 feet, and
KpacssaaL TaLsat ^mx wben ihe pi u mm i head falls below 5 feet?
:3Si A jiimn bns ihree smgjLeacting plungers 29^ inches dianwter
.srirniL W ,39aks 13 IJl> decEees wish e«ch o^^ The stroke is 5 feet and
7St> x^mbinr :f scrokes per minase 40. The soctkui is 16 feet and theloigtfa
':t ^re sxkrcara pipe b ±2 &«ec The dehTery pipe is 8 feet diametor and
^j »ii Irc^. Tbe hMa as the dellTerr Talve is 214 feet.
Fzzf i z ibe ziizi2.:im dsamecer of the suction pipe so that there is no
sep&ruoD:!!. 'Qg— ^^^ do air ressei and that separation takes place when
she rr^=sasir^ b^ccoes zenx
V The bcrsepower of the pump when there is an air Yeesel on the
delirerr vtry zear to the pomp. /^OOT.
^The ssciient should draw out three cosine curves differing in phase by
12i3 degrees. Then remembering that the pump is single acting, the
resultant curve of accelerations will be found to have Tn«.TimniTi poeitiTe
•V.A
and also negative values of ——^ every 60 degrees. The maximiun
acceleration head is then h^= ± — r— — •
For no separation, therefore, a » '^ .
(29) The piston of a doubleacting pump is 5 inches in diameter and
the stroke is 1 foot. The delivery pipe is 4 inches diameter and 400 feei
long and it is fitted with an air vessel 8 feet from the pump cylinder. Tbe
water is pumped to a height of 150 feet. Assuming that the motion of the
piston is simple harmonic, find the pressure per square inch on the pision
at the beginning and middle of its stroke and the horsepower of the pomp
when it makes 80 strokes per minute. Neglect the effect of the variaUe
quantity of water in the cylinder. Lond. Un. 1906.
PUMPS
483
(30) The plunger of a p«mp moves with simple harmonic motioo*
Eod the condition that Bepai^ation shall not take place on tlie suction
roke and show why the speed of the pump may he increased if an air
Nisel Is put in the suction pipe. Sketch an indic^itoT diagram showing
jpuc&tioD. Explain ^^ negative slip." Loud. Un. I9()6.
I) In & Bingleacting force pump, the diameter of the plunger ia
ties, stroke 6 inches, length of suction pipe 63 feet, diameter of suction
pe !2 inches, suction head 0'07 ft. When going at 10 revolutions per
innte, it is found that the average loss of head per ntroko between the
Letion tank and plunger cy Under is 0*28 ft. Assuming that the friction al
WSmB vary as the square of the speed, find the absolute head on the suction
da of thJe plunger at the two ends and at the middle of the stroke, the
ETolntions being 50 per minute, and the barometric head M feet. Draw a
a^Eftm of pressures on the plunger — aimplo hannonic motion being
Lond. Un 1906.
(S2) A singleactittg pump without an air vessel has a stroke of
\ inches. The diameter of the plunger is 4 inches and of the suction
tpe a inches. The lengtli of the suction pipe is 12 faet, and the centre
I ahe pmnp is ^ feet above the level in tlie sum^p.
Determine the number of single strokes per second at which theoretl
liiy separation will take place, and explain why separation will actually
ike place when the number of strokes is less tlian the calculated value.
(38) Hxpiain carefully the use of an air vessel in tlie delivery pipe of a
omp. The pump of question (32) makea 100 single strokes per minnte,
nd dehvers water to a lieight of 100 feet above the water in the well
brough a dehvery pipe 1000 feet long and 2 inches diameter. Large air
edeels being put on tlie suction and dehvery pipes near to the pamp.
On the assumption that aU losses of head other than by friction m
he delivery pip© are neglected, determine the horsepower of the pump,
Thefpe is no slip.
(34) A pump plunger has an acceleration of 8 feet per second per
leoond when at the end of the stroke, and Hie sectional area of the plunger
I twice the sectional area of the delivery pipe. The dehvery pipe is 152
bet bug. It runs from the pump horizontally for a lengtli of 45 feet, then
rertically for 40 feet^ then rises 5 feet, on a slope of 1 vertical to 8 hori*
kttstal, and finally nms in a horizontal direction.
Find whether separation will take place, and if so at whiuh section
of ihe pipe, if it be assumed tliat separation takes place when the prassnre
bead m the pipe becomes 7 feet.
S5) A pump of the duplet kind, Fig, 325, in which the steam piston is
Kunected directly to the pump piston, works against a head of h feet of
rtter, the head being supphed by a column of watec' in the dehvery pipe«
rhe piston area is Aot the pUmger area A, the delivery pipe area a, the
angth of the delivery pipe t and the constant steam pressure on the piston
^Jba, per square foot. The hydranho resistance may be represented by
ri V being the Telocity of the plunger and F a ooe^cieut.
^V— ^
484 HTDRAUUGS'
Show thai when the plunger has moTed a diatanoe x from the beguming
ol the stroke
r»«^(^*)(le^). Land. Un. 1906.
(96) A pomp Talye of Ivass has a specific gravity of 8 with a lift of
^ foot, the stroke ol the piston being 4 feet, the head of water 40 feet and
the ratio of the fall valye area to the pistcm area onefifth.
If the Talve is neither assisted nor meets with any reeistanoe to closing,
find the time it will take to dose and the ''slip" due to this gradnal closing.
7*6
Timetodoseisgrvod byfennnla, Ss^. /«g;gX822. Lond. Un. 1906.
CHAPTER XI.
HYDRAULIC MACHINES.
267. Joints and packings used in hydraulic work.
The high pressures used in hydraulic machinery make it
necessary to use special precautions in making joints.
Figs. 332 and 333 show methods of connecting two lengths of
pipe. The arrangement shown in Fig. 332 is used for small
Fig. 832.
Fig. 888.
Fig. 834.
486
HYDEATJLICS
WTOugbt^iroti pipes, no pacldtig being required. In Fig, 333 tho
packJDg material is a guttapercha ring, Pig< 336 siioT^rs an
ordinary siK*ket joint for a castiron hydraulic main. To maka
the joint, a few cords of hemp or tarred rope are driven into*
the socket. Clay is then put round the outside of the socket and
molten lead run in it* The lead is then jammed into the sockel
with a caulking tool. Fig. 33*5 showg various forms of packiag
l^rthersj the applications of which will be seen in the esampli
given of hydraulic machines.
Nffk letUher
Ring Utith^r
Cup leather
Fig. 835.
Fig. aa§.
Hemp twine, carefully plaited, and dipped in hot taUow,
makes a good packing, when used in suitably designed glanda
(see Fig. 339) and is also very suitable for pump bucketSi
Fig. 323. Metallic packings are also used as shown in Figs* 337
and 338,
^>^.
Fig. 337.
iig. ajs.
268, The accumulator.
The accumulator is a deWce used in connection with hydruulic
machinery for storing energy*
In the form generally adopted in practice it consists of a long
cylinder C^ Fig. 339, in which elides a ram R and into wliich water
is delivered from pumps. At the top of the ram is fixed a rigid
cross bead which carries, by means of the bolts, a large cylin*ler
which can be filled with slag or other heavy material, or it imj
be loaded with caetiron weights as in Fig. 340. The water h
HTDRAULIC MACHINES
487
tig. Sd'j. lljdrauUc Acctimulmtor,
=1
488
HTDRAtFLICS
admitted to the cylinder at any desired pressures through a pipe
connected to the cylinder by the flange shown dotted^ and tlie
weight is so adjusted that when the pressure per sq. inch in
the cylinder is a given amount the ram rises.
If d is the diameter of the ram in inches, p the pressure
in lbs. per sq. inch, and h the height in feet through which the
ram can be lifted, the weight of the ram and its load is
W = p/j<flbs.,
and the energy that can be stored in the accumulator is
E = p.d'./i foot lbs,
Tlie principal object of the accumulator is to allow hydraulic
machines, or lifta, which are being supplied \rith hydraulic power
from the pumps, to work for a short time at a much greater rata
than the pumps can supply energy. If the pumps are connect<Mi
directly to the machines the rate at which the pumps can supply
energy must be equal to the rate at which the machines are
working, together with the rate at which energy is being lost hf
friction, etc., and the pump must be of such a capacity as to supply
energy at the greatest rate required by the machineB, and tJm
frietional resistances. If the pump supplies wat^er to an acciuDU
lator, it can be kept working at a steady rate, and during the rinit;
when the demand is less than the pump supply, energy can he
stored in the accumulator*
In addition to acting as a storer of energy, the accumnlator
acts as a pressure regulator and as an automatic arrangement for
starting and stopping the pumps.
When the pumps are delivering into a long main, the demand
upon which is varjnng, the sudden cutting off of the whole or
a part of the demand may cause such a sudden rise in the pressure
as to cause breakage of the pipe line, or damage to the poinp.
With an accumulator on the pipe line, unless the ram »
descending and is suddenly brought to rest, the pressure camioi
rise very much higher than the pressure p which will lift the ram*
To start and stop the pump automaticallyj the ram as il
approaches the top of its stroke moves a lever connected 1*^
a chain wliich is led to a throttle valve on the steam pipe of tlie
pumping engine, and thuB shuts off steam. On the ram agtin
falling below a certain level, it again moves the lever and opem
the throttle valve* The engine is set in motion, pumping n
commences, and the accumulator rises.
HYDRAtTLlC MACHINES
489
^^am^te. A. bydjanlio er«tie working at & pf«Bimre of 100 Iba. per iq. ineh b&e
30 cwts. &t a rate of SOO feet per minute through a height of 50 feet, onoe
J 1 mliiute^. The effioieoaj of the crane ifl IQpet cent, aad aji a<K!i] mala tor
is piorided.
Fiiiii Ihf volame of the cjlinder af the cranes the minimum horBe^power for the
pomp, mud the minimum capacity of the accamulatar,
£^ A be the sectional are* of the mm of the orane cylinder m sq^ feet and L
\he length of the stroke in feet.
Then, I>a44. A. Lx 070 30x112x60',
! _ 30x112x50
H *^~ 70x144x700
^P =233 eubio feet.
Ihe rate of doing work in the lift e^lindtr k
112 :< 30x200
I
07
= 960,000 ft^ Iba. per minute.
lli« ve^rk done in lifting 50 feet ia 210,000 ft, lbs. Since thia has to be done
•rery one aud hiilf miuuteH, the work the pump munt supply in one and half
atee is at ka^st 240,000 ft. lbs., imd the minimym bort^epower is
240,000 ^
1*5
= 40,000 ft, lbs.
The work done by the pnmp while the crane h lifting is
B 240,000x025
efoergy stored in the aeoumuktor mn&t be, therefore^ at least 200,000 ft. Ibfk
!rhererore^ if T^ it its minimum capacity in cubic feet,
V^xTOOx 144 = 20O.0O0»
V^, = 2 cubic feet iieaily*
269. DifTeretitial accumulator ••
TweddelFa differential accamalator, shown in Fig. 340, has a
fixed ram, the lower part, of which is made slightly larger than
the upper by forcing a brass liner upon it. A cylinder loaded
ifith heavy caetiron weights elides upon the i*am, watertight
jomtB hein^ made by means of the cup leathers shown. Water
is pumped into the cylinder through a pipe, and a passage drilled
ajdally along the lower part of the ram.
Let p be the pressure in lbs, per sq, inch, d and di the dia
meters of the upper and lower parts of the ram respectively,
^ba weight lifted (neglecting friction) is then
r
if /t is the lift in feet, the energy stored is
Ep.^We?)feJootlbs,
The difference of the diameters dt and d being small, the pres
sure p can be very great for a comparatively small weight W.
The capacity of the accumulator is, however, very smaU.
33us is of advantage when being used in connection with
* Procetdingt Imt. Meeh. Engf,, 1874,
490
n"
J
^^■d
HYDRAULICS
V
<
\
Fig, 340.
1.
11
JJif
;:?
3
X
Fig, 841. Hydraulic Inteiisiflflr.
HYDRAULIC MACHINES 461
hydraolie rivetera, aB when a demand is made upon the ac
cttmalatorj the ram falls qaickly, hut is suddenly arrested when
the ram of the riveter comes to rest, and there m a consequent
increase in the pressure in the cylinder of the riveter which
clinches the rivet. Mr Tweddell estimatea that when the ac
cumulator is allowed to fall suddenly through a distance of from
IS to 24 inches, the pressure is increased by 50 per cent.
270* Air accumolatoir*
The air accumulator is simply a vessel partly filled with air and
into which the pumps^ which are supplying power to machinery,
deliver water while the machinery is not at work*
Such an atr vessel has already been considered in connection
with reciprocating pumps and an application is shown in connection
with a forging press. Fig, 343,
If V is the volume of air in the vessel when the pressure is
p pounds per sq. inch and a volume v of water ia pumped into
the vessel J the volume of air is (V — v),
AsBuming the temperature remains constant^ the pressure pi in
the vessel will now be
p.V
If V is the volume of air, and a volume of water v is taken out
of the vessel,
271, Intensiflers.
It is frequently diginible that special machines shall work at
a higher pressure than k available from the hydraulic mains. To
increase the pressure to the desired amount the intensifier is used.
One form is shown in Fig. 34L A large hollow ram works in
a fixed cylinder C, the ram being made watertight by means of a
§taflingbox. Connected to the cylinder by strong bolts is a cross
head which has a smaller hollow ram projecting fi*om it, and
Altering the larger ram^ in the upper part of which is made a
Btoffingbox. Water from the mains is admitted into the large
cylinder and also into the hollow i*am through the pipe and
the lower valve respectively shown in Fig. 342,
If p Iba. per aq. inch is the pressure in the main, then on
the underside of the large ram there is a total force acting
of p 7 D* pounds^ and the pressure inside the hollow ram rises to
D*
p ^ pounda per sq, inch, D and d being the external diameters
of the large ram and the small ram respectively.
492
HYDRAULICS
The form of iiiteiiBifier kere ahovm is used in connection with
a large flanging press. The cylinder of the press and the tipper
part of the intensitier are fiUed with water at 700 Ihs. per sq. inch
and the die brought to the work. Water at the same pressore i
admitted below the large ram of the intensifier and the preasti
in the upper part of the intenaifierj and thus in the press cylinder^
rises to 2000 lbs, per sq* inch, at which preesure the flanging
is finished*
JbSmaU
ptrsq. indL.
^
Tp^
iM
1
Tb Large ^Undjet at IntenrnJUF
Nofh Return Vcdvee for
Intensi/Ur
Fig. 342.
^ 1
aiioon
272, Steam intensifiers.
The large cylinder of an intensifier may be supplied wit
steam, instead of water, as in Fig* 343^ which shows a steam
tenaifier used in ^con junction with a hydraulic forging press, Thes©
intensitiers have also been used on board ship* in connection wit
hydraulic steering gears.
273, Hydraulic forging preaa, with steam intensifier and]
air accumulator.
The application of hydraulic power to forging presses is illu^ I
trated in Fig. 343. This press is worked in conjunction with ft]
steam intensifier and air accumulator to allow of rapid world^iij
The whole is controlled by a single lever K, and the pre«
capable of making 80 working strokes per minute.
When the lever K is in the mid position everything is at rest j
on moving the lever partly to the rights steam is admitted into th^
cylinders D of the press through a valve. On mo\nng the lever w
its extreme position, a finger moves the valve M and admits w»ler
* Prootedinffi IntU MecK Enff».^ 187i^
HTDRAITLIC MACHIKES
493
relay piston shown at the top of the figure, which opens
E at the top of the air vessel* In small presses the valve
by levers. The ram B now ascends at the rate of
foot per secondj the water in the cylinder c being forced
accnmiiiatorf On moving the lever K to the left^ a« soon
M passed the central position the valve L is opened to
494
WTDEATJLICS
exhauBt, and water from the sir veeael, assisted by gravity, forces!
down the ram B, the velocity acqtiired being about 2 feet perl
second, until the press head A touches the work. The movementj
of the lever K being continuedj a valve situated above the vah
J is opened, and steam is admitted to the inteneifier cylinder H;'
the valve E closes automatically, and a large pressure is exerted
on the work under the press head*
K only a very short stroke is requiredj the byepass valve L i
tamporarily discoimectedj so that steam is supplied eontiiitiousl]^
to the Hfting cylinders D* The lever K is then sim^ply used
admit and exhaust steam from the inteneifier H, and no water
enters or leaves the accamulator. An automatic controlling ge
is also fitted, which opens the valve J sufficiently early to preventj
the intensifier from overrunning its proper stroke.
BTDRAITLIC MACHINES
495
On© end of a wire rope, or chaiiij is fixed to a lug L on the
cylijiderj and the rope ia then passed alternately round the upper
and lower pulleys, and finally over the puJley on the Jib of the
crane. Fig. 346, In the crane shown there are three pulleys on
the ram, and neglecting friction, the pressure on the ram is equally
divided among the iix ropes. The weight lifted is therefore one
aixth of the pressure on the ram, but the weight is lifted a distance
equal to six times the movement of the ram.
Let the number of pulleys on the end of the ram of any crane
he J , arranged as in Fig. 347
The movement of the weight wiH then he n times that of
tlie ram.
Let p be the pressure in lbs. per sq. inch in the cylinder and
d the diameter of the ram in inches.
The pressure on the i*ara is
and the energy supplied to the crane per foot travel of the ram is
therefore P foot pounds,
kThe energy supplied per unit volume displacement is 144 * p.
The actnal weight lifted is
W = 6^^pd»lbs.,
being the efficiency.
^\'hen full load is being lifted e is between 07 and 0^8.
For a given lift of the weight, the number of cubic feet of water
aaed, and consequently the energy supplied, is the same whatever
th© load lifted, and at light loads the efficiency is very small.
275. Double power cranes.
To enable a crane designed for Iieavy work to lift light loads
iHth reasonable efficiency, two lifting rams of different diameters
art* employed, the smaller of which can be used at light loads.
A convenient arrangement is as shown in Fig^. 348 and 349,
the smaller ram R' working inside the large ram B*
^lien Ught loads are to be lifted j the large ram is prevented
from momkg by strong catches 0, and the volume of water used
IS only eciual to the diameter of the small mm into the length of
the stroke. For large loadsj the catches are released and the
two rama move together.
Another arrangement is shown in Fig, 350, water being ad
mitted to both faces of the piston when light loads are to be
lift^, and to the face A only when heavy loads are to be raised.
HTDRAtrtlC MACHINES
49?
For a given stroke B of the mm, the energy supplied in the
&l«t ease is
■ted in the second case
«r){D^^)ft. lbs.,
Fi« 350. AitnitroGg Double pow«r Hjdrftulio Crftne Cjlindar.
276, Hydraulic crane valves.
Figs. 361 and 352 show two forms of lifting and lowering
talves tLsed by Armstrong, Whitworth and Co. for hydraulic
In the arrangement shown in Fig, 351 there are two inde
pendent valves, the one on the left being the pressure, and that
cm the right the exhaust valve.
, 851, Ann stTQQg'Wli It worth
Hydraalie Cr&ne ValTe.
I u a.
Fig. SS2. Armstroog Whitworth
MjriirftuUe Cr&ae SMd« Valve.
4»
HTDBAtlUCS
tbowu in Fif . S52 a single D ^de vake i
llv Talvp chest duon^ the prefi&ure puaga
in the neiitrml podtion^ If llie Wve
enters the cylinder, but if it is rights
dbe ejriiixder tkr^u^h die port of Uie slids
faydmUlc pnn. Fig. 353 is a section tbroagh
kydmnlie preas^ naed for teotui^ springs.
is fitted widi m brsas linear, and axiaUf
a rod Fr ^* a platan P at thjb irsie end^
il ram is hoUow^ the umsr
id fmm the largt? cy lindur
m top of the cylinder ADif
1 cylmder ini^de the tsm ii
1]^ a hole drilled along tk
the water ts cotitinuooBlf
I the annnl^r ring RR.
I prerenled hf means d i '
G ^nd 1 rrae If^ther r. and leakage past the
i i i ri>c n P by cup leathers L and Lq.
«^ _ :1 vilve >p:r.dle is moved to the right, the port p is
tvcvi ^ >. :":ir rxhaust, and the ram is forced up by the
<>^ : "1 wjkTcr n the annular ring RR. On moving the
 :  V r : :he left, pressure water is admitted into the
  ' >" i :r.r ri::i :> f «n.>e^ down. Immediately the pressure
>,*:•;. ':t i::: . ni^s baok a^rain.
 r " * n "ir i::i meter of the ram, ti the diameter of the
:V   i:^:v.r:cr of the piston P, and p the water pressure
rve acting on the ram is
jT :"r.e ram when pivjsurf is released from the
..f»: = p^(iyrf.') lbs.,
F
r  I •1;  'Pi lbs.
.^" vA.ve spindle S, has a chamber C cast in it,
: >:x holes m each ring are drilled through
: the chamber. These rings of holes are at
.^wX^rt that, when the spindle is moved to the
i I ivssite to the exhaust and the other opposite
I Nvhti: the spindle is moved to the left, the holes
L Fig, a53. HydratUio Pr^aa with Lutlie Valve.
Qrdratilic riveter.
n through the cylinder and ram of a hydraulic riveter
Fig, 354.
600
HYDRAULICS
h III 1^ 1 I ■ 1
^ i Inlft Va/ie
Fig. 355. V&lTes for Hydraalio Biveter.
HYDRAULIC MACHtJ^ES
^01
e mode of working is exactly the same as that of the small
described in section 277.
^ enlarged section of the valves is shown in Fig* 355* On
g the lever L to the right, the inlet valve V 18 opened, and
\re water is admitted to the large cylinder^ forcing out
to Wlien the lever is in mid position, both valves are
, hy the springs S, and on moving the lever to the left, the
ftst valve V, is opened, allowing the water to escape from the
ier* The pressure acting on the annular ring inside the
ram then brings back the ram* Tlie methods of preventing
[© are clearly shown in the figures,
B, Hydraulic engiBea,
rdraulic power is admirably adapted for machines having a
CK^ting motion only, especially in those cases where the load
Really constant.
Fig, 356. Hydraulic Capstati.
501
HTDRAUUCS
II hmB masearer been successfully applied to the drivini
mdk WB capstaiLS and winches in which a reciprocal
i ocMif^ned into a rotarj motion.
Tlid kjn&miLtieeEigine shown in Figs, 356 and 357, haa H
fl^liiideri in one castixigi the axes of which meet on the axis of
on&k eliaft & Hie motion of the piston P is transmitted to
crmnk pin hf short eoDneetin^ rods R. Water is admitted
ejchausted tluxmgh m Tmlve V, and ports p.
The face of the valve is as shown in Fig. 358, E
exhansi port oonnected through the centre of the valve i^
exliaiisl pipe, luid KM the pressure port, connect^ to the msft
chamber H hy a small port through the side of the valve. 1
valve seating is generally made of lignnmvitae, and has A
circnlar ports as shown dotted in Fig. 358» The valve receiv^a
motion from a small auxiliary crank T, revolved by a project
from the cranX pin G. When the piston 1 is at the end of
stroke, Fig* 359^ the port p, should be just opening to the preA
port, and just closing to the exhaust port E* The port pi shoi
be fnDy open to pressure and port pa fully open to exhai
When the crank has turned through 60 degrees, piston 3 i
A
HYDRAULIC MACHINES
503
be at tlie inner end of its stroke, and the edge M of the pressure
port ahoold be just closing to the port p^. At the same instant the
edg^ 1^ of the exhaust port should be coincident with the lower
ed^e of the port pt. The angles QOM, and LON, therefore,
ahoald each be 60 degrees. A little lead may be given to the
valve i>orts, i.e. they may be made a little longer than shown in
the Fig" 358, so as to ensure full pressure on the piston when
commencing its stroke. There is no dead centre, as in whatever
position the crank stops one or more of the pistons can exert a
turning' moment on the shaft, and the engine will, therefore, start
in any position.
Fig. 358.
Fig. 359.
The crank* effort, or turning moment diagram, is shown in
Fig. 359, the turning moment for any crank position OK being
OM. The turning moment can never be less than ON, which is
the magnitude of the moment when any one of the pistons is at
the end of its stroke.
This tyi)e of hydraulic engine has been largely used for the
driving of hauling capstans, and other machinery which works
intermittently. It has the disadvantage, already pointed out in
connection with hydraulic lifts and cranes, that the amount of
^water supplied is independent of the effective work done by the
machine, and at light loads it is consequently very inefRcient.
There have been many attempts to overcome this diflSculty,
notably as in the Hastie engine t, and Bigg engine.
* See text book on Steam Engine.
t Proceedings Inst. Mech, Engs,, 1874.
lfTDRAlJt.lCS
i 280« Bigg hydranUc engine.
To adapt the quajitity of wat^sr nsed to the work done*, 1
hafi modified the three cylinder engine by fbdng the crank pin, and
aUawmg the cylinders to revolve about it aa centre.
The three pistons Pi, Pa and P* are connected to a disc,
Fig. 360, by three pins. Thii disc revolves about a fixed ceutre A.
The three cylinders rotate about a centre 6, which is capahle d
' being moved nearer or furtlier away from the point A as desired*
The stroke of the pietons is twice AG> whether the crank or the
cylinders revolve, and since the cylinders, for each stroke* have to
be filled with high pressure water, the quantity of water supplied
per revolution is clearly proportional to the length AGr.
Fig. 360. Higg Hydrnnlio Engine.
The alteration of the length of the stroke is effected by
of the subsidiary hydraulic engine, shown in Fig. 36L There
two cylinders C and d, in which slide a hollow doable ended
ram PPi which carries the pin G, Fig. S60. Cast in one piece iMth
the ram is a valve box B. R is a fixed ram, and through it wat^r
enters the cylinder Ci, in which the pressure is continuously
maintained. The difference between the etfectivB areas of P and
Pv when water is in the two cylinders, is clearly equal to th e at«»
of the ram head Ri .
See lilso Engimgrt Tgl. LXxxv» 189S.
HrDBAUUC MACHINES
From the cylinder Ci the water is led along the passages
^OTfn to the valve V» On opening this valve highpressure
^ter 19 admitted to the cylinder C. A second valve similar to
V, bat not shown, is used to regulate the exhaust from the
cylinder C. A?VTien this valve is opened, the ram PPi moves to
th left and carries with it the pin (t. Fig* 360. On the exhaust
hehig closed and the valve V opened, the full pressure acts upon
l»ctth ends of the ranij and since the effective area of P is great
th^n Pi it is moved to the right carrying the pin G, If botkj
ralves are closed, water cannot escape fron> the cylinder C and
the ram is locked in position by the pressure on the two ends*
Water
Fig. 361.
EXAMPLES.
ill The ram of & hydraulic crane is 7 inches diameter. Water is
flopplied to the crane at 700 Iba. per square inch. By suitable gearing the
load is lifted times ob quickly as the raw. Assuming the total efficiency
of tb© crane is 70 per eent., find the weight lifted.
(*2,t An accumulator has a stroke of 23 feet ; the diameter of the ram is
23 iJM!he<i; the working pressure is 700 lbs. per squaje inch. Find the
capacity ol the a^ocumalatoi* in horse power hour».
id) The total weight on the cage of an ammonition hoist is 8250 lbs.
The velocity ratio between the cage and the ram is eix, and the extra load
QOI ilie cage due to friction may be taken as 30 per cent, of the load on the
ea^. The steady spaed of the ram is 6 inches per second and the available i
re at the working valre is 700 lbs, per square inch.
B^ttimate the loss of head at the entrance to the ram cylinder^ and
ling this was to be doe to a smJden enlargement in passing through
^port to the cyhnder^ estimate* on the usual asaumption, tlie area of the
yorl. the ram cylinder being 9 inclies diameter. Lond, Un, 1906,
The eflective preaanre p *
4
506
HTTDRAITLICS
Lobs of head
Area of port
t?= velocity through the ir&lv6i.
4
(4) DeBcribe, with sketche^^ some form of hydraulic Rccumiilatot so
able for use in connection with riTstin^. Explain by tl>e aid of dia
if pOHsible, the general nature of the curve of pre^gxire on Uie riveter i
during the stroke ; and point out tlte re^aaona of the Tariation^. Lond Uij
1905, (See sections 262 and 269,)
(5) Describe with sketche*j a hydraulic intensifier.
An intejosifler is required to increase the pressure of 700 IbR, per squa
inch on the mains to SOOO lbs. per square inch. The stroke of the int
fier is to be 4 feet and ita capacity Uiree gallons. Detennine the (
of the rama. Inat. C, E. 1905.
(6) Sketch in good proporttou a section through a differential hydra
accumulator. What load would be necessary to produce a pressuFe of 1 1
ier square inch, if tlie diameters of the two rams are 4 incliea and 4 i
respectively ? Neglect the Mction of the packing. Give an instance of tlw'
use of auch a machine and state why accumulators are used.
(7) A Tweddeirs differential accumulator is supplying water to riTetung
machines. The diameters of the two rams are 4 inches and 4 icchef
respectively, and tlie pressure in the accumulator b 1 ton per square mrJi.
Suppoae when the valve is closed Uie accumulator is falling at a vekxity
of 5 feet per second, and the time taken to bring it to rest is 2 seconds. tioJ
the increase in pressure in the pipe.
(8) A hit weighing 12 tons is worked by water pressure, the pn
in tlie maiu at the accumulator being 1200 lbs. per square inch ; the lengtiil
of tlie supply pipe which is Sj inches in diameter is flOO yards, Wh^i >r
the approximate speed of ascent of this Ijftt on the assumption that lb* I
friction of the Btufiingbox, guides, etc. is equal to 6 per cent, of the j
load lifted and the ram is 8 inches diameter ?
(9) Explain what is meant by the " coefficient of hydraulic resiHtaEoe"
as applied to a whole system, and what assumption is usually made rt<gsnl*
ing it? A direct acting lift having a ram 10 inches diameter i«f '^nppll
from an accumulator working under a pressiu^e of 750 lbs. i>er si]
When carrying no load the ram moves through a distance of t^^ '
uniform speed, in one minute, the valves being fully open. Estiroatt i
coefficient of hydraulic resistance referred to the velocity of the ram, I
als^ how long it' would take to move the same distance when tli« i
carries a load of 20 tons. Loud. Vn. 1905.
^''^^ead 1 ost = ;^:;^^ . Assumption is made that resistance varies «a •'J
64
624
CHAPTER XII.
EESISTAJS'CE TO THE MOTION OF BODIES IK WATER.
281, Froude^s* expeiimenta to determine Motional re
istances of tliin boards when propelled in water.
It has been shown that the frictional resiatance to the flow of
gr along pipes is proportional to the velocity raised to some
BF n, which approximates to two, and Mr Froude's classical
ents, in connection with the resistance of ahips, show that
ince to motion of plane vertical boards when propelled
II w^ftter, follows a similar law.
^ft Fig. 303,
'^The experiments were carried out near Torquay in a parallel
lidted tank 278 feet longj 36 feet broad and 10 feet deep. A light
teilway on '* which ran a stout framed truck, suspended from the
\xlem of two pairs of wheels," traversed the whole length of the
^ok, about 20 inches above the water level* Tlie truck was pro
lelled by an endless wire rope wound on to a barrel, which could
le made to revolve at varjang speeds, so that the truck could
llftvirne the length of the tank at any desired velocity between
DO and 1000 feet per minute,
* Bril. Amu. Rep&tU, li^724*
508
HTDRAULIOS
Planes of wood, about ^ incli thick, the Bui^ces of widch were
covered with variona materials as set out in Table XXXIX, were
made of a uniform depth of 19 inches^ and when under experi
ment were placed on edge in the water, the uppefr edge hmg
ibont \j inches below the surface. The lengths were varied from
; to 50 feet.
T\\e ap^mratus as used hj Fronde is illustrafced and described
in the British A^sociution Reports for 1872.
A later adaptation of the apparatus as uaed at Haslar for
determining the resistance of ships* models is shown in Fig. 361
An arm L is connected to the model and to a frame beam, which
is carried on a double knife edge at H, A spring S is attached to
a knife edge on the beam and to a fisted knife edge N on the
frame of the truck* A link J connects the upper end of the beam
to a multiplying lever which moves a pen D over a recordiBg
cylinder. This cylinder is made to revolve by means of a worm
and wheel, the worm being driven by an endless belt from the ajcle
of the truck. The extension of the spring S and thus the move*
ment of the pen D is proportional to the resistance of the model,
and the rotation of the drum is proportional to the distance mOTedi
A pen A actuated by clockwork registers time on the cylinder.
The time taken by the truck to move through a given distance
can thus be determined.
To calibrate the spring Sj weights W are hung from a knife
edge, which is exactly at the same distance from H as the pointi
of attachment of L and the spring S.
From tlie results of these experiments, Mr Frond© mad© the
following deductions,
(1) The frictional resistance varies very nearly with tlie
square of the velocity.
(2) Tlie mean resistance per square foot of surface for leugdia
up to 50 feet diminishes as the length is increased, but is pmc*
tically constant for lengths gr^^at^r than 50 feet,
(3) The frictional resistance varies very couidderably wil
the roughness of the surface.
Expressed algebraically the frictional resistance to the motii
of a plane surface of area A wheii moving with a velocity o
per second is
/ being equal to
A
10 •
KESJSTAiCCE TO THE MOTION OF BODIES IN WATER
509
TABLE XXXIX,
Showing the result of Mr Froude'a experiments on the ftnctional ^
(resiivtatice to the motion of thin vertical boards towed through
in a direction parallel to its plane.
Width of boards 19 inches, thickness fV inch,
power or index of speed to which resistance is approxi*
^tely proportionah
f^^ tlie mean resistance in pounds per square foot of a surface,
length of which is that specified in the heading, when the
relocity is 10 feet per second.
/i = the resistance per square foot, at a distance from the
ing edge of the boards equal to that specified in the heading^
a velocity of 10 feet per second.
As an example, the resistance of the tinfoil surface per square
at 8 feet from the leading edge of the board, at 10 feet per
cond, is esttinated at 0^263 pound per square foot; the mean
resistance ia 0"278 pound per square foot.
Soffaoe
J^^irea with
'^ttmiah
Letigth of plameB
2 feet
2^
2^16
193
20
20
20
041
0*90
087
061
090
110
0890
0*295
0725
0690
0780
0880
8 feel
20 ie%i
199
I '92
20
2^0
2*0
0325
0278
0626
0583
0625
0714
0264
0*263
0*504
0400
0*488
0520
n f, /j
185
190
189
20
20
20
0278 0240
0*262 ' 0244
0*581 0447
0*480 0384
0584 0465
0588 0^490
50 feet
1*83
1*88
X87
206
2*00
0250
0246
0474
0*405
0486
0226
0282
0423
0*887
0456
The dirainution of the resist^nce per unit area, with the length,
is principally due to the relative velocity of the water and the
b<mrd not being constant throughout the whole length,
A& the board moves through the water the frictional resistance
of the first foot length, say, of the board, imparts momentum to
tlia water in contact with it, and the water is given a velocity in
til© direction of motion of the board. The second foot length will
tlierefore be rubbing against wat^r having a velocity in its own
direction, and the frictional resistance w411 be less than for the
fir&t foot* The momentum imparted to the water up to a certain
point, is accumulative, and the total resistance does not therefore
increase proportionally with the length of the board.
510
FTBIlAtJLfCS
282. Stream line theory of the resistance offered to the
motion of bodies in water.
Rem^tance of ships* In con^deriBg the motion of water along
ptpee and channels of uniform section, the water has been aesumed
to move in ** stream lines/' which have a relative motion to the
sides of the pipe or channel and to each other^ and the readstaiw
to the motion of the water has been considered as due to the
friction between the consecutive stream lines, and between the j
water and the surface of the channel, these frictional resistance^
above certain speeds being such as to cause rotational motions
the mass of the water.
theT
Fig. 363.
Fig. 8S4.
It has also been shown that at any sudden enlargement olj
stream, energy is loat due to eddy motions, and if bodies,
as are shown in Figs 110 and 111, be placed in the pipe, there \
a pressure acting on the body in the direction of motion of the
water. The origin of the remetance of ships is best realise*! by
the "stream line^' theory, in which it i^ assumed that relative to
the ship the water is moving in stream lines as shown in Fig*.
363, 364, consecutive stream lines also having relative motion.
lEISTANCE TO THE MOTION OF B0DI^1N^^^^»511
Lecordmg to this theory the resistance is divided into three
L) Frictional resistance due to the relative motions of con
scutive stream Unesj and of the stream lines and the surface
f the ship.
(2) Eddy motion resistances dae to the dissipation of the
aergy of the stream Hues, all of which are not gradually brought
St.
J) Wave making resistances due to wave motions set up at
i§tirface of the wat^er by the ship, the energy of the waves
leing' dissipated in the surrounding water.
According to the late Mr Froude, the greater proportion of
be resistance is due to frictionj and especially is this so in long
h]p<s^ Tftith ftue lines, that is the cross section varies very gradually
rtiTu the bow towards midsliipSj and again from the midships
DWards the stern. At speeds less than 8 knots, Mr Fronde has
huim that the frictional resistance of ships, the full sx>eed of
rhich is about 13 knots, is nearly 90 pt^r cent, of the whole
^eBistance, and at full speed it is not much less than 60 per cent,
3e has further shown that it is practically the same as that
■existing the motion of a thin rectangle, the length and area of
ihe two sides of which are equal to the length and immersed
urea respectively of the ship, and the surface of which has the
lame degree of roughness as that of the ship.
If A is the area of the immersed surface, / the coefficient of
friction, which depends not only upon the roughness but also
Upon the length, V the velocity of the ship in feet per second, the _
resistance due to friction is ^
^5 value of the index n approximating to 2»
The eddy resistatice depends upon the bluntness of the stem of
Ihe boatu, and can be reduced to a minimum by diminishing the
iMtion of the ship gradually, as the stem is approached, and by
^piding a thick stern and stern post.
As an extreme case consider a ship of the section shown in
JPig, 364j and suppose the stream lines to be as shown in the
figare. At the st«rn of the boat a sudden enlargement of the
Btn^m lines takes place, and the kinetic energy, which has been
g]¥en to the stream lines by the ship, is dissipated. The case is
analogous to that of the cylinder, Fig. lU, p* 169, Ehie to the
Io6s of energy, or head, there is a resultant pressure acting up<jn
the ahip in the direction of flow of the stream lines, and con
leqoently opposing ite motion*
512
HTtlRA0LlCS
If the ship baa fine lines towards the stem, as in Fi^, 363,
the velocitiei of tlie stream lines are diminished gradually and tk
logg of energy by eddy motions becomes very smalL In actiml
ships it is probably not more than 8 per cent, of the whde
resistance.
The wave making resistance depends upon the length and tli6
form of the ship, and especially upon the length of the '^entraiic*'*
and ^* run/' By the " entrance *' is meant the front pan of the
sliip, which gradually increases in section* until the middle body,
which is of uniform section, is reached, and by the *^ nin," the
hinder part of the ship, which diminishes in section from tie
middle hody to tlie stern post.
Beyond a certain speed, called the critical speed, the rate of
increase in wave making resistance is very much greater than
the rate of increase of speed. Mr Fronde found that for tb
S*S, " Merkara " the wave making resistance at 13 kmM, tlie
normal speed of the ship, was 17 per cent, of the whole, but at 19
knots it was 60 per cent. The critical speed was about 18 kiiDt**
An appro3ciinate formula for the critical speed V in knot^ is
L being the length of entrance, and Li the length of the rim b
feet.
The mode of the formation by the ship of waves can be partly
realised aa follows.
Suppose the ghip to be moving in smooth water, and the stT^affl
lines to be passing the ship as in Fig. 363. As the bow of tb
boat strikes the dead water in front there is an increase in
pressure, and in the horizontal plane SS the pressure mB k
greater at the bow than at some distance in front of it, and
consequently the water at the bow is elevated above the nofiBii
surface.
Now let AA, BE, and CC be three sections of the ship and ^
stream lines.
Near the midship section CC the stream lines will be m(0
closely pa<iked together, and the velocity of flow will
therefore, than at AA or BB* Assuming there is no 1(»
in a stream Hue between AA and BB and applying Hemotiilli'i
theorem to any stream line^
w 2g w 2g w 2g*
* See Sir W. While's Kami ATcMieeture, Tramaetmm ttf Nm^ Anhint^
1877 aad 1881,
RBSISTAKCE TO THE MOTtOK OF BODIES IN WATEB
id since TjL and t^ are less than i'^^
^ and — are greater than —
w
w
w
513
4
*rhe Burface of the water at AA and BB is therefore higher
%n at CC and it takes the form shown in Fig. 363,
jTwo sets of waves are thus fomiedj one by the advance of the
and the other by the stream hnes at the st^m, and tbesal
motions are transmitted to the surroanding water, where
energy is dissipated. This energy, as well as that lost in
idy motions, must of necessity have been given to the water by^
■" ship, and a corresponding amount of work has to be done by
:ie i^hip's propeller. The propelling force required to do work
t&al to the loss of energy by eddy motions is the eddy resist^^f
ice, and the force required to do work equal to the energy of™
waves set up by the ship is the wave resistance.
To reduce the wave resistance to a ndnimuin the sliip should
Qiade very long, and should have no parallel body, or the
sfcire length of the ship should be devoted to the entrance and
JH On the other hand for the frictional resistance to be small,
ne area of immersion must be small, so that in any attempt
design a ship the resistance of which shall be as small as
Dsgible, two conflicting conditions have to be met, and, neglecting^
eddy resistances, the problem resolves itself into making th^
of the frictional and M^ave resistances a minimum.
Total rensiance. K R is the total resistance in pounds, T/ the_
ictiotml resistance, n the eddy resistance, and r„ the wave
stance,
^^Tf + rt + r^.
be frictional resistance r/ can easily be determined when tl;
latitre of the surface is known. For painted steel ships / is'
[iractieally the same as for the varnished boards, and at 10 feet
Br second the frictional resistance is therefore about \ lb. per
tiare foot, and at 20 feet per second 1 lb, per square foot, The^
If satisfactory way to determine r^ and r„ for any ship is tijH
lake experiments upon a model, from which, by the principle of
ailarity, the corresponding resistances of the ship are deducec
be horsepower required to drive the ship at a velocity of V fe
sr second is
EV
the
a^v^l
th6
HP =
550'
To determine the total resistance of the model the apparatus
bown in Fig, 362 is used in the same way as in determining the
ictional resistance of thin boards.
L. a.
33
283. BetemiixLation of the resistance of a Bhlp firotn the
resistance of a model of the ship.
Tu obtain the resistance of the ship from tiie experimental
i^eaifltaiicQ of the model the priociple of sim^Uantyj as stated bf
Mr Froudej is used. Let the linear dimensions of the ship be D
tunes those of the modeL
Cmre^pofuUfig speeds. According to Mr Fronde's theor>% for
any speed V„ of the models the speed of the ship at which m
resistanoa mnst be compared with that of the model, or th
ocMn^ponding speed V* of the ship, is
CorrBsponJUng remstances. If lU is the resistance of the model
^ the velocity Vmt and it be assmmed that the coefficients d
'friction for the ship and the model are the same, the redstamoe &
of the ship at the corresponding speed V* is
Ab an example, snppose a model ouedxteentb of the siis
of the ship; the corresponding' speed of the ship will be fonr timei
the speed of the model, and the resistance of the ship at eon^
spondiBg speeds wOl be 1^ or 4006 times the resistance of tJie
Comadtbii for the difffsrente of the coeffim^nts of frietimi for HiM
mocbl OffM? 0hip. Tlie material of which the immersed surface
tlie model is made is not generally the same as that of the ship^
and conseqeentiy R, mnst be corrected to make allowance for the
difference of ronghnees of the surfaces. In addition the ship i
very much longer than the model, and the coefficient of frictiaii»
0f«ii if the surfaces were of the same degree of roughness, would
tbetBfore be less than for the model.
let Am he the immersed surfoce of the model and A.
the shipw
LeC fm be the coefficient of friction for the model and /, for
ahipi the values being made to depend not only upon the rougJiD*
bat also upon the length. If the resistance is assumed to vaiy
V*t Ihe frietioinal resistance of the model at the velocity Y^ is
•ai for Iks ship at the corresponding speed Y« the frietioQii
r,=/,A.V/.
A, = A»I>*
But
SBBISTANCS TO THE MOnOH OF BODIES IN WATER
515
and, therefore, r,=yiA«V«*D*
Then the Tesistance of the ship is
R.= (R,r.)D' + r.
= {R.^r.(^l)}D..
Determination of the cwrve of resistance of the ship from the
curve of resistance of the model. From the experiments on the
model a curve having resistances as ordinates and velocities as
abscissae is drawn as in Fig. 365. If now the coefficients of
friction for the ship and the model are the same, this curve, by
an alteration of the scales, becomes a curve of resistance for the
ship.
For example, in the figure the dimensions of the ship are
supposed to be sixteen times those of the model. The scale of
Telocities for the ship is shown on CD, corresponding velocities
being four times as great as the velocity of the model, and the
scale of resistances for the ship is shown at EH, corresponding
resistances being 4096 times the resistance of the model.
H
Fig. 365.
^r Froude^s method of correcting the curve for the difference of
the €X>€fficienfs of friction for the ship and the model. From the
formula
rm — JmAm V m ,
516
HTBRAUI.ICS
the frictional resistance of the model for several values of V,
is calculated, and the curve FF plotted on the same scale as aad
for the curve RR. The wave and eddy making resistance at an;
velcw::ity is the ordinate between FF and RR. At velocities c
200 feet per second for the model and 800 feet per second fi
the shipj for example^ the wave and eddy making resistance is bt
measured on the scale BG for the model and on the iK^ale EH t
the ship.
The frictional reaiBtance of the ship is now calculated from
form u la r, =/jA«W, and ordinate^ are set down from the c
FFj equal to r«j to the scale for ship rpsistance. A third curve
thus obtained J and at any velocity the ordinate between this cum
and RR is the resistance of the ship at that velocity. For exampl
when tlie ship has a velocity of 800 feet per second the resistanc
is aCj measured on the scale EH.
EXAMPLES.
(1) Taking skin Miction to be D'4 lb. per sq^uare foot at 10 faet per
aecondt find the skill resietaace of a sMp of 12^000 square feet immeraed
surface at 15 kxtots (1 kuot = I'd9 leet per second). Also tind the horsepQwt*r
to drive the ship against tliis resistance.
(2) If the skin friction of a ship is 05 of a pound per square foot otf
immersed surface at a speed of 6 knota» what horsepower will prob*blj
be required to obtain a speed of 14 knota* if the immersed surface is 18,0110
square feet ? You may assume the maximmn speed for which the ship Is
designed is 17 knots.
(8) The resistance of a vessel is deduced from that of a model V^ ^
Hnear size. The wetted surface of the model is 20*4 square feet, the ekii
friation per square foot^ in fresh water, at 10 feet per Beoond is 0*3 lb., asd
the index of velocity is 1'94. Tlie skin friction of tlie vessel in salt i
is 60 iba. per 100 square feet at 10 knots, and the index of velocity is !
The total resistance of the model in frei^h water at 200 feet per jimt
146 lbs. Estimate the total resistance of the vessel in salt water 4
speed corresponding to 200 feet per minute in ihe model. Lond. Uo. M
(4) How from model experiments may the resistance of a
inferred ? Point out what corrections have to be made. At a
800 feet per minute in fresh water, a model 10 feet in length with * '
akin of 24 square feet has a total resistance of 2*39 lbs*» 2 lbs. being dws ^
skin resistance^ and 39 lb, to wavemaking. What will be the total i
ance at the corresponding speed in salt water of a diip 25 tijnes tbe I
dimensions of the model, having given that the surface friction per tfp^
foot of the ship at that speed is 13 lbs, 7 Lond, Un. 1906*
CHAPTEK XIIL
STEEAM LINE MOTION.
284. Hele Shaw's experiments on the fLow of thin
slieetB of water,
lVofessor Hele Shaw* has very beautifully shown, on a small
e^le, the fomi of the stream lines in moving masses of water
imder varying circumstances^ and has exhibited the change from
stream line to sinuous, or rotational flowj by experiments on the
flow of water at varying velocities between two parallel glass
plnUft^. In some of the exi:)eriinents obstacles of various forma
were placed between the plates, past which the water had to flow,
and in others, channels of various sections were formed through
which tlie water was made to flow. The condition of the water
as it Howed between the plates was made visible hy mixing with
it a certain quantity of air, or else by allowing thin streams of
coloured water to flow between the plates along with the other
water. When the velocity of flow was kept sufficiently low,
whatever the form of the obstacle in the path of the water, or
the form of the channel along which it HoweJj the water persisted
ia stream line flow* When the channel between the plates was
tnade to enlarge suddenly, as in Fig. 58, or to pass through an
orifice, as in Fig. 59, and as long as the flow was in stream lines,
•no eddy motions were produced and there were no indications
Lof \ij8^ of head. Wien the velocity was sufficiently high for the
I flow to become sinuous, the eddy motions were very marked.
^len the motion was sinuous and the water was made to flow
l^st, ohstacles similar to those indicated in Figs. 110 and III, the
^ikter immediately in contact with the downstream face waa
*h<mn to be at rest. Similarly the water in contact with the
tttiiiulor ring surrounding a sadden enlargement appeared to be
'^t TL^i and the assumption made in section 51 was thus justified.
* Proc€tdinff» of Naval ArchiUcU, 1S9T &iid IS9B. Engineer, Aug. IS^T and
518
HTDRAUUCS
Wlien the Arm was dan^ channels and eiQUOCUS, ilie stnoonglf
movm^ iratrr appeared to be separated from the sides c( llie
ckannel bj a thm filiii of water, wbich Profeaoor Hele Sbaw
5a.«g^?>ted was mornig in stream lines, tke velocity of which ia
the ilm dii ": as tbe surface of the channel is approached,
Thie experi  miao indicaSfMi that a stmilar film surroimded
obc^t;kcIe^ of fiu{i4tke wad other forms placed in flowing wata't
And :r waa ^ reA by Pr^jfe^tsr Hele Shaw thatj snrrT^unding
a ioip ^s it ^ tkrom^ sdH waler, there ia a thin film moving
m. 5t7*?ikiiL b ely to the shi ''i© shearing forces between
wiiicii Ami .. : wOodiBS waiei ^ ap eddy motions which
.accv:cL:i: fan ^km faid i ci of th lip.
285. C^rrvd abeam Una m< i*
Ijtt A &ju9 oC feiil be moriii^ i trred ^ream lineSf and kl
JlKFx tr — ^^Mieoftheato fines.
A~ Aaj t tW rai&a Kfi aiun* of the etiieaiti iiBfi
O ci^iie \^. .^ riitB«faiB net&i o&dd finrrcrandnig ^B
Ic' '^ :tr jif T^r :x:i: f this element.
.: ':»r ^T Tr^scLT^ per z.ziiz ^reik on the face hd.
   > "!•: 7rf>«?cLre per uziit area on the face ae,
^ t: .:  jitivn t ine :;ing>rnt to the stream line at c
i'^i * fiw:i :* :~e fiice^s >f and a^.
: » ~7 f :i:»e <cr^az: line at •:,
c^^il:::! jjitf 5 .z i Tem:^. plane the forces acting
W
^^ ^rix^ "^ r>f ^ icrji^ alzng" the radius away
:r^ss^^ r r ictinx aI n^ :'~^ ra^iins towards the
>v>^ ^ i.
..■^""w...
= '~"r::.r • D.
STREAM LINE MOTION
519
TOtind the bend of a river, Oc is horizontal and the component of
"W along Oc is zero.
Then ^=^^ (2).
dr g r ^ ^
Int^prating between the limits R and Bi the difference of
p i Oflom pe on any horizontal plane at the radii R and Ri is
^"■^"I/r r^ ^^^'
"wliich can be integrated when v can be written as a function of r.
Now for any horizontal stream line, applying Bemouilli's
equation,
or
Differentiating
— + TT 18 constant,
to 2g
w 2g
Idp, vdv _ dH
w dr gdr ~~ dr
.(4).
Then
.(5).
Fig. 367.
Free vortex. An important case arises when H is constant for
all the stream lines, as when water flows round a river bend, or as
in Thomson's vortex chamber.
1 dp_ vdv
w dr~^ gdr
Substituting the value of ^ from (5) in (2)
— yyo dv_wt^
g dr~ g ' r'
from which rdv + vdr = 0,
and therefore by integration
vr = constant = C
620 HTDBAULIC8
Equation (3) now becomes
Pi  p C f^dr
w g JR f*
= C!/i_j_\
2g\B? RiV'
2sr
Forced vortex. If , 849 in the turbine wheel and centrifugal
pump, the angular velocities of all the stream lines are the same,
then in equation (3)
t? = ciir
w g jR
and tL^^ ^dr
g jR
Scou/ring of the banks of a river at the bends. When water
runs round a bend in a river the stream lines are practically
concentric circles, and since at a little distance from the bend the
surface of the water is horizontal, the head H on any horizontal
in the bend must be constant, and the stream lines form a free
vortex. The velocity of the outer stream lines is therefore less
than the inner, while the pressure head increases as the outer
bank is approached, and the water is consequently heaped up
towards the outer bank. The velocity being greater at the inner
bank it might be expected that it will be scoured to a greater
extent than the outer. Experience shows that the opi)osite effect
takes place. Near the bed of the river the stream lines have a
less velocity (see page 209) than in the mass of the fluid, and, as
Lord Kelvin has pointed out, the rate of increase of pressure near
the bed of the stream, due to the centrifugal forces, will be less
than near the surface. The pressure head near the bed of the
stream, due to the centrifugal forces, is thus less than near the
surface, and this pressure head is consequently unable to balance
the pressure head due to the heaping of the surface water, and
crosscurrents are set up, as indicated in Fig. 367, which cause
scouring of the outer bank and deposition at the inner bank.
ANSWERS TO EXAMPLES.
Chapter I.
<1) 8900 lbs. 9872 lbs. (2) 784 lbs. (8) 78*6 tons.
(4) 6880 lbs. (5) 17*1 feet. (6) 19800 lbs.
(7) P=865,e00 lbs. X=126ft. (8) 91 foot. (9) 089 in.
<10) 1595 lbs. per sq. ft. (11) 5400 lbs. (12) 87040 lbs. ^
Chapter II.
(1) 85,000 eft (8) 298 ft.
(4) Depth of C. of B. =2195 ft. BM= 14*48 ft. (5) 191 ft. 69 ft.
(6) Less than 18*8 ft. from the bottom. (7) 1*57 ft. (8) 2*8 ins.
Chapter m.
(1) 945. (2) 146 ft. per sec. 183 c. ft. per sec. (8) 2601 ft.
(4) 115 ft (5) 58*8 ft. per sec. (6) 63 c. ft. per sec.
(7) 44928 ft lbs. 136 h. p. 8*84 ft. (8) 862 ft. 114 ft. per sec.
(9) 1048 gallons.
Chapter IV, page 78.
(1) 8025. (2) 3906. (8) 37636. (4) 5 ins. diam.
(6) 8567 ins. (6) 763. (7) 86 ft. per sec. 115 ft.
(8) 806. (9) 895. (10) 058. (11) 1448 ft. per sec.
(12) 294 ins. (18) *60. (14) 572 gaUons. (15) 22464 lbs.
<16) 6206. (17) 558 eft. (18) 755. (19) 102 c. ft.
(90) 875 ft 186 lbs. per sq. foot 545 ft. lbs.
(21) 105 ins. 2985 ins. (22) •688 ft. (23) 452 minutes.
(24) 17*25 minates. (25) 629 sq. ft. (26) 142 hours.
Chapter IV, page 110.
(1) 18,170 eft. (2) 415 ft
(8) 699 c ft per sec. 129*8 c. ft. per sec. (4) 2685.
(6) 1828. (7) 483 c. ft. per sec. (8) 1676 ft.
(9) 892 ft (10) 222 ft. (11) 552 ft. (12) 23,500 c. ft.
(18) 24,250 eft (14) 105 minutes. (15) 284 h. p.
522 AN8WEB8 TO SXAJfPLES
Chapter V.
(1) 27*8 ft. (2) 142 ft. (4) 65. (5) 2888 ft
(6) 1076. 14 ft. 88 ft. 782 ft. "0961 ft
(8) 61 eft. 2854 ft. 258 ft. 9 ft. (9) 28 per oent
(10) 197. 21ft. 80 ft. 26 ft 24 ft 15 ft (11) 864 eft
(12) 808 eft. (18) 674 ft •267 ft 772 ft (14) 21 ft
(16) 186 c. ft per sec. (16) F=08181bB. /•006868.
(17) 1028. (18) 704. (19) 29 ft per sec
(20) 4*4 c. ft. per sec. (21) If pipe is clean 46 ft
(22) 28 ft. 786 ft. (28) Diri^ castiron 6'1 feet per mile.
(24) 818 feet (26) 1 f oot
(27) —A(\ — » ^ = friction per unit area at unit velocity.
(28) 108 H. p. (29) 1480 lbs. 1*08 ins. (80) 002825.
(81) A:=004286. n=l84. (82) (a) 940 ft. (b) 2871 h. p. (88) 045
(84) If d»9^ v^5 ft. per sec., and/=0066, ;i=102 and Hsl82.
(86) 1487x10*. Yes. (86) 6816 ft (87) lhour48nmL
(88) 46,260 gallons. Increase 17 per cent (89) 296*7 feet
(40) 6 pipes. 480 lbs. per sq. inch.
(42) Velocities 6*18, 608, 816 ft. per sec. Quantity toB=60c.ftperim:
Quantity to C —66*6 c. ft. per min. (46) *468 c. ft per sec
(46) Using formula for old castiron pipes from page 188, t? =8*62 ft per se
(47) 291 ft. (48) d=88ins. d,=84in8. d2=29 ins. d32'2ini
(49) Taking G as 120, first approximation to Q is 144 c. ft. per sec.
(61) d=4'18 ins. v=20*66 ft. per sec. p=840 lbs. per sq. inch.
(68) 7069 ft. 801ft. C,.=ll9. C^ for tubes « 606.
(64) Loss of head by friction ='78 ft.
A head equal to ^ will probably be lost at each bend.
(66) 43*9 ft. 936 in.
(67) 7i=68'. Taking 006 to be/in formula ^=y^ . v=166 ft. persec.
(68) Vi=88 ft. per sec. from A to P. V8=496 ft. per sec. from B to P
1^3= 1375 ft. per sec. from P to C.
Chapter VI.
(1) 886. (2) 1lft. diam.
(8) Value of m when discharge is a maximum is 1857. «■= 17*62. C=121
Q = 76 c. ft. per sec.
(4) 0136. (6) 16,250 c. ft. per sec. (6) 8 ft.
(7) Bottom width 15 ft. nearly. (8) Bottom width 10 ft. nearly.
(9) 630 c. ft. per sec. (10) 96,000 c. ft. per sec.
(11) Depth 785 ft. (12) Depth 10*7 ft.
(18) Bottom width 76 ft. Slope 00052. (17) C = 875.
ANSWERS TO EXAMPLES 523
Chapter Vm.
124*8 lbs. 456 H. p. (2) 628 lbs.
104 lbs. 58*7 lbs. 294 ft. lbs. (4) 960 lbs.
261 lbs. 47 H. p. (6) 218. (7) 57 lbs. (8)^ 194 lbs.
Impressed velocity=285 ft. per sec. Angle =57'. (10)* 181 lbs.
•98. 678. 68. (18) 192.
YeL into tank —84*8 ft. per sec. Vel. through the orifice =41*6 ft. per
sec. Wt. lifted » 108 tons. Increased resistance » 2880 lbs.
125 lbs. 8*4 ft. per sec. 1*91 h. p.
Work done, 575, 970, 1150, 1940 ft. lbs. Efficiencies ^, '50, f , 1.
1420 H. p. (18) 9875. (19) 82 H. p.
8666 lbs. 161 H. p. 62 per cent.
Chapter IX.
105H.P. (2) <^=29*. V^=7ft.persec
14*8 per min. ll"* from the top of wheel, ^s70^
1*17 eft (5) 4*14 ft. (8) 82^2'.
1025 ft. per sec. 1*7 ft. 5*8 ft. per sec. 11° to radios.
r= 247 ft. per sec. (18) </)=47''80'. a=27"20'.
79' 16'. 19** 26'. 58.
856 ft per sec 6'24'. 28^ ms. llfins. 12^89'. 16§ms. 82jins.
99 per cent. <^=78% a=18'. <^=120', a=18°.
</)=158'28'. H= 77*64 ft. h. p. =14116. Pressure head =48*58 ft.
d=l*22ft. D=2*14ft. Angles 12'' 45', 125*22', 16° 4'.
<^=184'58', ^=16'25', a=9n0'. H.p.=2760.
616. Heads by gauge,  14, 85*6, 81. U=51*5 ft. per sec.
<^ = 158"68',a = 25'. H. P. =29*8. Eff. = *957.
Blade angle 18° 80'. Vane angle 80° 25'. 8*92 ft. lbs. per lb.
At 2' 6" radius, 6^ 10°, <^ = 28° 45', a = 16° 24'. At 8' 8" radius, ^=12° 11'
(f> = 78° 47', a = 12° 46'. At 4' radius, $ = 15° 46', (f> = 152° 11', a = 10° 21 .
79° 80'. 21° 40'. 41° 30'.
58° 40'. 86°. 24°. 86*8 per cent. 87 per cent.
12° 45'. 62° 15'. 81° 45'.
r=45*85. U=77. V,.=44. v^=86. Ui=28. €=7875 per cent.
'86 ft. 40° to radius. (80) About 22 ft.
H. p. =80*8. Eff . = 92*5 per cent.
Chapter X.
47*4 H. p. (2) 26°. 681 ft. per sec. 94 ft. 60 ft.
55 per cent. (4) 52*5 per cent.
^^ = 106 ft. ^'=61 ft. ^1=56 ft.
g 2g w
11° 86'. 106 ft. 47*4 ft.
60 per cent. 161 h. p. 197 revs, per min.
700 revs, per min. 81 in. Radial velocity 14*2 ft. per sec.
15*6 ft. lbs. per lb. 806 ft. 14 ft. per sec.
624 ANSWERS TO EXAMPLES
(16) v=2864 ft per sec V=ll8.
(16) d=a9im8. D SB 19 ins. Revs, per min. 472 or higher.
(17) 15 H. p. 96 ins. diam. (18) 6*6 ft
(19) Vels. 128 and 241 ft per sec Max. accel. 2*82 and 4*55 ft per see.
per sec
(20) 898 ft lbs. Mean friction head=*0268, therefore work doe to frictiaa
is very small.
(21) 4*61 H. p. 1191 c ft per min. (22) 888.
4ii^QQi IT
(28) p^ — ^^' Acceleration is zero when ^=j(iii+2), m being any
integer.
(27) Separation in second case.
(29) 676 and 66*1 lbs. per sq. inch respectively, h. p. sd'14.
(81) 798 ft 25*8 ft 41*98 ft (82) 648. (88) *6.
(84) Separation in the sloping pipe.
Chapter XI.
(1) 8150 lbs. (2) 8*88 H.P. hours. (5) 4*7 ins. and 9*7 ins.
(6) 8*888 tons. (7) 175 lbs. per sq. inch.
(8) 2*8 ft. per sec. (9) 4*2 minutes.
Chapter Xn.
(1) 80,890 lbs. 1425 H. p. (2) 8500 H. p.
(8) 4575 lbs. (4) 25,650 lbs.
^ [All numb^i refer to pages.] ^^^^^^^^H
Abw>kte Tdoeilj 262
B«ods, loaa of bead due to 140 ^^M
Acoeleratiot] io pumps* effect of (ue
BemouilU'a theorem 39 ^^^
Eaciprocalmg piuup)
appUed to centrifugal pomps 4 IS, 1
Aecumxilatars
423, 437* 439 J
Air 4S1
applied to turbineB 334, 349 ^M
difleratiluU 482
examples on 46 ^^
bydnulic 486
eipen mental illtiBtrations of 41 ^^
Air gauge, mTeited
ext^n^ion of 4B ^^
Air vesflels on pumps 451 , 455
Borda'a mouthpiece 72 ^^B
Angtil&r momoutum S73
Boui^diucBq'B theory for discharge of a ■
Angaidir moment uirii rate of change of
weir 104 ^J
eqoAl to a conply 274
Boyden diffuser 314 ^^B
Appold eentrlfu^al pump 415
Brotherhood hyilraulic engine 501 ^^H
Aiimedueifl 1, 189, 195
Buoyancy of Hoating bodies 21 ^^M
eeclioDft of 216
centre of 23 ^^M
Archimedes* principle of 22
^^H
Canal boats, steering of 47 ^^H
497
Capstan, hydraulic aOl ^^U
AliDo«phede piretttire 8
Centre of buojanory 23 ^^H
Centre of prestjore 13 ^^
Bfteon 1
Centrifugal force, effect of in discbarge ^^^
:B*me« ftnd Goker 129
from water wheel 286 1
Centrifugal head 1
Sftsin's eiperitnents on
in oentrifugal pumps 405, 408, 409, 1
eftlih^tion of Piboi tnbe 245
419t 421 ^J
diiitribtilioQ of preflsiur^ in Ihe plane
m reaction turbtnea 303, 334 ^^M
of an ori^ee 59
Ceutrifugul pumpa, see Pumpe ^^H
' distiibatioD of velod^ in the orosa
CbanjislB V
leoUon of a ahaimel 206
circular, de£^th of flow for maitimum 1
diatributbii of vdoeity in the crous
discharge 221 ■
eectiOD of a pipe 144
circular, depth of flow for maiimum 1
iiatribntion of Telocity in the plane of
velocity 22U ^J
mn orifice 69, 244
coefficienta for, in formnlaa of ^H
1 flow uj chAnoeU 182
Bazin 166, 137 ^H
flow OTcr dama 102
Darcy and Basin 183 ^H
1 flow over weirs 89
QanguiUet and Kuttt^r 184 ^H
flow through ori^ces 56
coeffioientB for, in logarithmic for ^f
form of the jet from orifioea 63
muloe 300203 1
3«&iii'fl formulae for
ooefficienta, variation of 190 ^J
cbAoneLfl 182, 185
curves of veloc i ty and di seharj^ for 222 ^H
ori^ccfs aharp' edged 57, 51
dimensions of, for given flow deter* ^^H
¥«loeily al any depth in a Tertical
miued by approximation 225227 V
section of a channel 212
diameter of, for given maiimum die* M
▼docity at any point in the orosa
charge 224 ^H
teetion of a pipe 144
distribution of velocily in croaa i€^ ^^H
weir, Hat created 9*J
tion of 208 ^H
1 ueir, iharp*creeted 9799
earthy of trapezoidal form 2111 ^^M
L^ weir, sill of amaU thickneas 99
eroAion of earth 216 ^^M
t»DEX
in
of
US
of I7e
lor flow in
of SaS
lor mx^ SOI, SOT
i S3S
1*1 ohetbod c^f determining tlie
in 187
im
^ ' bller« 183, 184
It of 231
19t~lSe9CMI
of 119
iJBt, 101. lae. 197. flOQ, i06
183» ISA, 187, 195, 901
l€3, 184, 107. ISS. 195, 197,
1&4, 186, 187, 19a,
1^, 1^. im, 187, 901. 207
18a. 1§4
1^ lift. ISI, 906
18i, 186, 1S7. 2^
r m196
of, te prcD vdoaty
GuiMivifaiar m
Gummt meleia tSS
«alil»r»tioii of 940
aurlej 938
BiAkdl S40
Curred ttf«uii linia mottcm 51§
CyUndncal mouthpiece 79
Dftms, flow oyer 101
ej^i^zimeDtit on flow in ehonneb M
experun«Dt« on flow in pipei 12S
fortDQlA for flow in «*haJTLwak 1^
formolA for flow ld plpei 111
Deacon's w&Btewaler meter ^
Denftitj 3
of gmooline 11
of keroiltie 11
of m«t^ii]j 8
of pare water 4 11
Depth of e^Gtre of pr««»nie 13
Di&griim of prefi£me on a pkne uM
lU
Diagram of pressure on i vcrSioi] out^
16
Diagram of work done in a recipiooiUiig
pump 413, 459. 467
Diflbrenti&i aooDmalaior 489
IHJfiscBDtial ga.nee 8
Diftdivge
cocffii^ient o^ for oriflcei 60 if
Orifieefl)
ooeflkient of, for Vdntohweicr IS
of ft channel ITS {»et Olmnneii}
ov«T weirs 83 (<ee Weirs)
thrODgh notches S5 i*ct Noichet)
throngb oriflceB 50 {Met Orifioei)
thiongh pip^ 112 {$€e ¥ipmj
DiBtrihotion of ?eloeity on gtom MClka
of a channel 206
Dusribataon of veloeitj on crote aeite
of a pipe 148
DiTecgeol monthpitoes 7S
Dock cuBson 181, 192, 926
Doelis, floftling 31
Diowned nappes of weiis 9fi, 100
Drowned orificee 65
Drowned weirs 9$
appcoKunate fixmink for 301,
eoeffictenla for in Beam's fonei^
187
eoeffieienla for in Duej vid Ba^'i
f<»^nJa 183
noeflkientis for in Qanguillel
£utter*fi formula 184
smaoa of 216
Elbows, ktea of head due to 140
Ra^itee, ludxanlic mi
BnH^eriiood 501
BMlie S03
mm £04
of eextli channels 316
mDEX
527
Ximiapliii, $oltitiotta to wMah iire given
in the text —
P Boiler, time of omptjing Uirough a
moQthpieoe 78
Centrifugal pumpa^ determlaniioa of
preftmre heftd at Inl^ and outlet
^^^■Omtril^igal pampB, dimensiooa for a
^M given diaoharge 404
^FCetitrifugal pumpat seriea, number
f of whtida for a givua lift 435
CenCfifogal pi;mp», veiocitj at which
deliT«ij start? 412
Clu»iiitle» droukr diameter, for a
glvoi majrimiini discharge 224
^CMsnela, diameter of siphon pipes
to given same dboharge m an
aciitidaot 224
" aun«)a» dimensions of a canal for
i gtven flow and slope 2*25, 22t»» 227
dinchargo of an earth
225
flow ini for giren section
•nd slope 223
12, 489
P^lotttiog docket height of metacentre
of 34
^ floating doeks, water to be pumped
ftom S3
Bad of wa£«r 7
Hjdraolic ma<!hitiery, capacity of
aoecunQlator for working a hy
draulic crane 489
Hjdraulic motors variation of the
pressure on the plunger 470
IsifMcl on vanesj form of vane for
w«t«T to enter wit bout ^hmk and
leave in a given direction 271
ItDpat^t on vanest presanrc on a vane
when a jet tn contact with ia tamed
Ihnmgh a given angle 267
Impaot on vaneci, turbine wheel,
form of vanes on 272
Impact o» vanes, turbine wheel,
water leaving the vancf; of 269
trnpaot on vanes, work done on a
vane 271
Metaoentrei height of, for a floating
■ 34
Eitre, height of, for a ship 36
bpiece, dteeharg^ through^ into
[ eondenssr 76
rfehpieoe, time of emptying a
boiler by m^ui« of 76
oQlhpleoe^ time of emptying a
ttmxww hj means of 78
" ftar of, for a given die
charge 152. 153
I Pipes, discharge along pipe connecting
two T titter voir^ Idl, 154
KpipeB m parallel 154
>1^pet preavore at end of a service
pipe 1§1
dock
^tamplea {cunt.)
Pontoon^ dimifntionB for given da^
plaeement 29
Pressure on a flap valve 13
PreBsure on a masonry dam 13
Pressure on the end of a pontoon 18
Iteoiprocating pump htted with an
air vessel 470
Reciprocating pump, horsepower of,
with long delivery pipe 470
Eeciprocating pump, presBure iu an
air vessel 470
Eeciprocating pump. Beparation io^
diameter of suction pipe for no 469
Eeciprocating pump, separation in
the delivery pipe 464
Eeeiproeating pomp, aeparation in,
number of strokes at which sepa
ration takes place 458
Reciprocating pnmp, variation of
pleasure in, due to inertia fonies
470
Eeaerroira, time of emptying by weir
108
Beservoirs^ time of emptying through
orifice 78
Ship, height of metacentre of 26
Transmisaion of fiuid preasofe 12
Turbine, design of vanes and de
termiDAtion of effitfiency of, con
sidering friction 331
Torbine^ design of vaues and de
termination of eMciency of, frio
tion neglected 322
TurbiDt;, tlimeDMiQEis and form of
varies for given horse power 341
Turbine, double eompartment parallel
!iow 349
Turbine, form of vanes for an out
ward flow 311
Turbine, hammer blow in a supply
pipe 385
Turbine, velocity of the wheel for a
given he^d 321
Venturi meter 46
Water wheel, diameter of breast
wheel for given horaepower 290
Weir, CO rr fiction of cooflloient for
velocity of approach 1)4
Wtjir, dischargt^ of 94
Weir, disoharge of by approsimation
loa
Weir, time of emptying reservoir by
of 110
4
Fall of free level 51
Fire hose nozzle 73
Flap valve, prefigure on 18
centre of IS
Floatliig bodiofl
Archimedes, principle of 22
booyanoy of 21
oentre of buoyancj of 23
coaditions of eq.iiilibnTim of 21
cottt&inimg ^at«i, stubilitj of 29
exomplefl on 34* 516
inetaoentTe of 24
resifiiftnoe to the motion of 507
email di^plaoements of 24
HtAbility of eqoilibrium, ooodltion of
24
stability of flo&iing dock 33
eiabililj of reot&Dgular pontoon 26
stability of veeael i^oQt&miDg wat^r 29
stability of Teaad wholJy imtnerBed
30
weight of fitiid displaced 23
FloAtiBR docks 31
stability of 33
TioKis, doable 237
rod 239
iriirfac« 237
Flow of water
deinitions reladng to 36
enetrgy per pound of flowing vater 88
m QpetL obdnnek 176 (k^i* ChannelB)
over damB IQl (*« DaniB)
over weirs 81 [tite Weirs)
tbmugh notches 80 (#«« NotcheF)
through orificea 50 {s^e Orifices)
through pipes 112 {*ee Pipes)
FlnldB (iiquidj^)
U ttBt 319
exam pies on 19
ooin^resaible 3
deohity of 3
flow of, tkrough orifices 50
iijconipressible 3
in motion 37
presijnre in^ la the same in all direo
tionii 4
pi^sanre on an area in 12
pressnra on a horizontal plane in, is
constant 5
Bpecifio gravity of 3
steady motion of 37
stream line motion in 37, 517
trflDfimisaioii of pressure by 11
used in U tubea 9
Tlaeosi^ of 2
Forging press, bydrsnlic 492
Foumoyron turbine 307
Friction
coefficients of, for ships* surfaces 509,
615
effect of, on discharge of centrifagal
pump 421
e^ect ofp on velocity of exit from Im
pulse Turbine 373
effect of, on velocity of eiit from
Poncebt Wheel 297
Froude^a experiments on fimd 507
in centrifugal pumps 400
in cbanuela 180
in pipes 113, 113
Friction (fiwit.)
in reciprof!Ating pumps 449
m tnibinea 313, 321, 339, 37ft
Ganguillet and Kutter
coefficients in formula of 1%, lU
experiment* of 183
formula for channels 184
formula for pipe« 124
Oasoline, ffpeciSc granty of 11
Gauges, preeaure
differential H
inverted air 9
Inverted oil 10
Gauging the flow of water 234
by an orifice 235
by a weir 247
by chemical means 2B8
by floats 239 {*« Floats)
by meters 234, 2^1 {tee Met<
by Pitot tnbea 241
by weighing 2M
examples on 260
in open ehannela 236
in pipes 251
Glaz^ earthenware pipes 136
Gurley*B current meter 238
Hammer blow m a long pip€ 3S4
Haskell's cuirent meter 240 _
Hastie'fi engine 503
Head
position 39
preasure 7* 89
velocity 39
High pressure pump 471
Historical devetopment of pip
channel formulae 231
Hook gauge 248
Hydraulic aocumulator 4S6
Hydraulic capstan 501
Hydranh'o crane 494
double power 495
valves 497
Hydraulic differential accumulftti
Hydraulic engines 501
orank effort diagram for 509
Hydraulic forging press 402
Hydraulic gradient 115
Hydraulic inteusifler 491
Hydraulic machines 485
conditions which vanea of,
flatisfy 270
examples on 489, 505
joints for 484
ma^cimum efhclency of 29S
packings for 485
Hydraulic mean depth 119
Hydraulic motors, vwiationR of j
in, due to inertia forces 469
Hydraulic ram 474
Hydraulic riveter 499
Hydraulics, definition of 1
INDEX'
629
419
»r on ▼anes 261 {see Vanes)
. in hydranlio motors 469
} in reciprocating pumps
ent of 14
f^oge 9
;auge 9
lydranlic 491
valves for 492
5team 493
turbines 275, 818 {$ee
)
n hydraulic work 485
er 255
meter 253
!cific gravity of 11
hydraulic packiugs 486
formulae for flow
i 192
5
plottings
iB 195
27, 133
499
I 17
,vity of 8
barometer 7
U tubes 8
leight of 24
aste water 254
55
i
, 75, 251
lertia 14
ane of floating body 25
id law of 263
54
of discharge for
73
3
il 71, 76
le 78
of velocity for 71, 73
73
73
3
n 78
h, under constant pressure
d at entrance to 70
ptying boiler through 78
MonthpieoM (cofit.)
time of emptying reservoir throogh
78
Nappe of a weir 81
adhering 95
depressed 95
drowned or wetted 95
free 95
instability of the form of 97
Newton's second law of motion 268
Notation used iu connection wiUi vanes,
turbines and centrifugal pumps 278
Notches
coefficients for rectangular {ae Weizs)
coefficients for triangular 85
rectangular 80 ($ee Weirs)
triangular 80
Nozzle at end of a pipe 159
Nozzle, fire 74
Oil pressure gauge, inverted 10
calibration of 11
Oil pressure regulator for turbines 877
Orlflcea
Bazin's coefficients for 57, 61
Bazin's experiments on 56
coefficients of contraction 62, 56
coefficients of discharge 57, 60, 61»
63
coefficients of velocity 54, 57
contraction complete 58, 67
contraction incomplete or aappraiiecl
53, 63
distribution of pressure in plane of
59
distribution of velocity in plane of 59
drowned 65
drowned partially 66
examples on 78
flow of fluids through 50
flow of fluids through,^ under constant
pressure 75
force acting on a vessel when water
issues frcon 277
form of jet from 63
large rectangular 64
partially drowned 66
pressure in the plane of 59
sharpedged 52
time of emptying a lock or tank by
76, 77
Torricelli's theorem 51
velocity of approach to 66
velocity of approach to, effect on dis
charge from 67
Packings for hydraulic machines 485
Parallel flow turbine 276, 342, 868
Parallel flow turbiue pump 487
Pelton wheel 276, 377, 880
Piezometer fittings 139
Piezometer tubes 7
1H0KX
lomvi
C Itt
/liii=
la
'^
for «Mt koa^ ne* cni «M 190^
Itl, 121, IS, ttl
lor mml tiraied 121
133, isa
B«r ftad old ISS, 137«
Il§
i)M» 14s
riveted nm, IM
VDod l3S« ISB
wtohqJu ir^Mi l^'i, las. 13$
n In GingniUfi aud Ku tier's fi^nstUa
«tt«t iitm, new atid old 125
for elftsped caithentiriue 125
for stod riveted 184
for wood {ripet 125, 164
T&miioii of, vitfa 8erdoe 123
OOiUuciiiig llirie TCKTrmtv I06
oounectiag two r«eeir?oirs 149
oOimfifftJiig two teaer¥oirs, difljueter of
for ^ma ditcbarge 152
ootksl velocity in 128
DftrcT'a foimula fo(r 122
deilettmimtioD of the coeffioient C,
m giTcn in tables hj logarilhmio
pbtiing las
dtaiaptcr of, for git«a discharge
dianieler for miniitiiiin eont 158
diameter varjing 160
diTided into ivo brmnches 154
elbow4 for 111
empirical fortniila for head lost in 119
empirieal formula fot relocitj gf Sow
in lia
equation of flow ia 117
examples on flow in 149162, 170
expert uieoiai detertnmation of loss of
head by frictiot] m lid
experimentfi on djstributioii of velocity
m U4
expetimeny on flow in, Gjitici&ni of
136
fl^tednieiita on loes of head at bends
142
«ipfrimi>nts on loss of head in 122,
i2d, m, 131. laa, 136
experimentB on loss of head in,
criticism of 138
flow through 112
flow diminishuig at imifonn rate id
157
formula for
Ch^^y 119
Darcy 1^2
Fipia {oiut,)
fortnnia fqr (^oni.)
logarithmic 125, xBi^ las, m4
Beynolds ISl
ma&m%ry <4 146
f^docltj'&t anj poiat in 1 oroM
section of 143
Mctioo In, loss of bead bf 113
determination of 116
Gangoillet and Enttet*« tmmak kt
124
gmugmg the flow in Ml
hAmmer blow in ^1
bead lost St entrance of 70, lU
head loet by friclioD m US
head loat by friction in, emiukit
formula for 119
head loel by Mction Iil, exaaiplii «a
150^163, 170
bea^i loflt by friction in, logimUmiiO
formula for ItB^ 133
head rcquinjd lo ^ive tieioaity Vi
wai^F m the pip« 146
bead rvij aired to give velodtj to v^^
in the pipe, «{]rproximate ^lu iU
hydratilio gfadient lor 113
hydratilii; mean depth of lid
joints for 485
law of frictional resistaiioe foi n^D^
the crjtical velocity 130
law ol 5iotion&l xesistanoe fat, Wjjv
the critical velocity 125
Htniting diameteir of 165
logarithmic formula for 135
logarithm it; fomiola for, coeffideiiti
m 138
logarithmic formula, use of, foi pii^
tical ealculftiionB 136
logarithinic plotting^ for 136
nozzle at diichafige end of, ares of
when energy of jet ia a madmoiD
when momentum of jet is a matl*
mum 159
pie7,ometer fillings for in^
pre&snre on bt^nds of 160
ptesaure on a cyhnder in 169
preafiure on a plate in 1S8
problems 147 {*^^ Problema
pumping watc^r thjougb long p$»^
diameter of for m inimajn ooet lii
reg [stance to motion of Haid In Uf
TiBing iibove hydtHulle gndieat 111
short 153
siphon 161
temperBtore, elfeet of, ua
flow in 131, 140
tranBrnieeion of power along, h; l^
draulic pressure 162
Taluee of C in the formula r=0%^
for 120, 121
variation of C in the formola r=C^'i
for 123
IlfDEX
T^rimuon of the diftch«rge of» with
strviee 133
veltK^itj of £lo^ ullowable in 102
velocity, heftd riiqiiired to give Telocity
to water in I ■IB
velo^itj, varifition of, in a crosaMetion
of n pipe f43
virtual siope of 115
Pitot tube 241
CAlib ration of 245
Poacelet water wh^l 294
P'oDtooii, pfert«Tire on end of 18
P^tiou hei^ 29
hctsa, forging 493
PfttS, bydraulid 49S, 49B
pMinre
&t anj point in a 6uid 4
atmonphenc, In feet of w&ter B
ginges 8
head 7
mettinred in feet of water 7
on a horizontal piftue in a fluid 5
on A plate in a pipe IBS
on pipe bend!4 IHti
Principle of ArcliimedeB 19
Principle of siiniliirity 64
Problems, ^olutiooB of which ait given
in the t*ixt —
chiLnnelH
difljueter of, for a given maximnm
discharge ^i24
dimennionB of, for a given flow
««rth ditfchmge along, of given di*
mensioms and elope 22 \
_ flow in, of given section and Blope
■ n^
^L filope of^ for tninimttm coat 227
^m fioltitaons of, b) approxiination
^ 225^227
, ptpe^
acting as a siphon 161
connecting three Psaervoira 155
connecting two reflen^lri 149
diameter of, for a given diMchar^
152
divided info two branches 154
head lost in^ wlien flow diminiahea
ftt nniform rate 157
lam of hyad in, of varying diameter,
leo, lei
ptumplng water along, diameter of,
for minininro oo»t 1S8
with nozzle at the end 158, 159
E^pnlajon of shipa by water jeta 279
Pmmping water through long pipea 158
Pomp*
G«>ntdfngai 392
advantoges of 43y
Appold 415
Bemonilli's equation applied to
4U
Fiimpi (conL)
centrifugal {conL]
centrifugal head, effect of variation
of on discharge 421
tientrifugal head, impressed on the
water by the wheel 405
design of, for given diucharge 402
discharge, effect of the variation
of ttiu cen trifilgal head and loaa
by friction on 419
diflcharget head ^velocity curve at
^ero 409
diiicharge, varmtion of with the
h£ad at oo octant speed 410
discharge, variation of with speed
at constant heod 410
effideucies of 401,
efEoieneieft of; (experimental de>
tennination of 401
examples od 404» 412, 411, 418,
43o, 478
form of vanes 39B
friction, effect of on dbchorgt 419,
421
general equation for 421, 425, 428,
4B0
gross lift of 400
head discharge cMirve at constant
velocity 410, 412, 427
head loet in 414
head, variation of with discharge
and flpeed 418
bead velocity curve at constant
discharge 429
head velocity curve at zero dis
charge 409
kinetic energy of water at exit
a99
limiting height to which single
wheel pump will raiae water 4B1
liuiiting velocity of wheel 404
lo9He£t of head in 414
multi'fltafie 433
»eriea 433
spiral casing for 394, 429
starting of 395
suction at 431
Bulzer series 434
Thomson's vortei chamber 397, 407.
422
triangles of velocities at Inlet and
exit 397
vane angle at exit, effect of variation
of on the efficiency 415
velocity discharge curve at conatant
heiul 411,412,421.428
velocity, heaildischargf curve fof at
couBtant 410
velocity head, sp^^iil amngement
for con%'erting into pressure head
422
velocity, limiting, of rim of wheel
404
532
tKDKX
Telocity of whirl, ratio of, to fdocitj
of outlet e4g« of vmoe S98
TOftex chfijuber of 397, 407, 423
with whirlpool or vortex ch«imber
B97, ^37, in
work done on wmler by 9t7
oompTesHed air 477
duplfx 473
exAmpW tm 458, 464, 460» 47S
kigh pramtire 472
hjdrftnlic mm 476
{iiacking« for plangera of 47S 436
reciproeating 439
ftOOGle ration, elfeot of on preasura
in oylinder of a 446, 448
acce)er&tioD of the plaoger of 444
acceleration of the water id delivery
pipe of 448
acceleration of the water in inotiOD
pipe of 445
ur Teaael on dellverj pipe of 454
air veaiel on eactioD pipe of 451
air vessel on taction pipci effect of
on a©p«kration 462
eodfflcient of diaaharge of 442
diagram of work done by 443, 4S0,
U% 467
diaebftigtt coefficient of 443
di3pl«i 473
examples on 458, 464, 469i 470
480
frictino, variation of ptesattre in the
cylinder due to 449
head lost at auction, valve of 468
head lONt bj Mction in the auotion
and deUvery pipes 449
high pre»Bur6 plunger 471
preasufe in cylinder of when the
plnngoj move& with simple har
monic motion 446
preaisurt^ in the cylinder, variation
of dtie to friction 449
separation in delirerv pipe 463
separation dnring suction atroke
456
separation daring suction stroke
when pi anger moves with fiimple
harmonic motion 4o8* 461
alip of 442, 451
snetion stroke of 441
suction stroke, separation in 456,
4fll, 463
Tangye duplex HH
Tertlcal single acting 440
work done by 441
work done by, diagmm of 445, 459,
467
turbiue 396, 42€
heafl'disoharge currea at oonatant
speed 427
lor bine {ei>nt.\
head'Tdocity ctLrv@«
discharge 429
inward 6ow 439
multii^lage 433
parallel 6ow 437
veloei ly diBcharge carvei si eciif biat
head 42B
Worthington 432 J
work done by 44B H
work done by, diagram of (KiS
oiprocating pompfi)
work done by, series 43$
Heaotion turbines 301
limiting bead for 367
series 367
Beaction wheels 301
efficieney of 304
Beciprocating pumps 4S9 Ue*
Bectangular pontoon, etabihtv uf
Beotangalar sharp edged weir §1
Beclangular sluices 65
Rectangular wetr with end oostiM
tiona 88
Begnlation of turbines 306, 317, tU
328, 343
Regulators
oil pressure, for impulse tnrbine R71
water proisure, for impuife tur^i
379
Relative velocity 265
as a vector 2&%
Beservoirs, time of emptying
orifioo 76
ReservoirB, time of emptying over «
100
Reaiatance of ^hip 510
Bigg hvdrAulic engine 503
Rivera,' flow of 191, 207* 211
Rivera, acouring biinkB of *%2(>
Riveter, hydraulic 500
Scotch turbine 301
Second law of motion 2G3
Sepamtion (*i*f Pumpe)
Sbaq>edKed orificea
Bazm'a experiments on 66
di Kiribati on ofveiocity in the ptanj
pressure in the plane «f 59 m
table of coeffioltmta for, «bfll
traction is complete 57, 6^
table of coeffloi«nta for, whtn
traction it iuppreased 63
Sharp edged weir 81 {»et Weirs)
Shipa
ppopalsion of by water J eta
rcBistanoe of 510
resistance of, from model oXJ
streani line theory of the
of 510
Similarity, principle of M
J!
1
ITTOEI
S33
forming part of ^ueduct 31G
lei
pomiM U2t 461
65
gnl&ttng tuirbmes {ite Tarbinefl)
»lme 11
!Vien« 11
I, Tftriatton of^ with temperatiire
re water 4
iatioi] of, with tempemture 11
r ol
% hoij ^ 35
ig dock SI
ig ve&st^ QOQtuDiEig water 29
tgttlar pontoon 26
paodoD of fluids 37
nien&ifier 493
line motion 37, 1^3, 517
I SIS
Shawns eicperiments oti 384
line thaotj of resistaiioe of
m 510
oentrifagal piunp 431
redprooating pnmp 441
Hbe of tttrbine S06
sontnu^tiQn of a cuirent of
m
lar^ment of a current of
67
inltist&ge pamp 434
1 contracttiQH 53
K on dlsdmrge from ori6de
\ on diicharge of a w«ir @3
, i«wers and aitnedu^li, par
I of, and values of  in
ftU 4=*^; 195
Baofai
maxlmtim velooitiea of
^ 15
a and ^ in Bazln's formnla
EM of V atid i as deC^^rmmed
Ep£rimenlalij and >^ c&leuliit«d
oa lopintbniie tarmaia^ lUS,
»l208
tttita for duna 103
for shju^ed^ed oHfioe,
ocmmlete 57. 61
lbr «narp< edged orifiee,
duppre^sed 63
ahftrp^adged wetri 89,
for Venturi meters 46
intieli;, v^loeities aboTe which
takes place 216
minimum dopei for varjing valuea
of the hjdraultc mean depth of
brick £)lLivnneb that the veloci^
tnajf not be less than 2 ft, per
eeeond 215
momenta of Inertia 15
PeltOD wheels « particutt^m of 377
pipes
lead, slope of Rnd velocity of 6ow
in 128
reasonable valaes of y and n in
the formula ft = ^^ 188
Taloes of^ C in the formula
r^Qs^mi 120, 131
valiaea of f in the formula
2ffd
vaiiaefl of n in Oangaitlet and
Kutt^*s formula IMl iM
valuea of n and k in the formula
(=itr» 137
resistance to motion of boards in
Anids 509
turbines, peripheral Teloeiiiea and
heads of Inward and ouc^ard f ow
333
uBefnl data 3
Thomson f centrifagal pump, vortei
chamber for 397, 407, 423
principle of simihurity 63
turbine 323
Time of emptying tank or reservoir by
an orificp 76
Time of emptyinj;? a tank or reaervoir
by a weir 10<J
Torricelli^s theorem 1
proof of 51
Total pressure 12
Triaiiji^ular notches 80
dificlifljge through 85
TnrMnes
axial flow 276i 343
axial flowt impuUe 368
axial flow, prGHaure or reaction 343
axial fiow, section of tJie vane with
the variation of the rftdius 344
Bernuuilli*a equation a far 334
best peripheral velocit> for 329
central vent 320
oentrifuffal head impressed on watar
by wheel of 334
eone 'd&\t
design of vanes for 346
efficiency of 315, 331
examples on 311. 321, 333, 331, 341,
340, 385, 387
flow through, eflect of diminishing,
by means of moreable guide blades
363
flow through, effect of dimiuiabing
by means of aluices BM
si* INDEX ^^^^^^H
Turliiuti {€f>ni.)
Turbines {cotit,^ ^^^
Jlow ibitttigbf e£f60t of dttniniBhuig
ImB of head in 313, 3*21 ^^M
on velooHy O'f exit H63
imixed ^ow 350 ^^^1
FoDlaine, rcigiilAting slaioes 343
form of vaiipi of 355 ^^^
form of Timefl for 308, 347, Bti5
guide blade regulating gesaF for
l^oomeyroD B06
352354
general formula for 31
in open atneam Bi\0
geoeral forToalik, including frietion
Swain gate for 374
315
triangle:^ of velocltjea for 3S5—
guid^ btadefl for 320, 3S6, 048, 35S,
dm
362
wheel of 351
fuidd blades, «fleet c^f ehonging the
Hiagaju falln 313
direotton of 362
oil preiiiore regulator for 377
gotde blades, vaHatioti of the angle
outward fiow, 275, 306
of, for parallel flow turbil)«ii 344
Beroooilh^B equatini^B for 334«
hors« power, to develop a given
339
33y
b«st peripheral volooity far, at inlet
impnltte 3O0, 369384
329
AtM flow 363
Boy den 314
diffuser for 314
«Kampleft 387
for high beads 373
doable 316
form of Tanea for 371
examples on 311^ 3B7
Girard 369, 37D, 379
eiperimental deteniiinatioa of iha
best velocity for 329
bjdmiilic efRcieiicy of 37l» 378
in airtight chamber 370
Foiirneyrt>n 307
oil preBRure regoUtor for 377
lo{>&6B of head in 313
radiiLl flow 370
Niagara falls 318
tmngles of vetocitiei! for 37*2
Buctjoa tube of 308, 317
triaogleH of velocities tor conaideriog
triangles of velocities for 308
friction 373, 376
work done on Iho wheel per lb, d
water prii&gnra r^^lator for 379
water 310, 315
w&ter pressure reijulator, hjdraiili«
pamllel flow !^76, 34:^
valve for 3#2
adjustable guide bludes for MS
water presBiire regulator, water filter
Bernouilli'fl rquationa for S4S
for 383
deaipi of vanes for 344
work doue on wheel per Ibu of water
double compattmtut MB
272, 277» 323
example! on 349, 3B7
incliuation of vanes at inUt of wheel
regulation of the flow to 341*
308, 321, 344
triangle of velooitieai for 344
inclinatioo of vanea at outlet of whe^l
reaction 301
ms, 321, 345
aiial flow 276342
in open atream 360
cone 359 ^^H
inward iow 275, 318
inward flow 275, 318 ^^M
Btroooilli's eqiiatmiii for 334, 339
miied flow 350
be«t peripheral vdooity for, at
outward flow 306
inlet 329
paraUel flow 276342
central vent 320
Scotch 302
examples on 321, 931, 341, mi
aeries 368
Batpeiimeotal determination of the
regulation of 306, 317. 318, 323, $18,
best velocity for 329
S50. 352, 360, 362. BU
for low and variable fulls 328
Scotch 301
Franeia 320
Bluicea for 305, 307, 316. 317, 31»,
borieontal ajcis 337
327, 328, 348, 350, S61, Mi
lom&B in S21
auction tube of 306
Themaon 334
Bwain gale for 364
to develop a given horse* power
ThomBon's inward flow 323
339
to develop given boraepower 338
tnanglea of veboitiea for 322, 326,
trianglea of velocities at inlet an!
332
outlet of impulae 372, 376
work done on the wheel per lb. of
triangles of velocities at inlet and
water 321
outlet of inward flow 3U«
limiting head for reaction turbine
triangles of velocitieu at inlet mbi
867
outlet of nii:xed flow 356
INDEX
535
Tiiztiiii08 (ecmt.)
triangles of velocities at inlet and
outlet of oatward flow 844
triangles of velocities at inlet and
outlet of parallel flow 844
types of 800
vanes, form of
between inlet and outlet 866
for inward flow 321
for mixed flow 351, 356
for outward flow 311
for parallel flow 344
Velocity of whirl 278, 310
ratio of, to velocity of inlet edge
of vane 332
velocity with which water leaves 384
wheels, path of water through 312
wheels, peripheral velocity of 333
Whitelaw 302
work done on per lb. of flow, 275,
304, 815
Turning moment, work done by 273
Tweddell's differential accumulator 489
XJ tubes, fluids used in 9
Undershot water wheels 292
Valves
crane 497
hydraulic ram 476
intensifier 492
Luthe 499
pump 470472
Vanes
conditions which vanes of hydraulic
machines should satisfy 270
examples on impact on 269, 272, 280
impulse of water on 263
notation used in connection with
272
Pelton wheel 276
pressure on moving 266
work done 266, 271, 272, 275
Vectors
definition of 261
difference of two 262
relative velocity defined as vector
266
sum of two 262
Velocities, resultant of two 26
Velocity
coefficient of, for orifices 54
bead 89
of approach to orifices 66
of approach to weirs 90
relative 265
Venturi meter 44, 75, 251
Virtual slope 115
Viscosity 2
Water
definitions relating to flow of 38
Water (cont,)
density of 3
sjpecific gravity of 3
viscosity of 2
Water wheels
Breast 288
effect of centrifugal forces on water
286
examples on 290, 886
Impulse 291
Overshot 288
Poncelet 294
Sagebien 290
Undershot, with flat blades 292
Welra
Bazin's experiments on 89
Boussinesq's theory of 104
coefficients
Bazin's formula for
adhering nappe 98
depressed nappe 98
drowned nappe 97
flat crested 99, 100
free nappe 88, 98
Bazin's tables of 89, 93
for flatcrested 99, 100
for sharpcrested 88, 89, 93, 97,
98
for sharpcrested, curve of 90
Rafter's table of 89
Cornell experiments on 89
dams acting as, flow over 101
discharge of, by principle of simi
larity 86
discharge of, when air is not ad
mitted below the nappe 94
drowned, with sharp crests 98
examples on 93, 98, 108, 110
experiments at Cornell 89
experiments of Bazin 89
flatcrested 100
form of, for accurate gauging 104
formula for, derived from that of a
large orifice 82
Francis* formula for 83
gaugiug flow of water by 247
nappe of
adhering 93, 96
depressed 95, 98, 99
drowned 95, 96, 98
free 88, 95, 98
instability of 97
wetted 95, 96, 99
of various forms 101
principle of similarity applied to 86
rectangular sharpedged 81
rectangular, with end contractions
82
side contraction, suppression of 82
sill, influence of the height of, on
discharge 94
sill of small thickness 99
S36 on>KX
(eoMf.)
rvqvired to lomr wter in ^doeitj of ftpproaAh, effect of on
mai l oil tj meini of 109 dudbaige 90
wi:>si forss of 101 wide llfttcrested 100
w^txXT c< ftnxoeefa, eoneeuoo ol Wbitelaw tozbine 302
eot&eaemt for 9f Whole prenore 12
viuac^T of effTCMdL, eonectiofi of Worthington multistage pomp 433
■pies on M
CAvumssx : ruxrsp bt johx cx^t, m.a. at thx unitersitt press.
Mr. Edward Arnold's List of
Tecbnical & Scientific Publications
Electrical Traction.
Bv ERNEST WILSON, Whit. Sch. M.LE^E.p
F^tjfcssor or t^tectricat Engineering in the Siecncns LAbarAtot^v Ring's. CoUcf^Ci Lciiiciant
AND FRANCIS LVDALL, H.A., B.Sc,
New Edition. Rewritten and Greatlv Enlarged.
Two volumes, sold separately* Demy 8vo,, doth.
Vol L with about 300 Illustrations and Index* 15s. net*
Vol. IL, with about 170 Illustrations and Index, 15s. net.
In dealing with ihis everincrcaaingly important subject ihe authors have
divided 4he work into the two branches which are^ for chronological
and other reasons, most convenient^ namel}', the utilization of direct and
alternating currents respectively. Direct current traction taking ihe
first place, the first volume is devoted to electric tramways and direct
current electric railways. In the second volume the application
of threephase alternating currents to electric railway pronlems is
considered in detail, and finally the latest developments in single
phase alternating current traction are discussed at length,
A Text Book of Electrical Engineering.
BY Dr* ADOLF THOMALEN<
Translated by GEORGE W. O. HOWE, M*Sc., Whit. Sch*,
A*MJ*E*E*,
Lecturer in Elef^trtcn! Enffiwermg »l ibe Ceniral Technical College, Santb Kcnsuigiotn
With 454 Illustrations* Royal 8vo., cloth, 15s. net.
This translation of the** Kuoe Lehrbuch derEiectrotechnik "' is intended
to fill the gap which appears to exist between the elementary
icxibooks and the specialiied works on various branches of electrical
engineering. It includes additional matter which is to be introduced
into the third German edition^ now in preparation. The book is
concerned almost exclusively with principles, and does not enter
into details of the practical construction of apparatus and machines,
aiming rather at laying a thorough foundation which shall make the
study of works on the design of machinery more profitable. Only
the simplest elements of the higher mathematics are involved.
Alternating Currents.
A TesctBook for Students of En^neenng*
BY C, G. LAMB, M.A., B.Sc,
Clue Colkg«i, CAinbridgc ; A«a^ocUte Meiitber cf the In&iituUon ef Klectrical Ensineen;
AisQCJAtje of tbe City and GuiUji of Londlon, In^iitute.
viii + 325 pages. With upwards of 230 inustratjoni* Demy Svo* cloth,
IDS, 6d* neL
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Sciences Tripos at Cambridge*
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Technical and Scigntific PuUimtions
Organic Chemistry for Advanced Students*
I
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of Orgmiiic Cbemisiry in (tie Untvcfikj of [.eeds^ and A^ocwt^ ofOweM
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book is written for students who have already completed an
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and general outline of the more important branches of the subject.
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textbook*
The Chemical Synthesis of Vita! Products and
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L Lectures on Theoretical and Physical
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BY DR. J, H, VAN T HOFF,
■ Profcsjbor ot Cbembtry ai the University af Berltts.
\ Translated by R* A. LEHFELDT, D*Sa,
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Introduction
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A MaxiLiAl of the Natural History and ladustrial AppticatiQns of tb6
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'^rofeuor of Botmny axid Lcciurer on Fiufstfy in th^ Cliy of l^ndoti College, and fctrmerly in the
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