APPLIED THERMODYNAMICS
FOR ENGINEERS
BY
WILLIAM D. ENNIS, M.E.
MEMBER OF AMERICAN SOCIETY OF MECHANICAL ENGINEERS
PROFESSOR OF MECHANICAL ENGINEERING IN THE
POLYTECHNIC INSTITUTE OF BROOKLYN
316 ILLUSTRATIONS
FOURTH EDITION, CORRECTED
NEW YORK
D. VAN NOSTRAND COMPANY
25 PARK PLACE
1915
COPYRIGHT, 1910, BY
D. VAN NOSTRAND COMPANY
COPYRIGHT, 1913, BY
D. VAN NOSTRAND COMPANY
COPYRIGHT, 1915, BY
D. VAN NOSTRAND COMPANY
THE SCIENTIFIC PRESS
ROBERT DRUMMOND AND COMPANY
BROOKLYN, N, V.
PREFACE TO THE THIRD EDITION
THIS book was published in the fall of 1910. It was the first
new American book in its field that had appeared in twenty years.
It was not only new in time, it was new in plan. The present edition,
which represents a third printing, thus demands careful revision.
The revision has been comprehensive and has unfortunately
somewhat increased the size of the book a defect which further
time may, however, permit to be overcome. Such errors in statement
or typography as have been discovered have been eliminated.
Improved methods of presentation have been adopted wherever
such action was possible. Answers to many of the numerical prob
lems have now been incorporated, and additional problems set.
Expanded treatment has been given the kinetic theory of gases
and the flow of gases; and results of recent studies of the properties
of steam have been discussed. There will be found a brief study of
gas and vapor mixtures, undertaken with special reference to the use
of mixtures in heat engines. The gas engine cycle has been subjected
to an analysis which takes account of the varying specific heats of
the gases. The section on pressure turbines has been rewritten, as
has also the whole of Chapter XV, on results of engine tests the
latter after an entirely new plan. A new method of design of com
pound engines has been introduced. Some developments from the
engineering practice of the past three years are discussed such as
Orrok's condenser constants; Clayton's studies of cylinder action
(with application to the Hirn analysis and the entropy diagram),
the Humphrey internal combustion pump, the Stumpf uniflow
engine and various gasengine cycles. The section on absorption
systems of refrigeration has been extended to include the method
of computing a heat balance. Brief additional sections on applica
tions of the laws of gases to ordnance and to balloon construction
are submitted. A table of symbols have been prefixed to the text,
and a " reminder " page on the forms of logarithmic transformation
iii
iv PREFACE
may be found useful. The Tyler method of solving exponential
equations by hyperbolic functions will certainly be found new.
In spite of these changes ; the inductive method is retained to
the largest extent that has seemed practicable. The function of the
book is to lead the student from what is the simple and obvious
fact of daily experience to the comprehensive generalization. This
seems more useful than the reverse procedure.
POLYTECHNIC INSTITUTE OF BROOKLYN,
NEW YORK, 1913.
PREFACE TO THE FIRST EDITION
" APPLIED THERMODYNAMICS " is a pretty broad title ; but it is
intended to describe a method of treatment rather than unusual
scope. The writer's aim has been to present those fundamental
principles which concern the designer no less than the technical
student in such a way as to convince of their importance.
The vital problem of the day in mechanical engineering is that
of the prime mover. Is the steam engine, the gas engine, or the
turbine to survive? The internal combustion engine works with
the wide range of temperature shown by Carnot to be desirable;
but practically its superiority in efficiency is less marked than its
temperature range should warrant. In most forms, its entire charge,
and in all forms, the greater part of its charge, must be compressed
by a separate and thermally wasteful operation. By using liquid
or solid fuel, this complication may be limited so as to apply to the
air supply only ; but as this air supply constitutes the greater part
of the combustible mixture, the difficulties remain serious, and there
is no present means available for supplying oxygen in liquid or solid
form so as to wholly avoid the necessity for compression.
The turbine, with superheat and high vacuum, has not yet
surpassed the best efficiency records of the reciprocating engine,
although commercially its superior in many applications. Like the
internal combustion engine, the turbine, with its wide temperature
range, has gone far toward offsetting its low efficiency ratio ; where
the temperature range has been narrow the economy has been low,
and when running noncondensing the efficiency of the turbine has
compared unfavorably with that of the engine. There is promise
of development along the line of attack on the energy losses in the
turbine; there seems little to be accomplished in reducing these
losses in the engine. The two motors may at any moment reach
a parity.
vi PREFACE
These are the questions which should be kept in mind by the
reader. Thermodynamics is physics, not mathematics or logic.
This book takes a middle ground between those textbooks which
replace all theory by empiricism and that other class of treatises
which are too apt to ignore the engineering significance of their
vocabulary of differential equations. We here aim to present ideal
operations, to show how they are modified in practice, to amplify
underlying principles, and to stop when the further application of
those principles becomes a matter of machine design. Thermo
dynamics has its own distinct and by no means narrow scope, and
the intellectual training arising from its study is not to be ignored.
We here deal only with a few of its engineering aspects ; but these,
with all others, hark back invariably to a few fundamental princi
ples, and these principles are the matters for insistent emphasis.
Too much anxiety is sometimes shown to quickly reach rules of
practice. This, perhaps, has made our subject too often the barren
science. Rules of practice eternally change ; for they depend not
alone on underlying theory, but on conditions current. Our theory
should be so sound, and our grasp of underlying principles so just,
that we may successfully attack new problems as they arise and
evolve those rules of practice which at any moment may be best
for the conditions existing at that moment.
But if Thermodynamics is not differential equations, neither
should too much trouble be taken to avoid the use of mathematics
which every engineer is supposed to have mastered. The calculus
is accordingly employed where it saves time and trouble, not else
where. The socalled general mathematical method has been used
in the one application where it is still necessary ; elsewhere, special
methods, which give more physical significance to the things de
scribed, have been employed in preference. Formulas are useful
to the busy engineer, but destructive to the student; and after
weighing the matter the writer has chosen to avoid formal definitions
and too binding symbols, preferring to compel the occasionally
reluctant reader to grub out roots for himself an excellent exor
cise which becomes play by practice.
The subject of compressed air is perhaps not Thermodynamics,
but it illustrates in a simple way many of the principles of gases
PREFACE vii
and has therefore been included. Some other topics may convey
an impression of novelty; the gas engine is treated before the steam
engine, because if the order is reversed the reader will usually be
rusty on the theory of gases after spending some weeks with vapor
phenomena ; a brief exposition of multipleeffect distillation is pre
sented; a limit is suggested for the efficiency of the power gas
producer ; and, carrying out the general use of the entropy diagram
for illustrative purposes, new entropy charts have been prepared
for ammonia, ether, and carbon dioxide. A large number of prob
lems has been incorporated. Most of these should be worked with
the aid of the slide rule.
Further originality is not claimed. The subject has been written,
and may now be only represented. All standard works have been
consulted, and an effort has been made to give credit for methods
as well as data. Yet it would be impossible in this way*to fully
acknowledge the beneficial influence of the writer's former teachers,
the late Professor Wood, Professor J. E. Denton, and Dr. D. S.
Jacobus. It may be sufficient to say that if there is anything good
in the book they have contributed to it ; and for what is not good,
they are not responsible.
POLYTECHNIC INSTITUTE OP BROOKLYN",
NEW YORK, August, 1910.
CONTENTS
CHAPTER
PAGE
TABLE OF SYMBOLS xiii
I. THE NATURE AND EFFECTS OF HEAT 1
II. Ta^ HEAT UNIT: SPECIFIC HEAT: FIRST LAW OF THERMODYNAMICS 11
III. LAWS OF GASES: ABSOLUTE TEMPERATURE: THE PERFECT GAS . 19
IV. THERMAL CAPACITIES SPECIFIC HEATS OF GASES: JOULE'S LAW . 32
V. GRAPHICAL REPRESENTATIONS: PRESSUREVOLUME PATHS OF PER
FECT GASES 43
VI. THE CARNOT CYCLE 76
VII. THE SECOND LAW OF THERMODYNAMICS 84
VIII. ENTROPY 92
IX. COMPRESSED AIR 106
The cold air engine: cycle, temperature fall, preheaters, design of
engine: the compressor: cycle, form of compression curve,
jackets, multistage compression, intercooling, power consump
tion: engine and compressor relations: losses, efficiencies, en
tropy diagram, compressor capacity, volumetric efficiency,
design of compressor, commercial types: compressed air trans
mission.
X. HOTAIR ENGINES 145
XI. GAS POWER . 162
The producer: limit of efficiency: gas engine cycles: Otto, Car
not, Atkinson, Lenoir, Brayton, Clerk, Diesel, Sargent, Frith,
Humphrey: practical modifications of the Otto cycle: mixture,
compression, ignition, dissociation, clearance, expansion, scav
enging, diagram factor: analysis with variable specific heats
considered: principles of design and efficiency: commercial
gas engines: results of tests: gas engine regulation.
XII. THEORY OF VAPORS 230
Formation at constant pressure: saturated steam: mixtures:
superheated steam: paths of vapors: vapors in general: steam
cycles: steam tables.
is
x CONTENTS
CHAPTER PAOB
XIII. THE STEAM ENGINE ......... 298
Practical modifications of the Rankino cycle: complete and incom
plete expansion, wiredrawing, cylinder condensation, ratio of
expansion, the steam jacket, use of superheated steam, actual
expansion curve, mean effective pressure, back pressure, clear
ance, compression, valve action: the entropy diagram: cylinder
feed and cushion steam, Boulvin's method, preferred method:
multiple expansion: desirability of complete expansion, conden
sation losses in compound cylinders, Woolf engine, receiver
engine, tandem and cross compounds, combined diagrams,
design of compound engines, governing, drop, binary vapor
engine engine tests: indicators, calorimeters, heat supplied,
heat rejected, heat transfers: regulation: types of steam engine.
XIV. THE STEAM TURBINE ........ 363
Conversion of heat into velocity: the turbine cycle, effects of fric
tion, rate of flow, efficiency in directing velocities: velocity
compounding, pressure compounding: efficiency of the turbine:
design of impulse and pressure turbines: commercial types and
applications.
XV. RESULTS OF TRIALS OF STEAM ENGINES AND STEAM TURBINES 397
"Economy, condensing and noncondensing, of various commercial
forms with saturated and superheated steam: mechanical effi
ciencies.
XVI. THE STEAM POWER PLANT ....... 415
Fuels, combustion economy, air supply, boilers, theory of draft,
fans, chimneys, stokers, heaters, superheaters, economizers,
condensers, pumps, injectors,
XVII. DISTILLATION .......... 430
The still, evaporation in vacuo, multipleeffect evaporation.
FUSION :
Change of volume during change of state, pressuretemperature
relation, latent* hoat of fusion of ice.
LiQUKKAOTION OF
Preswure and cooling, critical temperature, cascade system, regen
erative apparatus.
XVTII. MECHANICAL RKFIUCIKUATION ....... 454
Air machines: reversed cycle, BellColeman machine, deiw air
apparatus, coefficient of performance, Kelvin warming machine:
vaporcompression machines: the cycle, choice of fluid, ton
nage rating, icemelting effect, design of compressor: the absorp
tion system, heat balance: methods and fiolde of application: ice
making; commercial efficiencies.
CHARACTERISTIC SYMBOLS
F = Fahrenheit;
C = Centigrade;
R = R6aumur;
 Radiation (Art. 25);
= gas constant for air = 53,36 fUb.
=00686B.t.u.;
= ratio of expansion;
P,p = pressure: usually Ib. per sq, in.
absolute;
V, v = volume, cu. ft: usually of 1 Ib. ;
= velocity (Chapter XIV);
Tj Z = temperature, usually absolute;
!T=heat to produce change of tem
perature (Art. 12);
E = change of internal energy;
7 = disgregation work;
Q,#=heat absorbed or emitted;
= total heat above 32 of 1 Ib. of
dry vapor;
h = heat emitted;
^heat of liquid above 32 F;
=head of liquid;
c= constant;
^specific heat;
s= specific heat;
r=gas constant (Art. 52);
= internal heat of vaporization;
ratio of expansion;
jrj
7= specific heat;
m
dT
'dH
T
== entropy;
34,5 Ibs. water per hour from and at 212
F.=l boiler E.P.;
42 42 B.t.u. per min. = 1 H.P.;
2545 B.t.u. per hour = 1 H.P,;
17.59 B.t.u. per minute =1 watt;
W) w= weight (Ib.);
W  external mechanical work;
S= piston speed, feet per minute;
A =piston area, square inch;
k specific heat at constant pres
sure;
I  specific heat at constant volume;
k
y=r>
n=polytropic exponent;
N,n= entropy;
e coefficient of elasticity;
^external work of vaporization;
pro =mean effective pressure;
J> = piston displacement (Art, 190);
r.p.m. revolutions per minute;
H.P, =horsepower;
d= density;
gf32.2;
778= mechanical equivalent of heat;
459.6(460)= absolute temperate at
Fahrenheit zero;
L=heat of vaporization;
x  dryness fraction ;
7 =factor of evaporation;
7fc= entropy of dry steam;
Tie = entropy of vaporization;
nw = entropy of liquid.
CHAPTER I
THE NATURE AND EFFECTS OF HEAT
1. Heat as Motive Power. All artificial motive powers derive their
origin from heat. Muscular effort, the forces of the waterfall, the wind,
tides and waves, and the energy developed by the combustion of fuel, may
all be traced back to reactions induced by heat. Our solid, liquid, and
gaseous fuels are storedup solar heat in the forms of hydrogen and carbon.
2. Nature of Heat. We speak of bodies as "hot" or "cold," referring
to certain impressions which they produce upon our senses. Common
experimental knowledge regarding heat is limited to sensations of temper
ature. Is heat matter, force, motion, or position ? The old " caloric "
theory was that "heat was that substance whose entrance into our bodies
causes the sensation of warmth, and whose egress the sensation of cold."
But heat is not a " substance " similar to those with which we are familiar,
for a hot body weighs no more than one which is cold. The calorists
avoided this difficulty by assuming the existence of a weightless material
fluid, caloric. This substance, present in the interstices of bodies, it was
contended, produced the effects of heat; it had the property of passing
between bodies over any intervening distance, Friction, for example, de
creased the capacity for caloric; and consequently some of the latter
" flowed out," as to the hand of the observer, producing the sensatiou of
heat. Davy, however, in 1799, proved that friction does not diminish the
capacity of bodies for containing heat, by rubbing together two pieces of
ice until they melted. According to the caloric theory, the resulting water
should have had less capacity for heat than the original ice : but the fact is
that water has actually about twice the capacity for heat that ice has ; or,
in other words, the specific heat of water is about 1.0, while that of ice is
0,504. The caloric theory was further assailed by Rumford, who showed
that the supply of heat from a body put under appropriate conditions was
so nearly inexhaustible that the source thereof could not be conceived as
being even an " imponderable " substance. The notion of the calorists
was that the different specific heats of bodies were due to a varying capac
ity for caloric ; that caloric might be squeezed out of a body like water
from a sponge. Kumford measured the heat generated by the boring of
cannon in the arsenal at Munich. In one experiment, a gun weighing
2
APPLIED THERMODYNAMICS
113,13 Ib. was heated 70 E., although the total weight of borings produced
was only 837 grains troy. In a later experiment, Rtimford succeeded in
boiling water by the heat thus generated. He argued that "anything
which any insulated body or system of bodies may continue to furnish tuithout
limitation cannot possibly be a material substance." The evolution of heat,
it was contended, might continue as indefinitely as the generation of
sound following the repeated striking of a bell (1).*
Joule, about 1845, showed conclusively that mechanical energy
alone sufficed for the production of heat, and that the amount of heat
generated was always proportionate to the
energy expended. A view of his apparatus
is given in Fig. 1, v and h being the verti
cal and horizontal sections, respectively, of
the container shown at <?. Water being
placed in 0, a rotary motion of the contained
brass paddle wheel was caused by the de
scent of two leaden weights suspended by
cords. The rise in temperature of the
FIG. 1. Arts. 2, 30. Joule's Apparatus,
water was noted, the expended work (by the falling weights) com
puted, and a proper correction made for radiation. Similar experi
ments were made with mercury instead of water. As a result of
his experiments, Joule reached conclusions which served to finally
overthrow the caloric theory*
3, Mechanical Theory of Heat. Various ancient and modern
philosophers had conceded that heat was a motion of the minute
particles of the body, some of them suggesting that such motion
* Figures in parentheses signify references grouped at the ends ot the chapters.
THE NATURE AND EFFECTS OF HEAT 3
was produced by an "igneous matter/' Locke denned heat as "a
very brisk agitation of the insensible parts of the object, which pro
duces in us that sensation from which we denominate the object
hot ; so [that] what in our sensation is heat, in the object is nothing
but motion." Young argued, "If heat be not a substance, it must
be a quality; and this quality can only be a motion." This is the
modern conception. Heat is energy : it can perform work, or pro
duce certain sensations ; it can be measured by its various effects.
It is regarded as " energy stored in a substance by virtue of the state
of its molecular motion" (2).
Conceding that heat is energy, and remembering the expression for energy,
I mv z , it follows that if the mass of the particle does not change, its velocity (molec
ular velocity) must change; or if heat is to include potential energy, then the
molecular configuration must change. The molecular vibrations are invisible, and
their precise nature unknown. Rankine's theory of molecular vortices assumes a
law of vibration which has led to some useful results.
Since heat is energy, its laws are those generally applicable to energy,
as laid down by Newton : it must have a commensurable value ; it must
be convertible into other forms of energy, and they to heat; and the
equivalent of heat energy, expressed in mechanical energy units, must be
constant and determinable by experiment.
4. Subdivisions of the Subject. The evolutions and absorptions
of heat accompanying atomic combinations and molecular decompo
sitions are the subjects of thermochemistry. The mutual relations of
heat phenomena, with the consideration of the laws of heat trans
mission, are dealt with in general physics. The relations between
heat and mechanical energy are included in the scope o applied engi
neering thermodynamics, which may be defined as the science of the
mechanical theory of heat. While thermodynamics is thus apparently
only a subdivision of that branch of physics which treats of heat, the
relations which it considers are so important that it may be regarded
as one of the two fundamental divisions of physics, which from this
standpoint includes mechanics dealing with the phenomena of
ordinary masses and thermodynamics treating of the phenomena
of molecules. Thermodynamics is the science of energy.
5. Applications of Thermodynamics. The subject has farreaching
applications in physios and chemistry. In its mechanical aspects, it deals
4 APPLIED THERMODYNAMICS
with matters fundamental to the engineer. After developing the general
laws and dwelling briefly upon ideal processes, we are to study the condi
tions affecting the efficiency and capacity of air, gas, and steam engines
and the steam turbine; together with the economics of air compression,
distillation, refrigeration, and gaseous liquefaction. The ultimate engi
neering application of thermodynamics is in the saving of heat, an appli
cation which becomes attractive when viewed in its just aspect as a saving
of money and a mode of conservation of our material wealth.
6. Temperature. A hot body, in common language, IB one whose
temperature is high, while a cold body is one low in temperature. Tem
perature, then, is a measure of the hottwss of bodies. From a riso in tem
perature, we infer an accession, of heat; or from a fall in temperature,
a loss of heat.* Temperature is not, however, a satisfactory measure of
quantities of heat. A pound of water at 200 contains very much more
hieat than a pound of lead at the same temperature ; this may be demon
strated by successively ooolmg the bodies in a bath to the same final tem
perature, and noting the gain of heat by the bath. Furthermore, immense
quantities of heat are absorbed by bodies in passing from the solid to the
liquid or from the liquid to the vaporous conditions, without any change*
in temperature whatever. Temperature defines a condition of heat only.
It is a measure of t7ie capacity of the body for coni'iniinimting heat to otlwr
bodies. Heat always passes from a body of relatively high temperature j
it never passes of itself from a cold body to a hot one. Wherever two
bodies of different temperatures are in thermal juxtaposition, an inter
change of heat takes place ; the cooler body absorbs heat from the hotter
body, no matter which contains initially the greater quantity of heat,
until the two are at the saine temperature, or in thermal (tquflibrhwH,.
Two bodies are at the same temperature when there is no tendency toward a
transfer of heat between them. Measurements of temperature ai'o in gen
eral based upon arbitrary scales, standardized by comparison with some
physically established " fixed " point. One of these fixed temperatures is
that minimum at which pure water boils when under normal atmospheric
pressure of 14.697 Ib. per square inch; viz. 212 F. Another is the
maximum temperature of melting ice at atmospheric pressure*, which is
32 F. Our arbitrary scales of temperature cannot be expressed iu terms
of the fundamental physical units of length and weight
7 Measurement of Temperature. Temperatures are measured by thermome
ters. The common type of instrument consists of a connected bulb and vertical
tube, of glass, in which is contained a liquid. Any change in temperature affects
* "... the change in temperature is the thing observed and ... the idea of heat
is introduced to account for the change. , ." Gtoodimough.
THE NATURE AND EFFECTS OF HEAT 5
the volume of the liquid, and the portion in the tube consequently rises or falls.
The expansion of solids or of gases is sometimes utilized m the design of thermom
eters, Mercury and alcohol are the liquids commonly used. The former freezes at
38 F. and boils at 675 F. The latter freezes at 203 F. and boils at 173 F.
The mercury thermometer is, therefore, more commonly used for high tempera
tures, and the alcohol for low (2a).
8. Thermometric Scales. The Fahrenheit thermometer, generally
employed by engineers in the United States and Great Britain,
divides the space between the "fixed points" (Art. 6) into 180
equal degrees, freezing being at 32 and boiling at 212. The
Centigrade scale, employed by chemists and physicists (sometimes
described as the Celsius scale), calls the freezing point and the
boiling point 100. On the Reaumur scale, used in Russia and a
few other countries, water freezes at and boils at 80. One de
gree on the Fahrenheit scale is, therefore, equal to  C., or to R.
In making transformations, care must be taken to regard the differ
ent zero point of the Fahrenheit thermometer. On all scales, tem
peratures below zero are distinguished by the minus ( ) prefix.
The Centigrade scale is unquestionably superior in facilitating arithmetical
calculations; but as most English papers and tables are published in Fahrenheit
units, we must, for the present at least, use that scale of temperatures.
9. High. Temperature Measurements. For measuring temperatures above
800 ' F., some form of pyrometer must be employed. The simplest of these is the
metallic pyrometer, exemplifying the principle that different metals expand to dif
ferent extents when heated through the same range of temperature. Bars of irou
and brass are firmly connected at one end, the other ends being free. At some
standard temperature the two bars are of the same length, and the indicator, con
trolled jointly by the two free ends of the bars, registers that temperature. When
the temperature changes, the indicator is moved to a new position by the relative
distortion of the free ends.
In the Le Chatelier electric pyrometer, a thermoelectric couple is employed. For
temperatures ranging from 300 C. to 1500 C., one element is made of platinum,
the other of a 10 per cent, alloy of platinum with rhodium. Any rise in tempera
ture at the junction of the elements induces a flow of electric current, which is con
ducted by wires to a galvanometer, located in any convenient position. The ex
pensive metallic elements are protected from oxidation by enclosing porcelain
tubes. In the Bristol thermoelectric instrument, one element is of a platinum
rhodium alloy, the other of a cheaper metal. The electromotive force is indicated
by a Weston millivoltmeter, graduated to read temperatures directly. The in
strument is accurate up to 2000 F. The electrical resistance pyrometer is based on
the law of increase of electrical resistance with increase of temperature. In Cal
lendar's form, a coil of fine platinum wiie i wound on a serrated mica fram*.
The instrument is enclosed in porcelain, and placed in the space the temperature
6 APPLIED THERMODYNAMICS
of which is to be ascertained. The resistance is measured "by a Wheatstone bridge,
a galvanometer, or a potentiometer, calibrated to read temperatures directly.
Each instrument must be separately calibrated.
Optical pyrometers are based oil the principle that the colors of bodies vary
with their temperatures (26). In the Morse thermogage, of this type, an incandescent
lamp is wired in circuit with a rheostat and a millivoltmeter. The lamp is located
between the eye and the object, and the current is regulated until the lamp be
comes invisible. The temperature is then read directly from the calibrated milli
voltmeter. The device is extensively used in hardening steel tools, and has been
employed to measure the temperatures in steam boiler furnaces.
10. Cardinal Properties. A cardinal or integral property of a
substance is any property which is fully defined by the immediate
state of the substance. Thus, weight, length, specific gravity, are
cardinal properties. On the other hand, cost is a noncardinal prop
erty ; the cost of a substance cannot be determined by examination
of that substance; it depends upon the previous history of the sub
stance. Any two or three cardinal properties of a substance may be
used as coordinates in a graphic representation of the state of the sub
stance. Properties not cardinal may not be so used, because such
properties do not determine, nor are they determinable by, the pres
ent state of the substance. The cardinal properties employed in
thermodynamics are five or six in number.* Three of these are pres
sure, volume, and temperature ; pressure being understood to mean
specific pressure, or uniform pressure per unit of surface, exerted by or
upon the body, and volume to mean volume per unit of weight. The
location of any point in space is fully determined by its three coordi
nates. Similarly, any three cardinal properties may serve to fix the
thermal condition of a substance.
The first general principle of thermodynamics is that if two of the
three named cardinal properties are known, these two enable us to calcu
late the third. This principle cannot be proved d priori ; it is to be justi
fied by its results in practice. Other thermodynamic properties than
pressure, volume, and temperature conform to the same general principle
(Art. 169) ; with these properties we are as yet unacquainted. A correlated
principle is, then, that any two of the cardinal properties suffice to fully
determine the state of the substance, For certain gases, the general prin
ciple may be expressed; PV= (f}T
*For gases, pressure, volume, temperature, internal energy, entropy; for wet
vapors, dryness is another*
THE NATURE AND EFFECTS OF HEAT 7
while for other gaseous fluids more complex equations (Art. 363) must be
used. In general, these equations are, in the language of analytical
geometry, equations to a surface. Certain vapors cannot be represented,
as yet, by any single equation between P, F, and T, although correspond
ing values of these properties may have been ascertained by experiment.
With other vapors, the pressure may be expressed as a function of the
temperature, while the volume depends both upon the temperature an<l
upon the proportion of liquid mingled with the vapor.
11. Preliminary Assumptions. The greater part of the subject
deals with substances assumed to be in a state of mechanical equilibrium,
all changes being made with infinite slowness. A second assumption
is that no chemical actions occur during the thermodynamic trans
formation. In the third place, the substances dealt with are assumed
to be so homogeneous, as to be in uniform thermal condition through
out : for example, the pressure property must involve equality of
pressure in all directions ; and this limits the consideration to the
properties of liquids and gases.
The thermodynamics of solids is extremely complex, because of the obscure
stresses accompanying their deformation (3), Kelvin (4) has presented a general
analysis of the action of any homogeneous solid body homogeneously strained.
12. The Three Effects of Heat. Setting aside the obvious un
classified changes in pressure, volume, and temperature accompanying
manifestations of heat energy, there are three known, ways in which
heat may be expended. They are :
(#) In a change of temperature of the substance.
(6) In a change of physical state of the substance.
(<?) In the performance of external work by or upon the substance.
Denoting these effects by T, I, and W, then, for any transfer of heat
JJ", we have the relation
H= T + I + W,
any of the terms of which expression may be negative. It should be
quite obvious, therefore, that changes of temperature alone are in
sufficient to measure expenditures of heat.
Items (#) and (6) are sometimes grouped together as indications
of a change in the INTERNAL ENERGY (symbol E) of the heated
substance, the term being one of the first importance, which it is
8 APPLIED THERMODYNAMICS
essential to clearly apprehend. Items (5) and (c) are similarly some
times combined as representing the total work.
13. The Temperature Effect. Temperature indications of heat activity are
sometimes refened to as " sensible heat." The addition of heat to a substance
may either raise or lower its temperature, in accordance \v ith the fundamental
equation of Art. 12.
The temperature effect of heat, from the standpoint of the mechanical
theory, is due to a change in the velocity of molecular motion, in conse
quence of which the kinetic energy of that motion changes.
This effect is therefore sometimes referred to as vibration work. Clausiua
called it actual energy.
14. External Work Effect. The expansion of solids and fluids, due to the supply
of heat, is a familiar phenomenon. Heat may cause either expansion or contraction,
which, if exerted against a resistance, may suffice to perform mechanical work.
15. Changes of Physical State. Broadly speaking, such effects
include all changes, other than those of temperature, within the sub
stance itself. The most familiar examples are the change between
the solid and the liquid condition, when the substance melts or
freezes, and that between the liquid and the vaporous, when it boils
or condenses ; but there are intermediate changes of molecular aggrega
tion in all material bodies which are to be classed with these effects
under the general description, disgregation work. The mechanical
theory assumes that in such changes the molecules are moved into
new positions, with or against the lines of mutual attraction. These
movements are analogous to the "partial raising or lowering of a
weight which is later to be caused to perform work by its own descent.
The potential energy of the substance is thus changed, and positive
or negative work is performed against internal resisting forces."
When a substance changes its physical state, as from water to steam, it
can be shown that a very considerable amount of external work is done, iu
consequence of the increase in volume which occurs, and which may be
made to occur against a heavy pressure. This external work is, however,
equivalent only to a very small proportion of the total heat supplied to
produce evaporation, the balance of the heat having been expended in the
performance of disgregation work.
The molecular displacements constituting disgregation work are exemplified in
THE NATURE AND EFFECTS OF HEAT .9
16. Solid, Liquid, Vapor, Gas. Solid bodies are those which resist tendencies
to change their form or volume. Liquids are those bodies which in all of their
parts tend to preserve definite volume, and which are practically unresistant to
influences tending to slowly change their figure. Gases are unresistant to slow
changes in figure or to increases in volume. They tend to expand indefinitely so
as to completely fill any space in which they are contained, no matter what the
shape or the size of that space may be. Most substances have been observed in
all three forms, under appropriate conditions ; and all substances can exist in any
of the forms. At this stage of the discussion, no essential difference need be
drawn between a vapor and a gas. Formeily, the name vapor was applied to
those gaseous substances which at ordinary temperatures were liquid, while a
" gas " was a substance never observed in the liquid condition. Since all of the
socalled "permanent" gases have been liquefied, this distinction has lost its force.
A useful definition of a vapor as distinct from a true gas will be given later
(Art. 380).
Under normal atmospheric pressure, there exist welldefined tempera
tares at which various substances pass from the solid to the liquid and
from the liquid to the gaseous conditions. The temperature at which the
former change occurs is called the melting point or freezing point; that of
the latter is known as the boiling point or temperature of condensation.
17. Other Changes of State. Although the operation described as boiling
occurs, for each liquid, at some definite temperature, there is an almost continual
evolution of vapor from nearly all liquids at temperatures below their boiling
points. Such "insensible" evaporation is with some substances nonexistent, or
at least too small in amount to permit of measurement: as in the instances of mercury
at 32 F. or of sulphuric acid at any ordinary temperature. Ordinarily, a liquid
at a given temperature continues to evaporate so long as its partial vapor pressure
is less than the maximum pressure corresponding to its temperature. The inter
esting phenomenon of sublimation consists in the direct passage from the solid to
the gaseous state. Such substances as camphor and iodine manifest this property.
Ice and snow also pass directly to a state of vapor at temperatures far below the
freezing point. There seem to be no quantitative data on the heat relations accom
panying this change of state (see Art. 382 6).
18. Variations in " Fixed] Points." Aside from the influence of pressure
(Arts. 358, 603), various causes may modify the positions of the "fixed points" of
the thermometric scale. Water may be cooled below 32 F. without freezing, if
kept perfectly still. If free from air, water boils at 270290 F. Minute particles
of air are necessary to start evaporation sooner; their function is probably to aid
in the diffusion of heat.
(1) Tyndall: Heat as a Mode of Motion. (2) Nichols and Franklin: The F,le
ments of Physics, I, 161. (2o) Heat Treatment of High Temperature Mercurial
Thermometers, by Dickinson; Bulletin of the Bureau of Standards, 2, 2. (2&) See
the paper, Optical Pyrometry, by Waidner and Burgess, Bulletin of the Bureau of
Standards, 1, 2. (3) See paper by J. E. Siebel: The Molecular Constitution of
10 APPLIED THERMODYNAMICS
Solids, in Science, Nov. 5, 1909, p. 654. (4) Quarterly Mathematical Journal,
April, 1855. (5) Darling: Heat for Engineers, 208.
SYNOPSIS OF CHAPTER I
Heat is the universal source of motive power.
Theories of heat : the caloric theory heat is matter; the mechanical theory heat
is molecular motion, mutually conveitible with mechanical energy.
THEBMOOHEMISTRY, THERMODYNAMICS.
Thermodynamics : the mechanical theory of heat ; in its engineering applications, the
science of heatmotor efficiency.
Heat intensity, temperature : definition of, measurement of ; pyrometers.
Thermometric scales: Fahrenheit, Centigrade, Reaumur; fixed points and their
variations
Cardinal properties : pressure, volume, temperature; PF=(/)!T.
Assumptions: uniform thermal condition ; no chemical action ; mechanical equilibrium,
Effects of heat : Bf T+I+ W\ T+I= E= "internal energy " ; J7= external work.
Changes of physical stale, perceptible and imperceptible: I=disgregation work.
Solid, liquid, vapor , gas: melting point, boiling point; insensible evaporation;
sublimation.
PROBLEMS
1. Compute the freezing points, on the Centigrade scale, of mercury and alcohol.
(Ans., mercury, 38.9: alcohol, 130.6,)
2. At what temperatures, RSaumur, do alcohol and mercury boil? (Ans., mer
cury, 285.8: alcohol, 62.7.)
3. The normal temperature of the human body is 98.6 F. Express in Centigrade
degrees, (Ana., 37 C.)
4. At what temperatures do the Fahrenheit and Centigrade thermometers read
alike? (Ans., 40.)
5. At what temperatures do the Fahrenheit and Rgaumur thermometers read
alike? (Ans., 25.6.)
a. Express the temperature 273 C. on the Fahrenheit and Reaumur scales.
3., 459.4 F.: 218.4 R.)
CHAPTER II
THE HEAT UNIT: SPECIFIC HEAT: FIRST LAW OF
THERMODYNAMICS
19. Temperature Waterfall Analogy. The difference between temperature
and quantity of heat may be apprehended from the analogy of a waterfall. Tem
perature is like the head of water ; the energy of the fall depends upon the head,
but cannot be computed without knowing at the same time the quantity of water.
As waterfalls of equal height may differ in power, while those of equal power may
differ in fall, so bodies at like temperatures may contain different quantities of
heat, and those at unequal temperatures may be equal in heat contents.
20. Temperatures and Heat Quantities. If we mix equal weights of
water at different temperatures, the resulting temperature of the mix
ture will be very nearly a mean between the two initial temperatures.
If the original weights are unequal, then the final temperature will be
nearer that initially held by the greater weight. The general principle of
transfer is that
The loss of heat by the hotter water will equal the gain of heat by the
colder.
Thus, 5 Ib. of water at 200 mixed with 1 Ib. at 104 gives 6 Ib. at
184; the hotter water having lost 80 " pounddegrees," and the colder
water having gained the same amount of heat. If, however, we mix the
5 Ib. of hot water with 1 Ib. of some other substance say linseed oil
the resulting temperature will not be 184, but 194.6, if the initial tem
perature of the oil is 104.
21. General Principles. Before proceeding, we may note, in addition to the
principle just laid down, the following laws which are made apparent by the ex
periments described and others of a similar nature :
(a) In a homogeneous substance, the movement of heat accom
panying a given change of temperature * is proportional to the
weight of the substance.
(J) The movement of heat corresponding to a given change of
* Not only the amount, but the method^ of changing the temperature must be
fixed (Art. 57).
11
12 APPLIED THERMODYNAMICS
temperature is not necessarily the same for equal intervals at all
parts of the thermoinetric scale ; thus, water cooling from 200 to
195 does not give out exactly the same quantity of heat as in cool
ing from 100 to 95.
<Y) The loss of heat during cooling through a stated range of
temperature is exactly equal to the gain of heat during warming
through the same range.
22. The Heat Unit. Changes of temperature alone do not measure heat quan
tities, because heat produces other effects than that of temperature change. If,
however, we place a body under " standard" conditions, at which these other
effects, if not known, are at least constant, then we may define a unit of quantity
of heat by reference to the change m temperature which at produces, understand
ing that there may be included perceptible or imperceptible changes of other
kinds, not affecting the constancy of value of the unit.
The British Thermal Unit is that quantity of heat which is expended in
raising the temperature of one pound of water (or in producing other effects
during this change in temperature) from 62 to 63 F.*
To heat water over this range of temperature requires very nearly the same
expenditure of heat as is necessary to warm it 1 at any point on the thermometric
scale. In fact, some writers define the heat unit as thab quantity of heat necessary
to change the temperature front 39.1 (the temperature of maximum density) to
40.1. Others use the ranges 32 to 33, 59 to 60, or 39 to 40. The range first
given is that most recently adopted.
23. French TTnits. The French or C. G. S. unit of heat is the
calorie, the amount of heat necessary to raise the temperature of one
kilogram of water 1 C. Its value is 2.2046 X f = 3.96832 B. t. u., and
1 B. t. u. = 0.251996 cal. The calorie is variously measured from 4 to
5 and from 14.5 to 15.5 (J. The gramcalorie is the heat required to
raise the temperature of one gram of water 1 C. The Centigrade heat
unit measures the heat necessary to raise one pound of water 1 G in
temperature.
24. Specific Heat. Eef erence was made in Art. 20 to the different heat
capacities of different substances, e.g. water and linseed oil. If we mix
a stated quantity of water at a fixed temperature successively with equal
weights of various materials, all initially at the same temperature, the
final temperatures of the mixtures will all differ, indicating that a unit
* There are certain grounds for preferring that definition which makes the B. t. u.
the yj^ part of the amount of heat required to raise the temperature of one pound of
water at atmospheric pressure from the freezing point to the boiling point,
THE HEAT UNIT. SPECIFIC HEAT 13
rise of temperature of unit weight of these various materials represents a
different expenditure of heat in each case.
The property by virtue of winch materials differ in this respect is
that of specific heat, which may be defined as the quantity of heat
necessary to raise the temperature of unit weight of a body through one
degree.
The specific heat of water at standard temperature (Art. 22) is, meas
ured in B. t. u., 1.0 ; generally speaking, its value is slightly variable, as is
that of all substances.
Rankine's definition of specific heat is illustrative : " the specific heat of any
substance is the ratio of the weight of water at or near 39.1 F. [626r3 F.] which
has its temperature altered one degree by the transfer of a given quantity of heat,
to the weight of the other substance under consideration, which has its temperature
altered one degree by the transfer of an equal quantity of heat."
25. Mixtures of Different Bodies. If the weights of a group of
mixed bodies be X, Y f Z, etc., their specific heats #, ?/, z, etc., their ini
tial temperatures t, u, v, etc., and the final temperature of the mixture
be m, then we have the following as a general equation of thermal equi
librium, in which any quantity may be solved for as an unknown:
ni + zZvm =0.
This illustrates the usual method of ascertaining the specific heat of any
body. When all the specific heats are known, the loss of heat to sur
rounding bodies may be ascertained by introducing the additional term,
+ Jf2, on the lefthand side of this equation. The solution will usually
give a negative value for R, indicating that surrounding bodies have
absorbed rather than contributed heat. The value of R will of course be
expressed in heat units.
26. Specific Heat of Water. The specific heat of water, according
to Rowland's experiments, decreases as the temperature is increased
from 39.1 to 80 P., at which latter temperature it reaches a minimum
value, afterward increasing (Art. 359, footnote). The variation in its
value is very small. The approximate specific heat, 1.0, is high as com
pared with that of almost all other substances.
27. Problems Involving Specific Heat. The quantity of heat re
quired to produce a given change of temperature in a body is equal
to the weight of the body, multiplied by the range of temperature
and by the specific heat.
Or, symbolically, using the notation of Art. 25,
14 APPLIED THERMODYNAMICS
If the body is cooled, then m, the final temperature, is less than t, and the sign of
H is  ; if the body is warmed, the sign of II is f , indicating a reception of heat.
28. Consequences of the Mechanical Theory. The Mechanical Equivalent
of Heat. Even before Joule's formulation (Art. 2), Eumford's ex
periments had sufficed for a comparison of certain effects of heat
with an expenditure of mechanical energy. The power exerted by the
Bavarian horses used to drive his machinery is uncertain ; but Alexander
has computed the approximate relation to have been 847 footpounds =
1 B.t.u. (1), while another writer fixes the ratio at 1034, and Joule cal
culated the value obtained to have been 849.
Carnot's work, although based throughout on the caloric theory, shows evident
doubts as to its validity. This writer suggested (1824) a repetition of Ruinford's
experiments, with provision for accurately measuring the force employed. Using
a method later employed by Mayer (Art. 29) he calculated that 0.611 units
of motive power" were equivalent to "550 units of heat"; a relation which
Tyndall computes as representing 370 kilogrammeters per calorie, or 676 foot
pounds per B. t. u. Montgolfier and Seguin (1839) may possibly have anticipated
Mayer's analysis.
29. Mayer's Calculation. This obscure German physician published in 1842
(2) his calculation of the mechanical equivalent of heat, based on the difference
in the specific heats of air at* constant pressure and constant volume, giving
the ratio 771.4= footpounds per B. t. u. (Art. 72). This was a substantially correct
result, though given little consideration at the time. Mayer had previously made
rough calculations of equivalence, one being based on the rise of temperature
occurring in the " beaters " of a paper mill.
30. Joule's Determination. Joule, in 1843, presented the first of his
exhaustive papers on the subject. The usual form of apparatus employed
has been shown in Fig. 1. In the appendix to his paper Joule gave 770 as
the best value deducible from his experiments. In 1849 (3) he presented
the figure for many years afterward accepted as final, viz. 772.
In 1878 an entirely new set of experiments led to the value 772.55, which
Joule regarded as probably slightly too low. Experiments in 1857 had given the
values 745, 753, and 766. Most of the tests were made with water at about 60 F.
This, with the value of g at Manchester, where the experiments were made, in
volves slight corrections to reduce the results to standard conditions (4),
31. Other Investigators. Of independent, though uncertain, merit, were the
results deduced by the Danish engineer, Colding, in 1843. His value of the
equivalent is given by Tyndall as 038 (5). Helmholtz (1847) treated the matter
of equivalence from a speculative standpoint. Assuming that "perpetual motion "
is impossible, he contended that there must be a definite relation between heat
energy and mechanical energy. As early as 1845, Holtzmann (6) had apparently
MECHANICAL EQUIVALENT OF HEAT 15
independently calculated the equivalence by Mayer's method. By 1847 the reality
of the numerical relation had been so thoroughly established that little more was
heard of the caloric theory. Clausius, following Mayer, in 1850 obtained wide
circulation for the value 758 (7) .
32. Hirn's Investigation. Joule had employed mechanical agencies in the
heating of water. Him, in 1865 (8), described an experiment by which he trans
formed into heat the work expended in producing the impact of solid bodies.
Two blocks, one of iron, the other of wood, faced with iron in contact with a lead
cylinder, were suspended side by side as pendulums. The iron block was allowed
to stnke against the wood block and the rise in temperature of water contained in
the lead cylinder was noted and compared with the computed energy of impact.
The value obtained for the equivalent was 775.
Far more conclusive, though less accurate, results were obtained
by Him by noting that the heat in the exhaust steam from an engine
cylinder was less than that which was present in the entering steam.
It was shown by Clausius that the heat which had ' disappeared was
always roughly proportional to the work done by the engine, the
average ratio of footpounds to heat units being 753 to 1. This was
virtually a reversal of Joule's experiment, illustrating as it did the
conversion of heat into work. It is the most striking proof we have
of the equivalence of work and heat.
33. Recent Practice. In 1876 a committee of the British Association for the
Advancement of Science reviewed critically the work of Joule, and as a mean
value, derived from his best 60 experiments, recommended the use of the figure
774.1, which was computed to be correct within ? fo. In 1879, Rowland, having
conducted exact experiments on the specific heat of water, carefully redetermined
the value of the equivalent by driving a paddle wheel about a vertical axis at
fixed speed, in a vessel of water prevented from turning by counterbalance weights.
The torque exerted by the paddle was measured. This permitted of a calculation
of the energy expended, which was compared with the rise in temperature of the
water, Rowland's value was 778, with water at its maximum density. This
was regarded as possibly slightly low (9). Since the date of Rowland's work, the
subject has been, investigated by Griffiths (10), who makes the value somewhat
greater than 778, and by Reynolds and Moorby (11), who report the ratio 778 as
the mean obtained for a range of temperature from 32 to 212 F. This they
regard as possibly 1 or 2 footpounds too low.
34. Summary. The establishing of a definite mechanical equivalent of
heat may be regarded as the foundation stone of thermodynamics. Accord
ing to Merz (12), the anticipation of such an equivalent is due to Poncelet
and Carnot ; Bumf ord's name might be added. " The first philosophical
generalizations were given by Mohr and Mayer j the first mathematical
16 APPLIED THERMODYNAMICS
treatment by Helmholtz; the first satisfactory experimental verification
by Joule." The constr action of the modern science on this foundation
has been the work chiefly of Kankine, Clausius, and Kelvin.
35. First Law of Thermodynamics. Heat and mechanical energy
are mutually convertible in the ratio of 778 footpounds to the British
thermal unit.
This is a restricted statement of the general principle of the conservation of
energy, a principle which is itself probably not susceptible to proof.
We have four distinct proofs of the first law :
(a) Joule's and Rowland's experiments on the production of
heat by mechanical work.
(J) Hirn's observations on the production of work by the ex
penditure of heat.
(V) The computations of Mayer and others, from general data.
(J) The fact that the law enables us to predict thermal proper
ties of substances which experiments confirm.
36. WormelPs Theorem. There cannot be two values of the mechanical
equivalent of heat. Consider two machines, A and B, in the first of which work
is transformed into heat, and in the second of which heat is transformed into
work. Let J be the mechanical equivalent of heat foi A, W the amount of work
which it consumes in pi educing the heat $; then W = JQ or Q = W /. Let
this heat Q be used to drive the machine B, in which the mechanical equivalent
of heat is, say K. Then the work done by B is V = KQ = KW  J. Let this
work be now expended in driving .4. It will produce heab R, such that JR = V
or R = F T /.* If this heat R be used in B, work will be done equal to KR ; but
KR = KVJ = (Y W.
Similarly, after n complete periods of operation, all parts of the machines occupy
ing the same positions as at the beginning, the work ultimately done by B will be
If K is less than 7, this expression will decrease as n increases; i.e. the system
will tend continually to a stale of rest, contiary to the first law of motion. If K
be greater than J, then as n increases the work constantly increases, involving the
assumed fallacy of perpetual motion. Hence K and / must be equal (13).
37. Significance of the Mechanical Equivalent. A very little heat is seen to be
equivalent to a great deal of work. The heat used in raising the temperature of
*The demonstration assumes that the value of the mechanical equivalent is con
stant for a given machine.
FIRST LAW OP THERMODYNAMICS 17
one pound of water 100 represents energy sufficient to lift one ton of water nearly
39 feet. The heat employed to boil one pound of water initially at 32 F. would
suffice to lift one ton 443 feet. The heat evolved in the combustion of one pound of
hydrogen (62,000 B. t. u.) would lift one ton nearly five miles.
(1) Treatise cm Thermodynamics, London, 1892. (2) Wohler and Liebig's
Annalen der Pharmncie : Bemerkungen iiber die Krafte der unbelebten Natur, May,
1842. (3) Phil. Trans., 1850. (4) Joule's Scientific Papers, Physical Society of
London, 1884. (5) Probably quoted by Tyndall from a later article by Colding, in
which this figure is given. Colding's original paper does not seem to be accessible.
(6) Ueber die Wdrme und Elasticitdt der Gase und Dampfe^ Mannheim, 1846.
(7) Poggendorff, Annalen, 1860. (8) Theorie Mecanique, etc , Paris, 1865 (9) Proc.
Amer. Acad. Arts and Sciences, New Series, VII, 187879. (10) Phil. Trans. Boy.
Soc , 1893. (11) Phil. Trans , 1897. (12) History of European Thought, II, 137.
(13) K. Wormell: Thermodynamics, 1886.
SYNOPSIS OF CHAPTEH H
Heat and temperature : heat quantity vs. heat intensity.
Principles : (a) heat movement proportional to weight of substance ; (&) temperature
range does not accurately measure heat movement ; (c) loss during cooling equals
gain during warming, for idntical ranges.
The British thermal unit: other units of heat quantity.
Specific heat : mixtures of bodies ; quantity of heat to produce a given change of tem
perature ; specific beat of water.
The mechanical equivalent of heat : early approximations. First law of thermody
namics : proofs \ oaly one value possible ; examples of the motive power of heat.
PROBLEMS
1. How many Centigrade heat units are equivalent to one calorie? (Ana.,
2.2046.)
2. Find the number of gramcalories in one B. t. u. (Ans., 252.)
3. A mixture is made of 5 Ib. of water at 200, 3 Ib. of linseed oil at 110, and
22 Ib. of iron at 220 (all Fahrenheit temperatures), the respective specific heats
being 1.0, 0.3, and 0.12. Find the final temperature, if no loss occurs by radiation.
(Ans., 196.7 F.)
4. If* the final temperature of the mixture in Problem 3 is 189 F., find the num
ber of heat units lost by radiation. (Ans., 65.7 B. t. u.)
5. Under what conditions, with the weights, temperatures and specific heats of
Problem 3, might the final temperature exceed that computed?
6. How much heat is given out by 7J Ib. of linseed oil in cooling from 400 F. to
32 F.? (Ans., 828 B. t. u.)
7. In a heat engine test, each pound of steam leaves the engine containing 125.2
B.t.u, less heat than when it entered the cy Under. The engine develops 155 horse
power, and consumes 3160 Ib. of steam per hour. Compute the value of the mechani
cal equivalent of heat. (Ans., 775.7.)
18 APPLIED THERMODYNAMICS
8. A pound of good coal will evolve 14,000 B, t. u. Assuming a train resistance
of 11 Ib. per ton of train load, how far should one ton (2000 Ib.) of coal burned in the
locomotive without loss, propel a tiam weighing 2000 tons? If the locomotive weighs
125 tons, how high would one pound of coal lift it if fully utilized?
(Ans., a, 187.2 miles, 6, 43.5ft.)
9. Find the number of kilogrammeters equivalent to one calorie, (1 meter = 39.37
in., 1 kilogram = 2.2046 Ib.) (Ans., 426.8.)
10. Transform the following formula (P being the pressure in kilograms per square
meter, V the volume in cubic meters per kilogram, T the Centigrade temperature
plus 273), to English units, letting the pressure be in pounds per square inch, the
volume in cubic feet per pound, and the temperature that on the Fahrenheit scale
plus 459.4, and eliminating coefficients in places where they do not appear in the
original equation :
P7=47.1 !TP(140.000002 P) I" 0.031 (~\ 3 0.0052~ .
L \ / I
!., PF=0.5962 TP(1+0.0014P)
11. There are mixed 5^ Ib. of water at 204, 3 J Ib. of linseed oil at 105 and 21 Ib.
of a third substance at 221, The final temperature is 195 and the radiation loss is
known to be 8.8 B. t. u. What is the specific heat of the third substance?
CHAPTER III
LAWS OF GASES: ABSOLUTE TEMPERATURE: THE PERFECT GAS
38. Boyle's (or Marietta's) Law. The simplest thermodynamie
relations are those exemplified by the socalled permanent gases.
Boyle (Oxford, 1662) and Mariotte (16761679) separately enun
ciated the principle that at constant temperature the volumes of gases
are inversely proportional to their pressures. In other words, the
product of the specific volume and the pressure of a gas at a given
temperature is a constant. For air, which at 32 F. has a volume
of 12.387 cubic feet per pound when at normal atmospheric pressure,
the value of the constant is, for this temperature,
144 x 14.7 x 12.387 = 26,221.
Symbolically, if c denotes the constant for any given tempera
ture,
pv = P t r or, pv = c.
"Figure 2 represents Boyle's law graphically, the ordinates being pres
sures per square foot, and the abscissas, volumes in cubic feet per pound.
The curves are a series of equilateral hyperbolas,* plotted from the second
of the equations just given, with various values of c.
39. Deviations from Boyle's Law. This experimentally determined principle
was at first thought to apply rigorously to all true gases. It is now known to be
not strictly correct for any of them, although very nearly so for air, hydrogen,
nitrogen, oxygen, and some others. All gases may be liquefied, and all liquids
may be gasified. When far from the point of liquefaction, gases conform with
Boyle's law. When brought near the liquefying point by the combined influences
of high pressure and low temperature, they depart widely from it. The four gases
just mentioned ordinarily occur at far higher temperatures than those at which they
will liquefy. Steam, carbon dioxide, ammonia vapor, and some other wellknown
gaseous substances which may easily be liquefied do not confirm the law even
approximately. Conformity with Boyle's law may be regarded as a measure of
the "perfectness" of a gas, or of its approximation to the truly gaseous condition.
* Kef erred to their common asymptotes as axes of P and V.
19
20
APPLIED THERMODYNAMICS
8000
10
20 30 40 50 60
FIG. 2. Arts. 38, 91. Boyle's Law.
40. Dalton's Law, Avogadro's Principle. Dalton has been credited (though
erroneously) with the announcement of the law now known as that of GayLussac
or Charles (Art. 41). What is properly known as Dalton's law may be thus
stated . A mixture of gases having no chemical action on one another exerts a pres
sure which is the sum of the pressures which would be exerted by the component
gases separately if each in turn occupied the containing vessel alone at the given
temperature.
The ratio of volumes, at standard temperature and pressure, in which two
gases combine chemically is always a simple rational fraction (J, J, J, etc.).
Taken in conjunction with the molecular theory of chemical combination, this
law leads to the principle of Avogadro that all gases contain the same number of
molecules per unit of volume, at the same temperature and pressure. Dalton's
law has important thermodynamic relations (see Arts. 52 6, 382 6).
41. Law of GayLussac or of Charles (1). Davy had announced that the
coefficient of expansion of air was independent of the pressure. GayLus
sac verified this by the apparatus shown in Pig. 3. He employed a glass
tube with a large reservoir A 9 containing tlie air ; which, had been previously
LAWS OF GASES
21
dried. An index of mercury mn separated the air from the external atmos
phere, while permitting it to expand. The vessel B was first filled with
melting ice. Upon applying heat, equal in
tervals of temperature shown on the ther
mometer were found to correspond with
equal displacements of the index mn. When
a pressure was applied on the atmospheric
side of the index, the proportionate expansion
of the air was shown to be still constant for
equal intervals of temperature, and to be equal
to that observed under atmospheric pressure.
Precisely the same results were obtained with FIG. 3 Arts. 41, 48. Verifica
other gases. The expansion of dry air was lono ar es w<
found to be 0.00375, or ^ of the volume at the freezing point, for each
degree C. of rise of temperature. The law thus established may be
expressed :
For all gases, and at any pressure, maintained constant, equal increments of
volume accompany equal increments of temperature.
42. Increase of Pressure at Constant Volume. A second statement
of this law is that all gases, when maintained at constant volume,
undergo equal increases of
pressure with equal increases
of temperature.
This is shown experimen
tally by the apparatus of Fig. 4.
The glass bulb A contains the
gas. It communicates with the
open tube manometer Mm,
which measures the pressure
P is a tube containing mercury,
in which an iron rod is submerged to a sufficient depth to keep the level
of the mercury in m at the marked point a, thus maintaining a constant
volume of gas.
43. Regnault's Experiments. The constant 0.00375 obtained by Gay
Lussac was pointed out by Rudberg to be probably slightly inaccurate.
Begnault, by employing four distinct methods, one of which was sub
stantially that just described, determined accurately the coefficient of
increase of pressure, and finally the coefficient of expansion at constant
pressure, which for dry air was found to be 0.003665, or j^ per degree
0., of the volume at the freezing point.
^
1
^
FIG. 4. Arts. 42, 48. Coefficient of Pressure.
22 APPLIED THERMODYNAMICS
44. Graphical Representation. In Fig. 5, let db represent the
volume of a pound of gas at 32 F. Let temperatures and volumes
be represented, respectively, by ordinates and
/* abscissas. According to Charles' Law, if the
/ pressure be constant, the volumes and tempera
_ v tares \vill increase proportionately ; the volume
ab increasing 3^ for each degree C. that the
temperature is increased, and vice versa. The
straight line cbe then represents the successive
relations of volume and temperature as the gas
FIG 5 Arts. 44, M. is heated or cooled from the temperature at b.
Charles' Law. ^ t t ] ie p O i n fc ^ where this line meets the a? 7 axis,
the volume of the gas will be zero, and its temperature will be 273 C.,
or 491.4 F., lelow tlie freezing point.
45. Absolute Zero. This temperature of 459.4 F. suggests
the absolute zero of thermodynamics. All gases would liquefy or
even solidify before reaching it. The lowest temperature as yet
attained is about 450 F. below zero. The absolute zero thus experi
mentally conceived (a more strictly alxsolute scale is discussed later,
Art. 156) furnishes a convenient starting point for the measure
ment of temperature, which will be employed, unless otherwise speci
fied, in our remaining discussion. Absolute temperatures a) c those
in which the zero point is the absolute zero. Their 'numerical values
are to be taken, for the present, at 459.4 greater than those of the cor
responding Fahrenheit temperature.
46. Symbolical Representation. The coefficients determined by GayLussac,
Charles, and Regnaulfc were those for expansion from an initial volume of 32 F.
If we take the volume at this temperature as unity, then letting T represent the
absolute temperature, we have, for the volume at any temperature,
V= r^. 40 1.4.
Similarly, for the variation in pressure at constant volume, the initial pressure
being unity, P = T^ 491.4. If we let a denote tho value 1 * 401.4, the first
expression becomes V  aT, and the second, P = aT. Denoting temperatures on
the Fahrenheit scale Ly t, we obtain, for an initial volume v at 32" and any other
volume F corresponding to the temperature , produced without change of pressure,
7=v[l + a(*32)].
Similarly, for variations in pressure at constant volume,
P,ppH"a(i82);.
LAWS OF GASES : ABSOLUTE TEMPERATURE 23
The value of a is experimentally determined to be very nearly the same for pres
sure changes as lor volume changes , the difference m the case of air being less
than \ of one per cent. The temperature interval between the melting ot ice and
the boiling of water being 180, the expansion of volume of a gtus between those
180 x 1
limits is = 0.365, whence Rankine's equation, originally derived from the
491.4
experiments of Regnault and Rudberg,
T= 1.365,
po
in which P, V refer to the higher temperature, and p> v to the lower.
47. Deviations from Charles 1 Law. The laws thus enunciated are now known
not to hold rigidly for any actual gases. For hydrogen, nitrogen, oxygen, air,
caibon monoxide, methane, nitric oxide, and a few others, the disagreement is
ordinarily very slight. For carbon dioxide, steam, and ammonia, it is quite pro
nounced. The leason for this is that stated in Art. 30. The first four gases named
have expansive coefficients, not only almost unvarying, but almost exactly identical.
They maybe legarded as our most nearly perfect gases. For air, for example,
Regnaulfc found over a range of temperature of ISO F., and a range of pressure
of from 109.72 mm. to 499i\0<) mm,, an extreme vai iation in the
coefficients of only 1 G7 per cent. For caibon dioxide, on the
other hand, with the same range of lempeiatures and a de
creased pressure range
of from 78o.47 mm.
to 4759.03 mm., the
variation was 4.72
per cent of the lower
value (2).
\
48. The Air Thermometer. The law of Charles sug
gests a form of thermometer far more accurate than the
ordinary mercurial instrument.
If we allow air to expand with
out change in pressure, or to
increase its pressure without
change in volume, then we have
by measurement of the volume
or of the pressure respectively a
direct indication of absolute tem
perature. The apparatus used
by GayLussac (Fig, 3), or,
equally, that shown in Fig. 4, is in fact an air ther
mometer, requiring only the establishment of a scale to fit it for practical
use. A. simple modern form of air thermometer is shown in Fig. 6. The
FIG. 6.
Art, 48. Air Ther
mometer,
FIG. 7. Art. 48.
Preston Air
Thermometer.
APPLIED THERMODYNAMICS
bulb A contains dry air, and communicates through a tube
of fine bore with the short arm of the manometer BB> by
means of which the pressure is measured. The level of the
mercury is kept constant at a by means of the movable
reservoir E and flexible tube m. The Preston air ther
mometer is shown in Fig. 7. The air is kept at constant
volume (at the mark a) by admitting mercury from the
bottle A through the cock B. In the Hoadley air ther
mometer, Fig. 8, no attempt is made to keep the volume
of air constant; expansion into the small tube below the
bulb increasing the volume so slightly that the error is com
puted not to exceed 5 in a range of 600 (o).
49. Remarks on Air Thermometers. Following Renault,
the instrument is usually constructed to measure pressures at
constant volume, using either nitrogen, hydrogen, or air as a
medium, thily one "fixed point" need be marked, that of Iho
temperature of melting ice. Having marked at 32 the atmos
pheric pressure registered at this temperature, the degrees aie
spaced so that one of them denotes an augmentation of pressure
of 14.7  4914 = 0.0290 Ib. per square inch. It is usually more
convenient, however, to determine the two fixed points as usual
and subdivide the intervening distance into 180 equal degrees.
The air thermometer readings differ to some extent from those of
the most accurate mercurial instruments, principally because of
the fact that mercury expands much less than any gas, and the
modifying effect of the expansion of the glass container is there
fore greater in its case. The air thermometer is itself nob a
E perfectly accurate instrument, since air doos not ewtetly follow
Charles' law (Art. 47). The instrument is used for standardizing
mercury thermometers, for direct measurements of temperatures
belov the melting point of glass (000800 F.) as in Regnault's
experiments on vapors; or, "by using porcelain "bulbs, for measur
ing much higher temperatures.
50. The Perfect Gas, If actual gases conformed pre
cisely to the laws of Boyle and Charles, many of their
thermal properties might be computed directly. The
slightness of the deviations which actually occur sug
gests the notion of a perfect gas, which would exactly
and invariably follow the laws,
Fia.8. Art 48.
Hoadley
Air Ther
Any deductions which might be made from these sym
 riii A np /wiirqA he rigorously true only
THE PERFECT GAS 25
for a perfect gas, which does not exist in nature. TJie current thermo
dynamic method is, however, to investigate the properties of such a gas, modi
fying the results obtained so as to make them applicable to actual gases,
rather than to undertake to express symbolically or graphically as a
basis for computation the erratic behavior of those actual gases. The
error involved in assuming air, hydrogen, and other "permanent 3 ' gases
to be perfect is in all cases too small to be of importance in engineering
applications. Zeuner (4) has developed an " equation of condition " or
"characteristic equation" for air which holds even for those extreme con
ditions of temperature and pressure which are here eliminated.
51. Properties of the Perfect Gas. The simplest definition is that
the perfect gas is one which exactly follows the laws of Boyle and
Charles. (Rankine's definition (5) makes conformity to Daltoa's
law the criterion of perfectness.) Symbolically, the perfect gas con
forms to the law, readily deduced from Art. 50,
in which R is a constant and T the absolute temperature. Consid
ering air as perfect, its value for Jt may be obtained from experi 7
mental data at atmospheric pressure and freezing temperature:
R = PV+ ^=(14.7 x 144 x 12.387)* 491.4 = 53.36 footpounds.
For other gases treated as perfect, fche value of R may be readily
calculated when any corresponding specific' volumes, pressures, and
temperatures are known. Under the pressure and temperature just
assumed, the specific volume of hydrogen is 178.83 ; of nitrogen,
12.75; of oxygen, 11.20. A useful form of the perfect gas equation
may be derived from that just given by noting that _PF"* 2 7 = B, a
constant : ]PV pv
IF** t '
52. Significance of ff. At the standard pressure and temperature
specified in Art. 51, the values of E for various gases are obviously
proportional to their specific volumes or inversely proportional to their
densities. This leads to the form of the characteristic equation some
* At the temperature < 1? let the pressure and volume be p 1? tv If the gas were
to expand at constant temperature, it would conform to Boyle's law, Pi&i=*c, or
<^ = Ci. Let the pressure be raised to any condition p z while the volume remains t? a ,
&i
the temperature now becoming t^. Theu by Charles' law, ^=7, jpa^Pi T> P&*
Pi &i *i
=P&i=Pii 4 2 =Ci<2, where ^ is a constant to which we give the symbol 5.
26
APPLIED THERMODYNAMICS
times given, PV= rT + M, in which Mia the molecular weight and r
a constant having the same value for all sensibly perfect gases.
TABLE : PROPERTIES OF THE COMMON GASES
APPROXIMATE
ATOMIC
WEIGHT
MOLECULAR
WEIGHT
SPECIFIC HEAT
AT CONSTANT
PRESSURE
SYMBOL
Hvdroeen
1
2
3.4
H
Nitrogen
Oxygen
Carbon dioxide
14
16
28
32
44
2438
0.217
215
N
C0 2
Alcohol
46
4534
C 2 H fl O
Carbon monoxide
Ether
28
74
2438
4797
CO
(C 2 H 6 ) 2
Ammonia
Sulphur dioxide . . .
Chlo'rofQTrn. . T . .
17
64
119
5
15
1567
NHa
S0 2
CHC1 3
Methane
16
5929
CEU
Olefiant gas
14
4040
CH 2
Air
2375
SteRTY) . . ,.,.,..
18
5
H 2 O
52a. Principles of Balloons. A body is in vertical equilibrium in a
fluid medium when its weight is equal to that of the fluid which it dis
places. In a balloon ; the weight supported is made up of (a) the car,
envelope and accessories, and (6) the gas in the inflated envelope. The
equation of equilibrium is
W=w+V(dd') 9
where TT = weight in Ibs., item (a), above;
w = weight of air displaced by the car, framework, etc., in Ibs.;
"F = volume of inflated envelope, cu. ft.;
d = density of surrounding air, Ibs. per cu. ft.;
d r = density of gas in envelope, Ibs. per cu. ft.
The term w is ordinarily negligible. The pressure of the gas in the
envelope is only a small fraction of a pound above that of the atmos
phere. When gas is vented from the balloon, the latter is prevented
from collapsing by pumping air into one of the compartments (ballonets),
so that the effect of venting is, practically speaking, to decrease the size
of the envelope.
If the balloon is not in vertical equilibrium, then W w 7(d d')
is the net downward force, or negatively the upward pull on an anchor
rope which holds the balloon down, A considerable variation in the
THE PERFECT GAS 27
conditions of equilibrium arises from variations in the value of d.
Atmospheric pressure varies with the altitude about ^as follows:
Altitude Normal Atmospheric
in Miles. Pressure, Lbs per Sq. In.
14.7
i 14.02
\ 13.33
 12.66
1 12.02
li 11.42
1J 10.88
2 9.80
52b. Mixtures of Gases. By Dalton's law (Art. 40), if wi, w z , w* be the weights
of the constituents of a mixture at the state V (volume of entire mixture, Tiot its
specific volume), T, P; and if the R values for these constituents be Ri, R*, Rz,
then
If W be the weight of the mixture =t0i+wfc+v\ then the equivalent R value
for the mixture is
PV r>
..
~ W
Then, for example,
P! _VRiTwi
If vi t v Z) v 3 denote the actual volumes of several gases at the conditions P, T' 3
and Wi, 102, w>3, their weights, then Pvi=wiRiT } Pv 2 ^w z RzTj Pvz=
V = U
PVWRT t
PVW&l, V ~WRTP
f
From expressions like the last we may deal with computations relating to mixed
gases where the composition is given by volume. The equivalent molecular weight
of the mixture is, of course, (Art. 52).
K
Dalton's law, like the other gas laws, does not exactly hold with any actual
gas: but for ordinary engineering calculations with gases or even with superheated
vapors the error is negligible.
53. Molecular Condition. The perfect gas is one in which the molecules move
with perfect freedom, the distances between them being so great in comparison
with their diameters that no mutually attractive forces are exerted. No per
formance of disgregation work accompanies changes of pressure or temperature.
28 APPLIED THERMODYNAMICS
Hirschfeld (6), in fact, defines the perfect gas as a substance existing in such a
physical state that its constituent particles exert no interattraction. The coefficient
of expansion, according to Charles 7 law, would be the exact reciprocal of the abso
lute tempeiature of melting 1 ice, for all pressures and temperatures. Zeuner has
shown (7) that as necessary consequences of the theory of perfect gases it can be
proved that the product of the molecular weight and specific volume, at the same
pressure and temperature, is constant for all gases; whence he derives Avogadro's
principle (Art. 40). Rankine (8) has tabulated the physical properties of the
"perfect gas."
54. Kinetic Theory of Gases. Beginning with Bernoulli! in 1738, various
investigators have attempted to explain the phenomena of gases on the basis of
the kinetic theory, which is now closely allied with the mechanical theory of heat.
According to the former theory, the molecules of any gas are of equal mass and
like each other. Those of different gases differ in proportions or structure. The
intervals between the molecules are relatively very great. Their tendency is to
move with uniform velocity in straight lines. Upon contact, the direction of mo
tion undergoes a change. In any homogeneous gas or mixture of gases, the moan
energy due to molecular motion is the same at all parts. The pressure of the gas
per unit of superficial area is proportional to the number of molecules in a unit of
volume and to the average energy with which they strike this area. It is there
fore proportional to the density of the gas and to the average of the squares of the
molecular velocities. Temperature is proportional to the average kinetic energy
of the molecules. The more nearly perfect the gas, the more infrequently do the
molecules collide with one another. When a containing vessel is heated, the mole
cules rebound with increased velocity, and the temperature of the gas rises; when
the vessel is cooled, the molecular velocity and the temperature are decreased.
" When a gas is compressed under a piston in a cylinder, the particles of the gas
rebound from the inwardly moving piston with unchanged velocity relative to
the piston, but with increased actual velocity, and the temperature of the gas con
sequently rises. When a gas is expanded under a receding piston, the particles of
the gas rebound with diminished actual velocity, and the temperature falls" (9).
Recent investigations iu molecular physics have led to a new terminology but
in effect serve to verify and explain the kinetic theory.
55. Application of the Kinetic Theory. Let w denote the actual molecular
velocity. Resolve this into components #, y, and z, at right angles to one another.
Then w a =; a 2 f y 2 + z*. Since the molecules move at random in all directions,
x = y = 2, and to 2 = 3 x 2 . Consider a single molecule, moving in an x direction
back and forth between two limiting surfaces distant from each other d, the x
component of the velocity of this particle being a. The molecule will make
(a 2 rf) oscillations per second. At each impact the velocity changes from + a
to a, or by 2 a, and the momentum by 2 am, if m represents the mass of the
molecule. The average rate of loss of momentum per single impact is 2 am X (a * 2 d)
=ma z +d; and this is the average force exerted per second on each of the limiting
surfaces. The total force exerted by all the molecules on these surfaces is then
equal to F = jN = ~JV =* [ N ' m w ^ c ^ ^ ^ tne totia ^ number of molecules
THE PERFECT GAS 29
in the vessel. Let q be the area of the limiting surface. Then the force per unit
of aurfawpF+g^ + a^.whencepB^gw*, in which ,
is the volume of the gas = gd and W is its weight in Ibs. (10). See Art. 127 a.
56. Applications to Perfect Gases. Assuming that the absolute temperature
is proportional to the average kinetic energy per molecule (Art. 54), this kinetic
energy being ^ miv 2 , then letting the mass be unity and denoting by R a constant
relation, we have pv = RT. The kinetic theory is perfectly consistent with Dai
ton's law (Art. 40). It leads also to Avogadro's principle. Let two gases be pres
ent. For the first gas, p  nmw 2  3, and for the second, P = N3IW 2  3. If
t T, mw 2 = MW 2 , and if p = P, then n = N. If M denote the mass of the gas,
M = mN 9 and pv = Mw 2 3, or w 2 = 3jow M 9 from which the mean velocity of
the molecules may be calculated for any given temperature.
For gases not perfect, the kinetic theory must take into account, (a) the effect
of occasional collision of the molecules, and (b) the effect of mutual attractions
and repulsions. The effect of collisions is to reduce the average distance moved
between impacts and to increase the frequency of impact and consequently the
pressure. The result is much as if the volume of the containing vessel were
smaller by a constant amoant, ft, than it really is. For w, we may therefore wiite
v b. The value of b depends upon the amount and nature of the gas.* The
effect of mutual attractions is to slow down the molecules as they approach the
walls. This makes the pressure less than it otherwise would be by an amount
which can be shown to be inversely proportional to the square of the volume of
the gas. For p, we therefore write p 4 (a i> 2 ), in which a depends similarly
upon the quantity and nature of the gas. We have then the equation of Van der
Waals,
(1) Cf. Verdet, Legons de Chemie et de Physique, Paris, 1862. (2) ReL des Exp.,
I, 111, 112. (3) Trans. A. S. M. E., VI, 282. (4) Technical Thermodynamics
(Klein tr.), II, 313. (5) "A perfect gas is a substance in such a condition that the
total pressure exerted by any number of portions of it, against the sides of a vessel in
which they are inclosed, is the sum of the pressures which each such portion would
exert if enclosed in the vessel separately at the same temperature.' 1 The Steam
Engine, 14th ed., p. 220. (6) Engineering Thermodynamics, 1007. (7) Op. cit., I,
104107. (8) Op. eft., 593595. (9) Nichols and Franklin, The Elements of Physics,
I, 199200. (10) Ibid., 199 ; Wormell, Thermodynamics, 167161. (11) Over de
Continuiteit van den Gas en Vloeistoestand, Leinden, 1873, 76 ; tr. by Roth, Leipsic,
1887.
SYNOPSIS OF CHAPTER 3H
Boyle's law, pw = PF: deviations.
Dalton's law, Avogadro's principle.
Law of GayLussac or of Charles: increase of volume at constant pressure; increase
of pressure at constant volume; values of the coefficient from 32 F.j deviations
with actual gases.
* Strictly, it depends upon the space between the molecules ; but Richardslsuggests
(Science, XXXIV, N, S., 878), that it may vary with the pressure and the temperature.
30 APPLIED THERMODYNAMICS
The absolute zero. 459.4 F , or 491 4 F. below the freezing point.
Air thermometers Preston's ; Hoadley's ; calibration } gases used.
The perfect gas, == , definitions; properties, values of R ; absence of inter
t j.
molecular action, the kinetic theory; development of the law PF^T there
from ; conformity with Avogadro's principle , molecular velocity.
Table ; the common gases j
Constants for gas mixtures . R~ l ^ * ' '
Balloons: weight weight of fluid displaced.
The Van der Waals equation for imperfect gases :
PROBLEMS
1. Find the volume of one pound of air at a pi ensure of 100 Ib. per square inch,
the temperature being 32 F., using Boyle's law only. (Ans,, 1.821 cu. ft.)
2. From Charles' law, find the volume of one pound of air at atmospheric pres
sure and 72 F. (Ans , 13.4 cu. ft.)
3. Find the pressure exerted by one pound of air having a volume of 10 cubic
feet at 32 F. (Ans,, 18.2 Ib. per sq. in.)
4. One pound of air is cooled from atmospheric pressure at constant volume from
32 F. to 290 F. How nearly perfect is the vacuum pioduced? (Ans., 65.5%.)
5. Air at 50 Ib. per square inch pressure at the freezing point is heated at con
stant volume until the temperature becomes 2900 F. Find its pressure after heating.
(Ans , 341,8 Ib. per sq. in.)
6. Five pounds of air occupy 50 cubic feet at a temperature of F. Find the
pressure. (Ans., 17.03 Ib. per sq. in.)
7. Find values of R for hydrogen, nitrogen, oxygen.
(Ans., for hydrogen, 770.3 ; for nitrogen, 54.9 ; for oxygen, 48.2.)
8. Find the volume of three pounds of hydrogen at 15 Ib. pressure per square
inch and 75 F. (Ans , 571.8 cu. ft.)
9. Find the temperature of 2 ounces of hydrogen contained in a 1gallon flask
and exerting a pressure of 10,000 Ib. per square inch. (Ans., 1536 F.)
10. Compute the value of r (Art. 52). (Ans , 1538 to 1544.)
11. Find the mean molecular velocity of l Ib. of air (considered as a perfect gas)
at atmospheric pressure and 70 F. (Ans., 1652 ft. per sec.)
12. How large a flask will contain 1 Ib. of nitrogen at 3200 Ib. pressure per square
inch and 70 F. ? (Ans., 0.0631 cu. ft.)
13. A receiver holds 10 Ib. of oxygen at 20 C. and under 200 Ib. pressure per
square inch. What weight of air will it hold at 100 F. and atmospheric pressure ?
14. For an oxyhydrogen light, there are to be stored 25 Ib. of hydrogen and
200 Ib. of oxygen. The pressures m the two tanks must not exceed 500 Ib. per
square inch at 110 F Fhul their volumes.
15. A receiver containing air at normal atmospheric pressure is exhausted until
the pressure is 0.1 inch of mercury, the temperature remaining constant. "What per
PROBLEMS 31
cent of the weight of air has been removed ? (14.697 Ib. per sq. in. =29. 92 ins.
mercury.)
16. At sea level and normal atmospheric pressure, a 60,000 cu. ft, hydrogen
balloon is filled at 14.75 Ib. pressure. The temperature of the hydrogen is 70 F.;
that of the external air is 60 F. The envelope, car, machinery, ballast, and occu
pants weigh 3500 Ib. Ignoring the term w, Art 52a, what is the upward pull on the
anchor rope ?
17. How much ballast must be discharged from the balloon in Prob. 16 in order
that when liberated it may rise to a level of vertical equilibrium at an altitude of 2
miles ?
18. In Problem 17, there are vented from the balloon, while it is at the 2mile
altitude, 10 per cent of its gas contents. If the ballonet which has been vented is
kept constantly filled with air at a pi ensure just equal to that of the external atmos
phere, to what approximate elevation will the balloon descend ? What is the net
amount of force available for accelerating downward at the moment when descent
begins ?
19. In Problem 17, while at the 2mile level, the temperature of the hydrogen
becomes 60 anil that of the surrounding air 0, without change in either internal or
external pressure. What net amount of ascending or descending force will be caused
by these changes ? How might tins be overcome ?
20. In a mixture of 5 Ib. of air with 1C Ib. of steam, at a pressure of 50 Ib. per
square inch at 70 IT., what is the value of R for the mixture ? What is its equiva
lent molecular weight ? The difference of k and I * The partial pressure due to
air only ?
21 A mixed gas weighing 4 Ib. contains, by volume, 35 per cent of CO, 16 per cent
of II and 3 poi cent of CH 4 , the balance being N. The pressure ib 50 Ib. per square
inch and the temperature 100 F. Find the value of E for the mixture, the partial
pressure due to each constituent, and the percentage composition by weight.
22. Compute (and discuss) values of R and y for gases listed in the table, page 26.
(See Arts. 69, 70.)
CHAPTER IV
THERMAL CAPACITIES : SPECIFIC HEATS OF GASES : JOULE'S LAW
57. Thermal Capacity. The definition of specific heat given in Art. 24 is,
from a thermodynainic standpoint, inadequate. Heat jtroducea other effects than
change of temperature. A definite movement of heat cun l>o estimated only \vlion
all of these effects are defined. For example, the quantity of heat necessary to
raise the temperature of air one degree in a constant volume air thermometer is
much less than that used in raising the temperature ono degree in the constant
pressure typo. The specific heat may be .satisfactorily defined only by referring
to the condition of the substance during the changu of tein)>e,raturo. Ordinary
specific heals assume constancy of jwYMWwrp, that oC tho atmosphere, whilo the
volume increases with the temperature in a ratio "which is determined by the coeffi
cient of expansion of the material. A specific heat determined in this way as
are those of solids and liquids generally I'M the specific heat at constant pressure.
Whenever the term te ^ecffia heat" 'fit 'tittcd without qualification, this yur
tictdar specific Jieat in intended. Heat may be absorbed during changes of
either pressure, volume, or temperature, while porno other of these proper
ties of the substance is kept constant. For a specific change of property,
the amount of heat absorbed represents a specific thermal capacity.
58. Expressions for Thermal Capacities. If 7/ represents heat absorbed,
c a constant specific heat, and (T ) a range of temperature, then, by
definition, H=c(Tf) and c? = //* (T f). If c be variable, then
H= \ cdT and c =* fl H * tl'F. If in place of c we wish to denote the
specific heat at constant pressure (k), or that nt constant volume (f), we may
apply subscripts to the differential coefficients ; thus,
and I '
the subscripts denoting the property which remains constant during the
change in temperature.
We have also the thermal capacities,
' \W>/r'
The first of these denotes the amount of heat necessary to increase the specific
volume of the substance by unit volume, while the temperature remains constant;
SPECIFIC HEATS OF GASES 33
this is known as the latent heat of expansion. It exemplifies absorption of heat
without change of temperature. "No names have been assigned for the other
thermal capacities, but it is not difficult to describe their significance.
59. Values of Specific Heats. It was announced by Dulong and Petit that the
specific heats of substances are inversely as their chemical equivalents. This was
shown later by the experiments of Regnault and others to be approximately,
though not absolutely, correct. Considering metals in the solid state, the product
of the specific heat by the atomic weight ranges at ordinary temperatures from 6.1
to 6.5. This nearly constant product is called the atomic heat. Determination of
the specific heat of a solid metal, therefore, permits of the approximate computa
tion of its atomic weight. Certain n on metallic substances, including chlorine,
bromine, iodine, selenium, tellurium, and arsenic, have the same atomic heat as
the metals. The molecular heats of compound bodies are equal to the sums of the
atomic heats of their elements ; thus, for example, for common salt, the specific
heat 0.219, multiplied by the molecular weight, 58.5, gives 12.8 as the molecular
heat ; which, divided by 2, gives 6.4 as the average atomic heat of sodium and
chlorine; and as the atomic heat of sodium is known to be 6.4, that of chlorine
must also be 6.4 (1).
60. Volumetric Specific Heat. Since the specific volumes of gases are in
versely as their molecular weights, it follows from Art. 59 that the quotient of the
specific heat by the specific volume is practically constant for ordinary gases. In
other words, the specific heats of equal volumes are equal. The specific heats of
these gases are directly proportional to their specific volumes and inversely pro
portional to their densities, approximately. Hydrogen must obviously possess the
highest specific heat of any of the gases.
61. Mean, "Real," and "Apparent" Specific Heats. Since all specific
heats are variable, the values given in tables are mean values ascertained
over a definite range of temperature. The mean specific heat, adopting
the notation of Art. 58, is c H^(T f); while the true specific heat, or
specific heat " at a point," is the limiting value c = dHs dT
Rankine discusses a distinction between the real and apparent specific heats ;
meaning by the former, the rate of heat absorption necessary to effect changes of
temperature alone, without the performance of any disgregation or external work
and by the latter, the observed rate of heat absorption, effecting the same change
of temperature, but simultaneously causing other effects as well. For example,
in heating water at constant pressure from 62 to 63 F., the apparent specific heat
is 1.0 (definition, Art. 22). To compute the real specific heat, we must know the
external work done by reason of expansion against the constant pressure, and the
disgregation work which has readjusted the molecules. Deducting from 1.0
the heat equivalent to these two amounts of work, we have the real specific heat,
that which is used solely for making the substance hotter. Specific heats determined
by experiment are always apparent; the real specific heats are known only by
computation (Art. 64).
34 APPLIED THERMODYNAMICS
62. Specific Heats of Gases. Two thermal capacities of especial
importance are used in calculations relating to gases. The first is
the specific heat at constant pressure, k, which is the amount of heat
necessary to raise the temperature one degree while the pressure is kept
constant; the other, the specific heat at constant volume, 1, or the
amount of heat necessary to raise the temperature one deyree while the
volume is kept constant.
63. Regnault's Law. As a result of his experiments on a large number of
gases over rather limited ranges of temperature, Regnault announced that the
specific heat of any gas at constant pressure is constant. This is now known not to
be rigorously true of even our most nearly perfect gases. It is not even approxi
mately true of those gases when far from the condition of perfectness, a'.e. at low
temperatures or high pressures. At very liigli temperatures, also, it is well known
that specific heats rapidly inci ease. This pailicular variation is perhaps due to
an approach toward that change of state described as dmocwtion. When near
any change of state, combustion, fusion, evaporation, dissociation, every sub
stance manifests erratic thermal properties. The specific heat of carbon dioxide
is a conspicuous illustration. Recent determinations by Holborn and Ilenning
(2) of the mean specific heats between and x C. give, for nitrogen, k = 0.255
+ 0.000019 x\ and for carbon dioxide, Jt = 0.201 + 0.0000742 j; 0.000000018^:
while for steam, heated from 110 to x C., *= 0.4000 0.0000108 x+ 0.000000044 a*
The specific heats of solids also vary. The specific heats of substances in general
increase with the temperature. Kegnault's law would hold, however, for a perfect
gas; in this, the specific heat would be constant under all conditions of tempera
ture, For our "permanent" gases, the specific heat is practically constant at
ordinary temperatures.
The table in Art. 52 shows that in general the specific heats at constant pressure
vary inversely as the molecular weights. Carbon dioxide, sulphur dioxide, ammonia,
and steam (which are highly imperfect gases) vary most widely from this law.
64. The Two Specific Heats. When a gas is heated at constant pressure,
its volume increases against that pressure, and external work is done 111
consequence. The external work may be computed by multiplying the
pressure by the change in volume. When heated at constant volume, no
external work is done ; no movement is made against an external resist
ance. If the gas be perfect, then, under this condition, no disgregation
work is done ; arid the specific heat at constant volume is a true specific
heat, according to Kankine's distinction (Art. 61). The specific heat at
constant pressure is, however, the one commonly determined by experi
ment. The numerical values of the two specific heats must, in a perfect
gas, differ by the heat equivalent to the external work done during heating
at constant pressure. Under certain conditions, as with, water at its
SPECIFIC HEATS OF GASES 35
maximum density, no external work is done when heating at constant
pressure ; and at this state the two specific heats are equal, if we ignore
possible differences in the disgregation work.
65. Difference of Specific Heats. Let a pound of air .at 32 F.
and atmospheric pressure be raised 1 F. in temperature, at constant
pressure. It will expand 12.3877491.4 = 0.02521 cu. ft., against
a resistance of 14.7 x 144 = 2116.8 lb. per square foot. The external
work which it performs is consequently 2116.8 x 0.02521 = 53.36 foot
pounds. A general expression for this external work is W=P V+ T\
and as from Art. 51 the quotient P V~ T is a constant and equal to
R, then IP" is a constant for each particular gas, and equivalent in
value to that of R for such gas. The value of TFfor air, expressed
in heat units, is 53.367778 = 0.0686. If the specific heat of air at
constant pressure, as experimentally determined, be taken at 0.2375,
then the specific heat at constant volume is 0.2375 0.0686 = 0.1689,
air being regarded as a perfect gas.
66. Derivation of Law of Perfect Gas. Let a gas expand at constant pres
sure P from the condition of absolute zero to any other condition F, T. The total
external work which it will have done in consequence of this expansion is PV.
The work done per degree of temperature is PF T. But, by Charles' law, this
is constant, whence we have PV=RT. The symbol R of Art. 51 thus represents
the external work of expansion during each degree of temperature increase (3).
67. General Case. The difference of the specific heats, while constant for any
gas, is different for different gases, because their values of R differ. But since
values of R are proportional to the specific volumes of gases (Art. 52), the differ
ence of the volumetric specific heats is constant for all gases. Thus, let , I be the
two specific heats, per pound, of air. Then k  I = r. Let d be the density of
the air; then, d(kT) is the difference of the volumetric specific heats. For any
other gas, we have similarly, K L = R and D(K L) ; but, from Art. 52
R:r d:D, or R  rd  D. Hence, K L = rd  D = (k  l)(d  Z>), or
D(K L) = d(k Z). The difference of the volumetric specific heats is for all
gases equal approximately to 0.0055 B. t. u. (Compare Art. 60.)
68. Computation of External Work. The value of JK given in Art. 52 and
Art. 65 is variously stated by the writers on the subject, on account of the
slight uncertainty which exists regarding the exact values of some of the con
stants used in computing it. The differences are too small to be of consequence
in engineering work.
69. Ratio of Specific Heats. The numerical ratio between the
two specific heats of a sensibly perfect gas, denoted by the symbol y,
is a constant of prime importance in thermodynamics.
36 APPLIED THERMODYNAMICS
For air, its value is 0.2375 ^0.1689 =1.4 +. Various writers, using other funda
mental data, give slightly different values (4). The best direct experiments (to
be described later) agree with that here given within a narrow margin. For
hydrogen, Lummer and Pringsheim (5) have obtained the value 1 408; and for
oxygen, 1 396. For carbon dioxide, a much leas perfect gas than any of these,
these observers make the value of y, 1.2961; while Dulong obtained 1.338. The
latter obtained for carbon monoxide 1.428. The mean value for the "permanent"
gases is close to that for air, viz.,
The value of y is about the same for all common gases, and is practically inde
pendent of the temperature or the pressure.
From Arts. 59, 60, 65, we have, letting m denote chemical equivalents and V
specific volumes,
m
~
where a and b are constants having the same value 'for all gases.
70. Relations of R and y. A direct series of relations exists
between the two specific heats, their ratio, and their difference. If
we denote the specific heats by Jc and ?, then in proper units,
k lR lkR iv * .y.
AZ.B. ik t. t y k _ E y
fFor air, this gives ' 237 ^ ^ = 1.402.)
9^7^ ^"Jt5v /
778
~k = Tcy yTJ. fcyJc = yR. k = R ^r
c/ t/ / ^ j_
71. Rankme's Prediction of the Specific Heat of Air. The specific heat of air
was approximately determined by Joule in 1852. Earlier determinations were
unreliable. Eankine, in 1830, by the use of the relations just cited, closely ap
proximated the result obtained experimentally by Reguault three years later.
Using the values y = 1.4, R = 53.15, Rankine obtained
y
Regnault's result was 2375.
= R  = (53.15  772) x (1.4  0.4) = 0.239.
y 1
SPECIFIC HEATS OF GASES
37
72. Mayer's Computation of the Mechanical Equivalent of Heat
Reference was made in Art. 29 to the computation of this constant
prior to the date of Joule's conclusive experiments. The method is
substantially as follows : a cylinder and piston having an area of one
square foot, the former containing one cubic foot, are assumed to hold
air at 32 F., which is subjected to heat. The piston is balanced, so
that the pressure on the air is that of the atmosphere, or 14.7 Ib.
per square inch ; the total pressure on the piston being, then,
144 x 14.7 = 2116.8 Ib. While under this pressure, the air is heated
until its temperature has increased 491.4. The initial volume
of the air was by assumption one cubic foot, whence its weight
was 1 4 12.387 = 0.0811 Ib. The heat imparted was therefore
0.0811 x 0.2375 x 491.4 = 9.465 B. t. u. The external work was
that due to doubling the volume of the air, or 1 x 14.7 x 144 = 2116.8
footpounds. The piston is now fixed rigidly in its original position,
so that the volume cannot change, and no external work can be done.
The heat required to produce an elevation of temperature of 491.4
is then 0.0811 x 0.1689 x 491.4 = (3.731 B. t, u. The difference
of heat corresponding to the external work done is 2.734 B. t. u.,
whence the mechanical equivalent of heat is 2116.8 5 2.734 = 774.2
footpounds.
73. Joule's Experiment. One of the crucial experiments of the science was
conducted by Joule about 1844, after having been previously attempted by Gay
Lussac.
Two copper receivers, A and B, Fig. 9, were connected by a tube
and stopcock, and placed in a water bath. Air was compressed in A
to a pressure of 22 atmospheres,
while a vacuum was maintained
in . When the stopcock was
opened, the pressure in each re
ceiver became 11 atmospheres, and
the temperature of the air and of
FIG. 9. Arts. 73, so. Joule's Experiment. the water bath remained practically
unchanged. This was an instance of expansion without the perform
ance of external work; for there was no resisting pressure against the
augmentation of volume of the air.
38 APPLIED THERMODYNAMICS
74. Joule's and Kelvin's Porous Plug Experiment. Minute observations
showed that a slight change of temperature occurred in the water bath.
Joule and Kelvin, in 1852, by their celebrated "porous plug" experiments,
ascertained the exact amount of this change for various gases. In all of
the permanent gases the change was very small ; in some cases the tem
perature increased, while in others it decreased ; and the inference is jus
tified that in a perfect gas there would be no change of temperature (Art.
156).
75. Joule's Law. The experiments led to the principle that
when a perfect gas expands without doing external work, and without
receiving or discharging heat, the temperature remains unchanged and
no disgregation work is done, A clear appreciation of this law is of
fundamental importance. Four thermal phenomena might have
occurred in Joule's experiment : a movement of heat, the performance
of external work, a change in temperature, or work of disgregation.
From Art. 12, these four effects are related to one another in such
manner that their summation is zero; (9"= T+I+ W). By means
of the water bath, which throughout the experiment had the same
temperature as the air, the movement of heat to or from the air was
prevented. By expanding into a vacuum, the performance of external
work was prevented. The two remaining items must then sum up
to zero, i.e. the temperature change and the disgregation work. But
the temperature did not change ; consequently the amount of disgre
gation wort must have been zero.
76. Consequences of Joule's Law. In the experiment described, the pres
sure and volume changed without changing the internal energy. !N"o dis
gregation work was done, and the temperature remained unchanged.
Considering pressure, volume, and temperature as three cardinal thermal
properties, internal energy is then independent of the pressure or volume
and depends on the temperature only, in any perfect gas. It is thus itself
a cardinal property, in this case, a function of the temperature. "A
change of pressure and volume of a perfect gas not associated with change
of temperature does not alter the internal energy. In any change of tem
perature, the change of internal energy is independent of the relation of
pressure to volume during the operation ; it depends only on the amount
by which the temperature has been changed" (6). The gas tends to cool
in expanding, but this effect is "exactly compensated by the heat which
JOULE'S LAW 39
is disengaged through the friction in the connecting tube and the im
pacts which destroy the velocities communicated to the particles of gas
while it is expanding" (7) TJiere is ^racfr'raZ/;/ no disgregation work in
heating a sensibly perfect gas; all of the interned energy is evidenced by
temperature alone. When such a gas passes from one state to another in
a variety of ways, the external work done varies; but if from the total
movement of heat the equivalent of the external work be deducted, then
the remainder is always the same, no matter in what way the change of
condition has been produced. Instead of H = T f 1 4 T7, we may write
#= T+W.
77. Application to Difference of Specific Heats. The heat absorbed dur
ing a change in temperature at constant pressure being H=Jc(T), and
the external work during such a change being W= P(Vv) = R(T ),
the gain of internal energy must be
H W=(kR)(Tt}.
The heat absorbed during the same change of temperature at constant
volume is H=l(T ). Since in this case no external work is done, the
whole of the heat must have been applied to increasing the internal energy.
But, according to Joule's law, the change of internal energy is shown by the
temperature change alone. In whatever way the temperature is changed
from T to f, the gain of internal energy is the same. Consequently,
t) = l(Tt) and fc J? = Z,
a result already suggested in Art. 65.
78. Discussion of Results. The greater value of the specific heat at
constant pressure is due solely to the performance of external work dur
ing the change in temperature. The specific heat at constant volume is
a real specific heat, in the case of a perfect gas ; no external work is done,
and the internal energy is increased only by reason of an elevation of tem
perature. There is no disgregation work. All of the heat goes to make
the substance hot. So long as no external work is done, it is not neces
sary to keep the gas at constant volume in order to confirm the lower
value for the specific heat; no more heat is required to raise the tempera
ture a given amount when the gas is allowed to expand than when the
volume is maintained constant. For any gas in which the specific heat at
constant volume is constant, Joule's law is inductively established ; for no
external work is done, and temperature alone measures the heat absorp
tion at any point on the thermometric scale. If a gas is allowed to expand,
doing external work at constant temperature, then, since no change of inter*
40 APPLIED THERMODYNAMICS
nal energy occurs, it is obvious from Art. 12 that the external work is equal
to the heat absorbed. Briefly, the important deduction from Joule's experi
ment is that item (6), Art. 12, may be ignored when dealing with sensibly
perfect gases.
79. Confirmatory Experiment. By a subsequent experiment, Joule
showed that when, a gas expands so as to perform external work, heat dis
appears to an extent proportional to the work done. Figure 10 illustrates
the apparatus. A receiver A, containing gas compressed to two atmos
pheres, was placed in the calorimeter B and connected with the gas holder
Of placed over a water tank. The gas passed
from A to G through the coil D } depressed the
water in the gas holder, and divided itself be
tween the two vessels, the pressure falling to
that of one atmosphere. The work done was
computed from the augmentation of volume shown Fl ^ 10 " Art> 7 a 9 ' ~~ J J oul A e ' s
T . . , . .,* . Experiment, Second Ap
by driving down the water in G against atnios pa ratus.
pheric pressure; and the heat lost was ascertained
from the fall of temperature of the water. If the temperature of the
air were caused to remain constant throughout the experiment, then the
work done at G would be precisely equivalent to the heat given up by
the water. If the temperature of the air were caused to remain constantly
the same as that of the water, then H= = T+ 1+ W, (T+ 1)=  W, or
internal energy would be given up by the air, precisely equivalent in amount
to the work done in (7.
80. Application of the Kinetic Theory. In the porous plug experiment referred
to in Art. 74, it was found that certain gases were slightly cooled as a result of the
expansion, and others slightly warmed. The molecules of gas are very much closer
to one another in A than in B, at the beginning of the experiment. If the mole
cules are mutually attractive, the following action takes place : as they emerge from
A, they are attracted by the remaining particles in that vessel, and their velocity
decreases. As they enter B, they encounter attractions theie, which tend to in
crease their velocity; but as the second set of attractions is feebler, the total effect
is a loss of velocity and a cooling of the gas. In another ga>s, in which the molecules
repel one another, the velocity during passage would be on the whole augmented,
and the temperature increased. A perfect gas would undergo neither increase nor
decrease of temperature, for there would be no attractions or repulsions between
the molecules.
(1) A critical review of this theory has "been presented by Mills The Specific
Heats of the Elements, Science, Aug. 24, 1908, p. 221. (2) The Engineer, January
13, 1908. (3) Throughout this study, no attention will be paid to the ratio 778 as
affecting the numerical value of constants in formulas involving both heat and work
SYNOPSIS OF CHAPTER IV PROBLEMS 41
53.36
quantities. R may by either 63.36 or ZZT The student should discern whether
778
heat units or footpounds are intended. (4) Zeuner, Technical Thermodynamics,
Klein tr., I, 121. (5) Ibid., loc. tit. (6) Ewing: The Steam Engine, 1906. (7)
Wormell, Thermodynamics.
SYNOPSIS OF CHAPTER IV
Specific thermal capacities; at constant pressure, at constant volume; other capacities.
Atomic heat = specific heat X atomic weight; molecular heat.
The volumetric specific heats of common gases are approximately equal.
* 77" (JTT
Mean specific heat = ; true specific heat = ; real and apparent specific heats.
T t aT
EegnauWs law : u the specific heat is constant for perfect gases."
Difference of the two specific heats E = 53.36 ; significance of R.
The difference of the volumetric specific heats equals 0.0055 B. t. u. for all gases.
Ratio of the specific heats : y = 1.402 for air ; relations between A', Z, ?/, J?.
Rankine's prediction of the value of k: Mayer 1 s computation of the mechanical equiva
lent of heat.
Joule* s Law : no disgregation work occurs in a perfect gas.
If the temperature does not change, the external work equals the heat absorbed.
If no heat is received, internal energy disappears to an extent equivalent to the
external work done.
The condition of intermolecular force determines whether a rise or a fall of temperature
occurs in the porous plug experiment.
PROBLEMS
1. The atomic weights of iron, lead, and zinc being respectively 56, 206.4, 65 ; and
the specific heats being, for cast iron, 0.1298 ; for wrought iron, 0.1138 ; for lead,
0.0314 ; and for zinc, 0.0956, check the theory of Art. 69 and comment on the results.
(Ans., atomic heats are: lead, 6.481; zinc, 6.214; wrought iron, 6.373; cast iron,
7.259.)
2. [Find the volumetric specific heats at constant pressure of air, hydrogen, and
nitrogen, and compare with Art, 60. ( k = 3.4 for H and 0.2438 for N.)
(Ans., air 0.01917; hydrogen 0.01901; nitrogen 0.01912.)
3. The heat expended in warming 1 ib. of water from 32 F. to 160 F. being 127.86
B, t. u., find the mean specific heat over this range. (Ans., 0.9989.)
4. The weight of a cubic foot of water being 59.83 Ib. at 212 F. and 62.422 Ib. at
32 ff F., find the amount of heat expended in performing external work when ont>
pound of water is heated between these temperatures at atmospheric pressure.
(Ans., 0,00189 B.t.u,)
5. (a) Find the specific heat at constant volume of hydrogen and nitrogen.
(Ans., 2.41; 0.1732.)
(6) Find the value of y for these two gases. (Ans*, 1.4108; 1.4080.)
6. Check the value 0.0055 B. t. u. given in Art. 67 for hydrogen and nitrogen.
(Ans., 0.00554; 0.00554.)
42 APPLIED THERMODYNAMICS
7. Compute the elevation in temperature, in Art. 72, that would, for an expansion
of 100 per cent, under the assumed conditions, and with the given values of k and Z,
give exactly 778 as the value of the mechanical equivalent of heat. What law of
gaseous expansion would be invalidated if this elevation of temperature occurred ?
(Ans , 489.05 F )
8. In the experiment of Art. 79, the volume of air in C mci eased by one cubic foot
against normal atmospheric pressure. The weight of water in B was 20 Ib The tem
perature of the air remained constant throughout the experiment. Ignoring radiation
losses, compute the fall of temperature of the water. {Ans., 0.13604 F.)
CHAPTER V
GRAPHICAL REPRESENTATIONS: PRESSURE VOLUME PATHS OF
PERFECT GASES
81. Thermodynamic Coordinates. The condition of a body being fully
defined by its pressure, volume, and temperature, its state may be repre
sented on a geometrical diagram in winch these properties are used as
coordinates. This graphical method of analysis, developed by Clapeyron,
is now in universal use. The necessity for three coordinates presupposes
the use of analytical geometry of three dimensions, and representations
may then be shown perspectively as related to one of the eight corners
of a cube; but the projections on any of the three adjacent cube faces are
commonly used ; and since any two of three properties fix the third when
the characteristic equation is known, a protective representation is suffi
cient. Since internal energy is a cardinal property (Arts. 10, 76), this also
may be employed as one of the coordinates of a diagram if desired.
82. Illustration. In Fig. 11 we have one corner of a cube
constituting an origin of' coordinates at O. The temperature of a
substance is to be represented by the distance upward from 0; its
pressure, by the distance to the right ; and its volume, by the dis
tance to the left. The lines forming the cube edges are correspond
ingly marked OT, OP, 0V* Consider the condition of the body to
be represented by the point A., within the cube. Its temperature is
then represented by the distance AB, parallel to TO, the point B
being in the plane VOP. The distance AD, parallel to PO, from A
to.the plane TO F", indicates the pressure; and by drawing AQ paral
lel to VO, being the intersection of this line witli the plane TOP,
we may represent the volume. The state of the substance is thus
fully shown. Any of the three projections, Figs. 1214, would equally
fix its condition, providing the relation between P, V, and T is
known. In each of these projections, two of the properties of the
substance are shown ; in the three projections, each property appear^
43
44
APPLIED THERMODYNAMICS
twice; and the corresponding lines AB, AC, and AD are always
equal in length.
1
1
FIG. 11
Perspe
gram.
A
xf
x^ o
)
Art. 82. FIG
ictive Dia
V t
C
...A. D .
V
\
! o <,
" B P C u B
.12. Art. 82. FIG. 13. Art. 82. FIG. 14. Art 82
TP Diagram. VP Diagram. TV Diagram.
83. Thermal Lines. In Tig. 15, let a substance, originally at A, pass
at constant pressure and temperature to the state JB ; thence at constant
temperature and volume to the state 0\ and thence at constant pressure
'D
B,C,
r
FIG. 15. Art. 83.
Perspective Ther
mal Line.
FIG. 16. Art. 83.
TP Path.
FIG. 17. Art. 83.
VP Path.
FIG. 18. Art. 83.
TV Path.
and volume to D. Its changes are represented by the broken line ABCD,
which is shown in its various projections in Figs. 1618. The thermal
line of the coordinate diagrams, Figs. 11 and 15, is the locus of a series of
successive states of the substance. A path is the projection of a thermal
line on one of the coordinate planes (Figs. 1214, 1618). The path of a
substance is sometimes called its process curve, and its thermal line, a
thermogram.
The following thermal lines are more or less commonly studied :
(a) Isothermal, in which the temperature is constant; its plane is
perpendicular to the O^axis.
(5) Isometric, in which the volume is constant ; having its plane per
pendicular to the OF' axis.
(c) Isopiestic, in which the pressure is constant; its plane being per
pendicular to the OP axis.
(d) Isodynamic, that along which no change of internal energy
occurs.
GRAPHICAL REPRESENTATIONS
45
(e) Adiabatic, that along which no heat is transferred between the
substance and surrounding bodies; the thermal line of an.
insulated body, performing or consuming work.
84. Thermodynamic Surface. Since the equation of a gas in
cludes three variables, its geometrical representation is a surface;
and the first three, at least, of the above paths, must be projections
of the intersection of a plane with such surface. Figure 19, from Pea
FIQ. 19. Arts, 84, 103. Thermodynamic Surface for a Perfect Gas.
body (1), admirably illustrates the equation of a perfect gas,
RT. The surface pmnv is the characteristic surface for a perfect gas.
Every section of this surface parallel to the PV plane is an equilat
eral hyperbola. Every projection of such section on the PV plane
is also an equilateral hyperbola, the coordinates of which express the
law of Boyle, PF"=(7. Every section parallel with the TV plane
gives straight lines pm, a?, etc., and every section parallel with the
TP plane gives straight lines vn, xy, etc. The equations of these
212
32
46 APPLIED THERMODYNAMICS
lines are expressions of the two forms of the law of Charles, their
appearance being comparable with that in Fig. 5.
85. Path of Water at Constant Pressure. Some such diagram as that
of Fig. 20 would represent the behavior of water in its solid, liquid, and
vaporous forms when heated at constant pressure.
The coordinates are temperature and volume. At
A } the substance is ice, at a temperature below
the freezing point. As the ice is heated from A
to B, it undergoes a slight expansion, like other
solids. At B, the melting point is reached, and
as ice contracts in melting, there is a decrease in
volume at constant temperature. At C, the sub
stance is all water; it contracts until it reaches the
FIG 20 Art 85 Water ^ ^ ' . , . . _ _ , T .
at Constant Pressure. temperature of maximum density, 39.1 F., at D,
then expands until it boils at E 9 when the great
increase in volume of steam over water is shown by the line EF. If the
steam after formation conformed to Charles' law, the path would con
tinue upward and to the right from F, as a straight line.
86. The Diagram of Energy. Of the three coordinate planes, the PV
is most commonly used. This gives a diagram corresponding with that
produced by the steam engine indicator (Art. 484). It is sometimes called
Watt's diagram. Its importance arises principally from the fact that it
represents directly the external work done during the movement of the
substance along any path. Consider a vertical cylinder filled with fluid,
at the upper end of which is placed a weighted piston. Let the piston be
caused to rise by the expansion, of the fluid. The force exerted is then
equivalent to the weight of the piston, or total pressure on the fluid ; the
distance moved is the movement of the piston, which is equal to the aug
mentation in volume of the fluid. Since work equals force multiplied by
distance moved, the external work done is equal to the total uniform pressure
multiplied by the increase of volume.
87. Theorem. On a PV diagram, the external work done along
any path is represented by the area included be p
tween that path and the perpendiculars from its
extremities to the horizontal axis.
Consider first a path of constant pressure, a5,
Fig. 21. From Art. 86, the external work is
equivalent to the pressure multiplied by the in FlG 21< Art 87 ~
f 1 r ^ z IJF yv 7 External Work at
crease of volume, or to ca x ab = cabd. General constant Pressure
CYCLES
case : let the path be arbitrary, ab, Fig. 22. Divide the area aide
into an infinite number of vertical strips, amnc, mopn, oqrp, etc.,
each of which may be regarded as a rectangle,
such that ac = mn, win = op, etc. The external
work done along am, mo, oq, etc., is then repre
sented by the areas amnc, mopn, oqrp, etc., and
the total external work along the path ab is repre
sented by the sum of these areas, or by aide. c L p r 
FIG 22 Arts 87,t>8
Corollary L Along a path of constant volume External Work,
no external work is done. y at '
Corollary II. If the path be reversed, i.e. from right to left, as
along ba, the volume is diminished, and negative work is done ; work
is expended on the substance in compressing it, instead of being per
formed by it.
88. Significance of Path. It is obvious, from Fig. 22, that the amount
of external work done depends not only on the initial and final states a and
b, but also on the nature of the path between those states. According to
Joule's principle (Art. 75) the change of internal energy (T+ 1, Art. 12)
between two states of a perfect gas is dependent upon the initial and final
temperatures only and is independent of the path. The external work
done, however, depends upon the path. The total expenditure ofJieatj which,
includes both effects, can only be known when the path is given. The
internal energy of a perfect gas (and, as will presently be shown, Art.
109, of any substance) is a cardinal property; external work and heat
transferred are not. They cannot be used as elements of a coordinate
diagram.
89. Cycle.
A series of paths forming a closed finite figure con
stitutes a cycle. In a cycle, the substance is brought
back to its initial conditions of pressure, volume,
and temperature.
Theorem. In a cycle, the net external work
done is represented on the PV diagram by the en
closed area.
Let abed, Fig. 23, be any cycle. Along abc, the
work done is, from Art. 87, represented by the
area abcef. Along cda, the negative work done is similarly repre
J
FIG. 23. Art. 89.
External Work in
Closed Cycle.
48 APPLIED THERMODYNAMICS
sented by the area adcef. The net positive work done is equivalent
to the difference of these two areas, or to abed.
If the volume units are in cubic feet, and the pressure units ^Q pounds
per square foot, then the measured area abed gives the work in footpounds.
This principle underlies the calculation of the horse power of an engine
from its indicator diagram. If the cycle be worked in a negative direction,
e.g. as cbad, Fig. 23, then the net work will be negative ; i.e. work will
have been expended upon the substance, adding heat to it, as in an air
compressor.
90. Theorem. la a perfect gas cycle, the expenditure of heat is
equivalent to the external work done.
Since the substance has been brought back to its initial tempera
ture, and since the internal energy depends solely upon the tempera
ture, the only 'heat effect is the external work. In the equation
#= 2 7 + Jh W, F+I= 0, whence H= W, the expenditure of heat
being equivalent to its sole effect.
If the work is measured in footpounds, the heat expended is calcu
lated by dividing by 778. (See Note 3, page 37.) Conversely, in a
reversed cycle, the expenditure of external work is equivalent to the gain of
heat.
91. Isothermal Expansion. The isothermal path is one of much
importance in establishing fundamental principles. By definition
(Art. 83) it is that path along which the temperature of the fluid
is constant. For gases, therefore, from the characteristic equation,
if T be made constant, the isothermal equation is
p v = RT  0.
Taking R at 53.36 and 2* at 491.4 (32 F.),
(753.36X491.4 = 26,221;
whence we plot on Fig. 2 the isothermal curve al> for this tempera
ture; an equilateral hyperbola, asymptotic to the axes of P and V.
An infinite number of isothermals might be plotted, depending upon
the temperature assigned, as cd, ef, gh, etc. The equation of the
isothermal may le regarded as a special form of the exponential
equation PV n = 0^ in whieh n = 1.
ISOTHERMAL EXPANSION
92. Graphical Method. For rapidly plotting curves of the form PV = C, the
construction shown in Fig. 24 is useful. Knowing the three corresponding prop
erties of the gas at any given
state enables us to fix one point
on the curve ; thus the volume x
12.387 and the pressure 2116.8
give us the point C on the
isothermal for 491.4 absolute.
Through C draw CM parallel
to 0V. From draw lines OD,
ON, OM to meet CM. Draw
CB parallel to OP. From tha
points 1, 5, 6, where OD, ON,
OM intersect CB, draw lines
1 2, 5 7, 6 8 parallel to 0V. From D, N, M, draw lines perpendicular to 0V.
The points of intersection 2, 7, 8 are points on the required curve. Proof : draw
EC, .F6, parallel to OV, and 8 A parallel to OP. In the similar tri
angles 0GB, OMA, we have 6 B : MA \\OB\OA, or 8 A : CB : : EC : FQ,
whence SA xF8=CBxEC,or P 8 F 8 = P c Vc
93. Alternative Method. In Fig. 25 let 6 be a known point on the
curve. Draw aD through & and lay off DA = ab. Then A is another
point on the curve. Additional points may be found by either of the
constructions indicated: e.g. by drawing dh and laying off hf=db,
or by drawing BK and laying off Kf= BA. These methods are prac
tically applied in the examination of the expansion lines of steam
engine indicator diagrams.
FIG. 24. Ait. 92, 93. Construction of Equilateral
Hyperbola.
94. Theorem: Along an isothermal path for a per
fect gas, the external work done is equivalent to
the heat absorbed (Art. 78).
~KT~ i a """""d v The internal energy
FIG. 25. Art. 93. Second Method for Plotting is Unchanged, as indi
Hyperbolas. cate d by Joule's law
(Art. 75) ; hence the expenditure of heat is solely for the performance
of external work. H=T+I+ W, l>ut 2^=0, T+I=Q, and H= W.
Conversely, we have Mayer's principle, that " the work done in compressing a
portion of gas at constant temperature from one volume to another is dynamically
equivalent to the heat emitted hy the gas during the compression" (2).
95. Work done during Isothermal Expansion. To obtain the ex
ternal work done under any portion of the isothermal curve, Fig. 24,
we must use the integral form,
50 APPLIED THERMODYNAMICS
in which v, "Fare the initial and final volumes. But, from the equa
tion of the curve, pv = P V, P = pv f V, and when p and v are given,
XV fiy JT Y p
V V V JL
The heat absorbed is equal to this value divided by 778.
96. Perfect Gas Isodynamic (Art. 87). Since in a perfect gas the
internal energy is fixed by the temperature alone, the internal energy
along an isothermal is constant, and the isodynamic and isothermal
paths coincide.
97. Expansion in General. We may for the present limit the
consideration of possible paths to those in which increases of volume
are accompanied by more or less marked decreases in pressure ; the
latter ranging, say, from zero to infinity in rate. If the volume in
CO.ST.NT PRESSURE n ~ o , creases without any fall in pressure, the
path is one of constant pressure ; if the
volume increases only when the fall of
pressure is infinite, the path is one of con
stant volume. The paths under considera
tion will usually fall between these two,
FIG. 26. Art. 97. Expansive like 5, aw, ad, etc., Fig. 2<3. The general
Paths  law for all of these paths is PV n s\> con
stant, in which the slope is determined by the value of the exponent n
(Art. 91). Foi"M=0, the path is one of constant pressure, ae, Fig. 2G.
For 7i= infinity, the path is one of constant volume.* The "steepness"
of the path increases with the value of n. (Note that the exponent
n applies to V only, not to the whole expression.)
98. Work done by Expansion. For this general case, the external
work area, adopting the notation of Art. 95, is,
5
8
But since pv n = PF", P = pu n F'*; whence, when p and v are given,
_
I n\ J n I n1
J. L
* F n =l, where w=0. If rt oc, we may write PccF=pa > y, or F=i;.
THE ADIABATIC 51
When F= infinity, P = 0, and the work is indeterminate by this expression; but
we may write W = ^~ (l  } = PL. [~l _ fHV" 1 ], in which, for V= in
nI \ pv J n 1 L \VI J
finity, W pv (n 1), a finite quantity, The work undev an exponential curve
(when n>l) is thus finite and commensurable, no matter how far the expansion
be continued.
99. Relations of Properties. For a perfect gas, in which  = H, we have
PVt= pvT.
If expansion proceeds according to the law P V n = pu n t we obtain, dividing the
first of these equations by the second,
Z^
V n v n
This result permits of the computation of the change in temperature following a
given expansion. We may similarly derive a relation between temperature and
pressure. Since
pv n = PV n , v(p*) n V(P) n . Dividing the expression pv T = PVt by this, we have
L ^1 /pN
n _//Z>N ? whence  = [ 
By interpretation of these formulas of relation, we observe that for
values of n exceeding unity, during expansion (i.e. increase of volume), the
pressure and temperature decrease, while external work is done. The
gain or loss of heat we cannot yet determine. On the other hand, during
compression, the volume decreases, the pressure and temperature increase,
and work is spent upon the gas. In the work expression of Art. 98, if
p, v } t are always understood to denote the initial conditions, and P, V, T 9
the final conditions, then the work quantity for a compression is negative.
100. Adiabatic Process. This term (Art. 83) is applied to any
process conducted without the reception or rejection of heat from or
to surrounding bodies by the substance under consideration. It is
by far the most important mode of expansion which we shall have to
consider. The substance expands without giving heat to, or taking
heat from, other bodies. It may Iqse heat, by doing work; or, in com
pression, work may be expended on the substance so as to cause it to
gain heat : but there is no transfer of heat between it and surrounding
bodies. If air could be worked in a perfectly nonconducting cylinder,
we should have a practical instance of adiabatic expansion. In
practice we sometimes approach the adiabatic path closely, by causing
expansion to take place with great rapidity, so that there is no time
52 APPLIED THERMODYNAMICS
for the transfer of heat. The expansions and compressions of the air
which occur in sound waves are adiabatic, on account of their rapidity
(Art. 105). In the fundamental equation H= T+ 1+ TF, the adi
abatic process makes JI= 0, whence W= (7+ J) ; or, the external
work done is equivalent to the loss of internal energy, at the expense of
which energy the work is performed.
101. Adiabatic Equation. Let unit quantity of gas expand adiabatically
to an infinitesimal extent, iucreasing its volume by dv, and decreasing its
pressure and temperature by dp and dt. As has just been shown,
TF (^4 1), the expression in the parenthesis denoting the change in
internal energy during expansion. The heat necessary to produce this
change would be Idt, I being the specific heat at constant volume. The ex
ternal work done is W=pdo\ consequently, pdv = Idt. Prom the
equation of the gas, pv = Rt, t =^ 9 whence, dt = =(pdv 4 vdp). Using
this value for dt, M H
pdv =  (pdv + vdp).
IL
But It is equal to the difference of the specific heats, or to & Z; so that
pdv =  (jpdv + vdp),
K t
ypdv pdv = pdv vdp,
=  E 9 giving by integration,
v p
ylog e v + log e p = constant,
or pv y = constant,
y being the ratio of the specific heats at constant pressure and con
stant volume (Art, 69.)
102. Second Derivation. A simpler, though less satisfactory, mode of
derivation of the adiabatic equation is adopted by some writers. Assum
ing that the adiabatic is a special case of expansion according to the law
PV n , the external work done, according to Art. 98, is
E(t  T)
ADIABATIC EXPANSION 53
During a change of temperature from t to T, the change in internal energy
is l(t T) } or from Art. 70, since I = R t(y 1), it is
Jffi  T)
yi
But in adiabatic expansion, f/te external icork done is equivalent to the
change in internal energy ; consequently
n y 1
rc = 2/, and the adiabatic equation ispu v = PFX For air, the adiabatic is
then represented by the expression ^(V) 1 ' 402 = a constant.
103. Graphical Presentation. Since along an adiabatic the external
work is done at the expense of the internal energy, the temperature must
fall during expansion. In the diagram of Fig. 19, this is shown by com
paring the line ab, an isothermal, with ae, an adiabatic. The relation of
p to v, in adiabatic expansion, is such as to cause the temperature to fall.
The projections of these two paths on the pv plane show that as
expansion proceeds from a, the pressure falls more rapidly along
the adiabatic than along the isothermal, a result which might have been
anticipated from comparison of the equations of the two paths. If an
isothermal and an adiabatic be drawn through the same point, the latter
will be the "steeper" of the two curves. Any number of adiabatics may
be constructed on the pv diagram, depending upon the value assigned to
the constant (ptf) ; but since this value is determined, for any particular
perfect gas, by contemporaneous values of p and v, only one adiabatic can
be drawn for a given gas through a given point.
104. Relations of Properties. By the methods of Art. 98 and
Art. 99, \ve find, for adiabatic changes,
During expansion, the pressure and temperature decrease, external work is done
at the expense of the internal energy, and there is no reception or rejection of heat.
105. Direct Calculation of ^the Value of y. The velocity of a wave in an
astic medium is, according to a fundamental proposition in dynamics, equal to
the square root of the coefficient of elasticity divided by the mass density:* that is,
w
* See, for example, Appendix A to Vol. HI of Nichols and Franklin's Elements of
Physics.
54 APPLIED THERMODYNAMICS
V being in feet per second and w in Ibs. per cubic foot. When a volume of gas
of crosssection =n and length I is subjected to the specific pressure increment dp,
producing the extension (negative compression) dl,
dp
e
dl+V
The volume of this gas is In v: so that y = and e = f* The pulsations
which constitute a sound wave are very rapid, hence adiabatic, so that pvv = constant,
and
ypvi  l dv = v
*
For 32 F. and p = 14. 697X144, w?=0.081. Taking ^ at 32.19 and V at the
experimental value of 1089,
1089X1089X0081
2/ "32.19X14.697Xl44
105 a. Velocity with Extreme Pressure Changes. The preceding computation
applies to the propagation of a pressure wave of very small intensity from a local
ized starting point. Where the pressure rises considerably say from JP to P, the
volume meanwhile decreasing from v to Fo, then
Now F (velocity) = V^~ and e =  ^ y ^ for finite changes. If v is the vol
ume of W Ib. of gas (not the specific volume), pv = Jft PF, PF = ^, and we have
for the velocity,
F =
ADIABATIC EXPANSION
For t = 530, p = 100, P = 400, this becomes
55
F =
32.2 x 53.36 x 530 x 800 x 144
=v^
(39,200,000,000
9060
= 2078 ft. per second.
This would he the velocity of the explosion in the cylinder of an internal com
hustion engine if the pressure were generated at all points simultaneously. As a
matter of fact, the combustion is local and the velocity and pressure rise are much
less than those thus computed (Art. 319).
106. Representation of Heat Absorbed. Theorem: The heat ab
sorbed on any path is represented on the PV diagram by the area en
closed between that path and the two adiabatics through its extremities,
indefinitely prolonged to the right.
Let the path be ab, Fig. 27. Draw the adiabatics an, IN. These
may be conceived to meet at an infinite dis
J> tance to the right, forming with the path the
closed cycle abNn. In such closed cycle,
the total expenditure of heat is, from Art.
V N 90, represented by the enclosed area ; but
_ v since no heat is absorbed or emitted along
FIG. 27 Arts. 106, 109. Rep the adiabatics, all of the heat changes in the
resentation of Heat Ab cycle must ] lave occurred along the path ab,
sorbed. J D
and this change of heat is represented by the
area abNn. If the path be taken in the reverse direction, i.e. from b
to a, the area abNn measures the heat emitted.
107. Representations of Thermal Capacities. Let ab, cd, Pig. 28, be two
isothermals, differing by one degree. Then efnN represents the specific
heat at constant volume, egmN the specific heat at
constant pressure, eN, fn } and gm being adiabatics.
The latter is apparently the greater, as it should
be. Similarly, if ab denotes unit increase of
volume, the area abMN represents the latent heat
of expansion. The other thermal capacities men
tioned in Art. 58 may be similarly represented.
** FIG. 28. Art. 107, Thermal
Capacities.
56
APPLIED THERMODYNAMICS
108. Isodiabatics. An infinite number of expansion paths is possible
through the same point, if the n values arc different. An infinite
number of curves may be dra\vn, having the name n value, if they do
not at any of their points intersect. Through a given point and with
a given value of n, only one curve can be drawn. When two or more
curves appear on the same diagram, each having the same exponent (n
^87 '
(i)
FIG. 29. Art. 108. Isodiabatics.
(c)
value), such curves are called isodiabatics. In most problems relating
to heat motors, curves appear in isodiabatic pairs. Much labor may be
saved in computation by carefully noting the following relations:
1. In Fig. 29 (a), let the isodiabatics pv Ml =const. be intersected
by lines of constant pressure at a, b, c and 4 d. Then
7111
nil
rn
m l b
(Art. 99).
n\ ;
2. In Kg. 29 (6), let the same isodiabatics be intersected by lines
of constant volume, determining points a, b, c and d. Then
"T c)
(Art. 99).
'14;
ISODIABATICS 57
3. In Fig. 29 (c), the same isodiabatics are intersected by isothermals
at a, b, c and d. Now
(Art. 99).
&T nr =Y
W * c J
= p^ and p a = p (I)
In this case, it is easy to show also, tnat
V a Va .__, V a V
(ID
d K c
but in this case (I) is not equal to (II) : the volume ratio is not equal to
the pressure ratio. Note also that in each of the three cases the equality
of ratios exists between properties other than that made constant along
the intersecting lines; thus, in (a), the pressure is constant, and the
volume and temperature ratio is constant.
109. Joule's Law. From the theorem of Art. 106, Rankine has
illustrated in a very simple manner the principle of Joule, that the
change of internal energy along any path of any substance depends
upon the initial and final states alone, and not upon the nature of the
path. In Fig, 27, draw the vertical lines ax, by. The total heat
absorbed along ab = nabNj the external work done = xaby. The
difference = nabN xaby = nzbN xazy, is the change in internal
energy; H = T + I + W, whence HW*=(T+r)} and the extent of
these areas is unaffected by any change in the path ab f so long as the
points a and b remain fixed.
58 APPLIED THERMODYNAMICS
110. Value of y. A method of computing the value of y for air has
been given in Art. 105. The apparatus shown in Pig. 30 has been used
by several observers to obtain direct values for various gases. The vessel
was filled with gas at P, F, and T, T being the temperature of the atmos
phere, and P a pressure somewhat in excess of that
r tQh* of the atmosphere. I>y opening the stopcock, a
sudden expansion took place, the pressure falling
to that of the atmosphere, and the temperature
falling to a point considerably below that of the
atmosphere. Let the state of the gas after this
adiabatic expansion be p, v, t. Then, since
y = 7j?' i T r
FIG. 30. Art, 110, De 2 log;? log P,
sormes' Apparatus. log F log V
After this operation, the stopcock is closed, and the gas remaining in the
vessel is allowed to return to its initial condition of temperature, T.
During this operation, the volume remains constant; so that the final
state is pa % T\ whence p z v = PF, or log F log v = logjp a log P. Sub
stituting this value of log F log v in the expression for y, we have
J io g y? 2 ~iogp'
so that the value of y may be computed from tlie pressure changes alone.
Clement and Desormes obtained in this manner for air, y = 1.3524 ; Gay
Lussac and Wilter found ?/ = 1 .3745. The experiments of Hirn, Weisbach,
Masson, Cazin, and Kohlrausch were conducted in the same manner. The
.method is not sufficiently exact.
11L Expansions in General. In adiabatic expansion, the external work
done and the change in internal energy are equally represented by the
expression P v ~~ 9 derived as in Art. 98. For expansion from p, v to
infinite volume, this becomes ' _. The external work done during any
" * rtrr
expansion according to the law pv n = PF n from pv to PF, is '
The stock of internal energy at p, v, is  = It ; at P, F, it is  = IT.
V 1 y 1
The total heat expended during expansion is equal to the algebraic sum
of the external work done and the internal energy gained. Then,
* The final condition being that of the atmosphere, all of the gas, both
within and without the vessel, is at the condition p, y, t. The change in quantity
(weight) of gas in the vessel during the expansion does not, therefore, invalidate the
equation.
POLYTROPIC PATHS 59
= Z( y)[ <y ~? )> i n which Z is the initial, and T the final temperature.
\n ly
This gives a measure of the net heat absorbed or emitted during any ex
pansion or compression according to the law po n = constant. When n
exceeds y, the sign of II is minus ; heat is emitted ; when n is less than y
but greater than 1.0, heat is absorbed : the temperature falling in both cases.
When ^=y, the path is adiabtitic, and heat is neither absorbed nor emitted.
112. Specific Heat. Since for any change of temperature involving
a heat absorption H 3 the mean specific heat is
* = T^?
we derive from the last equation of Art. Ill the expression,
,!=,
711
giving the specific heat along any path pv n = PV n . Since the values
of n are the same for isodidbatics, the specific heats along such paths are
equal (Art. 108).
113. Ratio of Internal Energy Change to External Work. For any given
value of n, this ratio has the constant value
n1
yi'
114. Polytropic Paths. A name is needed for that class of paths
following the general law pv PP 1 , a constant. Since for any
gas y and I are constant, and since for any particular one of these
paths n is constant, the final formula of Art. Ill reduces to
In other words, the rate of heat absorption or emission is directly pro
portional to the temperature change; the specific heat is constant. Such
paths are called polytropic. A large proportion of the paths exempli
fied in engineering problems may be treated as polytropics. The
polytropic curve is the characteristic expansive path for constant
weight of fluid.
60
APPLIED THERMODYNAMICS
115. Relations of n and 5. We have discussed such paths in which the
value of n ranges from 1.0 to infinity. Figure 31 will make the concep
tion ruore general. Let a represent the initial condition of the gas. If
FIG. 31. Art. 115. Poly tropic Paths.
it expands along the isothermal a& 5 n = 1, and s s the specific heat, is infi
nite ; no addition of heat whatever can change the temperature. If it
expands at constant pressure, along ae, n = Q, and the specific heat is finite
and equal to ly = k. If the path is ag, at constant volume, n is infinite
and the specific heat is positive, finite, and equal to ?. Along the isother
mal of (compression), the value of n is 1, and s is again infinite. Along
the adiabatic ah, n = 1.402 and s = 0. Along ai, n = and $ = k. Along
ad, n is infinite and s = L Most of these relations are directly derived
from Art. 112, or may in some cases be even more readily apprehended by
drawing the adiabatics, en, gN", fm, iM, dp, bP, and noting the signs of the
areas representing heats absorbed or emitted with changes in temperature.
Tor any path lying between ah and af or between etc and a&, the specific
heat is negative, i.e. the addition of heat cannot keep the temperature from fall
ing: nor its abstraction from rising.
116. Relations of Curves : Graphical Representation of n. Any number of
curves may be drawn, following the law pv n = C, as the value of C is changed.
RELATIONS OF n AND
61
In Fig. 32, let a&, ctf, e/be curves thus drawn. Their general equation is pv n = C,
whence
= or
civ v
If M TV is the angle made by
the tangent to one of the curves
with the axis 0V, and MOV
the angle formed by the radius
vector RM with the axis 0V,
then, since dp dv is the tan
gent of MTV, andjp v is the
tangent of MOV,
FIG. 32. Art. 116. Determination of Exponent.
 tan MTV = n tan MO V.
If the radius vector be produced as ItMNQ the relations of the angles made be
tween the OF axis and the successive tangents MT, NS, QU, are to the angle
MOV as just given; hence the various tangents
are parallel (4) .
Since tan MTV = Mg ^ gT and tan3/OF =
Mg r Of/, the preceding equation gives
whence n = Og = gT. (The algebraic signs of
0(j and ^T, measured from g, are different.) In
order to determine the value of n from a given
curve, we need therefore only draw a tangent
MT and a radius vector J/0, whence by drop
ping the perpendicular Mg the relation Og gT
is established. If we lay oif from the distance
OA as a unit of length, drawing A C parallel to
the tangent, and CB through C, parallel to the
c
FIG. 33.
Art 116. Negative
Exponent.
radius vector, then by similar triangles
OgigTiiOBiOA and Og r gT = 05= n.
Figure 33 illustrates the generality of this
method by showing its application to a
curve in which the value of n is negative.
117. Plotting of Curves: Brauer's
Method. The following is a simple method
for the plotting of exponential curves, in
cluding the adiabatic, which is ordinarily
a tedious process. Let the point Af,
Fig. 34, be given as one point on the re
quired curve. Draw a line OA making an
angle VOA with the axis OF, and a line
OB making an angle POB with the axis
FIG, 34, Art. 117. Brauer's Method.
62 APPLIED THERMODYNAMICS
OP. Draw the vertical line MS and the horizontal line MT. Also draw the
line TU making an angle of 45 with OP, and the line SJR making an angle of
45 with MS. Draw the vertical line EN through 11, and the horizontal line UN
through U. The coordinates of the point of intersection, JV, ot these lines, are
OR and RN. Let the coordinates of Jl/, TM (= OQ), and MQ be designated by
v, p ; and those of N, OR, and RN (= 0L), by F, P. Then tan J r 0. 1 = QS  OQ,
= Q,R^TM = (Vv)v', and tan 7>0JB = UL  = 7X NR = (pP) P;
whence 7= v (tan FO4 f 1) and jt? = P (tan PO5 H 1). If the law of the
curve through M and N is to be^y n = PF n , we obtain
P(tanP05 + !>= 7>{i'(fcan F6L4 + 1)3%
whence (tan POE 4 1)  (tan FCU + !)" If now, in the first place, we make the
angles POB, VGA such as to fulfill this condition, then the point N and others
similarly determined will be points on a curve following the law pv n = PF n .
118. Tabular Method. The equation pv n = P V n may be written p = P( j
or logjt? log P = n log (F i). Tf we express P as a definite initial pressure for
all P V n curves, then for a specific value of n and for definite ratios F v we may
tabulate successive values of log p and of p. Such tables for various values of n
are commonly used. In employing them, the final pressure ia found in terms of
the initial pressure for various ratios of final to initial volume.
119. Representation of Internal Energy. In Fig. 35, let An represent
an adiabatiu. Daring expansion from A to a, the external work done is
Aabc, which, from the law of the adiabatic, is
equal to the expenditure of internal energy. If
expansion is continued indefinitely, the adiabatio
An gradually approaches the axis OF, the area
below it continually representing expenditure of
internal energy, until with infinite expansion An
and OF coincide. The internal energy is then ex
35. Art. no. Repro h^usted. The total internal energy of a substance
sentation of Internal may therefore be represented by the area between
Ener gy the adiabatic through its state, indefinitely prolonged
to the right, and the horizontal axis. Representing this quantity by JB ; then
from Art. Ill,
where v is the initial volume, p the initial pressure, and y the adiabatio
exponent. This is a finite and commensurable quantity.
120. Representation by Isodynamic Lines. A defect of the preceding
representation is that the areas cannot be included on a finite diagram.
GRAPHICAL REPRESENTATIONS
63
In Fig. 36, consider the path. AB. Let BG be an adiabatic and AC an
isodynamic. It is required to find the change of internal energy between
A and B. The external work done daring adi
abatic expansion from B to G is equal to BCcb ;
and this is equal to the change of internal en
ergy between B and 0. But the internal energy
is the same at G as at A, because AC is an
isodynamic. Consequently, the change of in
ternal energy between A and B is represented
by the area BCcb; or, generally, by the area
included between the adiabatic through the final
state, extended to its intersection witli the iso
dynamic through the initial state, and the hori
zontal axis.
FIG. 36. Arts. 120, 121. In
ternal Energy, Second Dia
gram.
181.
FIG. 37 Art 121. External
Work and Internal Energy.
Source of External Work, If in Fig. 36 the path is such as to increase
the temperature of the substance, or even to keep its
temperature from decreasing as much as it would
along an adiabatio, then heat must be absorber! .
Thus, comparing the paths ad and ac, Fig. 37, aN
and cm being adiabatics, the external work done
along ad is adef, no heat is absorbed, and the internal
energy decreases by adef. Along ac, the external
work done is acef, of which arfe/was done at the ex
pense of the internal energy, and acd by reason of
the heat absorbed. The total heat absorbed was
Ncicm, of which acd was expended in doing external work, while Ndcm went
to increase the stock of internal energy.
122. Application to Isothermal Expansion. If the path is isothermal, Fig. 38,
line A B, then if BN t An are adiabatics, we have,
W + X = external work done,
X 4 Y = heat absorbed = W + X,
W f Z = internal energy at A,
Y 4 Z = internal energy at B,
W = work done at the expense of the in
ternal energy present at A,
X = work done by reason of the absorption
of heat along AB,
Z = residual internal energy of that originally
present at A ,
Y = additional internal energy imparted by
the heat absorbed;
and since in a perfect gas isothermals are isodynamics, we note that
FIG. 38, Art. 122. Heat and
Work in Isothermal Expansion,
64
APPLIED THERMODYNAMICS
123. Finite Area representing Heat Expenditure. In Fig. 39, let ab be any
path, In and aN adiabatics, and nc an isodynamic. The exteinal work done along
ab is abtle't while the increase of internal energy is
befit. The total heat absorbed is then represented by
the combined areas abcfe. If the path ab is iso
thermal, tins construction leads to the known result
that there is no gain of internal energy, and that th?
total heat absorbed equals the external work. If the
path be one of those de
scribed in Art. 115 as of
negative specific heat, \ye
may represent il as ag,
Fig. 40. Let Igm be an
adiabatic. The external
"S.
FIG. 39. Art 123 Represen
tation of Heat Absorbed.
m
V
FIG. 40. Art. 13.1. Nega
tive Specific Heat,
work done is ac/dc. The change of internal energy,
from Art. 120, is bydf, if ab is an isodynamic; and
this being a negative area, we note that internal en
ergy has been expended, although heat has been ab
sorbed. Consequently, the temperature has fallen. It
seems absurd to conceive of a substance as receiving heat while falling in tem
perature. The explanation is that it is cooling, "by doing external work, faster
than the supply of heat can warm it. Thus, H T+ /+ W', but //< W\ con
sequently, (T 4 7) is negative.
123 a. Ordnance. Some such equation as that given in Art. 105 a may apply
to the explosion of the charge in a gun. Ordinary gunpowder, unlike various de
tonating compounds now used, is scarcely a true explosive. It is merely a rapidly
burning mixture. A probable expression for the reaction with a common type of
powder is
4 KN0 3 + C 4 + S = K 2 C0 3 + K 3 S0 4 + N 4 + 2 CO 2 + CO.
It will be noted that a largo proportion of the products of combustion arc solids;
probably, in usual practice, from 55 to 70 per cent. As first formed, these may be
in the liquid or gaseous state, in which case they contribute large quantities of
heat to the expanding and cooling charge as they liquefy and solidify.
When the charge is first fired, if the projectile stands still, the temperature and
pressure will rise proportionately, and the rise of the former will be the quotient of
the heat evolved by the mean specific heat of the productw of combustion. Fortu
nately for designers, the projectile moves at an early stage of the combustion, so that
the rise of pressure and temperature is not instantaneous, and the shock is more or
less gradual. After the attainment of maximum pressure, the gases expand,
driving the projectile forward. Work is done in accelerating the latter, but the
process is not adiabatic because of the contribution of heat by the ultimately solid
combustion products. The temperature does fall, however, so that the expansion
is one between the isothermal and the adiabatic.
The ideal in design is to obtain the highest possible muzzle velocity, but this
should be accomplished without excessive maximum pressures. The more nearly
ORDNANCE 65
the condition of constant pressure can be approximated during the travel of the
projectile from breech to muzzle, the better. Both velocities and pressures during
this traverse have been studied experimentally; the former by the chronoscope,
the latter by the crusher gage.
The suddenness of pressure increase may be retarded by increasing the density
of the powder, and is considerably affected by its fineness and by the shape and
uniformity of the grains.
Suppose 1 + s Ib. of charge to contain 5 Ib. of permanently solid matter of spe
cific heat = c, and that the specific heat of the gaseous products of combustion,
during their combustion, is I. Let the initial temperature be F. Then the
temperature attained by combustion is
T
I + cs
where H is the heat evolved in combustion. During any part of the subsequent
expansion,
H= T + I + W=E + W,
dH = Idt f pdv.
The only heat contributed is that by the solid residue, and is equal to
dH  scdt = Idt +pdv,
so that  (sc + dt = pdv = Rt~,
and between the limits T and t,
where V is the initial and v the final volume of the charge. Now since p  = ,
iMo * T
= = f JY* st< The external work done during expansion is
W = (pdv =  ldt scdt = (sc + (T  t)
If we wish to include the effect due to the fact that a portion, say r, of the original
volume of charge forms nongaseous products, we may write or V, F(l r) ? and
for v, v rF, and the complete equation becomes
Suppose r = 4000 F., s = 0.6, c = 0.1, I = 0.18, fc = 0.25, 7=0.02, r = 0.6,
v = 0.20; then
66 APPLIED THERMODYNAMICS
007
W= 0.24 x 4000 x 778 { 1  (j^) "} = 1,000 ftlb.
If w be the weigh b of projectile, F the velocity imparted thereto, and / the
" factor of effect " to care for practical deductions from the computed value of W,
then
= V* and
which for our conditions, with w  5, /= 0.90, gives
= IG4.4 x 491,000 x 0.90 _
* 5
The maximum work possible would be obtained in a gun of ample length, the
products of combustion expanding down to their initial temperature, and would be,
for our conditions,
W= 778 H = 778 T(l + cs) = 778 x 4000 x 0.24 = 814,080 ft.lb.
The equation of the expansion curve is pv n = const., where n has the value
+ sg ; or, for our conditions " =13. nearly.
/ H sc 0.24
Viewing the matter in another way : the heat contributed by the solid residues
is that absorbed by the gases ; or
! = *!,
where $1 is the specific heat of the gases during expansion.
Then s, = I n ~~ V and n = + Jtc , as before.
1 n  1 1 + sc
The external work done during expansion is
from which the equation already given may be derived.
MODIFICATIONS IN IRREVERSIBLE PROCESSES
124. Constrained and Free Expansion. In Art. 86 it was assumed that
the path of the substance was one involving changes of volume against a
resistance. Such changes constitute constrained expansion. In this pre
liminary analysis, they are assumed to take place slowly, so that no
mechanical work is done by reason of the velocity with which they are
effected. When a substance expands against no resistance, as in Joule's
experiment, or against a comparatively slight resistance, we have what is
known as free expansion, and the external work is wholly or partly due
to velocity changes.
IRREVERSIBLE PROCESSES 67
125, Reversibility. All of the polytropic curves which have thus far
been discussed exemplify constrained expansion. The external and in
ternal pressures at any state, as in Art. 86, differ to an infinitesimal
extent only ; the quantities are therefore in finite terms equal, and the
processes may be worked at icill in either direction. A polytropic path
having a finite exponent is in general, then, reversible, a characteristic of
fundamental importance. During the adiabatic process which occurred
in Joule's experiment, the externally resisting pressure was zero while
the internal pressure of the gas was finite. The process could not be
reversed, for it would be impossible for the gas to flow against a pressure
greater than its own. The generation of heat by friction, the absorption
of heat by one body from another, etc., are more familiar instances of
irreversible process. Since these actions take place to a greater or less
extent in all actual thermal phenomena, it is impossible for any actual
process to be perfectly reversible. "A process affecting two substances is
reversible only when the conditions existing at the commencement of the
process may be directly restored without compensating changes in other
bodies."
126. Irreversible Expansion. In Fig. 41, let the substance expand
unconstrainedly, as in Joule's experiment, from a to &, this expansion
being produced by the sudden decrease in ex p
ternal pressure when the stopcock is opened.
Along the path ab, there is a violent movement of
the particles of gas ; the kinetic energy thus
evolved is transformed into pressure at the end
of the expansion, causing a rise of pressure to c.
The gain or loss of internal energy depends solely
upon the states a, c; the external work done does FIG. 41. Art, uo. irre
not depend on the irreversible path ab, for with versible Path,
a zero resisting pressure no external work is done. The theorem of Art. 86
is true only for reversible operations.
127. Irreversible Adiabatic Process, Careful consideration should be
given to unconstrained adiabatic processes like those exemplified in Joule's
experiment. In that instance, the temperature of the gas was kept up by
the transformation back to heat of the velocity energy of the rapidly
moving particles, through the medium of friction. We have here a special
case of heat absorption. No heat was received from without ; the gas
remained in a heatinsulated condition. While the process conforms to
the adiabatic definition (Art. 83), it involves an action not contemplated
when that definition was framed, viz , a reception of heat, not from, sur
68 APPLIED THERMODYNAMICS
rounding bodies, but from the mechanical action of the substance itself*
The fundamental formula of Art. 12 thus becomes
jy= r+ /+ w+ F,
in which V may denote a mechanical effect due to the velocity of the
particles of the substance. This subject will be encountered later in
important applications (Arts. 175, 176, 426, 513).
FUKTHEB APPLICATION OP THE KINETIC THEORY
127 a. The Two Specific Heats. The equation has been derived (Art, 55),
*?,*"+
in which p = the specific pressure exerted by a gas on its bounding surfaces ;
v = the aggregate volume (not the specific volume) of the gas,
W = the weight of the gas, whence r = its specific volume,
M its mass,
w = the average velocity of all of the molecules of the gas.
The kinetic theory asserts that the absolute temperature is proportional to the
mean kinetic energy per molecule. In a gas without intermolecular attractions
the application of heat at unchanged volume can only add to the kinetic energy of
molecular vibration. In passing between the temperatures ^ and t 9 then, the ex
penditure of heat may be written
M, 2N , A ^
2
If the operation is performed at constant pressure instead of constant volume
the expenditure of heat will be greater, by the amount of heat consumed in per
forming external work, jo(u 2 1^). From Charles' law,
The external work is then
and the total heat expended is
H k = A + B = (u>**  w^). (C)
If we divide C by JL, we obtain
M 10
FURTHER APPLICATION OF THE KINETIC THEORY 69
which would be the ratio of the specific heats for a perfect monatomic gas. In
such a gas, the molecules are relatively far apart, and move in straight lines. In
a polyatomic gas (in which each molecule consists of more than one atom), there
are interattractions and repulsions among the atoms which make up the molecule.
Clausius has shown that the ratio of the intramolecular to the " straight line" or
translational energy is constant for a given gas. If we call this ratio m, then for
the polyatomic gas
H k =
3/l +m 3 + 3m
If m = 0, this becomes J, as for the monatomic gas. The equation gives also,
m = 5 ""J/g For oxygen, with y = 1.4, m = 5 ~ ^ = = 0.667.
127 &. Some Applications. Writing the first equation given in the i'brm.
we have foi 1 Ib. of air at standard conditions
w a = V3 x 53.36 x 492 x 8J.2 ^ 1593 ft. per second,
the velocity of the air molecule. Noting also that w = (/) \/tJ under standard con
ditions, we obtain for hydrogen
w h = 1593Ji^ii = 6270 ft. per second.
* lli.OOY
These are mean velocities. Some of the molecules are moving more rapidly, some
more slowly.
The molecular velocities of course increase with the temperature and are
higher for the lighter gases. A mixture of gases inclosed in a vessel containing
an orifice, or in a porous container, will lose its lighter constituents first ; because,
since their molecular velocities are higher, their molecules will have briefer
periods of oscillation from side to side of the containing vessel and will more
frequently strike the pores or orifices and escape. This principle explains the com
mercial separation of mixed gases by the pi ocess of osmosis.
In any actual (polyatomic) gas, the molecules move in paths of constantly
changing direction, and consequently do not travel far. The diffusion or perfect
mixture of two or more gases brought together is therefore not an instantaneous
process. High temperatures expedite it, and it is relatively more rapid with the
lighter gases.
We may assume that intramolecular energy is related to a rotation of atoms
about some common center of attraction. The intramolecular energy has been
shown to be proportional to the temperature. A temperature may be reached at
which the total energy of an atomic system may be so greatly increased that the
70 APPLIED THERMODYNAMICS
system itself will be broken up, atoms flying off perhaps to form new bonds, new
molecules, new substances. This breaking up of molecules is called dissociation.
In forming new atomic bonds, heat may be generated ; and when this generation
of heat occurs with sufficient rapidity, the process becomes selfsustaining ; i.e. the
temperature will be kept up to the dissociation point without any supply of heat
from extraneous sources. If, as in many cases, the generation of heat is less rapid
than this, dissociation of the atoms will cease after the external source of heat has
been lemoved.
According to a theorem in analytical mechanics,* there is an initial velocity,
easily computed, at which any body projected directly upward will escape from the
sphei e of gravitational attraction and never descend. For earth conditions, this
velocity is, irrespective of the weight of the body, 6.95 miles per second .= 36,650 ft.
per second, ignoring atmospheric resistance. Now there is little doubt that some
of the molecules of the lighter gases move at speeds exceeding this ; so that it is
quite possible that these lighter gases may be gradually escaping from our planet.
On a small asteroid, where the gravitational atti action was less, much lower
velocities would suffice to liberate the molecules, and on some of these bodies
there could be no atmosphere, because the velocity at which liberation occurs is
less than the normal velocities of the nitrogen and oxygen molecules.
(1) Thermodynamics, 1907, p. 18. (2) Alexander, Treatise on Thermodynamics,
1893, p 105. (3) Wormell, Thermodynamics, 123, Alexander, Thermodynamics,
103; Rankine, The Steam Engine, 249, 321; Wood, Thermodynamics, 7177, 437.
(4) Zeuner, Technical Thermodynamics, Klein tr., I, 156. (5) Ripper, Steam Engine
Theory and Practice, 1895, 17.
SYNOPSIS OF CHAPTER V
Pressure, volume and temperature as therwodynanuc coordinates.
Thermal line, the locus of a series of successive states , path, a projected thermal line.
Paths : isothermal, constant temperature ; wodynamic, constant internal energy ;
adiabatic, no transfer of heat to or from surrounding bodies.
The geometrical representation of the characteristic equation is a surface.
The PV diagram: subtended areas represent external work; a cycle is an enclosed
figure ; its area represents external work ; it represents also the net expenditure of
heat.
The isothermal : pv n = c, in which n = 1, an equilateral hyperbola ; the external work
done is equivalent to the heat absorbed, = pv log e : with a perfect gas, it coin
cides with the isodynamic. v
Paths in general: pv" = c ; external work =^^T ; 1= (V~ n ; J= (\ n n\
The adiabatio ; the external work done is equivalent to the expenditure of internal
energy ; pvv=c ; y = 1.402 ; computation from the velocity of sound in air ; wave
velocities with extreme pressure changes.
The heat absorbed along any path is represented by the area between that path and
the two projected adiabatics ; representation of k and L
* See, for example, Bowser's Analytic Mechanics, 1908, p. 301.
SYNOPSIS 71
Isodiabatics : n^ = n^ ; equal specific heats ; equality of property ratios.
Rankine's derivation of Joule's law : the change of internal energy between two states
is independent of the path.
Apparatus for determining the value of y from pressure changes alone.
Along any path pv n = c, the heat absorbed is l(t !T)(^~ ? M ; the mean specific heat
is i n ~y. Such paths are called polytropics. Values of n and s for various paths.
n l
Graphical method for determining the value of n ; Brauer^s method for plotting poly
tropics ; the tabular method.
Graphical representations of internal energy ; representations of the sources of external
work and of the effects of heat ; finite area representing heat expenditure.
Poly tropic expansion in ordnance.
Irreversible processes: constrained and fiee expansion ; reversibility ; no actual proc
ess is reversible , example of irreversible process ; subtended areas do not repre
sent external work , in acliabatic action, heat may be received from the mechani
cal behavior of the substance itself; H=T + I+ W+V; further applications of
the kinetic theory.
Use of Hyperbolic Functions : Tyler's Method. Given x m = a, let x m = e'. Then
m logg x = s and x m = e mloSeX . Adopting the general forms
& = cosh t + sinh t,
e t cogn i _ s i n h ^
we have
x m = cosh (m log x) + sinh (m log fl or), where m log e x is positive ;
x m _. cogh ( m i O g p x } _ giuh ( m i O g e a;^ where m log a x is negative.
If now we have a table of the sums and differences of the hyperbolic functions,
and a table of hyperbolic logarithms, we may practically without computation ob
tain the value of x m . Thus, take the expression
Here x = 0.1281, m = 0.29, m log a x =  0.596, (cosh sinh) m log e x = 0.552.
The limits of value of x may be fixed, as in the preceding article, as and 1.0.
For x = 0, m log e x = oc, and the method would require too extended a table of
hyperbolic functions. But if we use the general form in which x > 1.0 and usually
<10.0, m loge ^ will rarely exceed 10.0, and the method is practicable.
For a fuller discussion, with tables, see paper by Tyler in the Polytechnic En
gineer, 1912.
o .
30103
written
30103
1
30103
written
1 30103
2
30103
written
2 30103
1
0.30103
wiitton
1 30103
72 APPLIED THERMODYNAMICS
NOTES ON LOGARITHMS
Definitions; log x or com log sn, where 10 n =z.
m where e m =x, e = 27183+.
= (2.3026) log x,
Characteristic and Mantissa, the log consists of a characteristic, integral and either
positive or negative; and a mantissa, a positive fraction or decimal Dividing or
multiplying a number by 10 or any multiple thereof changes the characteristic of
the log, but not its mantissa. Thus,
Characteristic Mantissa
log 2
log 20
log 200
log 0.2 =
and equivalent to 69897
log 02  2 30103 written 2 30103
and equivalent to 1,69897
Operations with logarithms.
log (aX&) =log a+log b. Remember also:
i
log (aj6) =log a log b. zfr^tyx.
1
log(a) n nlogo. x n =~.
Negative sign: the signs of negative characteristics must be carefully con
sidered. Thus, to find the value of 0.02~ 37 :
log 0,02=2 30103 = 2.0+0 30103.
0.37 log 0,02= 0,37(2 0+0,30103) =0,740.1114 = 0.6286.
log
When the final logarithm comes out negative, it must be converted into loga
rithmic form (negative characteristic and positive mantissa) by adding and sub
tracting 1. Thus 0.6286 = 1,3714 log 0.2352.
For example, to find the value of 0.02 37 :
log 0.02 = 2.30103= 2.0+0 30103
0.37 log 0.02 0.37( 2.0+0 30103) = 0.74+0.1114= 0.62861.3714
log (0.2352 =0.02' 37 ).
PROBLEMS 73
PEOBLEMS
1. On a perfect gas diagram, the coordinates of which are internal energy and
volume, construct an isodynamic, an isothermal, and an isometric path through E
(internal energy) =2, F=2.
2. Plot accurately the following: on the TV diagram,* an adiabatic through
T=270, F=10; an isothermal through T=300, F=20; on the TPf diagram, an
adiabatic through T=230, P = 5; an isothermal through T=190, P = 30. On the
JSV diagram,} show the shape of an adiabatic path through 22 = 240, F= 10.
3. Show the isometric path of a perfect gas on the PT plane ; the isopiestic, on
the FT plane.
4. Sketch the TV path of wax from to 290 F., assuming the melting point to
be 90, the boiling point 290, that wax expands m melting, and that its maximum
density as liquid is at the melting point.
5. A cycle is bounded by two isopiestic paths through P = 110, P = 100 (pounds
per square foot), and by two isometric paths through F= 20, F=10 (cubic feet).
Find the heat expended by the working substance. (Ans., 0.1285 B. t. u.)
6. Air expands isothermally at 32 F. from atmospheric pressure to a pressure of
6 Ib. absolute per square inch. Find its specific volume after expansion.
(Ans., 36.42 cu. ft.)
7. Given an isothermal curve and the 0V axis; find graphically the OP axis.
8. Prove the correctness of the construction described in Art. 93.
9. Find the heat absorbed during the expansion described in Problem 6.
(Ans., 36.31 B.t.u.)
10. Find the specific heat for the path PF 1  2 = c, for air and for hydrogen.
(An$., air, 0.1706; hydrogen, 2.54.)
11. Along the path PF 1 * 2 = c, find the external work done in expanding from
P=1000, F=10, to F=100. Find also the heat absorbed, and the loss of internal
energy, if the substance is one pound of air. Units are pounds per square foot and
cubic feet. (Ans., W= 18,450 f t.lb. ; J3T= 11,796 B. t. u. ; Jfy Jfc  11.8 B. t. u.)
12. A perfect gas is expanded from #=400, t?=2, = 1200, to P = 60, F=220*
Find the final temperature. (Ans., 19,800 aba.)
13. Along the path PF 1  2 = c, a gas is expanded to ten tjsnes its initial volume of
10 cubic feet per pound. The initial pressure being 1000, and the value of It 53.36,
find the final pressure and temperature. (See Problem 11.)
(Ans., p = 63.1 Ib. per sq. ft., t = 118.25 a"bs.)
14. Through what range of temperature will air "be heated if compressed to 10
atmospheres from normal atmospheric pressure and 70 F., following the law pi>i 3 =c ?
What will be the rise in temperature if the law is pW=c ? If it is # c ?
(Ans., Cf, 371.3 ; 6, 495 ; c, 0).
5 Absolute pressures are pressures measured above a perfect vacuum. The abso
lute pressure of one standard atmosphere is 14.697 Ib. per square inch,
74 APPLIED THERMODYNAMICS
15. Find the heat imparted to one pound of this air in compressing it as described
according to the lawjpw 1 3 = c, and the change of internal energy.
(Ana , ZT 2 A= 21. 6 B.t.u. , ^^ = 63.1 B.t.u.)
16. In Problem 14, after compression along the path pu 13 = c, the air is cooled
at constant volume to 70 F., and then expanded along the isodiabatic path to its
initial volume. Find the pressure and temperature at thu end of this expansion.
(Ans., p =8.64 Ib. per sq. in., t =311 abs.)
17. The isodiabatics ob, cd, are intersected by lines of constant volume ac, &d.
v *& *0 j * Q, * 6
Prove = 1 and r sr
18. In a room at normal atmospheric pressure and constant temperature, a
cylinder contains air at a pressure of 1200 Ib. per square inch. The stopcock on the
cylinder is suddenly opened. After the piessurc in the cylinder has fallen to that of
the atmosphere, the cock is closed, and the cylinder left undisturbed for 24 hours.
Compute the pressure in the cylinder at the end of this time.
(Ans , 51.94 Ib. per sq, in.)
19. Find graphically the value of n for the polytropic curve ob, Fig. 41.
20. Plot by Brauer's method a curve jp / u 1 S = 2G,200. Use a scale of 1 inch per
4 units of volume and per 80 units pressure. Begin the curve with p= 1000.
21. Supply the necessary figures in the following blank spaces, for ft = 1.8, and
apply the results to check the curve obtained in Problem 20. Begin with u = 6.12,
# = 1000.
=2.0, 2.25, 2.50, 3.0, 4.0, 5.0, 6.0, 7.0, S.O
P
P
P=
22. The velocity of sound in air being taken at 1140 ft. per second at 70 F. and
normal atmospheric pressure, compute the value of y for air. (Ans., 1.4293.)
23. Compute the latent heat of expansion (Art. 58) of air from atmospheric
pressure and at 32 F. (Ans., 2.615 B. t. u.)
24. Find the amount of heat converted into work in a cycle 1234, in which
P 1 = P 4 = 100, 7i = 5, 1?; = 1, Pj = 30 (all in Ib. per sq. ft.), and the equations of the
paths are as follows: for 41, PF = c; for 12, PF^cj'for 32, P7=c; for 43,
PY 1B = c. The working substance is one pound of air. Find the temperatures at
the points 1, 2, 3, 4.
(Ans.,2&= 1.386 B.t.u.; ^^9.37; ^ = 1.097; T 2 = 1.097; r 4 =1.874.)
25. Find the exponent of the polytropic path, for air, along which the specific
heat is k. Also that along which it is L Represent these paths, and the amounts
of heat absorbed, graphically, comparing with those along which the specific heats are
k and Z, and show how the diagram illustrates the meaning of negative specific heat.
(Ans., f or 3 = k, n = 1. 167 ; f or s = Z, n = 1.201.)
26. A gas, while undergoing compression, has expended upon it 38,900 ft. Ib. of
work, meanwhile, it loses to the atmosphere 20 B. t. XL of heat. What change occurs
in its internal energy?
PROBLEMS 75
27. One pound of air under a pressure of 150 Ib. per sq. in. occupies 4 cu. ft.
What is its temperature? How does its internal energy compare with that at atmos
pheric pressure and 32 F, 9
28. Three cubic feet of air expand from 300 to 150 Ib. pressure per square inch, at
constant temperature. Find the values of B", E and W.
29. How much work must be done to compress 1000 cu. ft. of normal air to a pres
sure of ten atmospheres, at constant temperature ? How much heat must be removed
during the compression?
30. Air is compressed in a waterjacketed cylinder from 1 to 10 atmospheres; its
specific volume being reduced from 13 to 2,7 cu. ft. How much work is consumed per
cubic foot of the original air?
31. Let p = 200, u = 3, P=100, 7=5. Find the value of n in the expression
jtt>"=P7.
32. Draw to scale the PT and TV representations of the cycle described in
Prob. 24.
33. A pipe line for air shows pressures of 200 and 150 Ib. per square inch and tem
peratures of 160 and 100 F., at the inlet and outlet ends, respectively. What is the
loss of internal energy of the air during transmission? If the pipe line is of uniform
size, compare the velocities at its two ends.
34. If air is compressed so that #i)i35=c, find the aonount of heat lost to the cyl
inder walls of the compressor, the temperature of the air rising 150 F. during com
pression.
CHAPTER VI
THE CARNOT CYCLE
128. Heat Engines. In a heat engine, work is obtained from
heat energy through the medium of a gas or vapor. Of the total
heat received by such fluid, a portion is lost by conduction from the
walls of containing vessels, a portion is discharged to the atmosphere
after the required work has been done, and a third portion disap
pears, having been converted into external mechanical work. By
the first law of thermodynamics, this third portion is equivalent to
the work done ; it is the only Jieat actually used. The efficiency of a
heat engine is the ratio of the net heat utilized to the total quantity of
heat supplied to the engine, or, of external work done to gross heat
^5 in which fi denotes the quantity of heat
W
absorbed; to
_Z
rejected by tlie engine, if radiation effects be ignored.
129. Cyclic Action. In every heat engine, the working fluid passes
through a series of successive states of pressure, volume, and temperature ;
and, in order that operation may be continuous, it is necessary either that
the fluid work in a closed cycle which may be repeated indefinitely, or
that a fresh supply of fluid be admitted to the engine to compensate for
such quantity as is periodically
discharged. It is convenient to
regard the latter more usual ar
rangement as equivalent 'to the
former, and in the first instance
to study the action of a constant
body of fluid, conceived to work
continuously in a closed cycle.
130. Forms of Cycle. The sev
eral paths described in. Art. 83, and
others less commonly considered, sug
gest various possible forms of cycle,
some of which are illustrated in Fig.
FIG. 42. Art. 130, Problem 2. Possible Cycles.
42. Many of these have been given names (1). The isodidbatic cycle, bounded by
two isothermals and any two isodiabatica (Art. 108), may also be mentioned.
76
THE CARNOT CYCLE 77
131. Development of the Carnot Cycle. Carnot, in 1824, by describing and
analyzing the action of the perfect elementary heat engine, effected one of the
most important achievements of modern physical science (2) Carnot, it is true,
worked with insufficient data. Being ignoiant of the fiist law of theimodynamics,
and holding to the caloric theoiy, he asserted that no heat was lost during the
cyclic process; but, though to this extent founded on error, his main conclusions
were correct. Before his death, in 1832, Carnot was led to a more just conception
of the true nature of heat; while, left as it was, his work has been the starting
point for nearly all subsequent investigations. The Cainot engine is the limit
and standard for all heat engines.
Clapeyron placed the arguments of Carnot in analytical and graphical form ;
Clausius expressed them in terms of the mechanical theory of heat ; James Thomp
son, Rankiue, and Clerk Maxwell corrected Carnot's assumptions, redescribed the
cyclic process, and redetennined the results ; and Kelvin (3) expressed them iu
their final and satisfactory modern form.
132. Operation of Carnot's Cycle. Adopting Kelvin's method,
the operation on the Carnot engine may be described by reference
to Fig. 48. A working piston moves in the cylinder c, the walls of
which are nonconduct
ing, while the head is
a perfect conductor.
The piston itself is
FIG. 43. Arts. 132, 138. Operation of the Carnot Cycle. , , ,
a nonconductor and
moves without friction. The body s is an infinite source of heat
(the furnace, in an actual power plant) maintained constantly at
the temperature T, 110 matter how much heat is abstracted from it.
At r is an infinite condenser, capable of receiving any quantity of
heat whatever without undergoing any elevation of temperature
above its initial temperature t. The plate f is assumed to be a per
fect nonconductor. The fluid in the cylinder is assumed to be
initially at the temperature T of the source.
The cylinder is placed on s. Heat is received, but the tempera
ture does not change, since both cylinder and source are at the
same temperature. External work is done, as a result of the recep
tion of heat ; the piston rises. When this operation has continued
for some time, the cylinder is instantaneously transferred to the non
conducting plate f. The piston is now allowed to rise from the expan
sion produced by a decrease of the internal energy of the fluid. It
continues to rise until the temperature of the fluid has fallen to ,
78 APPLIED THERMODYNAMICS
that of the condenser, when the cylinder is instantaneously trans
ferred to r. If eat is now yiven up by the fluid to the condenser, and
the piston falls ; but no change of temperature takes place. When this
action is completed (the point for completion will be determined
later), the cylinder is again placed on /, and the piston allowed to
fall further, increasing the internal energy and temperature of the
gas by compressing it. This compression is continued until the
temperature of the fluid is T and the piston is again in its initial
position, when the cylinder is once more placed upon s and the opera
tion may be repeated. No actual engine could be built or operated
under these assumed conditions.
133. Graphical Representation. The
first operation described in the preceding
is expansion at constant temperature. The
path of the fluid Is then an isothermal.
The second operation is expansion without
transfer of heat, external work being done
at the expense of the internal energy;
the path is consequently adiabatie. Dur FIG. 44. Arts. 133136, 1,38, 142.
ing the third operation, we have isothermal The Carnot c y cle
compression; and during the fourth, adiabatie compression. The
Carnot cycle may then be represented by abed. Fig. 44.
134. Termination of Third Operation. In order that the adiabatie compression
da may bring the fluid back to its initial conditions of pressure, volume, and tem
perature, the isothermal compression cd must be terminated at a suitable point d.
From Art. 99,
iv
lor the adiabatie da,
T I V \i
~ = ( .iJ? )
t \\ &/
T / T 7 " \ i~v
and  = ( ~5 J for the adiabatie Ic \
4gfc&
that is, the ratio of volumes during isothermal expansion in the first stage must be
equal to the ratio of volumes during isothermal compression in the third stage, if the
final adiabatie compression is to complete the cycle. (Compare Art. 108.)
135. Efficiency of Carnot Cycle. The only transfers of heat dur
ing this cycle occur along ab and cd. The heat absorbed along ab is
THE CARNOT CYCLE 7
f= RT\og e & Similarly, along cd, the heat rejected
r a 'a
is Rt log e  The net amount of heat transformed into work is the
y d
difference of these two quantities ; whence the efficiency, defined in
Art. 128 as the ratio of the net amount of heat utilized to the total
amount of heat absorbed, is
since ^ = ^, from Art. 134.
e
' a
136. Second Derivation. The external work done under the two adiabaties
, da is
yi yi
Deducting the negative work from the positive, the net adiabatic work is
but PO.VO, = PT>VI>, from the law of the isothermal al\ similarly, P^V* = P e V c , and
consequently this net work is equal to zero; and if we express efficiency by the
ratio of work done to gross heat absorbed, we need consider only the work areas
under the isothermal curves ab and cd, which are given by the numerator in the
expression of Art. 135.
The efficiency of the Carnot engine is therefore expressed by the
ratio of the difference of the temperatures of source and condenser to
the absolute temperature of the source.
137. Garnet's Conclusion. The computations described apply to any sub
stance in uniform thermal condition ; hence the conclusion, now universally
accepted, that the motive power of heat is independent of the agents employed tc
develop it ; it is determined solely by the temperatures of the bodies between which
the cyclic transfers of heat occur.
138. Reversal of Cycle. The paths which constitute the Carnot cycle,
Fig. 44, are polytropic and reversible (Art. 125); the cycle itself is rever
sible. Let the cylinder in Fig. 43 be first placed upon r, and the piston
allowed to rise. Isothermal expansion occurs. The cylinder is trans
ferred to /and the piston caused to fall, producing adiabatic compression,
The cylinder is then placed on s, the piston still falling, resulting In iso
thermal compression ; and finally onf, the piston being allowed to rise, s
as to produce adiabatic expansion. Heat has now been taken from the
80 APPLIED THERMODYNAMICS
condenser and rejected to the source. The cycle followed is dcbad, Fig. 44.
Work has been expended upon the fluid ; the heat delivered to the source s is
made up of the heat taken Jrom the condenser r, plus the heat equivalent of
the work done upon the fluid. The apparatus, instead of being a heat
engine, is now a sort of heat pump, ti an sf erring heat from a cold body to
one warmer than itself, by reason of the expenditure of external work.
Every operation of the cycle has been reversed. The same quantity of
heat originally taken from s has now been given up to it ; the quantity
of heat originally imparted to r is now taken from it; and the amount of
external work originally done by the fluid has now b^en expended upon
it. The efficiency, based on our present definition, may exceed unity ; it
is the quotient of lieat imparted to the source by work expended. Tho
cylinder c must in this case be initially at the temperature t of the con
denser r.
139. Criterion of Reversibility. Of all engines working between the
same limits of temperature, that which is reversible is the engine of maximum
efficiency.
If not, let A be a more efficient engine, and let the power which this
engine develops be applied to the driving of a heat pump (Art. 138),
(which is a reversible engine), and let this heat pump be used for restor
ing heat to a source s for operating engine A. Assuming that there is no
friction, then engine A is to perform just a sufficient amount of work to
drive the heat pump. In generating this power, engine A will consume
a certain amount of heat from the source, depending u^on its efficiency.
If this efficiency is greater than that of the heat pump, the latter will di$
charye more heat than the former receives (see explanation of efficiency,
Art. 138) ; or will continually restore more heat to the source than engine
A removes from it. This is a result contrary to all experience. It is
impossible to conceive of any selfacting machine which shall continually
produce heat (or any other form of energy) without a corresponding con
sumption of energy from some other source.
140. Hydraulic Analogy The absurdity may be illustrated, as by Heck (4),
by imagining a water motor to be used in driving a pump, the pump being em
ployed to deliver the water back to the upper level which supplies the motor.
Obviously, the motor would be doing its best if it consumed no more water than
bhe pump returned to the leservoir; no better performance can be imagined, and
with actual motors and pumps this performance would never even be equaled.
Assuming the pump to be equally efficient as a motor or as a pump (i.e. reversible),
the motor cannot possibly be more efficient.
141. Clausius' Proof. The validity of this demonstration depends upon the
3orrectness of the assumption that perpetual motion is impossible. Since the iui
THE CARNOT CYCLE 81
possibility of perpetual motion cannot be directly demonstrated, Cflausius estab
lished the criterion of reversibility by showing that the existence of a more effi
cient engine A involved the continuous transference of heat from a cold body to
one warmer than itself, without the aid of external agency : an action which is axio
matically impossible.
142. The Perfect Elementary Heat Engine. It follows from the analysis of
Art. 135 that all engines working in the Carnot cycle are equally efficient ; and
from Art. 139 that the Carnot engine is one of that class of engines of highest effi
ciency. The Carnot cycle is therefore described as that of the perfect elementary
heat engine. It remains to be shown that among reversible engines working be
tween equal temperature limits, that of Carnot is of maximum efficiency. Con
sider the Carnot cycle abed, Fig. 44. The external work done is abed, and the
efficiency, abed + nabN. For any other reversible path than &, like ae or fb,
touching the same line of maximum temperature, the work area abed and the heat
absorption area nabN are reduced by equal amounts. The ratio expiessing effi
ciency is then reduced by equal amounts in numeiator and denominator, and since
the value of this ratio is always fractional, its value is thus always reduced. For
any other reversible path than cd, like ch or gd, touching the same line of mini
mum temperature, the work area is reduced without any reduction in the gross
heat area nabN. Consequently the Carnot engine is that of maximum efficiency
among all conceivable engines worked between the same limits of temperature. A
practical cycle of equal efficiency will, however, be considered (Art. 257).
143. Deductions. The efficiency of an actual engine can therefore
never reach. 100 per cent, since this, even with the Carnot engine, would
require t in. Art. 135 to be equal to absolute zero. High efficiency is con
ditioned upon a wide range of working temperatures ; and since the mini
mum temperature cannot be maintained below that of surrounding bodies,
high efficiency involves practically the highest possible temperature of
heat absorption. Actual heat engines do not work in the Carnot cycle;
but their efficiency nevertheless depends, though less directly, on the tem
perature range. With many working substances, high temperatures are
necessarily associated with high specific pressures, imposing serious con
structive difficulties. The limit of engine efficiency is thus fixed by the
possibilities of mechanical construction.
Further, an ordinary steam boiler furnace may develop a maximum
temperature, during combustion, of 3000 F. If the lowest available
OQQQ
temperature surrounding is F., the potential efficiency is
=0.87. But in getting the heat from the hot gases to the steam the
temperature usually falls to about 350 F. Although 70 or 80 per
cent of the energy originally in the fuel may be present in the steam,
the availability of this energy for doing work in an engine has now been
82 APPLIED THERMODYNAMICS
Off /"i
decreased to ^ ft 4fi =0.43, or about onehalf. (A boiler is of course
not a heat engine.)
(1) Alexander, Treatise on Thermodynamics, 1893, 3840. (2) Garnet's Reflec
tions is available in Thurston's translation or in Magie's Second Law of TJiermody
namics. An estimate of his part in tlie development of physical science is given by
Tait, Thermodynamics, 18(18, 44. (3) Trans. Roy. Soc. Edinburgh, March, 1851 ;
Phil. Mag., IV, 1852 ; Math, and Phys. Papers, I, 174. (4) The Steam Engine, I,
50.
SYNOPSIS OF CHAPTER VI
Heat engines efficiency = heat utilized  heat absorbed = "I * =
Cyclic action . closed cycle , forms of cycle.
Carnot cycle: historical development; cylinder, source, insulating plate, condenser
graphical representation; termination of third operation, when  = J5; ^jl
rp j. YC rk
ciency =^
Carnot's conclusion : efficiency is independent of the working substance.
Reversal of cycle: the reversible engine is that of maximum efficiency; hydraulic
Carnot cycle not surpassed in efficiency by any reversible or irreversible cycle.
Limitations of efficiency in actual heat engines.
PROBLEMS
1. Show how to express the efficiency of any heatengine cycle as the quotient
of two areas on the PV diagram.
2. Draw and explain six forms of cycle not shown in Fig. 42.
3. In a Carnot cycle, using air, the initial state is P= 1000, F= 100. The pres
sure after isothermal expansion ia 500, the temperature of the condenser 200 F. Find
the pressure at the termination of the " third operation," the external work done along
each of the four paths, and the heat absorbed along each of the four paths. Units axe
cubic feet per pound and pounds per square foot.
Ans. p 3 =13.1; TF 12 = 69,237ft. lb.; ^=88,943. t.u. ;
TT 23 = 161,200 ft. lb.; Bi 3 =0;
W S t~ 24,368ft. lb.; #34 = 31,32 B. t. u.;
W* =1 61,200 ft lb.; J7 4l =0.
4 A nonreversible heat engine takes 1 B. t. u. per minute from a source and is
used to drive a heat pump having an efficiency (quotient of work by heat imparted to
source) of 0.70. What would be the rate of increase of heat contents of the source if
the efficiency of the heat engine were 0.80? (Ans., O.U3 B. t. u. per min<)
5. Ordinary noncondensing steam engines use steam at 325 F. and discharge it
to the atmosphere at 215 F. What is their maximum possible efficiency?
(An$., 0,14,)
PROBLEMS 83
6. Find the limiting efficiency of a gas engine in which a maximum temperature
of 3000 F. is attained, the gases being exhausted at 1000 F. (Ans^ 0.578.)
7. An engine consumes 225 B. t. u. per indicated horsepower (33,000 footpounds)
per minute. If its temperature limits are 430 F and 105 F., how closely does its
efficiency approach the "best possible efficiency? (Ans., 51.59 per cent.)
8. How many B. t. u. per indicated horse power per hour would be required by a
heat engine haying an efficiency of 15 per cent?
9. A power plant uses 2 Ib. of coal (14,000 B. t.u. per Ib.) per kilowatthour.
(1 kw. = 1.34 h,p.) What is its efficiency from fuel to switchboard?
10. A steam engine working between 350 F. and 100 F. uses 15 Ib. of steam con
taining 1050 B. t. u. per Ib,, per indicated horse power per hour. "What proportion
of the heat supplied was utilized by the engine? How does this proportion, compare
with the highest that might have been attained?
11. Determine as to the credibility of the following claims for an oil engine:
Temperature limits, 3000 F. and 1000 F.
Fuel contains 19,000 B. t u. per Ib. Engine consumes 0.35 Ib. per kw.hr.
Loss between cylinder and switchboard, 20 per cent.
12. If the engine in Problem 3 is doubleacting, and makes 100 r.p.m., what is its
horse power?
CHAPTER VII
THE SECOND LAW OF THERMODYNAMICS
144. Statement of Second Law. The expression for efficiency of
the Cariiot cycle, given in Art. 135, is a statement of the second law
of thermodynamics. The law is variously expressed ; but, in general,
it is an axiom from which is established the criterion of reversibility
(Art. 139).
With Clausius, the axiom was,
(a) " Heat cannot of itself pass from a colder to a hotter body; " while the
equivalent axiom of Kelvin was,
(6) " It is impossible, by means of inanimate material agency, to derive
mechanical effect from any portion of matter by cooling it below the tempera
ture of ike coldest of surrounding objects"
With Carnot, the axiom was that perpetual motion is impossible; while Ran
kine's statement of the second law (Art. 151) is an analytical restatement of the
efficiency of the Carnot cycle.
145. Comparison of Laws. The law of relation of gaseous properties (Art. 10)
and the second law of thermodynamics aie justified by their results, while thejirst
law of thermodynamics is an expression of experimental fact. The second law is a
" definite and independent statement of an axiom resulting from the choice of one
of the two propositions of a dilemma" (1). For example, in Carnot's form, we
must admit either the possibility of perpetual motion or the criterion of reversi
bility ; and we choose to admit the latter. The second ]aw is not a proposition to
be proved, but an. "axiom commanding universal assent when its terms are
understood."
146. Preferred Statements. The simplest and most satisfactory statement of
the second law may be derived directly from inspection of the formula for effi
ciency, (T  t) ^ T (Art. 135). The most general statement,
(c) rt The availability of heat for doing work depends upon its temperature" leads
at once to the axiomatic forms of Kelvin and Clausius j while the most specific of
all the statements directly underlies the presentation of Rankine :
(c?) " If all of the heat be absorbed at one temperature, and
rejected at another lower temperature, the heat transformed to
84
THE SECOND LAW OF THERMODYNAMICS 85
external work is to the total heat absorbed in the same ratio as that
of the difference between the temperatures of absorption and rejec
tion to the absolute temperature of absorption ;" or,
H h = T t
H T '
in which H represents heat absorbed ; and 7i, heat rejected.
147. Other Statements. Forms (a), (ft), (c), and (d) are those usually given
the second law. In modified forms, it has been variously expressed as follows
(e) "All reversible engines working between the same uniform tem
peratures have the same efficiency."
(/) " The efficiency of a reversible engine is independent of the nature
of the working substance."
(g) " It is impossible, by the unaided action of natural processes,
to transform any part of the heat of a body into mechanical work, except
by allowing the heat to pass from that body into another at lower
temperature. "
Qi) "If the engine be such that, when it is worked backward, the
physical and mechanical agencies in every part of its motions are reversed,
it produces as much mechanical effect as can be produced by any therm o
dynamic engine, with the same source and condenser, from a given quan
tity of heat."
148. Harmonization of Statements. It has been asserted that the state
ments of the second law by different writers involve ideas so diverse as,
apparently, not to cover a common principle. A moment's consideration
of Art. 144 will explain this. The second law, in the forms given in (a),
(&), (c), ({/), is an axiom, from ichich the criterion of reversibility is estab
lished. In (r?), (e) (/), it is a simple statement of the efficiency of the Car
not cycle, with which the axiom is associated ; while in (7i), it is the
criterion of reversibility itself. Confusion may be avoided by treating
the algebraic expression of (VZ), Art. 146, as a sufficient statement of
the second law, from which all necessary applications may be derived.
149. Consequences of the Second Law. Some of these were touched upon in
Art. 143. The first law teaches that heat and work are mutually convertible,
the second law shows how much of either may be converted into the other under
stated conditions. Ordinary condensing steam engines work between tempera
tures of about 350 F. and 100 F. The maximum possible efficiency of such
engines is therefore
350  100
350 + 459.4
= 0.31.
86 APPLIED THERMODYNAMICS
The efficiencies of actual steam, engines range from 2J to 25 per cent, with an
average probably not exceeding 7 to 10 per cent. A steam engine seems therefore
a most inefficient machine ; but it must be remembered that, of the total heat
supplied to it, a large prupoition is (by the second law) unavailable for use, and
must be refected when its temperature falls to that of surrounding bodies. We can
not expect a water wheel located in the mountains to utilize all of the head of the
water supply, measured down to &ea level. The available head is limited by the
elevation of the lowest of surrounding levels. The performance of a heat engine
should be judged by its approach to the efficiency of the Carnot cycle, rather than
by its absolute efficiency.
Heat must be regarded as a " low unorganized " form of energy, which pro
duces useful work only by undergoing a fall of temperature. All other forms of
energy tend to completely transform themselves into heat. As the universe slowly
settles to thermal equilibrium, the performance of work by heat becomes impossible
and all energy becomes permanently degenerated to its most unavailable form.*
150. Temperature Fall and Work Done. Consider the Carnot cycle, abed,
Fig. 45, the total heat absorbed being nabNaxKl the efficiency abcd^nabN
Draw the isothermals
, ij, successively differing by equal
temperature intervals ; and let the tem
peratures of these isothermals be T 19
T 2 , T s Then the work done in cycle
abfe is nabN x (T T^) * T >, that in
cycle abhg is nabNx(TT 2 )T; that
in cycle abji is nabNx(T T$*~T.
As (TT 3 ) = 3(T2 r7 1 ) and (TT 3 )
= 2(!T2 7 1 )> abji = 3(abfe) and abhg
= 2(a&/e); whence abfe = efhy = glvjL
FIG. 45. Arts. 150, 153, 154, i.w. Second In otlier wor ^ s th e external work
Law of Tuermodynamus. avai i able f rom a definite temperature fan
is the same at all' parts of tlie thermometric scale. The waterfall analogy of
Art. 149 may again be instructively utilized.
151. Rankine's Statement of the Second Law. " If the total actual heat of a
uniformly hot substance be conceived to be divided into any number of equal parts,
the effects of those parts in causing work to be performed are equal. If we re
member that by "total actual heat" Rankine means the heat corresponding to ab
solute temperature, his terse statement becomes a form of that just derived, dependent
solely upon the computed efficiency of the Carnot cycle.
152, Absolute Temperature. It is convenient to review the steps by which
the proposition of Art. 150 has been established. We have derived a conception
of absolute temperature from the law of Charles, and have found that the effi
ciency of the Carnot cycle bears a certain relation to definite absolute temperatures.
* *' Each time we alter our investment in energy, we have thus to pay a commis
sion, and the tribute thus exerted can never be wholly recovered by us and must be
regarded, not as destroyed, but as thrown on the wasteheap of the Universe." Griffiths,
KELVIN'S ABSOLUTE SCALE 87
Our scale of absolute temperatures, practically applied, is not entirely satisfactory ;
for the absolute zero of the air thermometer, 459.4 F., is not a true absolute
zero, because air is not a perfect gas. The logical scale of absolute temperature
would be that in "which temperatures were denned by reference to the work done
by a reversible heat engine Having this scale, we should be in a position to com
pute the coefficient of expansion of a perfect gas.
153. Kelvin's Scale of Absolute Temperature. Kelvin, in 1848, was led
by a perusal of Carnot's memoir to propose such, a scale. His first defini
tion, based on the caloric theory, resulted only in directing general atten
tion to Carnot's great work ; his second definition is now generally adopted.
Its form is complex, but the conception involved is simply that of Art. 150:
" The absolute temperatures of two bodies are proportional to the quanti
ties of heat respectively taken in and given out in localities at one temperature
and at the other, respectively, by a material system subjected to a complete
cycle of perfectly reversible thermodynamic operations, and not allowed to part
with or take in heat at any other temperature." Briefly,
" The absolute values of two temperatures are to each other in the propor
tion of the quantities of heat taken in and rejected in a perfect thermodynamic
engine, working with a source and condenser at the higher and the lower of
the temperatures respectively." Symbolically,
This relation may be obtained directly by a simple algebraic trans
formation of the equation for the second law, given in Art. 146, (d).
154. Graphical Representation of Kelvin's Scale. He turning to Fig. 45,
but ignoring the previous significance of the construction, let ab be an iso
thermal and an, bN adiabatics. Draw isothermals ef, gh, ij, such that the
areas abfe, efhg, ghji are equal. Then if we designate the temperatures
along ab, ef, gh, ij by T, T 19 T 2 , T s , the temperature intervals T T l9
TI T 2J T 2 T 3 are equal. If we take ab as 212 F., and cd as 32 F.,
then by dividing the intervening area into 180 equal parts, we shall have
a true Fahrenheit absolute scale. Continuing the equal divisions down
below cd, we should reach a point at which the last remaining area be
tween the indefinitely extended adiabatics was just equal to the one next
preceding, provided that the temperature 32F. could be expressed in an
even number of absolute degrees.
155. Carnot's Function. Carnot did not find the definite formula for effi
ciency of his engine, given in Art. 135, although he expressed it as a function of
the temperature range (T t). We may state the efficiency as
88 APPLIED THERMODYNAMICS
z being a factor having the same value for all gases. Taking the general expres
sion for efficiency, f ^ (Art. 128), and making H= h + d7i, we have
H
^' "^ ^ ~~\ f ?h'
~ h + (111 ~~ A + rlh
Tor e = z(T f)> we ^aJ write e zdt or s = f, giving
tn
* = 7 ^ 7   <ft, equivalent to ^L
But = (Art. 153) ; whence ^ = ^t and = , and t = ^ = .
t h t h t h (Ih z
Then z =  and e =  =  ~ in finite terniS; as already found. The factor z
is known as Camofs function* It is the reciprocal of the absolute temperature*
156. Determination of the Absolute Zero. The porous plug experiments con
ducted by Joule and Kelvin (Art. 74) consisted in forcing various gases slowly
through an orifice. The fact has already been mentioned that when this action
was conducted without the performance of external work, a barely noticeable
change in temperature was observed ; this being with some gases an increase, and
with others a decrease. When a reMbting pressure was applied at the outlet oC the
orifice, so as to cause the performance of some external work during tho flow of
gas, a fall of temperature was observed ; and tin's fall wan different for dijicrcnt #<7,se,s*.
The "porous plug" was a wad of silk fibers placed in the orifice for the purpose
of reconverting all energy of velocity back to heat. Assume a slight hill of tem
perature to occur iu passing the plug, the velocity energy being reconverted to
heat at the decreased temperature, giving the equivalent paths w/, rfc, Fig. 45.
Then expend a sufficient measured quantity of work to bring the substance back
to its original condition a, along cba. By the second law,
, and  = 
nefN nabN  abfe' T^ nal)Nabfe'
T T = T ( rcafrJV _ j \ __ rn (life
1 L \nal>N  altfe 1
_ __
\nal>N  altfe 1 x nabN  altfe '
If (T T^) as determined by the experiment = a, and nabN be put equal to unity,
rp _ aCl alfe)
A  abft '
In which abfe is the work expended in bringing the gas back to its original tem
perature. This, in outline, was the Joule and Kelvin method for establishing a
location for the true absolute zero the complete theory is too extensive for pres
entation here (2). The absolute temperature of inciting ice is on this scale
491.58 F. or 273.1 C.
The agreement with the hydrogen or the air thermometer is close.
The correction for the former is generally less than yj^ 0., and that for
THE SECOND LAW OF THERMODYNAMICS 89
the latter less than j^ C. Wood has computed (3) that the true absolute
zero must necessarily be slightly lower than that of the air thermometer.
According to Alexander, (4) the difference of the two scales is constant for
all temperatures. The Kelvin absolute scale establishes a logical defini
tion of temperature as a physical unit. Actual gas thermometer tempera
tures may be reduced to the Kelvin scale as a final standard.
In the further discussion^ the temperature 459.6 J?. will be regarded
as the absolute zero. (5)
(1) Peabody, Thermodynamics, 1907, 27. (2) Phil Trans., CXVTV, 349. (3),
Thermodynamics, 1905, 116. (4) Treatise on Thermodynamics, 1892, 91. (5) See
the papers, On the Establishment of the Thermodynamic Scale of Temperature by
Means of the Constant Pressure Thermometer, by Buckingham; and On the Standard
Scale of Temperature in the Interval to 100 C., by Waidner and Dickinson;
Bulletin of the Bureau of Standards, 3, 2; 3, 4. Also the paper by Buckingham,
On the Definition of the Ideal Gas, op. cit., 6, 3.
SYNOPSIS OF CHAPTER VII
Statements of the second law an axiom establishing the criterion of reversibility ;
jg h __ T t Of h _ _ a statement of the efficiency of the Carnot cycle ; the cri
H ~~ T H~~ T terion of reversibility itself.
The second law limits the possible efficiency of a heat engine.
The fall of temperature determines the amount oE external work done.
Temperature ratios defined as equal to ratios of heats absorbed and emitted.
The Carnot function for cyclic efficiency is the reciprocal of the absolute temperature.
The absolute zero, based on the second law, is at 459,6 F.
PROBLEMS
1. Illustrate graphically the first and the second laws of thermodynamics. Frame
a new statement of the latter.
2. An engine works in a Carnot cycle between 400 F. and 280 F., developing
120 h.p. If the heat rejected by this engine is received at the temperature of rejection
by a second Carnot engine, which works down to 220 F., find the horse power of the
second engine. (Ans., 60).
3. Find the coefficient of expansion at constant pressure of a perfect gas. What
is the percentage difference between this coefficient and that for air ?
(Ans. t 0.0020342 ; percentage difference, 0.03931.)
4. A Carnot engine receives from the source 1000 B. t. u., and discharges to the
condenser 500 B. t. u. If the temperature of the source is 600 F., what is the tem
perature of the condenser ? (.4ns., 70.2 F.)
5. A Carnot engine receives from the source 190 B.t. u. at 1440.4 F., and dis
charges to the condenser 90 B.t.u. at 440.4 F. Find the location of the absolute
zero. (Ans., 459.6 F.)
6. In the porous plug experiment, the initial temperature of the gas being that of
90 APPLIED THERMODYNAMICS
melting ice, and the fall of tempeiature being T J ff of the range from melting to boiling
of water at atmospheric pressure, the work expended in restoring the initial tempera
ture was 1.5S footpounds. Find the absolute temperature at 32 F. (Ans., 492.39.)
7. The temperature range in a Camot cycle being 400 F., and the work done
being equivalent to 40 pei cent of the amount of heat rejected, find the values of T
and t.
REVIEW PROBLEMS, CHAPTERS I VII
1. State the precise meaning, or the application, o the t olio wing expressions :
k 778 I (} =  H = T+I+W r y E 53.36 PV = RT R 459.6 F.
\P/v t
n1
I P V T pv logg y ( J *=
pi) n =c 42.42 pijy c 2545 pv c s = Z r
n 1
2. A heat engine receives its fluid at 350 F. and discharges it at 110 F. It con
sumes 200 B. t. u. per Ihp. per minute. Find its efficiency as compared with that of
the corresponding Carnot cycle. (Ans., 0.712.)
3. Given a cycle a&c, in which P a =P 6 = 100 Ib. per sq. in., V a  1, ^rr= 6 (cu. ft.),
YO,
PfiVj, 1 8 =P c V c 1 ' B ,P a V a P c Y ct find the pressure, volume, and temperature at c if the
substance is 1 Ib. of air.
4. Find the pressure of 100 Ib. of air contained in a 100 cu.ft. tank at 82 F.
(Ans., 28,900 Ib. per sq. ft.)
5. A heat engine receives 1175.2 B. t. u. in each pound of steam and rejects
1048.4 B. t. u. It uses 3110 Ib. of steam per hour and develops 142 lip. Estimate the
value of the mechanical equivalent of heat. (Ans., 712.96.)
6. One pound of air at 32F . is compressed from 14.7 to 2000 Ib. per square inch,
without change of temperature. Find the percentage change of volume.
(Ana., 99.3%.)
7. Prove that the efficiency of the Carnot cycle is ^.
8. Air is heated at constant pressure from 32 F. to 500 F. Find the percentage
change in its volume. (Ans., 95.2 % increase.)
9. Prove that the change of internal energy in passing from a to 6 is independent
of the path ab.
"P V ~P "\7"
10. Given the formula for change of internal energy, & & , prove that
11. Given It for air=53.36, V= 12,387; and given F= 178.8, fc=3.4 for hydro
gen : find the value of y for hydrogen. (Ans., 1.412.)
12. Explain isothermal, adiabatic, isodynamic, isodiabatic.
13. Find the mean specific heat along the pathpvi8 =c for air (2=0.1689).
(Ans., 0.084.)
PROBLEMS 91
14. A steam engine discharging its exhaust at 212 F receives steam containing
1100 B, t. u. per pound at 500 F. What is the minimum weight of steam it may use
per Ihp.hr. ? (Arts , 7.71 Ib.)
15. A cycle is bounded by polytropic paths 12, 23, 13. We have given
P i =P 2 100,000 Ib. per sq. ft.
V 2 = V z =40 cubic feet per pound.
T 1= :3000 F.
PiFxP.F,.
Find the amount of heat converted to work in the cycle, if the working substance is
air. (4ns., 4175 B.t.uJ
CHAPTER VIII
ENTROPY
157. Adiabatie Cycles. Let abdc, T?ig. 46, be a Carnot cycle, an and bJ$
the projected adiabatics. Draw intervening adiabatics em, g^f } etc., so
located that the areas naem, megM', M<jl)N, are equal. Then since the effi
ciency of each of the cycles aefc, eyhf, gbdJi, is (T t) = T, tJie work areas
represented by these cycles are all equal. To measure these areas by mechani
cal means would lead to approximate results only.
158. Rectangular Diagram. If the adiabatics and isothermals
were straight lines, simple arithmetic would suffice for the measure
ment of the work areas of Fig. 46. We
have seen that in the Carnot cycle,
bounded by isothermals and adiabatics,
= (Art. 158). Applying this for
mula to Rankine's theorem (Art. 106),
we have the quotient of an area and a
length as a constant. If the area h is
a part of .fiT, then there must be some
constant property, which, when, multi
plied by the temperatures T or , will
FIG. 40. Arts. 157, 158, 15<>, 100.
Adiabatie Cycles.*
710
050
600
650
1191 G
give the areas H or h. Let us conceive
of a diagram in which only one coor
dinate will at present be named. That
coordinate is to be absolute temperature.
Instead of specifying the other coordi
nate, let it be assumed that subtended
areas on this diagram are to denote
quantities of heat absorbed or emitted,
just as such areas on the JPV diagram
represent external work done. As an
example of such a diagram, consider
Fig, 47. Let the substance be one
* The adiabatics are distorted for clearness. In reality they are asymptotic. Many
of the diagrams throughout the "book are similarly u out of drawing" for the same
reason.
92
FIG. 4:7* Arts. 158, 163, ]71. En
tropy Diagram.
ENTROPY 93
pound of water, initially at a temperature of 32 F., or 491.6 abso
lute, represented by the height #5, the horizontal location of the
state b being taken at random. Now assume the water to be heated
to 212 F., or 671.6 absolute, the specific heat being taken as con
stant and equal to unity. The heat gained is 180 B. t. u. The
final temperature of the water fixes the vertical location of the
new state point cZ, i.e. the length of the line cd. Its horizontal lo
cation is fixed by the consideration that the area subtended between
the path bd and the axis which we have marked ON shall be
180 B. t. u. The horizontal distance ac may be computed from the
properties of the trapezoicl abdc to be equal to the area abdc divided
by [(a& f cd) + 2] or to 180 f [(491.6 + 671.6) f 2] = 0.31. The
point d is thus located (Art. 163).
159. Application to a Carnot Cycle. Ordinates being absolute
temperatures, and areas subtended being quantities of heat absorbed
or emitted, we may conclude that an isothermal must be a straight
horizontal line ; its temperature is constant, and a finite amount of
heat is transferred. If the path is from left to right, heat is to be
conceived as absorbed; if from right to
left, heat is rejected. Along a (re
versible) adiabatic, no movement of heat
occurs. The only line on this diagram
T which does not subtend a finite area is
a straight vertical line. Adiabatics are
1 consequently vertical straight lines. (But
see Art. 176.) The temperature must
N constantly change along an adiabatic.
FIG. 48. Arts. 169, 160, 161, ics, The lengths of all isothermals drawn be
106. Adiabatic Cycles, Entropy tween fc h e game two adiabatipq a pnnal
UWCCll UL1C &ClJ.llt5 u \V \J LLLJ.CtUc1i LlUo GiL\3 dJ LiCuJ..
Diagram.
The Carnot cycle on this new diagram
may then be represented as a rectangle enclosed by vertical and hori
zontal lines ; and in Fig. 48 we have a new illustration of the cycles
shown in Fig. 46, all of the lines corresponding.
160. Physical Significance. The new diagram is to be conceived
as so related to the P V diagram of Fig. 46 that while an imaginary
\,
M
94
APPLIED THERMODYNAMICS
pencil is describing any stated path on the latter, a corresponding
pencil is tracing its path on the former. In the PV diagram, the
subtended areas constantly represent external work done by or on the
substance; in the new diagram they represent quantities of heat ab
sorbed or rejected. (Note, however, Art. 176.) The area of the
closed cycle in the first case represents the net quantity of work done;
iu the second, it represents the net amount of heat lost^ and conse
quently, also, the net work done. But subtended areas under a single
path on the PV diagram do not represent heat quantities, nor in the
new diagram do they represent work quantities. The validity of the
diagram is conditioned upon the absoluteness of the properties chosen as
coordinates. We have seen that temperature is a cardinal property,
irrespective of the previous history of the substance ; and it will be
shown that this is true also of the horizontal coordinate, so that we
may legitimately employ a diagram in which these two properties
are the coordinates.
161. Polytropic Paths. For any path in which the specific heat
is zero, the transfer of heat is zero, and the path on this diagram is
consequently vertical, an adiabatic. For specific heat equal to
infinity, the temperature
cannot change, and the
path is horizontal, an iso
thermal. For any positive
value of the specific heat,
heat area and temperature
will be gained or lost
simultaneously; the path
will be similar to ai or #/,
Fig. 48. If the specific
heat is negative, the tem
perature will increase with
rejection of heat, or de
crease with its absorption, as along the paths ak, al, Fig, 48. These
results may be compared with those of Art. 115. Figure 49 shows
on the new diagram the paths corresponding with those of Fig. 31.
It may be noted that, in general, though not invariably, increases of
FIG. 49. Arts. 101, 1 05. Polytropic Paths on
Entropy Diagram.
ENTROPY
95
volume are associated with increases of the horizontal coordinate of
the new diagram.
162. Justification of the Diagram. In the PV diagram of Fig. 50, consider
the cycle ABCD. Let the heat absorbed along a portion of this cycle be repre
sented by the infinitesimal strips nabN,
NbcM, Mcdm, formed by the indefinitely
projected adiabatics. In any one of these
strips, as nabN, we have, in finite terms,
nabN _ T Qr
negN t'
nabN _ neqN
T t
Considering the whole series of strips
from A to C, we have
nabN __ ^ neqN
v or, using the symbol H for heat trans
ferred,
FIG. 50. Art. 162. Entropy a Cardinal
Property. S ^7T = >
in which T expresses temperature generally.
Let the substance complete the cycle ABCD A] we then have, the paths leing
reversible,
JP P A
dH_ I riff I rlH__~
^r~ \* "F + P "F'
C/^i *Jo
while for the cycle ADCDA,
whence,
The integral f thus has the same value whether the path is A DC or ABC,
or, indeed, any reversible path between A and C; its value is independent of the
path of the substance. Now this integral will be shown immediately to be the most
general expression for the horizontal coordinate of the diagram under discussion.
Since it denotes a cardinal property, like pressure or temperature, it is permissible
to use a diagram in which the coordinates are T and fm
96
APPLIED THERMODYNAMICS
163. Analytical Expression. Along any path of constant tem
perature, as al> Fig. 48, the horizontal distance nN may be computed
from the expression, nN=H+ T, in which S represents the quan
tity of heat absorbed, and T the temperature of the isothermal. If
the temperature varies, the horizontal component of the path during
the absorption of dH units of heat is dn = dffi T. For any path
along which the specific heat is constant, and equals 7c, ?cdT=
dn = , and = k. = k log, .
If the specific heat is variable, say Jc a + IF, then
The line Id of Fig. 47 is then a logarithmic curve, not a straight
line ; and the method of finding it adopted in Art. 158 is strictly
accurate only for an infinitesimal change of temperature. Writing
the expression just derived in the form n = &log e (jF* 1) and remem
bering that T= PVr 72, while t = pv * 72, we have
n = k log e (P V+ pv) .
The expression Jclog e (Tr) is the one most
commonly used for calculating values of the hori
zontal coordinate for polytropic paths. The
expression dn = dHt T is general for all re
versible paths and is regarded by Ranldne as
the fundamental equation of thermodynamics.
164. Computation of Specific Heat. If at any FlG . 51 . Art . 16 L_ Graphi _
point on a reversible path a tangent be drawn, the cal Determination of
length of the subtangent on the JVaxis represents the Specific Heat,
value of the specific heat at
that point. In Fig. 51, draw the tangent nm to the
curve AB at the point nand construct the infinitesimal
"~ triangle dtdn. From similar triangles, mr : nr : : dn : dt,
or mr = Tdn  dt = dH  dt = s (Art. 112).
165. Comparison of Specific Heats. If a gas is
heated at constant pressure from a, Pig. 52, it will
gain heat and temperature, following some such
path as ab. If heated at constant volume,
through an equal range of temperature, a less
FIG. 52. Art. 165. Com
parison of Specific Heats.
ENTROPY 97
quantity of heat will be gained ; i.e. the subtended area aefd will be less
than the area abed. In general, the less the specific heat, the more
nearly vertical will be the path. (Compare Fig. 49.) When & == 0, the
path is vertical ; when 7c = oo, the path is horizontal.
166. Properties of the Carnot Cycle. In Fig. 48, it is easy to see that
since efficiency is equal to net expenditure of heat divided by gross ex
penditure, the ratio of the areas abdc and abNn expresses the efficiency,
and that this ratio is equal to (T ?) H T. The cycle abdc is obviously
the most efficient of all that can be inscribed between the limiting iso
thermals and adiabatics.
167. Other Deductions. The net enclosed area on the TN diagram
represents the net movement of heat. That this area is always equivalent
to the corresponding enclosed area on the PV diagram is a statement of
the first law of thermodynamics. Two statements of the second law have
just been derived (Art. 166). The theorem of Art. 106, relating to the
representation of heat absorbed by the area between the adiabatics, is
accepted upon inspection of the TN diagram. That of Art. 150, from which
the Kelvin absolute scale of temperature was deduced, is equally obvious.
168. Entropy. The horizontal or N coordinate on the diagram
now presented was called by Clausius the entropy of the body. It
may be mathematically defined as the ratio n = ^  Any physical
/ J
definition or conception should be framed by each reader for himself.
Wood calls entropy " that property of the substance which remains
constant throughout the changes represented by a [reversible] adia
batic line." It is for this reason that reversible adiabatics are called
isentropics, and that we have used the letters H, JT in denoting
adiabatics.
169. General Formulas. It must be thoroughly
understood that the validity of the entropy diagram is
dependent upon the fact that entropy is a cardinal prop
erty, like pressure, volume, and temperature. For this
reason it is desirable to become familiar with compu
tations of change of entropy irrespective of the path
pursued, Otherwise, the method of Art, 163 is usually
FIG. 54. Arts. 169, 329a. _ more convenient.
ange o n opy. Consider the states a and b } Fig. 54. Let the
substance pass at constant pressure to c and thence at constant volume
98 APPLIED THERMODYNAMICS
T T
to &. The entropy increases by 7c log e ^ {I log a * (Art. 163), 7c and I
LO, J c
in this instance denoting the respective special values of the specific
heats. An equivalent expression arises from Charles 5 law :
n = k log e Z* + i log. = k log e *+ 1 log, J 6 , (A)
r *c r a * a
in which last the final and initial states only are included.
We may also write,
Z
= i io go
*V+ ot* T7"
= Z log a ^ + (ft  log, >, Arts. 51, 65 : (B)
'a ' a
and further,
The entropy is completely detei mined by the adiabatic through the state point.
T
In the expression n^=.k^ log e , the value of n L apparently depends upon that of k^
which is of course related to the path ; along another path, the gain or loss of
T
entropy might be n 2 = & 3 log, > a different value ; but although the temperatures
would be the same at the beginning and end of both processes, the pressures or
volumes would differ. The states would consequently be different, and the values
of n should therefore differ also.
A graphical method for the transfer of perfect gas paths from the PFto the
TN plane has been developed by Berry (1).
169a. Mixtures of Liquids. When m Ib. of water are heated from
32 to t absolute, the specific heat being taken at unity, the gain, of
entropy is
Let m Ib. at t be mixed with n Ib. at 1, the resulting temperature
of (m+n) Ib. being (from Art. 25), without radiation effects,
yy
This, if heated from 32 to Z ; would have acquired the entropy
(m+ri) log* ^~2,
and the change in aggregate entropy due to the mixture is
i t , i t\ f , \ i / nti + mt \
m log, m +n log e m  (+) log, m(m+n} )
The mixing of substances at different temperatures always in
creases the aggregate entropy. Thus, let a body of entropy n, at
the temperature t, discharge a small amount, H, of its heat to an adjacent
body of entropy N and temperature T. The aggregate entropy before
the transfer is n + N; after the transfer it is
TT rj
and since t>T f <TF and the loss of entropy is less than the gain:
t j.
or
170, Other Names for n. Rankine called n the thermodynamic func
tion. It has been called the " heat factor." Zeuner describes it as
" heat weight." It has also been called the " mass " of heat. The
letters T, N, which we have used in marking the coordinates, were
formerly replaced by the Greek letters theta and phi, indicating abso
lute temperatures and entropies; whence the name, thetaphi diagram.
The TN diagram is now commonly called the temperatureentropy
diagram, or, more briefly, the entropy diagram.
171. Entropy Units. Thus far, entropy has been considered as a
horizontal distance on the diagram, without reference to any particular
zero point. Its units are B. t. u. per degree of absolute temperature.
Strictly speaking, entropy is merely a ratio, and has no dimensional
units. Changes of entropy are alone of physical significance. The
choice of a zero point may be made at random ; there is no logical zero of
entropy. A convenient starting point is to take the adiabatic of the
substance through the state P =2116.8, T=32 F., as the OT axis, the
entropy of this adiabatic being assumed to be zero, as in ordinary tables.
100 APPLIED THERMODYNAMICS
Thus, in Fig. 47 (Art. 158), the OT axis should be shifted to pass through
the point b, which was located at random horizontally.
172. Hydraulic Analogy. The analogy of Art. 140 may be extended to illus
trate the conception of entropy. Suppose a certain weight of water W to be
maintained at a height H above sea level; and that in passing through a motor
its level is reduced to h. The initial potential energy of the water may be
regarded as WH, the final residual energy as Wh, the energy expended as
W(H A). Let this operation be compared with that of a Carnot cycle, taking
in heat at T and discharging it at t. Eegarding heat as the product of N and T,
then the heat energies corresponding to the water energies just described are NT,
Nt, and N(T t) ; N being analagous to W, the weight of the water.
173. Adiabatic Equation. Consider the states 1 and 2, on an adiabatic
path, Fig. 55. The change of entropy along the constant volume path 13
D rp
is I log e 3 ; that along the constant pressure path 32
T
is Jc log fl ^ The difference of entropy between
* i
1 and 2, irrespective of the path, is
.
V\
FIG. 55. Art. 173. ^ or a reversible adiabatic process, this is equal to
Adiabatic Equation. zero; whence
e or y lo & Fi + lo & A = y log, F! + log.P,,
L
from which we readily derive P^Vf = P a F^ #ie equation of the adiabatic.
174. Use of the Entropy Diagram. The intelligent use of the entropy
diagram is of fundamental importance in simplifying thermodynamic con
ceptions. The mathematical processes formerly adhered to in presenting
the subject have been largely abandoned in recent textbooks, largely on
account of the simplicity of illustration made possible by employing the
TN coordinates.
Belpaire was probably the first to appreciate their usefulness. Gibbs, at about
the same date, 1873, presented the method in this country and first employed as
coordinates the three properties volume, entropy, and internal energy. Linde,
Schroeter, Hermann, Zeuner, and Gray used TN diagrams prior to 1890. Cotterill,
Ayrton and Perry, Dwelshauvers Dery and Ewing have employed them to a con
siderable extent. Detailed treatments of this thermodynamic method have been
given by Boulvin, Reeve, Berry, and Golding (2). Some precautions necessary in
its practical application are suggested in Arts. 45i458.
IRREVERSIBLE PROCESSES
101
FIG. 56.
Art. 175. Irreversible
Cycle.
IRREVERSIBLE PROCESSES
175. Modification of the Entropy Conception. It is of importance to distinguish
between reversible and irreversible processes in relation to entropy changes.
The significance of the term reversible, as ap
plied to a path, was discussed in Art. 125. A
process is reversible only when it consists of a
series of successive states of thermal equilib
rium. A series of paths constitute a reversible
process only when they foim a closed cycle,
each path of which is itself reversible. The
Carnot cycle is a perfect example of a reversible
process. As an example of an irreversible cycle,
let the substance, after isothermal expansion,
as in the Carnot cycle, be transferred directly
to the condenser. Heat will be abstracted, and
the pressure may be reduced at constant vol
ume, as along be, Fig. 56. Then allow it to compress isothermally, as in the
Carnot cycle, and finally to be transferred to the source, where the temperature
and pressure increase at constant volume, as along da. This cycle cannot be
operated in the reverse order, for the pressure and temperature cannot be reduced
from a to d while the substance is in communication with the source, nor increased
from c to b while it is in communication with the condenser.
176. Irreversibility in the Porous Plug Experiment. We have seen that in this
instance of unresisted expansion, the fundamental formula of Art. 12 becomes
H= T + I + W + V (Art. 127). Knowing H = 0, W = 0, we may write
(T + I) = V, or velocity is attained at the expense of the internal energy. The
velocity evidences kinetic energy ; mechanical work is made possible ; and we might
expect an exhibition of % such work and a fall of internal energy, and consequently
of temperature. But we find no such utilization of the kinetic energy of the rapidly
flowing jet; on the contrary, the gas is gradually brought to rest and the velocity
derived from, an expenditure of internal energy is reconverted to internal energy,
The process was adiabatic, for no transfer of heat occurred ; it was at the same
time isothermal, for no change of temperature occurred ; and while both adiabatic
and isothermal, no external work was done, so that the PV diagram is invalid.
Further : the adiabatic path here considered was not isen tropic, like an ordinary
adiabatic. The area under the path on the TN diagram no longer represents heat
absorbed from surrounding bodies. Neither does dn = , for H is zero, while
n is finite. The expression for increase of entropy, C f , along a reversible path, does
not hold for irreversible operations.
In irreversible operations, the expression C ( r ceases to represent a cardinal
property. "We have the following propositions :
102 APPLIED THERMODYNAMICS
(a) In a reversible operation, the sum of the entropies of the participating substances
is unchanged. During a reversible change, the temperatures of the heatabsorbing
and heatemitting bodies must differ to an infinitesimal extent only; they are in
finite terms equal. The heat lost by the one body is equal to the heat gained by
the other, so that the expression f '" denotes both the loss of entropy by the one
substance and the gain by the othei , the total stock of enti opy remaining constant.
(1} During meuersible operations, the aggregate entropy increases. Consider two
engines working in the Carnut cycle, the first taking the quantity of heat H : from
the souice, and dischaiging the quantity H to the condenser; the second, acting
as a heat pump (Art. 130), taking the quantity II j from the condenser and restoring
H^ to the source. Then if the work produced by the engine is expended in driving
the pump, without loss by friction,
HIHI = UJHI.
If the engine is irreversible, H^ > ///, or IT l  If/ > 0, whence, H 2  H 2 ' > 0. Tf
we denote by a a positive finite value, H l = HJ + a and H 2 = H 2 ' +a. But
^L = , or y ~ ^  0, and consequently
<H$ J'i J i ^2
PL a H*a n , H } H, (1 1
~  =Q an<i = "
Since T, > T* 1  < 0, or > , or, generally, < 0. The value of
C ( UL j s thus, for irreversible cyclic operations, negative.
Now let a substance work irreversibly from A to JS, thence revemlly from B to
A. We may write.
(irruv ) Ciev ) (irrev ) (lev)
r B dii
But the cJiange of entropy of the substance in passing from A toBi$N B N' A = I ,
JA *
(IE being the amount of heat absorbed along any reversible path, while the change
of entropy of the source which supplies the substance with heat (reversibly) is
jyy Njf = C 7, the negative sign denoting that heat has been abstracted,
Jj. V
We have then, from equation (A),
i.e. the sum of the entropies of the participating substances increases when the
process is irreversible.
(c) The loss of work due to irrerentibtlity is ptopoitional to the increase of entropy.
Consider a partially completed cj cle : one which might be made complete were all
of the paths reversible. Let the heat absorbed be Q, at the temperature !T, in
creasing the entropy of the substance by ,; and let its entropy be further increased
IRREVERSIBILITY 103
by N f N during the process The total increase of entropy is then n = A T ' A 7 + y,,
whence Q = T(n  N' + A 7 ) T \ he work done may be written as // H ' 4 Q, in
which H and H' are the initial and final heat contents respectively. Calling this
W, we have
W = //  H 1 4 T(n  JV' + A 7 ).
In a reversible cycle J = n , whence W R = H H' + 2"(JV"  A"') and
^  W = Tn.
(A careful distinction should be made at this point between the expression
j TT
and the term entropy. The former is merely an expression for the latter
under specific conditions In the typical irreversible process furnished by the
porous plug experiment, the entropy increased; and this is the case generally with
such processes, in which dn > Internal transfers of heat may augment the
entropy even of a heatinsulated body, if it be not in uniform thermal condition.
Perhaps the most general statement possible for the second law of thermody
namics is that all actual processes tend to increase the entropy ; as we have seen, this
keeps possible efficiencies below those of the perfect reveisihle engine. The prod
uct of the increase of entropy by the temperature is a measure of the waste of
energy (3).)
Most operations in power machinery may without serious error be analyzed
as if reversible ; unrestricted expansions must always be excepted. The entropy
diagram to this extent ceases to have " an automatic meaning."
(1) Tlie TemperatureEntropy Diagram, 1008. (2) See Berry, op. cit. (3) The
works of Preston, Swinburne, and Planck may be consulted by those interested in this
aspect of the subject. See also the paper by M'Cadie, in the Journal of Electricity,
Power and Gas, June 10, 1911, p. 505.
SYNOPSIS OF CHAPTER VIII
It is impracticable to measure PFheat areas "between the adiabatics.
The rectangular diagram : ordinates = temperature; areas = heat transfers.
Application to a Carnot cycle : a rectangle.
1?he validity oj the diagram is conditioned upon the absoluteness of the horizontal
coordinate.
The slope of a path of constant specific heat depends upon foe value of the specific heat.
The expression C has a definite value for any reversible change of condition,
regardless of the path pursued to effect the change.
fj FT T T*
dn = , or n = Tc log e for constant specific heat = k, or n = a log e + &( T for
T t ' t
variable specific heat = a + & T.
The value of the specific heat along a poly tropic is represented "by the length of the sub
tangent.
Illustrations : comparison of k and I ; efficiency of Camot cycle ; the first law j the
second law ; heat area between adiabatics ; Kelvin's absolute scale.
101 APPLIED THERMODYNAMICS
Entropy units are B. t. u.per degree absolute. The adiabatic for zero entropy is at
.
= nog c ^^
The mixing of substances at different temperatures increases the aggregate entropy.
Hydraulic analogy ; physical significance of entropy ; use of the diagram.
Derivation of the adiabatic equation.
Irreversible Processes
A reversible cycle is composed of reversible paths ; example of an irreversible cycle.
Joule's experiment as an example of irreversible operation.
Heat generated by mechanical friction of particles ; the path both isothermal and adia
batic, but not isen tropic.
S T+I + W+
For irreversible processes, d?i is not equal to ~ 3 the subtended area does not repre
sent a transfer of heat ; nonisentropic adiabatics.
In reversible operations, the aggregate entropy of the participating substances is
unchanged.
During irreversible operations, the aggregate entropy increases, and J <0.
The loss of work due to increase of entropy is nT\ du>d.
T
PROBLEMS
1. Plot to scale the TJVpath of one pound of air heated (a) at constant pressure
from 100 F. to 200 F., then (Z>) at constant volume to 300 F. The logarithmic
curves may be treated as two straight lines.
2. Construct the entropy diagram for a Carnot cycle for one pound of air in which
T= 400 F., t = 100 F., and the volume in the first stage increases from 1 to 4 cubic
feet. Do not use the formulas in Art 169.
3. Plot on the TJV diagram paths along which the specific heats are respectively
0, oo,* 3.4, 0.23, 0.17, 0.3, 10.4, between T = 459.0 and T= 910.2, treating the
logarithmic curves as straight lines.
4. The variable specific heat being 0200.0004 T 0.000002 T 2 (T being in
Fahrenheit degrees), plot the TF path from 100 F. to 140 F. m four steps, using
mean values for the specific heat in each step.
Find by integration tlie change of entropy during the whole operation.
5. What is the specific heat at T=40 (absolute) for a path the equation of
which on the TN diagram is TN= c = 1200 ? (Ans., 32.)
6. Confirm Art. 134 by computation from the TN diagram.
7. Plot the path along which 1 unit of entropy is gained per 100 absolute,
What is the mean specific heat along this path from to 400 absolute? Begin at 0.
8. What is the entropy measured above the arbitrary zero per pound of air at
normal atmospheric pressure in a room at 70 F.? (Ans. t 0.01766.)
PROBLEMS 105
9. Find the arbitrary entropy of a pound of air in the cylinder of a compressor
at 2000 Ib pressure per square inch and 142 F. (Ans., 0.301.)
10. Find the entropy of a sphere of hydrogen 10 miles in diameter at atmospheric
pressure and 175 F. (Ans., 289,900,000,000.)
11. The specific heat being 0.24 f 0.0002 T, find the increase in entropy between
459.6 and 919.2 degrees, all absolute. What is the mean specific heat over this
range ? (Ans., increase of entropy 0.25809 ; mean specific heat, 0.378.)
12. In a Carnot cycle between 500 and 100, 200,000 ft. Ib. of work are done.
Find the amount of heat supplied and the variation in entropy during the cycle.
13. A Carnot engine works between 500 and 200 and between the entropies
1.2 and 1.45. Find the ft. Ib. of work done per cycle.
14. To evaporate a pound of water at 212 F. and atmospheric pressure, 970 4 B. t. u.
are required If the specific volume of the water is 0.016 and that of the steam
26.8, find the changes in internal energy and entropy during vaporization.
15 Five pounds of air in a steel tank are cooled from 300 F. to 150 F. Find
the amount of heat emitted and the change in entropy. (I for air =0.1689.)
16. Compare the internal energy and the entropy per pound of air when (a) 50
cu. ft. at 90 F. are under a pressure of 100 Ib. per sq. in., and (&) 5 cu. ft. at 100
F. are subjected to a pressure of 1200 Ib. per sq. in.
17. Air expands from p=100, u = 4 to P=40, F=8 (Ib. per sq in. and cu. ft. per
Ib.). Find the change in entropy, (a) by Eq (A) Art. 169, (&) by the equation
n 2  Hi =s log e j, where s=l
n
18 A mixture is made of 2 Ib. of water at 100, 4 Ib. at 160, and 6 Ib. at
90 (all Fahr.). Find the aggregate entropies before and after mixture.
CHAPTER IX
COMPRESSED AIR (1)
177. Compressed Air Engines. Engines are sometimes used in which the
working substance is cold air at high pressure. The pressure is previously pro
duced by a separate device ; the air is then transmitted to the engine, the latter
being occasionally in the form of an ordinary steam engine. This type of motor
is often used in mines, on locomotives, or elsewheie where excessive losses by con
densation would follow the use of steam. For small powers, a simple form of
rotary engine is sometimes employed, on account of its convenience, and in spite
of its low efficiency. The absence of heat, leakage, danger, noise, and odor makes
these motors popular in those cities where the public distribution of compressed
air from central stations is practiced (la). The exhausted air aids in ventilating
the rooms in which it is used.
178. Other Uses of Compressed Air. Aside from the driving of engines, high
pressure air is used for a variety of purposes in mines, quarries, and manufac
turing plants, for operating hoists, forging and bending machines, punches, etc,,
for cleaning buildings, for operating "steam" hammers, and for pumping water
by the ingenious "air lift" system. In many works, the amount of power trans
mitted by this medium exceeds that conveyed by belt and shaft or electric wire.
The air is usually compressed by steam power, and it is obvious that a loss must
occur in the transtormation. This loss may be offset by the convenience and ease
of transmitting air as compared with steam ; the economical generation, distribu
tion, and utilization of this form of power have become matters of first importance.
The first applications were made during the building of the Mont Cenis tun
nel through the Alps, about 18GO (2). Air was there employed for operating
locomotives and rock drills, following Colladon's mathematical computation of
the small loss of pressure during comparatively long transmissions, A general
introduction in mining operations followed. Twostage compressors with inter
coolers were in use in this country as early as 1881. Among the projects sub
mitted to the international commission for the utilization of the power of Niagara,
there were three in which distribution by compressed air was contemplated. Wide
spread industrial applications of this medium have accompanied the perfecting of
the small modern interchangeable "pneumatic tools."
179. Air Machines in General. In the type of machinery under consideration,
a considerable elevation of pressure is attained. Centrifugal fans or paddlewheel
blowers, commonly employed in ventilating plants, move large yolumes of air at
very slight pressures, usually a fraction of a pound, and the thermodynamio
106
THE AIR ENGINE
107
relations are unimportant. Rotary blowers are used for moderate pressures, up
to 20 lb., but they are generally wasteful of power and are principally employed
to furnish blast for foundry cupolas, forges, etc. The machine used for coin
pressing air for power purposes is ordinarily a piston compressor, mechanically
quite similar to a reciprocating steam engine. These compressors are sometimes
employed for comparatively low pressures also, as " blowing engines."'
rr
3 \g \^
c.
THE AIR ENGINE
180. Air Engine Cycle. In Fig. 57, ABOD represents an ideal
ized air engine cycle. AB shows the admission of air to the cylin
der. Since the latter is smull as compared with the transmitting
pipe line, the specific volume and pres P
sure of the fluid, and consequently
its temperature as well, remain un A
changed. BO represents expansion
after the supply from the mains is
cut off. If the temperature at B is that F
of the external atmosphere, and ex
pansion proceeds slowly, so that any
fall of temperature along BC is offset u
by the transmission of heat from the
outside air through the cylinder walls,
this line is an isothermal. If, however,
expansion is rapid, so that no transfer
of heat occurs, BO will be an adidbatic. In practice, the expansion
line is a polytropic, lying usually between the adiabatic and the
isothermal. CD represents the expulsion of the air from the cyl
inder at the completion of the working stroke. At _Z), the inlet
valve opens and the pressure rises to that at A. The volumes
shown on this diagram are not specific volumes, but volumes of air in
the cylinder. Subtended areas, nevertheless,*represent external work.
181. Modified Cycle. The additional work area LMC obtained by ex
pansion beyond some limiting volume, say that along onf, is small. A
slight gain in efficiency is thus made at the cost of a much larger cylin
der. In practice, the cycle is usually terminated prior to complete expan
sion, and has the form ABLMD, the line LM representing the fall of
pressure which occurs when the exhaust valve opens.
FIG. 57. Arts. 180183, 189, 222, 223,
226, Prob. 6. Air Engine Cycles.
108 APPLIED THERMODYNAMICS
182. Work Done. Letting p denote the pressure along AB, P
the pressure at the end of the expansion, q the "back pressure"
along MD (slightly above that of the atmosphere), and letting <Q
denote the volume at B, and Fthat at the end of expansion, both
volumes being measured from OA as a line of zero volumes, then,
for isothermal expansion, the work done is
V T7
e qV\
and for expansion such that pv n = PV n , it is
(In these and other equations in the present chapter, the air will
be regarded as free from moisture, a sufficiently accurate method of
procedure for ordinary design. For air constants with moisture
effects considered, see Art. 3S2&, etc.)
183. Maximum Work. Under the most favorable conditions, expan
sion would be isothermal and "complete"; i.e. continued down to the
backpressure line CD. Then, q = P~pv+ F, and the work would be
pv log e (F4 v). For complete adiabatic expansion, the work would be
y
PF= 0* PF)
184. Entropy Diagram. This cannot be obtained by direct transfer from the
PV diagram, because we are dealing with a varying quantity of air. The method
of deriving an illustrative entropy diagram is explained in Art. 218.
185. Fall of Temperature. If air is received by an engine at
P, F, and expanded to p, t, then from Art, 104, if P+p= 10, and
T 529 absolute, with adiabatic expansion, t = 187 F.
This fall of temperature during adiabatic expansion is a serious matter.
Low final temperatures are fatal to successful working if the slightest
trace of moisture is present in the air, on account of the formation of ice
in the exhaust valves and passages. This difficulty is counteracted in
various ways: by circulating warm air about the exhaust passages; by
specially designed exhaust ports 5 by a reduced range of pressures; by
avoidance of adiabatic expansion (Art. 219) ; and by thoroughly drying
the air. The jacketing of the cylinder with hot air has been proposed.
Unwin mentions (3) the use of a spray of water, injected into the air
while passing through a preheater (Art. 186). This reaches the engine
as steam and condenses during expansion, giving up its latent heat of
PREHEATERS
109
vaporization and thus raising the temperature. In the experiments on
the use of compressed air for street railway traction in ]N~ew York, stored
hot water was employed to preheat the air. The only commercially suc
cessful method of avoiding inconveniently low temperatures after expan
sion is by raising the temperature of the inlet air.
186. Preheaters. In the Paris installation (4), small heaters were
placed at the various engines. These were double cylindrical boxes of
cast iron, with an intervening space through which the air passed in a
circuitous manner. The inner space contained a coke fire, from which
the products of combustion passed over the top and down the outside of
the outer shell. For a 10hp. engine, the extreme dimensions of the
heater were 21 in. in diameter and 33 in. in height.
187. Economy of Preheaters. The heat used to produce elevation of
temperature is not wasted. The volume of the air is increased, and the
weight consumed in the
engine is correspondingly
decreased. Kennedy esti
mated in one case that
the reduction in air con
sumption due to the in
crease of volume should
have been, theoretically,
0.30; actually, it was 0.25.
The mechanical efficiency
(Art. 214) of the engine
is improved by the use of
preheated air. In
one instance, Ken
nedy computed a
saving of 225 cu. ft. of
"free" air (i.e. air at at
mospheric pressure and tem
perature) to have been ef
fected at an expenditure
of 0.4 Ib. of coke. Unwin
found that all of the air
used by a 72hp. engine
could be heated to 300 F.
by 15 Ib. of coke per hour.
Figure 58 represents a
modern form of preheater. FIG. 58. Art. 187. Band Air Pieheater.
110 APPLIED THERMODYNAMICS
188. Volume of Cylinder. If n be the number of single strokes per
minute of a doubleacting engine, V the cylinder volume (maximum vol
ume of fluid), W the number of pounds of air used per minute, v the
specific volume of the air at its lowest pressure p and its temperature
t, N the horse power of the engine, and U the work done in footpounds
per pound of air, then, ignoring clearance (the space between the piston
and the cylinder head at the end of the stroke), the volume swept
t
through by the piston per minute = Wv=nV = WR f whence
P
T , WRt , . TTrrr 00 nAr ^ SSOOON , SSQQQNRt
7=  ; and since TFE/=33,00(W, W = , and V =  ^ 
np U ' nup
189. Compressive Cycle. For quiet running, as well as for other
reasons, to be discussed later, it is desirable to arrange the valve
movements so that some air is gradually compressed into the clear
ance space during the latter part of the return stroke, as along JSa,
Fig. 57. This is accomplished by causing the exhaust valve to close
at jE, the inlet valve opening at a. The work expended in this com
pression is partially recovered during the subsequent forward stroke,
the air in the clearance space acting as an elastic cushion.
190. Actual Design. A singleacting 10hp. air engine at 100 r. p. m.,
working between 114.7 and 14.7 lb. absolute pressure, with an " appar
ent " (Art. 450) volume ratio during expansion of 5 : 1 and clearance equal
to 5 per cent of the piston displacement, begins to compress when the
return stroke of the piston is ^ completed. .The expansion and compres
sion curves are PV 13 c. Assuming that the actual engine will give 90
per cent of the work theoretically computed, find the size of cylinder
(diameter = stroke) and the free air consumption per Ihp.hr.
In Fig. 59, draw the lines ab and cd representing the pressure limits. "We are
to construct the ideal PV diagram, making its enclosed length represent, to any
convenient scale, the displacement of the piston per stroke. The extreme length
of the diagram from the oP axis will be 5 per cent greater, on account of clear
ance. The limiting volume lines ef and gh are thus sketched in ; EC is plotted,
making ^ = 5 ; the point E is taken so that =^? = 0.9, and EF drawn. Then
ABCDEF is the ideal diagram. We have, putting Di = D t
P A = P = H4.7.
V c = V D = 1.05 D.
=0.15 D.
DESIGN OF AIR ENGINE
111
= 61.31.
Work per stroke =jABi + iBCm  EDmk jFEk
D ,r T. x . P*V*PoVG r> (V V\ PrVrP*V*
= PA( I B t A) \ ~[ ^s( \ D VE) w _ ]_
= 144[(114.7 y, 0.20 D) + ^ L ' x "*** ' ~^
 (14 7 x 0.9 D)  f 81 " X  5 ^ C 1 " X  1S J) J
= 5803.2 D footpounds.
The actual engine will then give 0.9 x 5803.2 D = 5222.88 D footpounds per stroke
or 5222.88 D x 100 footpounds per minute, which is to be made equal to 10 hp., or
b 114.7
17.75
FIG. 59. Art. 190. Design of Air Engine.
to 10 x 33,000 footpounds. Then 522,288 Z> = 330,000 and D = 0.63 cu. ft. Since
the diameter of the engine equals its stroke, we write 0.7854 rf 2 x d 0,63 x 1728,
where d is the diameter in inches; whence d = 11.15 in.
To estimate the air consumption : at the point .B, the whole volume of air is
0.25 D. Part of this is clearance air, used repeatedly, and not chargeable to the
engine. The clearance air at E had the vulueie V s and the pressure P E . If its
112 APPLIED THERMODYNAMICS
behavior conforms to the law PF LS = c, then at the pressure of 1147 Ib. (point G)
we would have _i
The volume of fresh air brought into the cylinder per stroke is then
0.25 D  0.0309 D = 0.2191 D
or, per hour, 0.2191 x 0.63 x 100 x 60 = 828 cu. ft. Reduced to free air (Art. 187),
this would be 828 x ^jy = 6450 cu. ft., or C45 cu, ft. per Ihp.hr. (Compare
Art. 192.) l
191. Effect of Early Compression. If compression were to begin at a suffi
ciently early point, so that the pressure were raised to that in the supply pipe
before the admission valve opened, the fresh air would find the clearance space
already completely filled, and a less quantity of such fresh air, by 0.05 D, instead
of 0.0309 D, would be required.
192 Actual Performances of Air Engines. Kennedy (5) found a con
sumption of 890 cu. ft. of free air per Ihp.hr., in a small horizontal steam
engine. Under the conditions of Art. 183, the theoretical maximum work
which this quantity of air could perform is 1.27 hp. The cylinder effi
ciency (Art. 215) of the engine was therefore 1.0r 1.27 = 0.79. With
small rotary engines, without expansion, tests of the Paris compressed air
system showed free air consumption rates of from 1946 to 2330 cu. ft.
By working these motors expansively, the rates were brought within
the range from 848 to 1286 cu. ft. A good reciprocating engine with, pre
heated air realized a rate of 477 cu. ft., corresponding to 36,4 lb. ? per
brake horse power per hour. The cylinder efficiencies in these examples
varied from 0.368 to 0.876, and the mechanical efficiencies (Art. 214) from
0.85 to 0.92.
THE AIR COMPRESSOR
193. Action of Piston Compressor. Figure 60 represents the
parts concerned in. the cycle of an air compressor. Air is drawn
from the atmosphere through the spring check
valve a, Ming the space Q in the cylinder. This
inflow of air continues until the piston has
reached its extreme righthand position. On the
return stroke, the valve a being closed, compres
sion proceeds until the pressure is slightly greater
than that in the receiver D. The balanced outlet
FIG co. Art. 103 valve 5 then opens, and air passes from Q to D
Piston Compressor. J __ , , \\r\ ,,
at practically constant pressure. vV hen the pis
THE AIR COMPRESSOR
113
ton reaches the end of its stroke, there will still remain the clear
ance volume of air in the cylinder. This expands during the early
part of the next stroke to the right, but as soon as the pressure of
this air falls slightly below that of the atmosphere, the valve a again
opens.
194. Cycle. An actual diagram is given,
as ADCB) Fig. 61. Slight fluctuations in
pressure occur, on account of fluttering through
the valves, during discharge along AD and
during suction along CB; the mean discharge FIG. ci. Art. 194. Cycle
pressure must of course be slightly greater ir om P ressor 
than the receiver pressure, and the mean suction pressure slightly
less than atmospheric pressure. Eliminating these irregularities and
the effect of clearance, the ideal diagram is adcb.
195. Form of Compression Curve. The remarks in Art. 180 as to
the conditions of isothermal or adiabatic expansion apply equally to the
compression curve BA. Close approximation to the isothermal path is the
ideal of compressor per
formance. Let A, Fig. 62,
be the point at which
compression begins, arid
let DE represent the
maximum pressure to be
attained. Let the cycle
be completed through the
states F, #. Then the
work expended, if com
pression is isothermal, is
v ACFG; if adiabatic, the
FIG. 62. Arts. 195, 197, 2^18. Forms of Compression work expe nded is 45^G.
The same amount of air
has been compressed, and to the same pressure, in either case; the area
AEG represents, therefore, needlessly expended work. Furthermore, dur
ing transmission to the point at which the air is to be applied, in the
great majority of cases, the air will have been cooled down practically
to the temperature of the atmosphere ; so that even if compressed adia
batically, with rise of temperature, to B, it will nevertheless be at the
state C when ready for expansion in the consumer's engine. If it there
il
APPLIED THERMODYNAMICS
again expand adiabatically (along GH} instead of isothermally (along
CA) 9 a definite amount of available power will have been lost, repre
sented by the area CI1A. t During compression, we aim to have the work
area small ; during expansion the object is that it be large j the adiabatic
path prevents the attainment of either of these ideals.
The loss of power by adiabatic compression is so great that various
methods are employed to produce an approximately isothermal path. As
a general rule, the path is consequently intermediate between the iso
thermal and the adiabatic, a polytropic, pv n = 0. The relations derived
in Arts. 183 and 185 for adiabatic expansion apply equally to this path,
excepting that for y we must write n, the value of n being somewhere
between 1.0 and 1.402, The effect of water in the cylinder, whether in
troduced as vapor with the air, or purposely injected, is to somewhat
reduce the value of n, to increase the interchange of heat with the walls,
and to cause the line FG, rig. 62, to be straight and vertical, rather than
an adiabatic expansion, thus slightly increasing the capacity of the com
pressor, as shown in Art. 222.
196. Temperature Rise. The rise of temperature due to compression may be
computed as in Art. 185. Un'der ordinary conditions, the air leaves the com
pressor at a tempeiature higher than that of boiling water. Without cooling
devices, it may leave at such a temperature as to make the pipes red hot. It is
easy to compute the (not very extreme) conditions under which the rise in tern
perature would be* sufficient to melt the castiron compressor cylinder.
197. Computation of Loss. The uselessly expended work during adiabatic
(and similarly, during any other than isothermal) compression may be directly
computed from the difference of the work areas CAKI and CBAKI, Fig, 62.
The work under the isothermal is (jo, u, referring to the point C, and P, V, to
the point 4), pv log e (V v) = pv log c (p P) ; while if Q is the volume at B,
the work under ABC is
= PV* and Q = F() V ;
so that the percentage of loss corresponding to any ratio of initial and final pres
sures and any terminal (or initial) volume may be at once computed.
198. Basis of Methods for Improvement. Any value of n exceeding 1.0 for
the path of compression is due to the generation of heat as the pressure rises,
faster than the walls of the cylinder can transmit it to the atmosphere. The high
temperatures thus produced introduce serious difficulties in lubrication. Economi
cal compression is a matter of air cooling; while, on the consumer's part, economy
depends upon air heating.
COMPEESSION CURVE
115
199. Air Cooling. In certain applications, where a strong draft is available,
the movement of the atmosphere may be utilized to cool the compressor cylinder
walls and thus to chill the working air during compression. While this method
of cooling is quite inadequate, it has the advantage of simplicity and is largely
employed on the air " pumps " which operate the brakes of railway trains.
200. Injection of Water. This was the method of cooling originally em
ployed at Mont Cenis by Colladon. Figure 63 shows the actual indicator card
(Art. 484) from one of the older Colladon
compressors. EP> CD is the coi responding
ideal card with isothermal compression.
The cooling by stream injection was evi
dently not very effective. Figure 61 rep
resents another diagram from a compressor
in which this method of cooling was em
ployed ; oh representing the isothermal an*.
ac the adiabatic. The exponent of the
actual curve ad was 1.36; the gain over
adiabatic compression was very slight. B/
introducing ths
FIG. 03.
FIG.
Art. 200. Cooling by Jet
Injection.
Art. 20Q. Card from Colladon
Compressor,
water in a very
fine spray, a somewhat lower value of the exponent
was obtained in the compressors used by Colladon on
the St. Gothard tunnel. Ganse and Post (6) have re
duced the value of n to 1.2G by an atomized spray.
Figme 65 shows one of their diagrams, ab oeing the
isothermal and ac the adiabatic. In all cases,
spray injection is better than solid stream in
jection. The value n = 1.3(5, above given,
was obtained when a solid jet of halfinch
diameter was used. It is estimated that errors
of the indicator may introduce an uncer
tainty amounting to 0.02 in the value of n. Piston leakage would cause an
apparently low value. The comparative
efficiency of spray injection is sho\vn from
the fairly uniform temperature of dis
charged air, which can be maintained even
with a varying speed of the compressor.
In the Gause and Post experiments, with
inlet air at 81 F., the temperature of dis
charge was 95 F. Spray injection has the
objection that it fills the air with vapor, and
it has been found that the orifices must be
so small that they soon clog and become
inoperative. The use of either a spray or
FIG. 65. Art. 200. Cooling by Atomized
Spray.
a solid jet causes cutting of the cylinder and piston by the gritty substances carried
in the water. In American practice the injection of water has been abandoned.
116
APPLIED THERMODYNAMICS
201. Water Jackets. These reduce the value of n to a very slight ex
tent only, but are generally employed "because of their favorable influence
on cylinder lubrication. Gause and
Post found that with inlet air at
81 F , and jackets on the barrels of
the cylinders only (not on the heads),
the temperature of the discharged air
was 320 F. Cooling occurred dur
ing expulsion rather than during com
pression. The cooling effect depends
largely upon the heat transmissive
power of the cylinder walls, and the
value of n consequently increases at
cards
With
FIG.
Art. 201.CooliB by Jackets.
are given in Fig. 66 ; ab being the isothermal and ac the adiabatic.
more thorough cooling, jacketed
heads, etc., a lower value of n
may be obtained ; but this value
is seldom or never below 1.3.
Figure 67 shows a card given
by Unwinfrom a Cockerill com
pressor, D O indicating the ideal
isothermal curve. At the
higher pressures, air is appar
ently more readily cooled; its
own heatconducting power is
increased.
D 1
FIG. 67. Art. 201. Cockerill Compressor with
Jacket Cooling.
202. Heat Abstracted. In
Fig. 68, let AB and AC be the
adiabatic and the actual paths,
An and CN adiabatics ; the heat to be abstracted is then equivalent to
NO An = IAOL + nAIE  NCLK
>vPF , PV
. nAIEi = .
2/1
FIG. 68. Arts. 202, 203. Heat Ab This is the heat to be abstracted per
stracted by Cooling Agent. volume Fat pressure P > compressed to
MULTISTAGE COMPRESSION
117
p, expressed in footpounds. For isothermal compression; as along
AD, IACL=pv log e (Fsfl), and the total heat to be abstracted is measured
by this area. If the path is adiabatic, AB, n = y, and the expression for
heat abstraction becomes zero.*
203. Elimination of v. It is convenient to express the total area NCAn in
tei ins of p, Pj and V only. The area
FATT pv pv  i PV
W(i= ,I,^rT(^
Also,
y1 01
whence MM = 1 [() V  1 ] + ^  ^(^.
204. Water Required. Let the heat to be abstracted, as above com
puted, be H 9 in heat units. Then if S and s are the final and initial
temperatures of cooling water, and Q the weight of water circulated, we
have C=Hr(S s), the specific heat of water being taken as 1.0. In
practice, the range of temperature of the cooling water may be from 40
to 70 F.
205. Multistage Compression. The effective method of securing a
low value of n is by multistage operation^ the principle of which is
illustrated in Fig. 69. Let A be the
state at the beginning of compres
sion, and let it be assumed that the
path is practically adiabatic, in spite
of jacket cooling, as AB. Let AC
be an isothermal. In multistage
compression, the air follows the path
AB up to a moderate pressure, as at
), and is then discharged and cooled
Art. 205. Multistage Com
pression.
at constant pressure in an external *' G
vessel, until its temperature is as
nearly as possible that at which it was admitted to the cylinder.
The path representing this cooling is DE. The air now passes to
* More simply, as suggested by Chevalier, the specific heat along AC is s = 1 1^1^.
(Art. 112) ; the heat to be abstracted is then, per Ib. of air n l
^B^!h working airatid cushion air must be cooled.
118
APPLIED THERMODYNAMICS
a second cylinder, is adiabatically compressed along HF, ejected and
cooled along ]?G/, and finally compressed in still another cylinder
along GH. The diagram illus
trates compression in three
" stages " ; but two or four stages
are sometimes used. The work
saved over that of single stage
adiabatic compression is shown
by the irregular shaded area
HGrFUDB, equivalent to a re
duction in the value of n, under
good conditions, from 1.402 to
FIG. 70. Azts 205, 206. Twostage Com
pressor Indicator Diagram.
about 1 .25. Figure 70 shows the diagram from a twostage 2000 hp.
compressor, in which solid water jets were used in the cylinders.
The cooling water was at a lower
temperature than the inlet air,
causing the point h to fall inside
the isothermal curve AB. The
compression curves in each cyl
inder give w = 1.36. Figure 71
is the diagram for a twostage
Biedler compressor with spray in
jection, AB being an isothermal
and A an adiabatic.
JFio 71. Arts. 205, 214. Twostage Kledler
Compressor Diagram.
206. Interceding. Some work is always wasted on account of the friction of
the air passing through the intercooling device. In early compressors, this loss
often more than outweighed the gain due to compounding. The area ghij, Fig.
70, indicates the work wasted from this cause. In this particular instance, the
loss is exceptionally small. Besides this, the additional air friction through two
or more sets of valves and ports, and the extra mechanical friction due to a multi
plication of cylinders and reciprocating parts must be considered. Multistage
compression does not pay unless the intercooling is thoroughly effective. It seldom
pays when the pressure attained is low. Incidental advantages in multistage
operation arise from reduced mechanical stresses (Art. 462), higher volumetric
efficiency (Art. 226), better lubrication, and the removal of moisture by precipita
tion during the intercooling.
207. Types of Intercoolers. The " external vessel " of Art. 205 is called the
iatercooler. It consists usually of a riveted or castiron cylindrical shell, with cast
INTEKCOOLESTG
119
iron heads. Inside are straight tubes of brass or wrought iron, running between
steel tube sheets. The back tube sheet is often attached to a stiff castiron inter
FIG. 72. Art. 207. AllisChalmers Horizontal Intercooler
nal head, so that the tubes, sheet, and head
are free to move as the tubes expand
(Fig. 72). The air entering the shell sur
rounds the tubes and is compelled by baffles
to cross the tube space on its way to the out
let. Any moisture precipitated is drained
off at the pipe a. The water is guided to
the tubes by internally projecting ribs on
the heads, which cause it to circulate from
end to end of the intercooler, several times.
If of ample volume, as it should be, the
intercooler serves as a receiver or storage
tank. The one illustrated is mounted in
a horizontal position. A vertical type is
shown in Pig. 73. The funnel provides a
method of ascertaining at all times whether
water is flowing.
i
208. Aftercoolers. In most
manufacturing plants, the pres
ence of moisture in the air is ob
jectionable, on account of the
difficulty of lubrication of air
tools, and because of the rapid de
struction of the rubber hose used
for connecting these tools with
the pipe line. To remove the
moisture (and some of the oil) p^. Ta . Art. 207. IngersollSeigeant Vertical
present after the last stage of com Intercooler,
120 APPLIED THERMODYNAMICS
pression, and by cooling the air to decrease the necessary size of transmitting pipe,
aftercoolers are employed. They are similar in design and appearance to mter
coolers. The cooling of the air deci eases its capacity for holding water vapor,
and the latter is accordingly precipitated where it may be removed before the air
has reached its point of utilization. An incidental advantage arising from the
use of an aftercooler is the decreased expansive stress on the pipe line following
the introduction of air at a more nearly noimal temperature.
209. Power Consumed. From Art. 98, the work under any curve
pv n =PV n is, adopting the notation of Art. 202,
pv J
}
The work along an adiabatic is expressed by the last formula if we make
n = y = 1.402. The work of expelling the air from the cylinder after com
pression is pv. The work of drawing the air into the cylinder, neglecting
AP\~"~
clearance, is PV=pv( }  The net work expended in the cycle is the
algebraic sum of these three quantities, which we may write,
It is usually more convenient to eliminate v } the volume after compres
sion. This gives the work expression,
If pressures are in pounds per square inch, the footpounds of work per
minute will be obtained by multiplying this expression by the number of
working strokes per minute and by 144; and the theoretical horse power
necessary for compression may be found by dividing this product by
33,000. If we make F=l, P=14.7, we obtain the power necessary to
compress one cubic foot of free air. If the air is to be used to drive a
motor, it will then, in most cases have cooled to its initial temperature
(A.rt. 195), so that its volume after compression and cooling will be
PV^p. The work expended per cubic foot of this compressed and
cooled air is then obtained by multiplying the work per cubic foot of free
air by ^ 
210. Work of Compression. In some textbooks, the work area under the
compression curve is specifically referred to as the work of compression. This ig
not the total work area of the cycle.
RECEIVER PRESSURE 121
211. Range of Stages in Multistage Compression. Let the lowest pres
sure be g, the highest p, and the pressure during interceding P. Also let
intercooling be complete, so that the air is reduced to its initial tempera
ture, so that the volume V after intercooling is ^, in which r is the
volume at the beginning of compression in the first cylinder. Adopting
the second of the work expressions just found, and writing z for n ~~ , we
have n
Work in first stage = 21 j /TV _ 1 j .
Work in second stage = T( (&}'  1 } = 2T ( (. Y_ 1 } .
* \\PJ J * \\PJ J
Total
Differentiating with respect to P, we obtain
w
dP
q\q
Por a minimum value of W, the result desired in proportioning the pres
sure ranges, this expression is put equal to zero, giving
P 2 =pq, or P = Vpq, or = f
An extension of the analysis serves to establish a division of pressures
for fourstage machines. From the pressure ranges given, it may easily
be shown that in the ideal cycle the condition of rmnimmn work is that
the amounts of work done in each of the cylinders be equal. The number
of stages increases as the range of pressures increases; in ordinary prac
tice, the twostage compressor is employed, with final pressures of about
100 Ib. per square inch above the pressure of the atmosphere. In low
pressure blowing engines,the loss due to a high exponent for the compres
sion curve is relatively less and these machines are frequently single stage.
For threestage machines, working between the pressures pi (low)
and p 2 (high), with receiver pressures of PI (low) and P 2 (high), the
conditions of minimum work are P2 ^PIP2 2 &&& Pi~^p2pi 2 ,
the amounts of work done in the three cylinders will be equal, and the
cylinder volumes will be inversely as the suction pressures.
122 APPLIED THERMODYNAMICS
ENGINE AND COMPRESSOR RELATIONS
212. Losses in Compressed Air Systems. Starting with mechanical power
delivered to the compressor, we have the following losses
(a) friction of the compressor mechanism, affecting the mechanical
efficiency ;
(b) thermodynamic loss, chiefly from failure to realize isothermal com
pression, but also from friction and leakage of air, clearance, etc.,
indicated by the cylinder efficiency;
(c) transmissive losses in pipe lines ;
(c?) thermodynamic losses at the consumer's engine, like those of (&) ;
(e) friction losses at the consumer's engine, like those of (a).
213. Compressive Efficiency. While not an efficiency in the true sense of the
term, the i elation of work geueiated during expansion iii the engine to that ex
pended during compression in the compressor is sometimes called the compressive
efficiency. It is the quotient of the areas FCTIG and FBA (9, Fig. 62. From the
expression in Art. 209 for work under a polytropic plus work of discharge along
BF or of admission along PC, we note that, the values o P andp being identical
for the two paths, AB and CH< in question, the total work under either of these
paths is a direct function of the volume V at the lower pressure P. In this case,
providing the value of n be the same for both paths, the two work areas have the
ratio V x, where Fis the volume at J, and x that at H. It follows that all the
ratios of volumes LN  LIT, OQ  OP, etc , are equal, and equal to the ratio of
areas. The compressive efficiency, then, = = T  t, where t is the temperature
at A (or that at C% and I* that at II. For isothermal paths, T= t, and the com
pressive efficiency fs unity. In various testa, the compressive efficiency has ranged
from 0.488 to 898. It depends, of course, on the value of n, increasing as n decreases.
214. Mechanical Efficiency. For the compressor, this is the quotient of work
expended in the cylinder by work consumed at the flywheel; for the engine, it
is the quotient of work delivered at the fly wheel by work done in the cylinder.
Friction losses in the mechanism measure the mechanical inefficiency of the
compressor or engine. With no friction, all of the power delivered would be ex
pended in compression, and all of the elastic force of the air would be available
for doing work, and the mechanical efficiency would be 1.0. In practice, since
compressors are usually directly driven from steam engines, with piston rods in
common, it is impossible to distinguish between the mechanical efficiency of the
compressor and that of the steam engine. The combined efficiency, in one of the
best recorded tests, is given as 0.92. For the compressor whose card is shown in
Fig. 71, the combined efficiency was 0.87. Kennedy reports an average figure of
0,845 (7). Uuwin states that the usual value is fiom 0.85 to 0.87 (8). These
efficiencies are of course determined by comparing the areas of the steam and air
indicator cards.
215. Cylinder Efficiency. The true efficiency, thermodynamically speaking,
is indicated by the ratio of areas of the actual and ideal PV diagrams. For the
PLANT EFFICIENCY 123
compressor, the cylinder efficiency is the ratw of the work done in the ideal cycle,
without clearance, drawing in air at atmospheric pre sure, compressing it isothermally
and discharging it at the constant receiver pressure, to the work done in the actual cycle
of the same maximum volume It measures item (6) (Art. 212). It is not the "com
pressive efficiency " of Art. 213 For the engine, it is the ratw of the work done in
the actual cycle to the work of an ideal cycle without clearance, with isothermal expan
sion tp the same maximum volume from the sameinitial volume, and with constant pressures
during reception and discharge , the former leing that of the pipe line and the latter that
of the atmosphere. Its value may range from TO to 0.00 in good machines, in gen
eral increasing as the value of n decreases. An additional influence is fluid fric
tion, causing, in the compressor, a fall of pressure through the suction stroke and
a rise of pressure during the expulsion stroke ; a id in the engine, a fall of pressure
during' admission and excessive Lack pressme during exhaust. All of these condi
tions alter the area of tlie PV cycle. In welldesigned machines, these losses
should be small. A large capacity loss in the cylinder is still to be considered.
216. Discussion of Efficiencies. Considering the various items of loss sug
gested in Art. 212, we find as average values under good conditions,
(V) mechanical efficiency, 0.85 to 0.90; say 0.85;
(5) cylinder efficiency of compressor, 0.70 to 0.90; say 0.80;
(<?) transmission losses, as yet undetermined ;
(d) cylinder efficiency of air engine, 0.70 to 90.0; say 0.70;
(e) mechanical efficiency of engine, 0.80 to 0.90; say 0.80.
The combined efficiency from steam cylinder to work performed at the con
sumer's engine, assuming no loss by transmission, would then be, as an average,
0.85 x 0.80 x 0.70 x 0.80 = 0.3808.
For the Paris transmission system, Kennedy found the overall efficiency (includ
ing pipe line losses, 4 per cent) to be 26 with cold air or 0.384 with preheated
air, allowing for the fuel consumption in the preheaters (9).
217. Maximum Efficiency. In the processes described, the ideal efficiency in
each case is unity. We are here deahng not with thermodynamic transformations
between heat and mechanical energy, but only with transformations from one form
of mechanical energy to another, in part influenced by heat agencies. No strictly
thermodynamic transformation can have an efficiency of unity, ou account of the
limitation of the second law.
218. Entropy Diagram. Figure 62 may serve to represent the com
bined ideal PV diagrams of the compressor (GABF) and engine (FGHGT).
The quotient  is the compres&ive efficiency. The area representing
net expenditure of work, that is, waste, is CBAH, bounded ideally by two
124
APPLIED THERMODYNAMICS
adiabatics or in practice by two polytropics (not ordinarily isodiabatics)
and two paths of constant pressure. This area is now to be illustrated
on the TN coordinates.
For convenience, we reproduce the essential features of Fig. 62
in Fig. 74. In Fig. 75, lay off the isothermal T, and choose the
point A at random. Now
if either T B or T H be
given, we may complete
the diagram. Assume
that the former is given ;
then plot the correspond
ing isothermal in Fig. 75.
Draw AB, an adiabatic,
BO and AS as lines
of constant pressure
FIG.
Art. 218. Engine and Compressor Diagrams (n = k log e  J, the point
O falling on the isothermal F. Then draw OB, an adiabatic, de
T T
termining the point S\ or, from Art. 213, noting that ^ = 4, we
may find the point H di
rectly. If the paths AB
and OH are not adia
batics, we may compute
the value of the specific
heat from that of n and
plot these paths on Fig.
75 as logarithmic curves ;
but if the values of n are
different for the two
paths, it no longer holds
+1 f %B _ Zj. Tl FlG 75f Arts ' 218 ' 219 221. Compressed Air System,
tnat J.ne area Entropy Diagram.
OBAH in Fig. 75 now represents the net work expenditure in
heat units.
219. Comments. As the exponents of the paths AB and OH decrease,
these paths swerve into new positions, as AE, CD, decreasing the area
representing work expenditure. Finally, with n = 1 9 isothermal paths,
the area of the diagram becomes zero ; a straight line, OA. Theoretically,
ENTROPY DIAGRAMS
125
with water colder than the air, it might be possible to reduce the tempera
ture of the air during compression, giving such a cycle as AICDA, or even,
with isothermal expansion m the engine, AICA; in either case, the net
work expenditure might be nega
tive; the cooling water accomplish p
ing the result desired.  ( j c E B
220. Actual Conditions. Under
the more usual condition that the
temperature of the air at admission
to the engine is somewhat higher
than that at which it is received by
the compressor, we obtain Figs.
76, 77. T, T c and either T B or T H
must now be given. The cycle in
which the temperature is reduced
during compression now appears FIG. 76. Art 220. Usual Combination of
as AICDA or AIJA.
220. Usual Combination
Diagrams.
FIG. 77. Ait. 220. Combined Entropy Diagrams.
221. Multistage Operation. Let the ideal pv path be DECBA, Fig. 78.
The "triangle" ABC of Fig. 75 is then replaced by the area DECBA,
Fig. 79, bounded by lines of constant pressure and adiabatics. The area
p
F
FIG. 78. Art. 221. Threestage Com
pression and Expansion.
FIG. 79. Art. 221. Entropy Diagram,
Multistage Compression.
126
APPLIED THERMODYNAMICS
saved is BFEC, which approaches zero as the pressure along CE } Pig. 78,
approaches that along AB or at I), and becomes a maximum at an inter
mediate position, already determined in
Art. 211. With inadequate inter cooling,
the area representing work saved would be
yFEx. Figures 80 and 81 represent the
ideal pv and nt diagrams respectively for
compressor and engine, each threestage,
with perfect intercooling and aftercooling
and preheating and with no drop of pres
sure in transmission. BbA and AliB
would be the diagrams with singlestage
acliabatic compression and expansion.
6
Fm NO Art 22] Threestage
Compression and Expansion.
FIG 81. Art 221. Thieestage Compression and Expansion.
COMPRESSOK CAPACITY
222. Effect of Clearance on Capacity. Lei A BCD, Fig. 57, be the ideal pv dia
gram of a compressor without cleaiance. If there is clearance, the diagram will
be aBCE; the air left in the cylinder at a will expand, nearly adiabatically, along
, so that its volume at the intake pressure will be somewhat like DE. The
total volume of fresh air taken into the cylinder cannot be DC as if there were no
clearance, but is only EC. The ratio EC (VcV a ) is called the volumetric
efficiency. It is the ratio of free air drawn in to piston displacement.
223. Volumetric Efficiency. This term is sometimes incorrectly applied to the
factor 1 c, in which c is the clearance, expressed as a fraction of the cylinder
volume. This is illogical, because this fraction measures the ratio of clearance air
at final pressure, to inlet air at atmospheric pressure (Aa DC, Fig. 57) ; while
the reduction of compressor capacity is determined by the volume of clearance air
at atmospheric pressure. Jf the clearance is 3 per cent, the volumetric efficiency is
much lew than 97 per cent.
224. Friction and Compressor Capacity, If the intake ports or pipes are small,
an excessive suction will he necessary to draw in the charge, and the cylinder will
VOLUMETRIC EFFICIENCY
127
!G
be filled with air at less than atmospheric pressure. Its equivalent volume at
atmospheric pressure "will then be less than that of the cylinder. This is shown
in Fig. 82. The line of atmospheric pressure is DP, the capacity is
reduced by FG, and the volumetric efficiency is DP HG. The capacity
may be seriously affected from this cause, in the case of a badly designed
machine.
225. Volumetric Efficiency ; Other Factors. Where jackets or water jets
are used, the air is often
somewhat heated during
the intake stroke, increas
ing its volume, and thus,
as in Art. 224, lowering
the volumetric efficiency.
The effect is more notice
able with jacket cooling,
FIG 82 Art. 224. Effect of Suction Friction. with which the cylinder
walls often remain con
stantly at a temperature above that of boiling water. Tests have shown a loss
of capacity of 5 per cent, due to changing from spray injection to jacketing. A
high altitude for the compressor results in its being supplied with rarefied air, and
this decreases the volumetric efficiency as based 011 air under standard pressure.
At^an elevation of 10,000 ft. the capacity falls off 30 per cent. (See table, Art. 52a.)
This is sometimes a matter of importance in mining applications also. Volumetric
efficiency, in good designs, is principally a matter of low clearance. The clearance
of a cylinder is practically constant, regardless of its length; so that its percentage
is less in the case of the longer stroke compressors. Such compressors are com
paratively expensive. When water is injected into the cylinder, as is often the case
in European practice, the clearance space may be practically filled with water at
the end of the discharge stroke. Water does not appreciably expand as the pressure
is lowered; so that in these cases the volumetric efficiency may be determined
by the expression 1 c of Art. 223, being much greater than in cases where water
injection is not practiced.
226. Volumetric Efficiency in Multistage Compression. Since the
effect of multistage compression is to reduce the pressure range, the
expansion of the air caught in the clearance space is less, and the dis
tance DE, Fig. 57, is reduced. This makes the volumetric efnciencjr,
EC+ (V c V a ), greater than in singlestage cylinders. If FGH repre
sent the line of intermediate pressure, the ratio JE * (V c 7 a ) is the
gain in volumetric efficiency.
227. Refrigeration of Entering Air. Many of the advantages following multi
stage operation and intereooling have been otherwise successfully realized by the
plan of cooling the air drawn into the compressor. This of course increases the
density of the air at atmospheric pressure, and greatly increases the volumetric
efficiency. Incidentally, much of the moisture is precipitated. At the Isabella
furnace of the Carnegie Steel Company, at Etna, Pennsylvania, a plant of this
128
APPLIED THERMODYNAMICS
kind has been installed. An ordinary ammonia refrigerating machine cools the
air from 80 to 28 F. This should decrease the specific volume in the ratio
(450 + 28) (459. G f 80) = 0.90. The free air capacity should consequently
be increased in about this ratio (10).
228. Typical Values. Excluding the effect of clearance, a loss in ca
pacity of from 6 to 22 per cent has been found by Uiiwin (11) to be due
to air friction losses and to heating of the entering air. Heilemann (12)
finds volumetric efficiencies from 0.73 to 0,919. The volumetric efficiency
could be precisely determined only by measuring the air drawn in and
discharged.
229. Volumetric and Thermodynamic Efficiencies. The" volumetric effi
ciency is a measure of the capacity only. It is not an efficiency in the sense
of a ratio of " effect " to " cause." In Pig. 83 the solid lines show an actual
compressor diagram, the dotted lines, EGHB, the corresponding perfect
diagram, with clearance and isothermal compression. In the actual case
we have the wasted work areas,
HLJQ, due to friction in discharge ports ;
GQKD 9 due to nonisothermal compression;
DFMC, due to friction during the suction of the air.
At BHC, there is an area representing, apparently, a saving in work
expenditure, due to the expansion of the clearance air; this saving in
work has been accomplished, however,
with a decreased capacity in the pro
portion BCtBE, a proportion which
is greater than that of BHC to the total
work area. Further, expansion of the
clearance air is made possible as a result
of its previous compression along 1PDK\
and the energy given up by expansion
can never quite equal that expended in
compression. The effect of excessive
FIG. 8'X Art. 229. Volumetric and , . ,. , . ... , . ,,
Thenuodynamic Efficiencies. friction during suction, reducing the
capacity in the ratio DE r J3E, is
usually more marked on the capacity than on the work. Both suction
friction and clearance decrease the cylinder efficiency as well as the
volumetric efficiency, but the former cannot be expressed in terms of
the latter. In fact, a low volumetric efficiency may decrease the work
expenditure absolutely, though not relatively. An instance of this is found
in the case of a compressor working at high altitude. Friction during dis
charge decreases the cylinder efficiency (note the wasted work area
HLJQ), but is practically without effect on the capacity.
COMPRESSOR DESIGN 129
COMPRESSOR DESIGN
230. Capacity. The necessary size of cylinder is calculated much as in
Art. 190. Let p, v, t, be the pressure, volume, and temperature of dis
charged air (v meaning the volume of air handled per minute), and P, F, T,
those of the inlet air. Then, since jPFs T ' =f>v s t, the volume drawn
into the compressor per minute is V=pvTt Pt } provided that the air is
dry at both intake and delivery. If n is the number of revolutions per
minute, and the compressor is doubleacting, then, neglecting clearance,
the piston displacement per stroke is V* 2 u =
This computation of capacity takes no account of volumetric losses.
In some cases, a rough approximation is made, as described, and by
slightly varying the speed of the compressor its capacity is made equal to
that required. Allowance for clearance may readily be made. Let the
suction pressure be P 9 the final pressure p, the clearance volume at the
final pressure of the piston displacement. Then, if expansion in the
m
clearance space follows the law pv n = PV n , the volume of clearance air
at atmospheric pressure is
of the piston displacement For the displacement above given, we there
fore write,
zTi+i /ivm
2n '[_ m \mJ\Pj J
This may be increased 5 to 10 per cent, to allow for air friction, air
heating, etc.
231. Design of Compressor. The following data must be assumed :
(a) capacity, or piston displacement,
(Z>) maximum pressure,
(c) initial pressure and temperature,
(d) temperature of cooling water,
(e) gas to be compressed, if other than air.
Let the compressor deliver 300 cu. ft. of compressed air, measured
at 70 F., per minute, against 100 Ib. gauge pressure, drawing its supply at
14.7 Ib. and 70 F., the clearance being 2 per cent. Then, ideally, the free
air per minute will be 300 x (114.7 r 14.7) = 2341 cu. ft., or allowing 5
per cent for losses due to air friction and heating during the suction,
2341 r 0.95 = 2464 cu. ft. To allow for clearance, we may use the ex
pression in Art. 230, making the displacement, with adiabatic expansion
of the clearance air,
130
APPLIED THERMODYNAMICS
2464+ [10.02
+ 0.02] = 2640 cu. ft.
Assuming for a compressor of this capacity a speed of 80 r. p. m., the
necessary piston displacement for a doubleacting compressor is then
2640 r (2 x 80) = 16.5 cu. ft. per stroke, or for a stroke of 3 ft., the piston
area would be 792 sq. in. (13). The power expended for any assumed
compressive path may be calculated as in Art. 190, and if the mechanical
efficiency be assumed, the power necessary to drive the compressor at
once follows. The assumption of clearance as 2 per cent must be justified
in the details of the design. The elevation in temperature of the air may
be calculated as in Art. 185, and the necessary amount of cooling water
as in Art. 203, the exponents of the curves being assumed.
232. Twostage Compressor. From Art. 211 we may establish an inter
mediate pressure stage. This leads to a new correction for clearance, and
to a smaller loss of capacity due to air heating. Using these new values,
we calculate the size of the firststage cyliuder. For the second stage, the
maximum volume may be calculated on the basis that intereoolitig is com
plete, whence the cylinder volumes are inversely proportional to the suc
tion pressures. The clearance correction will be found to be the same as
in the lowpressure cylinder. The capacity, temperature rise, water con
sumption, power consumption, etc., are computed as before. A considera
ble saving in power follows the change to two stages.
233. Problem. Find the cylinder dimensions and power consumption of a
doubleacting singlestage air compressor to deliver 4000 cu. ft. of free air per
minute at 100 Ib. gauge pres
sure at 80 r. p. m., the intake
air being at 13.7 Ib. absolute
pressure, the piston speed
640 ft. per minute, clearance
4 per cent, and the clearance
expansion and compression
curves following the law
FIG. 84 Art. 233. Design of Compressor.
Lay off the distance Gff t
Fig. 84, to repiesent the (un
known) displacement of the
piston, which we will call D.
Since the clearance is 4 per cent, lay off GZ = 0.04 D 9 determining as a
coordinate axis. Draw the lines TU, VW, YX 7 representing the absolute pres
sures indicated. The compression curve 1 CE may now be drawn through C, and
the clearance expansion curve DI through D. The ideal indicator diagram is
CEDL We have,
COMPRESSOR DESIGN
131
(P\07
P)
V a = ( ^+= \ 1.04 D = 0.2158 D.
4 0.04 D = 0.1829 D,
1.04 D = 0.9872 ZX
But j4B = FB FA = 0. 8043 D is the volume of free air drawn into tlie cylinder :
AB f7.E?"= 0.8043 is the volumetric efficiency:* to compress 4000 cu. ft. of free air per
minute the piston displacement must then be 4000 0.8043 = 407^ cu. ft. per minute.
Since the compiessor is doubleacting, the necessary cylinder area is the quotient
of displacement by piston speed or 4973 640, giving 7.77 sq. ft., or (neglecting
the loss of area due to the piston rod), the cylinder diameter is 37.60 in. From the
conditions of the problem, the strolce is 640 (2 x SO) = 4 ft.
For the power consumption, we have
W = GDEF + FECH  JICH  GDIJ
035
035
= 144*[(114.7x0.1758 )+
(13.7 x O.S473)
^ J
0.35
= 144 Z>[20.16 + 30.01  11.61  5.59] = 144 D x 32,97.
This is the work for a piston displacement = D cubic feet. If we take D at 4973
per mmute, the horse power
consumed in compression is
144 x 32.97 x 4073
3300U
' = 715.
48.7
234. Design of a Two
stage Machine. With condi
tions as in the preceding, con
sider a twostage compressor
with complete interceding and
a uniform friction of one pound
between the stages. Here the
combined diagrams appear as
in Fig. 85. For economy of '
power, the intermediate pres FIG. 85. Art. 23. Design of Twostage Compressor.
*This is not quite correct, because the air at J5 is not "free" air, i.e., air at
atmospheric temperature. There is a slight rise in temperature between C and B,
If T R is the atmospheric temperature, and b =  a = , the volumetric efficiency
is TR l= =~\ . If there is no cooling during discharge (along ED\ T A =T&, and
\ & A/
the volumetric efficiency becomes ^(ba).
132 APPLIED THERMODYNAMICS
sure is V114.7 x 13.7 = 39.64, whence the firststage discharge pressure and the
secondstage suction pressure, corrected for friction, are respectively 40.14 and
39.14 Ib. For thejirst stage, Fig. 85,
P P = P Q = 40.14, P A = P = 14.7, P q = P M = 13.7, V H = 104 D, V F = 0.04 D.
/ p \ V4 / 1 ^ 7 \ 0.74
or V G = / V s = (^j 1 04 D = 0.4701 D*
' 74 0.04 D = 0.08864 tf.
V P = (j'0 04 /> =0.08412 />,
74
/ 73 \ n.74 / 1 o fr V 74
P, JV = PjrIV" or 7, = (JJ*) F* = (i J 1.04 D =0.987 D,
The volumetric efficiency is jiJ3  D = (V  FJ D = 0.987  0.08412 = 0.90288.
The piston displacement per minute is 4000 0.003 = 44SO. The piston diameter
is V(4430  040) x 144  0.7854 = 35.6 in. for a stroke of 640  (2 x 80) = 4 ft.
The power consumptive for this first stage is,
W =
^ 1 w 1
= [40.14(0.4701  0.04) + f*M*x <U701)(18.7 x 1.M) ,
I O.oo
 13.7(1.04  0.0886)  C 40 ' 14 x  M) " 3 ( 5 13  7 X  0886 >]l44 D
= 2348.64 D f ootpounda or 10,404,475 footpounds per minute, equivalent to
315.3 horse power.
SECOND STAGE
Complete interceding means that at the beginning of compression in the sec
ond stage the temperature of the air will be as in the first stage, 70 F. The
p
volume at this point will then be V z = ifV n = ~ 1.04 D =. 0.364 D. We thus
Jr z oy.lJ.
locate the point Z^ Fig. 85, and complete the diagram ZCE1, making V B = 0,04
(FsFj?) =0.0141), Pc = <P# = 114.7, Pj=aP^=39.14, and compute as follows:
y, = '0.3645= 0.3574 D.
= 0.1642 D.
or Fj= ' K, = T' 0.014 2)= 0.0305 fl.
\/ jl viU.!*/
or F/ = ^' 74 VB = (r^ ^ 0.014 D= 0.0311 D.
*Note that ^ very nearly; so that = ?^^; an approximation
Pv. PH VH VQ v z YI
which makes only one logarithmic computation necessary.
COMPRESSOR DESIGN 133
The piston displacement is Vz VE = 035 D; the volumetric efficiency is the quo
tient of (Vx Vj) =0 3269 D by this displacement, or 0.934. For a stroke of 4 ft,
the cylinder diameter is \/[(0.35 D = 1550) ^640] X 144 i 0.7854=21 .05 in. The
power consumption for this stage is
qi47X0.1642)f39.14X0.364>
0.35
 (39.14X0 3329; ,^47X0 OWgQ 14XO.OB11)]
=816 5 horse power.
The total horse power for the twostage compressor is then 631 8 and (within
the limit of the error of computation) the work is equally divided between the stages.
235. Comparisons. We note, then, that in twostage compression, the saving
irt e ftQO
in power is ^ "^ = 12 of the power expended in singlestage compression;
that the lowpressure cylinder of the twostage machine is somewhat smaller than
the cylinder of the singlestage compressor; and that, in the twostage machine,
the cylinder areas are (approximately) inversely proportional to the suction pressures.
The amount of cooling water required will be found to be several times that neces
sary in the singlestage compressor.
236. Power Plant Applications. On account of the ease of solution of air in
water, the boiler feed and injection waters in a power plant always carry a con
siderable quantity of air with them. The vacuum pump employed in connection
with a condenser is intended to remove this air as well as the water. It is esti
mated that the waters ordinarily contain about 20 tunes their volume of air at
atmospheric pressure. The pump must be of size sufficient to handle this air
when expanded to the pressure in the condenser. Its cycle is precisely that of any
ah* compressor, the suction stroke being at condenser pressure and the discharge
stroke at atmospheric pressure. The water present acts to reduce the value of the
exponent n, thus permitting of fair economy.
237. Dry Vacuum Pumps. In some modern forms of high vacuum apparatus,
the air and water are removed from the condenser by separate pumps. The
amount of air to be handled cannot be computed from the pressure and tempera
ture directly, because of the water vapor with which it is saturated. From Dai
ton's law, and by noting the temperature and pressure in the condenser, the pressure
of the air, separately considered, may be computed. Then the volume of air, cal
culated as in Art. 236, must be reduced to the condenser temperature and pressure,
and the pump made suitable for handling this volume (14) .
COMMERCIAL TYPES OF COMPRESSING MACHINERY
238. Classification of Compressors. Air compressors are classified according
to the number of stages, the type of frame, the kind of valves, the method of
driving, etc. Steamdriven compressors are usually mounted as one unit with the
steam cylinders and a single common fly wheel. ^Regulation is usually effected by
varying the speed. The ordinary centrifugal governor on the steam cylinder im
poses a maximum speed limit; the shaft governor is controlled by the air pressure,
which automatically changes the point of cutoff on the steam cylinder. Power
driven compressors may be operated by electric motor, belt, water wheel, or in
APPLIED THERMODYNAMICS
TYPES OF COMPRESSOR
135
other ways. They are usually regulated by means of an " unloading valve," which
either keeps the suction valve closed during one or more strokes or allows the air
to discharge into the atmosphere whenever the pipe lines aie fully supplied. In
air lift practice, a constant speed is sometimes desire d, irrespective of the load.
In the "variable volume" type of machine, the delivery of the compiessor is
varied by closing the suction valve before the completion of the suction stroke.
The air in the cylinder then expands below atmospheric pressure.
239. Standard Forms. The ordinary small compressor is a singlestage
machine, with poppet air valves on the sides of the cylinder. The frame is of the
" fork " pattern, with bored guides, or of the " duplex " type, with two singlestage
cylinders. These machines maybe either steam or belt driven. The "straight
line" compressors differ from the duplex in having all of the cylindeis in one
straight line, regardless of their number.
For highgrade service, in large units, the standard form is the crosscompound
twostage machine, the lowpressure steam and air cylinders being located tandem
beside the high pressure cylinders, and the air cylinders being outboard, as in
Fig. 86. Ordinary standard machines of this class are built in capacities ranging
up to 6000 cu. ft. of free air per minute. The other machines are usually con
structed only in smaller sizes, ranging down to as small as 100 cu. ft. per minute.
Some progress has been made in the development of rotary compressors for
direct driving by
steam turbines. The
efficiency is fully as
high as that of an
ordinary reciprocat
ing compressor, and
the mechanical losses
are much less. A
paper by Rice (Jour.
A. S. M. E. xxxiii,
3) describes a 6stage
turbo  machine at
1650 r p. m., direct
connected to a 4
stage steam turbine.
With the low dis
charge p r e s
sure (15 Ib.
gauge), num
erous stages
and intercool
FIG. 87. Art. 240. Sommeiller Hydraulic Piston Compressor,
ers, compression is practically isothermal.
240. Hydraulic Piston Compressors: Sommeiller's. In Fig. 87, as the piston F
moves to the right, air is drawn through C to G, together with cooling water
from B. On the return stroke, the air is compressed and discharged through D
and A. Indicator diagrams are given in Fig. 88.
136
APPLIED THERMODYNAMICS
The value of n is exceptionally low, and clearance expansion almost elimi
nated. This \vas the first commercial piston compressor, and it is still used to a
PIG 88. Art, 240. Variable Discharge Pressure Indicator Diagrams, Sommeiller
Compressor.
limited extent in Europe, the large volume of water present giving effective! cool
ing. It cannot be operated at high speeds, on account of the inertia of the
water.
The Leavitt hydraulic piston compressor at the Calumet and Hecla copper
mines, Michigan, has doubleacting cylinders GO by 12 m., and runs at 25 i evolu
tions per minute, a compaiatively
high speed. The value of n from the
card shown in Fig. 89 is 1.23. ~~
241. Taylor Hydraulic Compressor.
"Water is conducted through a vei tical
shaft at the necessary head (2 3 ft. per
pound pressure) to a separating cham
FIG. 89.
Art. 248 Cards from Leavitt
Compressor.
FIG. 90.
Art. 241. Taylor Hydraulic
ComDresssor.
TYPES OF COMPRESSOR
137
her. The shaft is lined with a riveted or a castiron cylinder, and at its top is a
dome, located so that the water flows downward around the inner circumference
of the cylinder. The dome is so made that the water alternately contracts and
expands during its passage, producing a partial vacuum, by means of which air is
drawn in through numerous small pipes. The air is compressed at the tempera
ture of the water while descending the shaft. The separating chamber is so
large as to permit of separation of the air under an inverted bell, from which it is
led by a pipe. The efficiency, as compared with that theoretically possible in
isothermal compression, is 60 to 70, some air being always carried away in
solution. The initial cost is high, and the system can be installed only where
a head of water is available. Figure 90 illustrates the device (15). The head of
water must be at least equal to that corresponding to the pressure of air.
The "cycle" of this type of compressor may be regarded as made up of two
constant pressure paths and an isothermal, there being no clearance and no "valve
friction."
242. Details of Construction. The standard form of cylinder for large machines
is a twopiece casting, the working barrel being separate from the jacket, so that
the former may be a good wearing metal and may be quite readily removable.
Access to the jacket space is provided through bolt holes.
On the smaller compressors, the poppet type of valve is frequently used for both
inlet and discharge (Fig. 91). It is usually considered best to place these valves
FIG. 9L Art. 242. Compressor Cylinder with Poppet Valves.
(Clayton AJr Compressor Works.)
in the head, thus decreasing the clearance. They are satisfactory valves for auto
matically controlling the point of discharge, excepting that they are occasionally
138
APPLIED THERMODYNAMICS
noisy and uncertain in closing, and if the springs are made stiff for tightness, a con
siderable amount of power may be consumed in opening the valves. Poppet valves
work poorly at very low pressures, and are not generally used for conti oiling the intake
of air. Some form of mechanically opei a ted valve is preferably employed, such as the
semirocking type of Fig. 92, located at the bottom of the cyhnder, which has poppet
valves for the discharge at the top. For large units, Corliss inlet valves are usually
employed, these being
rocking cylindrical valves
running crosswise. As in
steam engines, they are so
diiven from an eccentric
and wrist plate as to give
rapid opening and closing
of the port, with a com
paratively slow interven
ing movement. They are
not suitable for use as
discharge valves in single
stage compressors, or in
the highpressure cylin
ders of multistage com
pressors, as they become
fully open too late in the
stroke to give a suffi
ciently free discharge.
In Fig. 93 Corliss valves
SUCTJON
FIG. 92. Art 242 Compressor Cylinder with Rocking Inlet
Valves. (Clayton Air Comprobsor Woiks )
are used for both inlet
and discharge. The
auxiliary poppet shown
is used as a safety valve.
FIG. 93. Art. 242. Compressor Cylinder with Corliss Yalves. (AUisOhalmers Oo.)
COMPRESSED AIR TRANSMISSION 139
A gear sometimes used consists of Corliss inlet valves and mechanically operated
discharge valves, which latter, though expensive, are free from the disadvantages
sometimes experienced with poppet valves The closing only of these valves is
mechanically controlled. Their opening is automatic,
A common rule for proportioning valves and passages is that the average velocity
of the air must not exceed 6000 ft. per minute.
COMPRESSED Am TRANSMISSION
243. Transmissive Losses. The air falls in temperature and pressure in the
pipe line. The fall in temperature leads to a decrease in volume, which is farther
reduced by the condensation of water vapor; the fall in pressure tends to increase
the volume. Early experiments at Mont Cenis led to the empirical formula
F = 0.00000936 (n z l d), for a loss of pressure F in a pipe d inches in diameter,
I ft. long, in which the velocity is n feet per second (1C).
In the Paris distributing system, the main pipe was 300 mm. in diameter, and
about f in. thick, of plain end cast iron lengths connected with rubber gaskets.
It was laid partly under streets and sidewalks, and partly in sewers, involving the
use of many bends. There were numerous drainage boxes, valves, etc., causing
resistance to the flow ; yet the loss of pressure ranged only from 3.7 to 5.1 lb., the
average loss at 3 miles distance being about 4.4 lb., these figures of course including
leakage. The percentage of air lost by leakage was ascertained to vary from 0.38
to 1.05, including air consumed by some small motors which were unintentionally
kept running while the measurements were made. This loss would of course be
proportionately much greater when, tlie load was light.
244. TTnwin's Formula. Unwinds formula for terminal pressure after long
transmission is commonly employed in calculations for pipe lines (17). It is.
in which p = terminal pressure in pounds per square inch,
P = initial pressure in pounds per square inch,
f au experimental coefficient,
u = velocity of air in feet per second,
L = length of pipe in feet,
d = diameter of (circular) pipe in feet,
T = absolute temperature of the air, F.
A simple method of determining/is to measure the fall of pressure under known
conditions of P, , T, , and d 9 and apply the above formula. Unwin has in this
way rationalized the results of Riedler's experiments on the Paris distributing
system, obtaining values ranging from 0.00181 to 00449, with a mean value
/= 0.00290. For pipes over one foot in diameter, he recommends the value 0.003 ;
for 6inch pipe,/= 0.00435; for 8inch pipe,/ = 0.004.
Biedler and Gutermuth found it possible to obtain pipe lengths as great as
10 miles in their experiments at Paris. Previous experiments had been made, on
140 APPLIED THERMODYNAMICS
a smaller scale, by Stookalper. For castiron pipe, a harmonization of these
experiments gives /= 0.0027(1 f 0.3 e?), d being the diameter of the pipe in feet.
The values of f for ordinary wrought pipe are probably not widely different. In
any welldesigned plant, the pressure loss may be kept very low.
245. Storage of Compressed Air. Air is sometimes stored at very high pres
sures for the operation of locomotives, street cars, buoys, etc. An important con
sequence of the principle illustrated in Joule's porous plug experiment (Art. 74)
here comes into play. It was remarked in Art. 74 that a slight fall of temperatuie
occurred during the reduction of pressure. This was expressed by Joule by the
formula
in which F was the fall of temperatuie in degrees Centigrade for a pressure
drop of 100 inches of mercury when T was the initial absolute temperature
(Centigrade) of the air. For air at 70 F., this fall is only l F., but when stored
air at high pressure is expanded through a reducing valve for use in a motor, the
pressure change is frequently so great that a considerable reduction of tempera
ture occurs. The efficiency of the process is very low ; Peabody cites an instance
(IS) in. which with a reservoir of 7o cu. ft. capacity, carrying 450 Ib. pressure,
with motors operating at 50 Ib. pressure and compression in three stages, the
maximum computed plant efficiency is only 0.29. An element of danger arises in
compressed air storage plants from the possibility of explosion of minute traces
of oil at the high temperatures produced by compression.
246. Liquefaction of Air ; Linde Process (19). The fall of temperature accom
panying a reduction of pressure has been utilized by Linde and others in the
manufacture of liquid air. Air is compressed to about 2000 Ib. pressure in a
threestage machine, and then delivered to a cooler. This consists of a double
tube about 400 ft. long, arranged in a coil. The air from the compressor passes
through the inner tube to a small orifice at its farther end, where it expands into
a reservoir, the temperature falling, and returns through the outer tube of the
cooler back to the compressor. At each passage, a fall of temperature of about
37J C. occurs. The effect is cumulative, and the air soon reaches a temperature
at which the pressure will cause it to liquefy (Art. 610).
247. Refrigeration by Compressed Air. This subject will be more particularly
considered in a later chapter. The fall of temperature accompanying expansion
in the motor cylinder, with the difficulties which it occasions, have been men
tioned in Art. 185. Early in the Paris development, this drop of temperature was
utilized for refrigeration. The exhaust air was carried through flues to wine
cellars, where it served for the cooling of their contents, the production of ice, etc.
In some 1 cases, the refrigerative effect alone is sought, the performance of wort
during the expansion being incidental.
(1) As text books on the commercial aspects of this subject Peele's Compressed Air
Plant (John Wiley & Sons) and Wightman/s Compressed Air (American School of
COMPRESSED AIR 141
Correspondence, 1909), may be consulted, (la) Riedler, Neue Erfahrungen uber
die Kraftversorgung von Pans dmch Druckluft, Berlin, 1891. (2) Pernolet (L'Air
Compnme) is the standard reference on this work. (3) Experiments upon Trans
mission, etc. (IdeU ed ), 1903, 98. (4) Unwin, op. at , 18 et seq. (5) Unwin,
op. cit., 32 (6) Graduating Thesis, Stevens Institute of Technology, 1891. (7)
Umvin, op. ait , 48. (8) Op cit , 109. (9) Unwin, op at., 48, 49; some of the
final figures are deduced from Kennedy's data. (10) Power., February 23, 1909,
p 382. (11) Development and Transmission of Power, 182 (12) Engineering News,
March 19, 1908, 325. (13) Peabody, Thermodynamics, 1907, 378. (14) Ibid.,
375. (15) Hiscox, Compressed Air, 1903, 273. (16) Wood, Thermodynamics, 1905,
306. (17) Transmission by Compressed Air, etc , 68; modified as by Peabody.
(18) Thermodynamics, 1907, 393, 394 (19) Zeuner, Technical Thermodynamics
(Klein); II, 303313: Trans. A. S. M. E. t XXI, 156.
SYNOPSIS OF CHAPTER IX
The use of compressed cold air for power engines aud pneumatic tools dates from I860.
The Air Engine
The ideal air engine cycle is bounded by two constant pressure lines, one constant
volume line, and a polytropic. In practice, a constant volume drop also occurs
after expansion.
Work formulas :
rr ,. _ PIT TT / , \
pv + pv log, ! gF; pv +^ ^  qV t pv log e  ; OPF) 2 ) .
v n i v \y ly
Preheaters prevent excessive drop of temperature during expansion ; the heat em
ployed is not wasted.
Cylinder volume = 33,000 NItt %n Up, ignoring clearance.
To ensure quiet running, the exhaust valve is closed early, the clearance air acting as a
cushion. This modifies the cycle.
Early closing of the exhaust valve also reduces the air consumption.
Actual figures for free air consumption range from 400 to 2400 cu. ft. per Uiphr.
Vie Compressor
The cycle differs from that of the engine in having a sharp "toe 17 and a complete clear
ance expansion curve.
Economy depends largely on the shape of the compression curve. Close approximation
to the isothermal, rather than the adiabatic, should be attained, as during expan
sion in the engine. This is attempted by air cooling, jet and spray injection of
water, and jacketing. Water required^ C=
Multistage operation improves tfo compression curve most notably and is in other
respects beneficial.
Intercooling leads to friction losses but is essential to economy; must be thorough.
142 APPLIED THERMODYNAMICS
Work, neglecting clearance (single cylinder), = T ^ r =];
The area under the compression curve is called the ioork of compression.
Minimum work, in twostage compression, ih obtained when P 2 = qp.
Engine and Compressor Relations
Compressive efficiency : ratio of engine work to compressor work ; 0.5 to 0.9.
Mechanical efficiency : ratio of work in cylinder and work at shaft , 80 to 0.90.
Cylinder efficiency ratio of ideal diagram area and actual diagram area ; 0.70 to 0.90
Plant efficiency . ratio of work delivered by air engine to work expended at compressor
shaft; 0.25 to 045 , tlieoietical maximum, 1.00.
The combined ideal entropy diagram is bounded by tan constant pressure curves and
two pulytropics. The economy of thorough mtercooling with multistage operation
is shown , as is the importance of a low exponent for the polytropics. With very
cold water, the net power consumption might be negative.
Compressor Capacity
Volumetric efficiency =ratio of free air drawn in to piston displacement; it is decreased
by excessive clearance, suction friction, heating during suction, and installation at
high altitudes. Long stroke compressors have proportionately less clearance.
Water may be used to Jill the clearance space: multistage operation makes
clearance less detrimental; refrigeration of the entering air increases the volumet
ric efficiency. Its value ranges ordinarily from 70 to 0.02. Suction friction
and clearance also decrease the cylinder efficiency, as does discharge friction.
Compressor Design
Theoretical.pzstoft displacement per stroke ~ or including clearance,
to be increased 5 to 10 per cent in practice.
In a, multistage compressor with perfect interceding, the cylinder volumes are inversely
as the suction pressures.
The power consumed in compression may be calculated for any assumed compressive
path.
A typical problem shows a saving of 12 per cent by twostage compression,
The " vacuum pump" used with a condenser is an air compressor.
Commercial Types of Compressing Machinery
Classification is by number of stages, type of frame or valves, or method of driving.
Governing is accomplished by changing the speed, the suction, or the discharge pressure.
Commercial types include the single, duplex, straight line and crosscompound twostage
forms, the last having capacities up to 6000 cu, ft. per minute. Some progress has
been made with turbo compressors.
Hydraulic piston compressors give high efficiency at low speeds.
The Taylor hydraulic compressor gives efficiencies up to 0.60 or 0.70.
PROBLEMS 143
Cylinder barrels and jackets are separate castings. Access to water space must be
provided.
Poppet, mechanical inlet, Corliss, and mechanical discharge valves are used.
Compressed Air Transmission
Loss in pressure = 0.00000936 nlrd
In Paris, the total loss in 3 miles, including leakage, was 4.4 Ib. ; the percentage of leak
age was 0.3S to 1,05, including air unintentionally supplied to consumers.
Unwin'*sformula; p = P\ l_j^L_ 2 . Mean value of /= 0. 0029 /= 0.0027(1+ 0.3d).
(9*79 ITV >
I
Stored high pressure air may be used for driving motors, but the efficiency is low.
The fall of temperature induced by throttling may be used cumulatively to liquefy air,
The fall of temperature accompanying expansion m the engine may be employed f or
refrigeration.
PROBLEMS
1. An air engine works between pressures of 180 Ib. and 15 Ib. per square inch,
absolute. Find the work done per cycle with adiabatic expansion fioni v = 1 to F 4,
ignoring clearance. By what percentage would the work be increased if the expansion
curve were PF 1 3 =c ? (Ans., 44,800 ft. Ib, 4.3 %.)
2. The expansion curve is PF 1 3 = c, the pressure ratio during expansion 7 : 1, the
initial temperature 100 F. Find the temperature after expansion. To what tempera
ture must the entering air be heated if the final temperature is to be kept above 32 F. ?
(Ans., 103 F., 310 F.)
3. Find the cylinder dimensions for a doubleacting 100 hp. air engine with clear
ance 4 per cent, the exhaust pleasure being 15 Ib. absolute, the engine making 200
r. p. m., the expansion and compression curves being PF 135 c, and the air being
received at 160 Ib. absolute pressure. Compression is carried to the maximum pres
sure, and the piston speed is 400 ft. per minute. A 10lb. drop of pressure occurs at
the end of expansion. (Allow a 10 per cent margin over the theoretical piston dis
placement.) (Ans., 13.85 ins. by 12.0 ins.)
4. Estimate the free air consumption per Ihp.hr. in the engine of Problem 3.
(Ans., 612cu.ft.)
5. A hydrogen compressor receives its supply at 70 F. and atmospheric pressure,
and discharges it at 100 Ib. gauge pressure. Find the temperature of discharge, if the
compression curve is PF 1 32 = c. (Ans., 412 F.)
6. In Problem 5, what is the percentage of power wasted as compared with iso
thermal compression, the cycles being like CBAD, Fig. 57 ?
7. In Problem 3, the initial temperature of the expanding air being 100 F., find
what quantity of heat must have been added during expansion to make the path
PF 1 36 c rather than an adiabatic. Assuming this to be added by a water jacket, the
water cooling through a range of 70, find the weight of water circulated per minute.
8. Find the receiver pressures for minimum work in two and fourstage compres
sion of atmospheric air to gauge pressures of 100, 125, 150, and 200 Ib.
9. What is the minimum work expenditure in the cycle compressing free air at
70 F. to 100 Ib. gauge pressure, per pound of air, along a path PF 1  35 = c, clearance
being ignored ? (Ans., 76,600 ft. Ib.)
10. Find the cylinder efficiency in Problem 3, the pressure in the pipe line being
165 Ib. absolute. (Ans., 62.5%.)
11. Sketch the entropy diagram for a fourstage compressor and twostage air
144: APPLIED THERMODYNAMICS
engine, in which n is 1.3 for the compressor and 1.4 for the engine, the air is inade
quately mtercooled, perfectly af tercooled, and inadequately preheated between the
engine cylinders. Compaie with the entropy diagram for adiabatic paths and perfect
interceding and such preheating as to keep the temperature of the exhaust above 32 F,
12. Find the cylinder dimensions and theoretical power consumption of a single
acting smglestage air compressor to deliver SOOO cu. ft. of free air per minute at
ISO Ib. absolute pressure at GO r. p. m , the intake air being at 13 Ib. absolute press
ure, the piston speed 640 ft. pel minute, clearance 3 per cent, and the expansion and
compression curves following the law PV 1 31 = c. (Ans , 80 by 64 in.)
13. "With conditions as in Problem 12, find the cylinder dimensions and power
consumption if compression is in two stages, intercooling is perfect, and 2 Ib. of f ric
tiun loss occurs between the stages. (Ans., 74 by 38 by 64 in.)
14. The cooling water rising from *6S F. to 89 F. in temperature, in Art. 233,
find the water consumption in gallons per minute.
15. Find the water consumption for jackets and intercoolmg in Art. 234 t the range
of temperature of the water being from 47 to 68 F.
16. Find the cylinder volume of a pump to maintain 26" vacuum when pumping
100 Ib. of air per hour, the initial temperature of the air being 110 F , compression
and expansion curves PT ri28 c, clearance 6 per cent., and the pump having two
doubleacting cylinders., The speed is 60 r. p. m. Pipe friction may be ignored.
17. Compare the liiedler and Gutermuth formula for / (Art. 244) with Unwin's
values. What apparent contradiction is noticeable m the variation of / with d ?
18 In a compressed air locomotive, the air is stored at 2000 Ib. pressure and de
livered to the motor at 100 Ib. Find the temperature of the air delivered to the
motor if that of the air in the reservoir is 80 F., assuming that the value of F (Art.
245J is directly proportional to the pressure drop.
19. \Vith isothermal curves and no friction, transmission loss, or clearance, what
would be the combined efficiency from compiessor to motor of an air storage system
m. which the storage pressure was 450 Ib. and the motor pressure 50 Ib.? The tem
perature of the air is 80 F. at the motor reducing valve. (Assume that the f ormula in
Art. 245 holds, and that the temperature drop is a direct function of the pressure drop.)
20. Find by the Mont Gems formula, the loss of pressure in a 12m. pipe 2 miles
long 111 which the air velocity is 32 ft. per second. Compare with Unwin's formula,
using the Eiedlcr and Gutermuth value for/, assuming P = 80, 2 T =70 F.
21. Find the free air consumption per Jhp,hr. if the action of the engine in Art.
190 is modified as suggested in Ait. 191.
22. Find under what initial pressure condition, in Art. 183, an output of 1.27
Ihp. may theoretically be obtained from 890 cu. ft. of free air per hour, the exhaust
pressure being that of the atmosphere, and the expansive path being (a) isothermal,
(b) adiabatic. (/i?is., (a), 56 Ib. absolute )
23. A compressor having a capacity of 500 cu. ft. of free air per minute (p= 14.7,
t = 70) is requiied to fill a 700 cu. ft. tank at a pressure of 2500 Ib. per square inch.
How long will this require, if the temperature in the tank is 140 at the end of the
operation, and the discharge pressure is constant?
24. In Problem 10, what is the theoretical minimum amount of power that might
be consumed, with no clearance and no abstraction of heat during compression? How
does this compare vvith the power consumption in the actual case?
25. A Taylor hydraulic compressor (Art. 241), with water at 40 Q , compresses air
to 50 Ib. gauge pressure. If the efficiency is 0.65 of that possible in isothermal compres
sion, rind the horse power consumed in compressing 4000 cu. ft. of free air per minute,
CHAPTER X
HOTAIR ENGINES
248. General Considerations. From a technical standpoint, the class of
air engines includes all heat motors using any permanent gas as a working
substance. For convenience, those engines in which the fuel is ignited
inside the cylinder are separately discussed, as internal combustion or gas
engines (Chapter XI). The air engine proper is an external combustion
engine, although in some types the products of combustion do actually
enter the cylinder; a point of mechanical disadvantage, since the corro
sive and gritty gases produce rapid wear and leakage. The air engine
employs, usually, a constant mass of working substance, i.e., the same
body of air is alternately heated and cooled, none being discharged from
the cylinder and no fresh supply being brought in; though this is not
always the case. Such an engine is called a " closed " engine. Any
fuel may be employed; the engines require little attention; there is
no danger of explosion.
Modern improvements on the original Stirling and Ericsson forms of
air engine, while reducing the objections to those types, and giving
excellent results in fuel economy, are, nevertheless, limited in their
application to small capacities, as for domestic pumping service. The
recent development of the gas engine (Chapter XI) has further served
to minimize the importance of the hotair cycle.
In air, or any perfect gas, the temperature may be varied independ
ently of the pressure ; consequently, the limitation referred to in Art. 143
as applicable to steam engines does not necessarily apply to air engines,
which may work at much higher initial temperatures than any steam en
gine, their potential efficiency being consequently much greater. When
a specific cycle is prescribed, however, as we shall immediately find, pres
sure limits may become of importance.
249. Capacity. One objection to the air engine arises from the ex
tremely slow transmission of heat through metal surfaces to dry gases.
This is partially overcome in various ways, but the still serious objection
is the small capacity for a given size. If the Carnot cycle be plotted for
one pound of air, as in Fig. 94, the enclosed work area is seen to be very
small, even for a considerable range of pressures. The isothermals and
adiabatics very nearly coincide. For a given output, therefore, the air en
gine must be excessively large at anything like reasonable maximum pres
sures. In. the Ericsson engine (Art. 269), for example, although the cycle
was one giving a larger work area than that of Carnot, four cylinders
were required, each having a diameter of 14 ft. and a stroke of 6 ft.; it
145
115
APPLIED THERMODYNAMICS
was estimated that a steam engine of equal power would have required
only a single cylinder, 4 ft in diameter and of 10ft. stroke, running at 17
revolutions per minute and using 4 Ib. of coal per horse power per hour.
The air engine ran at 9 r p. m v and its great bulk and cost, noisiness and
rapid deterioration, overbore the advantage of a much lower fuel con
sumption, 1.S7 Ib. of coal per hp.hr. At the present time, with increased
 g  3~ 4 5 o 7 8 10
FIG 94. Arts. 249, 250. Carnot Cycle for Air.
steam pressures and piston speeds, the equivalent steam engine would
be still smaller.
250. Carnot Cycle Air Engine. The efficiency of the cycle shown in
Fig 94 has already been computed as (T t) * T (Art. 135). The work
done per cycle is, from Art. 135,
0 log, 2
t) log. .
K4
Another expression for the work, since
POLYTROPIC CYCLE
147
But from Art. 104 ? *=(y\ whence
Pa=J>t j and Tr  B ( z 'oi*.
This can have a positive value only when  1 ( }** exceeds unity ; which
s
~P f T*\ y
is possible only when =i exceeds ( \ y ~ L . Now since P l and P 3 are the
*s \t J
limiting pressures in the cycle, and since for air y f (y 1) = 3.486, the
minimum necessary ratio of pressures increases as the 3.486 power of the ratio
of temperatures.* This alone makes the cycle impracticable. In Eig. 94,
the pressure range is from 14.7 to 349.7 Ib. per square inch, although tlie
temperature range is only 100.
Besides the two objections thus pointed out large size for its
capacity and extreme pressure range for its efficiency the Carnot engine
would be distinguished by a high ratio of maximum to average
pressure; a condition which would make friction losses excessive.
251. Polytropic Cycle. In Fig. 05, let T, t be two isothermals, el and dft\vo
like polytropic curves, following the law pv n =. c, and ed arid bf two other like
polytropic curves, following the law pu m = c.
Then ebfd is a polytropic cycle. Let T, t, P b , P e
nl
be given. Then T e = T\ " . In the en
tropy diagram,
Fig. 96, locate the
isothermals T, t,
T e . Choose the
point e at i andom.
From Art. Ill, the
specific heat along
a path pv n c is
and
FIG. 96. Arts. 251, 256. Poly
tropic Cycle.
FIG. 95. Arts. 251, 256, Prob. 4a.
Polytropic Cycle.
from Art. 163, the increase of entropy when the
specific heat is s, in passing from e to &, is
T
N = s log,s . This permits of plotting the curve
*It has been shown that ^= (^ ) *~ . But P 3 <P^ if a finite work area is to
v
~P fT\ vT
be obtained; hence ^< ( ] . The efficiency of the Carnot cycle may of course
be written as 1
"ia
i/i
148
APPLIED THERMODYNAMICS
el in successive short steps, in Fig, 96. Along ed, similarly, s 1 = /  ^) and
rn \m I/
N^ = s, log e ^ between d and e. We complete the diagram by di awing bf and
2d
df, establishing the point of intersection which determines the temperature at /.
We find T f \ T b : : T d : T e . The efficiency is equal to
ne of N
, or to
[nebx
 ydfN  nedy~\  [nebx
,

f  T r j)
the negative sign of the specific heat s x being disregarded.
252. Lorenz Cycle. In Fig. 97 let ^ and bh be adiabatics, and let the curves
gb and <Zfe be polytropics, but unlike, the former having the exponent n, and the
latter the exponent q. This constitutes the cycle of Lorenz. We find the tempera*
FIG. 97. Arts. 252, 256, Prob, 5.
Lorenz Cycle.
FIG. 98. Arts. 252, 256. Lorenz Cycle,
Entropy Diagram.
tare at g as in Art. 251, and in the manner just described plot the curves gb and
dh on the entropy diagram, Fig. 98, P g , P b , T b , T d , n and q being given, dg and
bh of course appear as vertical straight lines. The efficiency is
253. Reitiinger Cycle. This appears as aug, Figs. 99 and 100. It is bounded
by two isothermals and two like polytropics (isodiabatics). The Carnot is a special
example of this type of cycle. To plot the entropy diagram, Fig, 100, we assume
the ratio of pressures or of volumes along ai or cj. Let V a and 7^ be given. Then
the gain of entropy from a to i is (p a V a lo&]r) +T. The curves ic and aj are
JOULE AIR ENGINE
149
plotted for the given value of the exponent n. This is sometimes called the isodia
batic cycle. Its efficiency is
f TJ J_ JT 77" TT \ /" 77" _L 7" \
\ n <u ~r "ic fljc Haj) (/"ai r Miejj
which may be expanded as in Arts. 251, 252.
p
FIG. 99. Arts. 233, 256. Reit
linger Cycle.
FIG. 100. Arts 253, 250, 257, 258 f
259. Reitlmger Cycle, Entropy
Diagram.
254. Joule Engine. An air engine proposed by Ericsson as early as
1833, and revived by Joule and Kelvin in 1851, is shown in Fig. 101. A
chamber contains air kept at a low temperature t by means of circulating
water. Another chamber A contains hot air in a state of compression,
the heat being supplied at a constant temperature T by means of an ex
ternal furnace (not shown). M is a pump cylinder by means of which air
Fia. 101. Arts. 254, 255, 275. Joule Air Engine.
may be delivered from C to A, and ^T is an engine cylinder in which air
from A may be expanded so as to perform work. The chambers A and
are so large in proportion to M and N that the pressure of the air in these
chambers remains practically constant.
150
APPLIED THERMODYNAMICS
The pump M takes air from (7, compresses it adiabatically, until its
pressure equals that in A, then, the valve v being opened, delivers it to A
at constant pressure. The cycle
is fdoe, Fig. 102. In this special
modification of the polytropic
cycle of Art. 251, fd represents
the drawing in of the air at con
stant pressure, do its adiabatic
compression, and oe its discharge
to A. Negative work is done,
equal to the area fdoe. Concur
rently with this operation, hot
FIG. 102. Arts. 254, 255, 256. Joule Cycle
air has been flowing from A to
through the valve u, then expand
ing adiabatically while u is closed ; finally, when the pressure has fallen
to that in C, being discharged to the latter chamber, the cycle being ebqf,
Fig. 102. Positive work has been done, and the net positive work per
formed by the whole apparatus is ebqf fdoe = obqd.
255. Efficiency of Joule Engine. We will limit our attention to the net
cycle obqd. The heat absorbed along the constant pressure line ob is
Hj J ='k(r T }. The heat rejected along qd is H qd = k(T q t). But
fp rp rp _ t t
from Art. 251, 2 = , whence, =^ = = =, , and the efficiency is
t JL JL JLo
t
q _^ _
TT, T ~
T
This is obviously less than the
efficiency of the Carnot cycle
between T and t. The entropy
diagram may be readily drawn
as in Tig. 103. The atmos
phere may of course take the
place of the cold chamber C,
a fresh supply being drawn in
by the pump at each stroke, and
the engine cylinder likewise
discharging its contents to the
atmosphere. The ratio fd * fq,
FIG. 103. Arts. 255, 256, Joule Cycle, Entropy
Diagram.
in Pig. 102, shows the necessary ratio of volumes of pump cylinder and
engine cylinder. The need of a large pump cylinder would be a serious
drawback in practice ; it would make the engine bulky and expensive, and
REGENERATOR 151
would lead to an excessive amount of mechanical friction. The Joule
engine has never been constructed.
256. Comparisons. The cycles just described have been grouped
in a single illustration in Fig. 104. Here we have, between the
temperature limits T and , the Oarnot cycle, abed ; the polytropic
cycle, debfi the Lorenz
cycle, dglh ; that of Reit
linger^ aicj ; and that of
Joule, obqd. These illus
trations are lettered to
correspond with Figs.
95100, 102, 103. A
graphical demonstration
that the. Carnot cycle is
the one of maximum
efficiency suggests itself.
We now consider the
most successful attempt
yet made to evolve a cycle
having a potential effi
ciency equal to that of
Carnot.
257. Regenerators.
By reference to Fig. 100,
it may be noted that the
heat areas under aj and
ic are equal. The heat
absorbed along the one
path is precisely equal to
that rejected along the
other. This fact does
not prevent the efficiency
from being less than that
of the Carnot cycle, for
efficiency is the quotient
of work done by the gross
heat absorption. If, however, the heat under ic were absorbed
not from the working substance, and that under ja were rejected
FIG. 104. Arts. 256, 266. Hotair Cycles.
152 APPLIED THERMODYNAMICS
not to the condenser ; but if some intermediate body existed having a
storage capacity for heat, such that the heat rejected to it along ja
could be afterward taken up from it along ic, then we might ignore
this quantity of heat as affecting the expression for efficiency, and the
cycle would be as efficient as that of Carnot. The intermediate body
suggested is called a regenerator.
258. Action of Regenerators. Invented by Robert Stirling about 1816, and
improved by James Stirling, Ericsson, and Siemens, the present foim of regener
ator may be regarded as a long pipe, the walls of which have so large a capacity
for heat that the temperature at any point remains practically constant. Through
this pipe the air flows in one diiection when working along iY, Fig. 100, and
in the other direction while working along ja. The air encounters a gradually
changing temperature as it traverses the pipe.
Let hot exhaust air, at i, Fig. 100, be delivered at one end of the regenerator.
Its temperature begins to fall, and continues falling, so that when it 'leaves the
regenerator its temperature is that at c, usually the temperature of the atmosphere.
The temperatuie at the inlet end of the regenerator is then T, that at its outlet t.
During the admission of fresh air, along;//, it passes through the regenerator in
the opposite direction, gradually increasing in temperature from t to T 9 without
appreciably affecting the temperature of the regenerator. Assuming the capacity of
the regenerator to be unlimited, and that there are no losses by conduction of heat
to the atmosphere or along the material of the regenerator itself, the process is
strictly reversible. We may cause either the volume or the pressure to be either
fixed or variable according to some definite law, during the regenerative move
ment. Usually, either the pressure 01 the volume is kept constant.
As actually constructed, the regenerator consists of a mass of thin perforated
metal sheets, so arranged as not to obstruct the flow of air. Some waste of heat
always accompanies the regenerative process; in the steamer Ericsson, it was 10
per cent of the total heat passing through. Siemens appears to have reduced the
loss to 5 per cent.
259. Influence on Efficiency. Any cycle bounded by a pair of
isothermals and a pair of like polytropics (Reitlinger cycle), if worked
with a regenerator, lias an efficiency ideally equal to that of the
Carnot cycle. To be sure, the heated air is not all taken in at T,
nor all rejected at t; but the heat absorbed from the source is all
at I 7 , and that rejected to the condenser is all at t. The regenerative
operations are mutually compensating changes which do not affect
the general principle of efficiency under such conditions. The heat
paid for is only that under the line ai, Fig. 100. The regenerator
thus makes the efficiency of the Carnot cycle obtainable by actual heat
engines.
THE STIRLING ENGINE
153
As will appear, the cycles in which a regenerator is commonly employed are
not otherwise particularly efficient. Their chief advantage is in the large work
area obtained, which means increased capacity of an engine of given dimensions.
For highest efficiency, the regenerator must be added.
260. The Stirling Engine. This important type of regenerative air engine
was covered by patents dated 1827 and 1840, by Robert and James Stirling. Its
action is illustrated in Fig. 105. G is the engine
cylinder, containing the piston H, and receiving
heated air through the pipe F from the vessel A A
in which the air is alternately heated and cooled.
The vessel A A is made \vith hollow walls, the inner
lining being marked aa. The hemispherical lower
portion of the lining is perforated ; while from A A
up to CC the hollow space constitutes the regener
ator, being filled with strips of metal or glass. The
plunger E fits loosely in the machined inner shell
aa. This plunger is hollow and filled with some
nonconducting material. The spaces DD contain
the condenser, consisting of a coil of small copper
pipe, through which water is circulated by a sepa
rate pump. An air pump discharges into the pipe
F the necessary quantity of fresh air to compensate
for any leakage, and this is utilized in some cases
to maintain a pressure which is at all stages con
siderably above that of the atmosphere. The furnace is built about the
ABA of the heating vessel.
FIG. 105. Arts. 260, 361, 262,
263, 264. Stirling Engine.
bottom
261. Action of the Engine. Let the plunger E and the piston H be in their
lowest positions, the air above E being cold. The plunger E is raised, causing
air to flow from X downward through the regenerator to the space 6, while H
remains motionless. The air takes up heat from the regenerator, increasing its
temperature, say to T, while the volume remains constant. After the plunger has come
to rest, the piston H is caused to rise by the expansion produced by the absorption
of heat from the furnace at constant temperature, the air reaching H by passing
around the loosefitting plunger E, which remains stationary. H now pauses in
its "up" position, while E is lowered, forcing air through the regenerator from
the lower space & to the upper space X, this air decreasing in temperature at con
stant volume. While E remains in its "down" position, H descends, forcing the
air to the condenser D, the volume decreasing, but the temperature remaining con
stant at t. The cycle is thus completed.
The working air has undergone four changes : (a) increase of pressure
and temperature at constant volume, (&) expansion at constant tempera
ture, (c) a fall of pressure and temperature at constant volume, and (d)
compression at constant temperature.
154
APPLIED THERMODYNAMICS
262. Remarks. With action as described, the piston II and the plunger E
(sometimes called the " displacer pistou '') do not move at the same time , one is
always nearly stationary, at or near the end of its stroke, while the other moves.
In practice, uniform rotative speed is secured by modifying these conditions, so
that the actual cycle merely approximates that described. The vessel A A is
sometimes referred to as the "leceiver." It is obvious that a certain residual
quantity of air is at all times contained in the spaces between the piston H and
the plunger E. This does not pass through the regenerator, nor is it at any time
subjected to the heat of the furnace. It serves merely as a medium for transmit
ting pressure from the "working air" to 77; and in contradistinction to that
working substance, it is called " cushion air.*' Being at all times in communica
tion with the condenser, its temperature is constantly close to tlie minimum attained in
the cycle. This is an important point in facilitating lubrication.
263. Forms of the Stirling Engine. In some types, a separate pipe is carried
from the lower part of the receiver to the working cylinder G, Fig. 105. This
removes the necessity for a loosefitting plunger; in doubleacting engines, each
end of the cylinder is connected with the hot (lower) side of the one plunger and
with the cold (upper) side of the other. In other forms, the regenerator has been
a separate vessel ; in still others, the displacer plunger itself became the regen
erator, being perforated at the top and bottom and filled with wiie gauze. The
LaubereauSchwartzkopfl: engine (1) is identical in principle with the Stirling,
excepting that the regenerator is omitted.
The maintenance of high minimum pressure, as described in Art. 260, and the
low ratio of maximum to average pressure, while not necessarily affecting the theo
retical efficiency, greatly increase the capacity, and (since friction losses are practi
cally constant) the mechanical efficiency as well.
P
\
\
FIG. 106. Arts. 204, 205, 267. Stirling Cycle.
264. PressureVolume Diagram. The cycle of operations described in
Art. 261 is that of Fig. 106, ABCD* Considering the cushion air, the
THE STIRLING ENGINE
155
actual diagram which would be obtained by measuring the pressures and
volumes is quite different. Assume, for example, that the total volume
of cushion air at maximum pressure (when E is at the top of its stroke
and H is just beginning to move) is represented by the distance NE.
Then if AT" be laid off equal to NE, the total volume of air present is NL
Draw an isothermal EFHG, representing the path of the cushion air 3 sep
arately considered, while the temperature remains constant. Add its vol
umes, PF, ZH } QG, to those of working air, by laying off BK= PF,
DM=ZH, CL=QG 9 at various points along the stroke. Then the
cycle IKLM is that actually experienced by the total air, assuming the
cushion air to remain at constant temperature throughout (Art. 262).
The actual indicator diagrams obtained in tests are roughly similar to the
cycle IKLM, Fig. 10G ; but the corners are rounded, and other distortions may
appear on account of nonconformity with the ideal paths, sluggish valve action,
errors of the indicating instrument, and various other causes.
265. Efficiency. The heat absorbed from the source along AB, Fig.
106, is
e^ That rejected to the condenser along CD is
P^Folog^^' The work done is the difference of these two quantities,
YD
and the efficiency is
Tt
T '
that of the Carnot cycle. Losses through the regenerator and by imper
fection of cycle reduce this in prac
tice.
266. Entropy Diagram. This is
given in Fig. 108. T and t are the
limiting isothermals, DA and BO
the constant volume curves, along
each of which the increase of en
tropy is n s= llQ%,(T*rt\ I being the
specific heat at constant volume.
The gain of entropy along the iso
thermals is obtained as in Art. 253. Ignoring the heat areas EDAF and
GCBH, the efficiency is ABCD + FABH, that of the Carnot cycle. The
Stirling cycle appears in the PV diagram of Fig. 104 as dkbl.
FIG. 108. Art. 266. Stirling Cycle,
Entropy Diagram.
156
APPLIED THERMODYNAMICS
267. Importance of the Regenerator. Without the regenerator, the non
reversible Stirling cycle would have an efficiency of
(P. P,)^ log. ?
*'vl
This is readily computed to be far below that of the corresponding Carnot
cycle. The advantage of the regenerative cycle lies in the utilization of
the heat rejected along J3<7, Fig. 106, thus cancelling that item in the
analysis of the cycle. Another way of utilizing this heat is to be
described ; but while practical difficulties, probably insurmountable, limit
progress in the application of the air engine on a commercial scale, the
regenerator, upon which has been founded our modern metallurgical in
dustries as well, has offered the first possible method for the realization
of the ideal efficiency of Carnot (2).
268. Trials. As early as 1847, a 50hp. Stirling engine, tested at the Dun
dee Foundries, was shown to operate at a thermal efficiency of 30 per cent, esti
mated to be equivalent, considering the rather low furnace efficiency, to a coal con
sumption of 1.7 Ib. per hp.hr. This latter result is not often surpassed by the aver
age steam engines of the present day. The friction losses in the mechanism were
only 11 per cent (3). A test quoted by Peabody (4) gives a coal rate of 1.66 Ib.,
but with a friction loss much greater, about 30 per cent. There is no question
as to the high efficiency of the regenerative air engine.
269. Ericsson's Hotair Engine. In 1833, Ericsson constructed an unsuccess
ful hotair engine in London. About 1855, he built the steamer Ericsson, of 2200
tons, driven by four immense hotair engines. After the abandonment of this
experiment, the same designer in 1875 introduced a third type of engine, and more
recently still, a small pumping engine, which has been extensively applied.
The principle of the engine of
1855 is illustrated in Fig. 109. B is
the receiver, A the displacer, H the
furnace. The displacer A fits loosely
in B excepting near its upper portion,
where tight contact is insured by
means of packing rings. The lower
portion of A is hollow, and filled
with a nonconductor. The holes
aa admit air to the upper surface
of A. D is the compressing pump,
with piston (7, which is connected
FIG. 109. Arts. 2(>9, 270, 275. Ericsson Engine, with A by the rods dd. E is a pis
ton rod through which the de
veloped power is externally applied. Air enters the space above C through
the check valve c, and is compressed during the up stroke into the magazine F
ERICSSON ENGINE
157
through the second check valve e. G is the regenerator, made up of M'ire gauze.
The control valves, worked from the engine mechanism, are at b and f. \Vhen
b is opened, air passes from F through G to B, raising A. Closing of b at part
completion of the stroke causes the air to work expansively foi the remainder of
the stroke. During the return stroke of A, air passes through G, /, and g to the
atmosphere.
270. Graphical Illustration. The PV diagram is given in Fig. 110. EBCF
is the network diagram, ABCD being the diagram of the engine cylinder, AEFD
that of the pump cylinder. Beginning with A in its lowest position, the state point
in Fig. 110 is, for the engine (lower side of .4), at
A, and for the pump (upper side of C), at F.
During about half the up stroke, the path in the
engine is AB, air passing to B from the re
generator through s, and being kept at constant
pressure by the heat from the furnace. During
the second half of this stroke, the supply of air
from the regenerator ceases, and the pressure falls
rapidly as expansion occurs, but the heat im
parted from the furnace keeps the temperature
practically constant, giving the isothermal path
BC. Meanwhile, the pump, receiving air at the
pressure of the atmosphere, has been fiist compressing it isothermally, or as
nearly so as the limited amount of cooling surface will permit, along FE, and
then discharging it through e at constant pressure, along EA, to the receiver F.
On the down stroke, the engine steadily expels the air, now expanded down to
atmospheric pressure, along the constant pressure line CD, while the pump simi
larly draws in air from the atmosphere at constant pressure along DF. At the end
of this stroke, the air in F } at the state A^ is admitted to the engine. The ratio of
pump volume to engine volume is FD DC, or
FIG. 110. Arts. 270, 272, 273.
Ericsson Cycle.
T
FIG. 111. Art. 271. Ericsson Cycle,
Entropy Diagram.
271. Efficiency. The Ericsson cycle be
longs to the same class as that of Stirling,
being bounded by two isothermals and two
like polytropics ; but the polytropics are in
this case constant pressure lines instead of
constant volume lines. The net entropy
diagram EBUF, Fig. Ill, is similar to that
of the Stirling engine, but the isodiabatics
swerve more to the right, since Jc exceeds l>
while the efficiency (if a regenerator is employed) is the same as that
rp
of the Stirling engine, 
272. Tests, As computed by Rankine from Norton's tests, the effi
ciency of the steamer Ericsson's engines was 26.3 per cent; the efficiency
of the furnace was, however, only 40 per cent. The average effecti v e pres
158 APPLIED THERMODYNAMICS
sure (EBCFr XC, Fig. 110) was only 2.12 Ib. The friction losses were
enormous. A small engine of this type tested by the writer gave a con
sumption of 15.64 cu. ft. of gas (652 B. t. u. per cubic foot) per Ihp.hr. ;
equivalent to 170 B. t. u. per Ihp.minute; and since 1 horse power
= 33,000 footpounds =33,000 * 778 = 42.45 B. t. u. per minute, the
thermodynamic efficiency of the engine was 42.45 f 170 = 0.25.
273. Actual Designs. In order that the lines FC and EB, Fig. 110, may be
horizontal, the engine should be triple or quadruple, as in the steamer Ericsson, in
which each of the four cylinders had its own compressing pump, but all were con
nected with the same receiver, and with a single crank shaft at intervals of a
quarter of a revolution. Specimen indicator diagrams are given in Figs. 107, 112.
FIG 107. Art. 273. Indicator FIG. 112. Art. 273. Indicator
Card from Ericsson Engine. Diagram, Ericsson Engine.
274. Testing Hotair Engines. It is difficult to directly and accurately meas
ure the limiting temperatures in an air engine test, so that a comparison of the
actually attained with the computed ideal efficiencies cannot ordinarily be made.
Actual tests involve the measurement of the fuel supplied, determination of its
heating value, and of the indicated and eifective horse power of the engine
(Art. 487). These data permit of computation of the thermal and mechanical
efficiencies, the latter being of much importance. In small units, it is sometimes
as low as 0.50.
275. The Air Engine as a Heat Motor. In nearly every large application, the
hotair engine has been abandoned on account of the rapid burning out of the
heating surfaces due to their necessarily high temperature. Napier and Rankine
(5) proposed an " air heater," designed to increase the transmissive efficiency of
the heating surface. Modern forms of the Stirling or Ericsson engines, in small
units, are comparatively free from this ground of objection. Their design permits
of such amounts of heattransmitting surface as to give grounds for expecting a
much less rapid destruction of these parts. It has been suggested that exceLSsive
bulk may be overcome by using higher pressures. (Zeuner remarks (6) that the
bulk is not excessive when compared with that of a steam engine with its auxiliary
boiler and furnace). Rankine has suggested the introduction of a second com
pressed air receiver, in Fig. 109, from which the supply of air would be drawn
through GJ and to which air would be discharged through/. This would make the
engine a "closed" engine, in which the minimum pressure could be kept fairly
high ; a small air pump would be required to compensate for leakage. A " con
denser " would be needed to supplement the action of the regenerator by more
HOTAIR ENGINES 159
thoroughly cooling the discharged air, else the introduction of " back pressure "
would reduce the working range of temperatures. The loss of the air by leakage,
and consequent waste of power, would of course increase with increasing pressures.
Instead of applying heat externally, as proposed by Joule, in the engine shown
in Fig. 101, there is no reason why the combustion of the fuel might not proceed
within the hot chamber itself, the necessary air for combustion being supplied by
the pump. The difficulties arising from the slow transmission of heat would thus
be avoided. An early example of such an engine applied in actual practice was
Cayley's (7), later revived by AVenham (8) and Buckett (9). In such engines,
the working fluid, upon the completion of its cycle, is discharged to the atmos
phere. The lower limit of pressure is therefore somewhat high, and for efficiency
the necessary wide range of temperatures involves a high initial pressure in the
cylinder. The internal combustion air engine even in these crude forms may be
regarded as the forerunner of the modern gas engine.
(1) Zeuner, Technical Thermodynamics (Klein), 1907, I, 340. (2) The theoreti
cal basis of regenerator design appears to have been treated solely by Zeuner, op. cit,,
I, 314323. (3) Rankme, The Steam Engine, 1897, 368. (4) Thermodynamics of the
Steam Engine, 1907, 302. (5) The Steam Engine, 1897, 370. (0) Op. eft., I, 381.
(7) Nicholson's Art Journal, 1807; Min. Proc. Inst. C. E., IX. (8) Proc. Inxt.
Mech. Eng., 1873. (9) Inst. Civ. Eng^ Heat Lectures, 18831884; Min. Proc. Inst.
C. E., 1845,1854.
SYNOPSIS OF CHAPTER X
The hotair engine proper is an external combustion motor of the open or closed type.
The temperature of a permanent gas may be varied independently of the pressure ; this
makes the possible efficiency higher than that attainable in vapor engines.
3486
; the Carnot cycle leads to either excessive pressures or an enormous
)
cylinder.
The poly tropic cycle is bounded by two pairs of isodiabatics.
The Lorenz cycle is bounded by a pair of adiabatics and a pair of unlike polytropics.
The Eeitlinger (isodiabatic) cycle is bounded by a pair of isothermals and a pair of
isodiabatics.
The Joule engine works in a cycle bounded by two constant pressure lines and two
adiabatics ; its efficiency is ~" .
The regenerator is a "fly wheel for heat." Any cycle bounded by a pair of iso
thermals and a pair of like polytropics, if worked with a regenerator, has an ideal
efficiency equal to that of the Carnot cycle ; the heat rejected along one poly tropic
is absorbed by the regenerator, which in turn emits it along the other polytropic,
the operation being subject to slight losses in practice.
The Stirling cycle, bounded by a pair of isothermals and a pair of constant volume
curves : correction of the ideal PV diagram for cushion air : comparison with indi
cator card ; the entropy diagram ; efficiency formulas with and without the regen
erator ; coal consumption, 1,7 Ib. per hp.hr.
The Ericsson cycle, bounded by a pair of isothermals and a pair of constant pressure
curves : efficiency from fuel to power, g$ per cent.
160 APPLIED THERMODYNAMICS
By designing as "closed" engines, the minimum pressure may "be raised and the
capacity of the cylinder increased.
The air engine is unsatisfactory in large sizes on account of the rapid "burning out of
the heating surfaces and the small capacity for a given "bulk.
PROBLEMS
(NOTE. Considerable accuracy in computation will he found necessary in solving Prob
lems 4 a and 5).
1. How much greater is the ideal efficiency of an air engine working "between tem
perature limits of 2900 F. and 600 F. than that of the steam engine described in Prob
lem 5, Chapter YI ?
2 Plot to scale (1 inch = 2 cu. ft. = 40 Ih. per square inch) the P 7" Carnot cycle
for r=GOO, = 500 (both absolute) the lowest pressure being 14.7 Ib. per square
inch, the substance being one pound of air, and the volume ratio during isothermal
expansion being 12 C.
3. In Problem 2, if the upper isothermal be made 700 absolute, what will be the
maximum pressure ?
4 a. Plot the entropy diagram, and find the efficiency, of a polytropic cycle for air
between 000 F. and 500 F , in which m = 1.3, n =  1.3, the pressure at d (Fig. 95)
is 18 Ib. per square inch, and the pressure at e (Tig 95) is 22 Ib. per square inch.
4 6. In Art. 251, prove that 7> T b : : T d : T e , and also that P d P e : : P f : P io
5. Plot the entropy diagram, and find the efficiency, of a Lorenz cycle for air
between 600 F. and 500 F., in which n = ~ 1.3, q = 0.4, the highest pressure being
80 Ib. per square inch and the temperature at g, Fig. 97, being 550 F.
6. Plot the entropy diagram, and find the efficiency, of a Reitlinger cycle between
000 F. and 500 F., when n = 1.3, the maximum pressure is 80 Ib. per square inch, the
ratio of volumes during isothermal expansion 12, and the working substance one
pound of air.
rji rp
7. Show that in the Joule engine the efficiency is ^, Art. 255.
8. Plot the entropy diagram, and find the efficiency, of a Joule air engine working
between C00 F. and 200 F., the maximum pressure being 100 Ib. per square inch,
the ratio of volumes during adiabatic expansion 2, and the weight of substance 2 Ib.
9. Plot PFand NT diagrams for one pound of air worked between 3000 F. and
400 F. : (a) in the Carnot cycle, (&) in the Ericsson cycle, (c) in the Stirling cycle, the
extreme pressure range being from 50 to 2000 Ib. per square inch.
10. Find the efficiencies of the various cycles in Problem 9, without regenerators.
11. Compare the efficiencies in Problems 4 a, 5, and 6, with that of the correspond
ing Carnot cycle.
12. Au air engine cylinder working in the Stirling cycle between 1000 F. and
2000 F., with a regenerator, has a volume of 1 cu ft. The ratio of expansion is 3.
By what percentages will the capacity and efficiency be affected if the lower limit of
pressure is raised from. 14.7 to 85 Ib. per square inch ?
18. In the preceding problem, one eighth of the cylinder contents is cushion air, at
1000 F, Plot the ideal indicator diagram for the lower of the two pressure limits, cor
rected for cushion air.
HOTAIR ENGINES 161
14. In Art. 268, assuming that the coal used in the Dundee foundries contained
14,000 B. t. u. per pound, what was the probable furnace efficiency? In the Peahody
test, if the furnace efficiency was 80 per cent, and the coal contained 14,000 B. t. u.,
what was the thermal efficiency of the engine ?
15. What was the efficiency of the plant in the steamer Ericsson ?
16. Sketch the TJVand PF diagrams, within the same temperature and entropy
limits, of all of the cycles discussed in this chapter, with the exception of that of Joule.
"Why cannot the Joule and Ericsson cycles be drawn between the same limits ? Show
graphically that in no case does the efficiency equal that of the Carnot cycle.
17. Compare the cycle areas in Problem 9.
18. In Problem 2, what is the minimum possible range of pressures compatible
with a finite work area ? Illustrate graphically.
19. Derive a definite formula for the efficiency of the Eeitlinger cycle, Art. 253.
20. Derive an expression for the efficiency of the Ericsson cycle without a
regenerator.
CHAPTER XI
GAS POWER,
THE GAS PRODUCER
276, History. The bibliography (1) of internal combustion engines is exten
sive, although their commercial development is of recent date. Coal gas was dis
tilled as early as 1691 , the waste gases from blast furnaces were first used for
heating in 1809. The first English patent for a gas engine approaching modern
form was granted iu 1794. The advantage of compression was suggested as early
as 1801 , but was not made the subject of patent until 1838 in England and 1861 in
France. Lenoir, in 1800, built the first practical gas engine, which developed a
thermal efficiency of 0.04. The now familiar polyti opic " Otto " cycle was pro
posed by Beau de Rochas at about this date. The same inventor called attention,
to the necessity of high compression pressures in 1862 ; a principle applied in
practice by Otto in 1874. Meanwhile, in 1S70, the first oil engine had been built.
The fourcycle compressive Otto "silent" engine was brought out in 1876, show
ing a thermal efficiency of 0.15, a result better than that then obtained in the best
steam power plants.
If the isothermal, isometric, isopiestic, and adiabatic paths alone are considered,
there are possible at least twentysix different gas engine cycles (2). Only four
of these have had extended development; of these four, only two have survived.
The Lenoir (3) and Hugon (4) noncompressive engines are now represented only
by the Bischoff (5) . The Barsariti " free piston " engine, although copied by
Grilles and by Otto and Langen (1866) (6), is wholly obsolete. The variable vol
ume engine of Atkinson. (7) was commercially unsuccessful.
Up to 1885, illuminating gas was commonly employed, only small engines
were constructed, and the high cost of the gas prevented them from being com
mercially economical. Nevertheless, six forms were exhibited in 1887. The
Priestman oil engine was built in 1888. ' With the advent of the Dowson process,
in 1878, with its possibilities of cheap gas, advancement became rapid. By 1897,
a 400hp. fourcylinder engine was in use on gas made from anthracite coal. At
the present time, doubleacting engines of 5400 hp. have been placed in operation ;
still larger units have been designed, and a few applications of gas power have
been made even, in marine service.
Natural gas is now transmitted to a distance of 200 miles, tinder 300 Ib. pres
sure. Illuminating gas has been pumped 52 miles. Martin (8) has computed that
coal gas might be transmitted from the British coalfields to London at a delivered
cost of 15 cents per 1000 cu. ft. His plan calls for a 25inch pipe line, at 500 Ib.
initial pressure and 250 Ib. terminal pressure, carrying 40,000,000,000 cu. ft. of
162
GAS POWER 163
gas per year. The estimated 46,000 hp. required for compression "would be derived
from the waste heat of the gas leaving the retorts.
Producer gas is even more applicable to heating operations than for power
production. It is meeting with extended use in ceramic kilns and for ore roast
ing, and occasionally even for firing steam boilers.
277, The Gas Engine Method. The expression for ideal efficiency,
(T t) r T, increases as T increases. In a steam plant, although boiler fur
nace temperatures of 2500 F. or higher are common, the steam passes to
the engine, ordinarily, at not over 350 IT. This temperature expressed in
absolute degrees limits steam, engine efficiency. To increase the value of
T 9 either very high, pressure or superheat is necessary, and the practicable
amount of increase is limited by considerations of mechanical fitness to
withstand the imposed pressures or temperatures. In the internal com
bustion engine, the working substance reaches a temperature approximat
ing 3000 F. in the cylinder. The gas engine has therefore the same ad
vantage as the hot air engine, a wide range of temperature. Its working
substance is, in fact, for the most part heated air. The fuel, which may
be gaseous, liquid, or even solid, is injected with a proper amount of air,
and combustion occurs within the cylinder. The disadvantage of the ordi
nary hot air engine has been shown to arise from the difficulty of trans
mitting heat from the furnace to the working substance. In this respect,
the gas engine has the same advantage as the steam engine, large capa
city for its bulk, for there is*no transmission of heat; the cylinder is
the furnace, and the products of combustion constitute the working sub
stance. A high temperature of working substance is thus possible, with
large work areas on the pv diagram, and a rapid rate of heat propagation.
In the gas engine, then, certain chemical changes which constitute the pro
cess described as combustion, must be considered ; although such changes are in gen
eral not to be included in the phenomena of engineering thermodynamics,
278. Fuels, (See Arts. 561, 561 a.) The common fuels are gases or oils. In*
so;ne sections, natural gas is available. This is high in heating value, consisting
mainly of methane, CH 4 . Carbureted water gas, used for illumination, is nearly as
high in heating value, consisting of approximately equal volumes of hydrogen,
carbon monoxide, and methane, with some methylene and traces of other substances.
.Uncarbureted (blue) water gas is almost wholly carbon monoxide and hydrogen.
Its heating value is less than half that of the carbureted gas. Both water gas and
coal gas are uneconomical for power production; in the processes of manufacture,
large quantities of coal are left behind as coke. Coal gas, consisting principally of
hydrogen and methane, is slightly lower in heating value than carbureted water
gas. It is made by distilling soft coal in retorts, about two thirds of the weight
of coal becoming coke. Coke oven gas is practically the same product; the main
output in its case being coke, while in the former it is gas.
Producer gas (" Dowson " gas, " Mond " gas, etc.) is formed by the
164 APPLIED THERMODYNAMICS
partial combustion of coal, crude oil, peat or other material, in air.
It is essentially carbon monoxide? diluted with large quantities of nitro
gen and consequently low in heating value. Its exact composition
varies according to the fuel from which it is made, the quantity of air
supplied, etc. When soft coal is used, or when much steam is fed to
the producer, large proportions of hydrogen are present.
It is of no value as an illuminant. Blast furnace gas is producer gas
obtained as a byproduct on a large scale in metallurgical operations. It contains
less hydrogen than ordinary producer gases, since steam is not employed in its
manufacture, and is generally quite variable in its composition on account of the
exigencies of furnace operation. Acetylene, C 3 H 2 , is made by combining calcium
carbide and water. It has an extremely high heating and illuminating value.
All hydiocarbonaceous substances maybe gasified by heating in closed vessels;
gases have in this way been produced from peat, sawdust, tan bark, wood, garbage,
animal fats, etc.
279. Oil Gases. Many liquid hydrocarbons may be vaporized by appropriate
methods, under conditions which make them available for gas engine use. Some
of these liquids must be vaporized by artificial heat and then immediately used, or
they will again liquefy as their temperatures fall. The vaporizer or gt carburetor "
is therefore located at the engine, where it atomizes each charge of fuel as required.
Gasoline is most commonly used ; its vapor has a high heating value. Kerosene,
and, more recently, alcohol, have been employed. By mixing gasoline and air in
suitable proportions, a saturated or " carbureted " air is produced. This acts as
a true gas, and must be mixed ^ ith more air bo permit of combustion. A gas
formed in the proportion of 1000 cu. ft. of air to 2 gallons of liquid gasoline, for
example, does not liquefy. A thiid form of oil gas is produced by heating certain
hydrocarbons without air; the "cracking" process produces, first, less dense
liquids, and, finally, gaseous bodies, which do not condense. The process must be
carried on in a closed retort, and arrangements must be made for the removal of
residual tar and coke.
280. Liquid Fuels. These have advantages over solid or gaseous fuels, aris
ing from the usually large heating value per unit of bulk, and from ease of trans
portation. All animal and vegetable oils and fats may be reduced to liquid fuels;
those oils most commonly employed, however, are petroleum products. Crude
petroleum maybe used; it is more customary to transform this to "fuel oil" by
removing the moisture, sulphur, and sediment; and some of these "fuel oils*' are
used in gas engines. Of petroleum distillates, the gaaolires are most commonly
utilized in this country. They include an 80 liquid, too dangerous for commer
cial purposes; the 74 "benzine," and the 69 naphtha. "Distillate," an impure
kerosene, from which the gasoline has not been removed, is occasionally used.
Both grain alcohol (C 2 H 6 0) and wood alcohol (CH 4 0) have been used in gas en
gines (9). Various distillates from brown and hard coal tars have been employed
in Germany. Their suitability for power purposes varies with different types of
engines. The benzol derived from coal gas tar has been successfully used ; the
brown coal series, C n H 2n , C n H 2n+2) C n H 2n _ 2 , contains many useful members (10).
THE GAS PRODUCER
165
281. The Gas Producer. This essential auxiliary of the modern gas
engine is made in a large number of types, one of which is shown in Fig.
113. This is a bricklined cylindrical shell, set over a watersealed pit P,
on which the ash bed rests. Air is forced in by means of the steam jet
blower A, being distributed by means of the conical hood B, from which
FIG. 113. Art, 281. The Amsler Gas Producer.
it passes up to the redhot coal bed above. Here carbon dioxide is formed
and the steam decomposes into hydrogen and oxygen. Above this " com
bustion zone" extends a layer of coal less highly heated. The carbon
dioxide, passing upward, is decomposed to carbon monoxide and oxygen.
The hot mixed gases now pass through the freshly fired coal at the top of
the producer, causing the volatile hydrocarbons to distill off, the entire
product passing out at C. The coal is fed in through the sealed hopper D.
166 APPLIED THERMODYNAMICS
At E are openings for the bars used to agitate the fire. At F are peep
holes.
An automatic feeding device is sometimes used at D. The air may
be forced in by a blower, or sucked through by an exhauster, or by the
engine piston itself, displacing the steam jet blower A. The fuel may
be supported on a solid grate, or on the bottom of a producer without the
water seal; grates may be either stationary or mechanically operated.
Mechanical agitation may be employed instead of the poker bars inserted
through E, Sometimes water gas, for illumination, and producer gas, for
power, are made in the same plant. Two producers are then employed,
the air blast being applied to one, while steam is decomposed in the other.
Provision must be made for purifying the gas, by deflectors, wet and dry
scrubbers, filters, coolers, etc. For the removal of tar, which would be seriously
objectionable in engines, mechanical separation and washing are useful, but the
complete destruction of this substance involves the passing of the gas through a
highly heated chamber; this may be a portion of the producer itself, as in
" underfeed," " inverted combustion/ 7 or " downdraft " types : causing the trans
formation of the tar to fixed gases. On account of the difficulty of tar removal,
anthracite coal or coke or semibituminous, noncaking coal must generally be used
in power plants. The air supplied to the producer is sometimes preheated by the
sensible heat of the waste gases, in a " recuperator." The " regenerative " prin
ciple heating the air and gas delivered to the engine by means of the heat of
the exhaust gases is inapplicable, for leasons which will appear.
282. The Producer Plant. The ordinary producer operates under a slight
piessure; in the suction type, now common in small plants, the engine piston
draws air through the producer in accordance with the load requirements. Pres
sure producers have been used on extremely low grade fuels : Jahn, in Germany,
has, it is reported, gasified mine waste containing only 20 per cent of coal. Suc
tion producers, requiring much less care and attention, are usually employed only
on the better grades of fuel. Most producers require a steam blast; the steam
must be supplied by a boiler or " vaporizer," which in many instances is built as a
part of the producer, the superheated steam being generated by the sensible heat
carried away in the gas. Automatic operation is effected in various ways: in
the Amsler system, by changing the proportion of hydrogen in the gas, involving
control of the steam supply ; in the Pintsch process, by varying the draft at the
producer by means of an inverted bell, under the control of a spring, from beneath
which the engine draws its supply; and in the Wile apparatus, by varying 1 the
drafb by means of valves operated from the holder. Figure 114 shows a complete
producer plant, with separate vaporizer, economizer (recuperator), and holder for
storing the gas and equalizing the pressure.
283. Byproduct Recovery. Coal contains from 0.5 to 3 per cent of nitrogen,
about 15 per cent of which passes off in the gas as ammonia. The successful
development of the Mond process has demonstrated the possibility of recovering
this in the form of ammonium sulphate, a valuable fertilizing agent.
THE GAS PRODUCER
167
168 APPLIED THERMODYNAMICS
284. Action in the Producer. Coal is gasified on the producer
grate. In suction producers, the rate of gasification may be anywhere
between 8 and 50 Ib. per sq ft. of grate per hour. Anthracite pro
ducers are in this country sold at a rating of 10 to 15 Ib. Ideally,
the coal is carbon, and leaves the producer as carbon monoxide,
4450 B. t. u. per pound of carbon having been expended 111 gasification.
Then only 10,050 B. t. u. per pound of carbon are present in the gas, and
the efficiency cannot exceed 10,050 * 14,500 = 0.694. The 4450 B. t. u. con
sumed m gasification are evidenced only in the temperature of the gas.
With actual conditions, the presence of carbon dioxide or of free oxygen
is an evidence of improper operation, further decreasing the efficiency. By
introducing steam, however, decomposition occurs in the producer, the tem
perature of the gas is reduced, and available hydrogen is carried to the
engine ; and this action is essential to producer efficiency for power pur
poses, since a high temperature of inlet gas is a detriment rather than a
benefit in engine operation. The ideal efficiency of the producer may thus
be brought up to something over 80 per cent; a limit arising when the
proportion of steam introduced is such as to reduce the temperature of the
gas below about 1800 F., when the rate of decomposition greatly decreases.
The proportion of steam to air, by weight, is then about 6 per cent, the
heating value of the gas is increased, the percentage of nitrogen decreased,
and nearly 20 per cent of the total oxygen delivered to the producer has
been supplied by decomposed steam. A similar result may be attained by
introducing exhausted gas from the engine to the producer. The carbon
dioxide in this gas decomposes to monoxide, which is carried to the engine
for further use. This method is practiced in the Mond system, and has
had other applications. To such extent as the coal is hydrocarbonaceous,
however, the ideal efficiency, irrespective of the use of either steam or
waste gas, is 100 per cent. Figure 115 shows graphically the results com
puted as following the use of either steam or waste gases with pure car
bon as the fuel. The maximum ideal efficiency is about 3 per cent greater
when steam is used, if the temperature limit is fixed at 1800 F., but the
waste gases give a more uniform (though less rich) gas. The higher ini
tial temperature of the waste gases puts their use practically on a parity
with that of steam. Either system tends to prevent clinkering. The
maximum of producer efficiency, for power gas purposes, is ideally from
5 to 10 per cent less than that of the steam boiler. High percentages of
hydrogen resulting from the excessive use of steam may render the gas
too explosive for safe use in an engine (10 a) (25).
285. Example of Computation. Let 20 per cent of the oxygen necessary for
gasifying pure carbon be supplied by steam. Each pound of fuel requires 1J Ib.
of oxygen for conversion to carbon monoxide. Of this amount, 0.20 x !$= 0.2666 Ib.
will then be supplied by steam ; and the balance, 1.0667 Ib., will be derived from
PRODUCER EFFICIENCY
169
the air, bringing in with it Jxi 0667=3 57 Ib. of nitrogen. The oxygen derived
from steam will also carry with it 4X02666=0.0333 Ib. of hydrogen. The pro
duced gas will contain, per pound of carbon,
2 33 Ib. carbon monoxide,
3 57 Ib. nitrogen,
0.0333 Ib. hydrogen.
Waste GassupphedjPercentageof Fuel gasified by Weight _g
109 202 256 382
J I I I
34 5 6 7 8 9 10 It tZ 13 M 15 16 17
Percentage of Steam by Weight.
FIG. 115. Art. 284. Reactions in the Producer.
The heat evolved in burning to monoxide is 4450 B. t. u. per pound. A por
tion of this, however, has been put back into the "gas, the temperature having been
lowered by the decomposition of the steam. Under the conditions existing in the
170 APPLIED THERMODYNAMICS
producer, the heat of decomposition is about 62,000 B t u per pound of hydrogen.
The net amount of heat evolved is then 4450  (0,0333 X 62,000) = 2383 B. t. u.,
and the efficiency is ' ~" = 0.84. The rise in temperatme is computed as
li,t)UU
follows : to heat the gas 1 F. there are required
SPECIFIC HEAT
For carbon monoxide, 2.33 X 0.2479 = 0.378 B. t. u.
For nitrogen, 3.57 X 0.2438 = 0.800 15. t u.
For hydrogen, 0.0333 x 3.4 = 113 B. t. u.
a total of 1.500 B. t. u.
The 2383 B. t. u. evolved will then cause an elevation of temperature of
. 2 3?3 = 1527 F.
1.560
With pure air only, used for gasifying pure carbon, the gas would consist of
2J Ib. of carbon monoxide and 4.45 Ib. of nitrogen ; the percentages being 34.5
and 65.5. For an actual coal, the ideal gas composition may be calculated on the
assumptions that the hydrogen and hydrocarbons pass oif unchanged, and that the
carbon requires 1J times its own weight of oxygen, part of which is contained in
the fuel, and part derived from steam or from the atmosphere, carrying with it
hydrogen or nitrogen. Multiplying the weight of each constituent gas in a pound
by its calorific value, we have the heating value of the gas. As a mean of 54
analyses, Fernald finds (11) the following percentages ly volume :
Carbon monoxide (CO) ............ 19.2
Carbon dioxide (COj) ............. 9.5
Hydrogen (H) ............... 12.4
Marsh gas and ethylene (CH 4 , C 2 H 4 ) ....... 3.1
Nitrogen (N) ................ 55.8
100.0
285 a. Practical Study of Producer Reactions. This subject has presented
unexpected complications. Tests made by Allcut at the University of Birming
ham (Power, July 18, 1911, page 90) call attention to three characteristic processes :
C + H 3 = CO + H a , (A)
C f 2 H 2 = C0 2 + 2 H 2 , (5)
CO + H 3 O = C0 3 + H 2 . (C)
Of these, (4) takes place at temperatures above 1832, is endothermic, and
results m the absorption of 4300 B. t, u. per pound of carbon. The corresponding
figure for reaction (), also endothermic, which occurs at temperatures below 1112,
is 2820 B. t. u. The former of the two is the reaction desired, and is facilitated
by high temperatures. The operation (C) is chemically reversible j taking place
as stated at temperatures above 932, but gradually reversing to the opposite (and
preferred) transformation when the temperature reaches 1832.
The tests show that increasing proportions of C0 2 may be associated with
increasing proportions of steam introduced. The maximum decomposition reached
was 0.535 Ib. of steam per pound of anthracite pea coal, at 1832 F. The maxi
THE GAS ENGINE
171
FIG. 116. Art. 287. Singleacting Gas Engine, Four Cycle.
(Prom " The Gas Engine, 1 ' by Cecil P. Poole, with the permission of the Hill Publishing Company )
FIG. 117. Art. 288. Piston Movements, Otto Cycle.
(From "The Gas Engine," by Cecil P. Poole, with the permission of the Hill Publishing Company.}
172 APPLIED THERMODYNAMICS
mum heat value iu the gas was obtained when 0.72 Ib. of steam was introduced
(only 0.52 Ib. of which was decomposed) per pound of coal. If we take the ratio
of air to coal by weight at 9 Ib., the ratio of steam decomposed to air supplied at
highest heat value and heat efficiency is 0.52 r 9.0 = 0.058 ; approximately 6 per
cent, as in Art. 284.
An interesting study of the principles involved may be found in Bulletins of
the University of Illinois ; vi, 16, by J. K. Clement, On the Rate of Formation of
Carbon Monoride in Gas Producers, and is., 2i, by Garland and Kratz, Tests of a
Suction Gas Producer.
286. Figure of Merit. A direct and accurate determination of efficiency is
generally impossible, on account of the difficulties in gas measurement (12). For
comparison of results obtained from the same coals, the figure of merit is sometimes
used. This is the quotient of the heating value per pound of the gas by the
weight of carbon in a pound of gas : it is the heating value of the gas per pound of
carbon contained. In the ideal case, for pure carbon, its value would be 10,050 B. t. u.
For a hydrocarbonaceous coal, it may have a greater value.
GAS ENGINE CYCLES
287. Fourcycle Engine. A gas engine of one of the most commonly used
types is shown in Fig. 116. This represents a singleacting engine; i.e. the gas is
in contact with one side of the piston only, the other end being open. Large en
gines of this type are frequently made doubleacting, the gas being then con
tained on both sides of a piston moving in an entirely closed cylinder, exhaust
occurring on one side while some other phase of the cycle is described on the
other side.
288. The Otto Cycle. Figure 117 illustrates the piston move
ments corresponding to the ideal pv diagram of Fig, 118. The
cycle includes five distinctly marked paths. During the out stroke
of the piston from position A to position jB, Fig. 117, gas is sucked
in by its movement, giving the line
5, Fig. 118. During the next in
ward stroke, B to 9 the gas is com
pressed, the valves being closed,
along the line Ic. The cycle is not
yet completed : two more strokes
are necessary. At the beginning
Fio.ll*. Arts.SSS.mTheOttoCrcle.
being at c, Fig. 118, the gas is ignited and practically instantaneous
combustion ogqurs at constant volume, giving the line <?(7, An out
THE TWOCYCLE ENGINE
173
stroke is produced, and as the valves' remain closed, the gas expands,
doing work along Cd, while the piston moves from Q to D, Fig. 117.
At d) the exhaust valve opens, and during the fourth stroke the
piston moves in from D to J?, expelling the gas from the cylinder
along de, Fig. 118. This completes the cycle. The inlet valve has
been open from a to 5, the exhaust valve from d to e. During the
remainder of the stroke, the cylinder was closed. Of the four
strokes, only one was a " working " stroke, in which a useful effort
was made upon the piston. In a doubleacting engine of this type,
there would be two working strokes in every four.
FIG. 110. Arts. 1289201, TO, 3#J. Twocycle Gas Engine.
(From "The Gas Engine," t>y Cecil P Poole, with the permission of the Hill Publishing Company )
289. Twostroke Cycle. Another largely used type of engine is shown
in Fig. 119. . The same five paths compose the cycle ; but the events are
now crowded into two strokes. The exhaust opening is at E ; no valve
is necessary. The inlet valve is at A, and ports are provided at C, and
/. The gas is often delivered to the engine by a separate pump, at a
pressure several pounds above that of the atmosphere ; in this engine, the
otherwise idle side of a singleacting piston becomes itself a pump, as
will appear. Starting in the position shown, let the piston move to the left.
It draws a supply of combustible gas through A, B and the ports into
the chamber D. On the outward return stroke, the valve A closes, and the
gas in D is compressed. Compression continues until the edge of the piston
passes the port I, when this high pressure gas rushes into the space F : at
174 APPLIED THERMODYNAMICS
practically constant pressure. The piston now repeats its first stroke.
Following the mass of gas which we have been considering, we find that
it undergoes compression, beginning as soon as the piston closes the ports
E and /, and continuing to the end of the stroke, when the piston is in its
extreme lefthand position. Ignition there takes place, and the next out
stroke is a working stroke, during which the heated gas expands. Toward
the end of this stroke, the exhaust port E is uncovered, and the gas passes
out, and continues to pass out until early on the next backward stroke this
port is again covered.
390. Discussion of the Cycle. We have here a twostroke cycle ; for
two of the four events requiring a perceptible time interval are always
taking place simultaneously. On the first stroke to the left, while gas is
entering D, it is for a brief interval of time also flowing from 7 to F, from
F through E, and afterward being compressed in F. On the next stroke
to the right, while gas is compressed in Z>, ignition and expansion occur in
F] arid toward the end of the stroke, the exhaust of the burned gases
through E and the admission of a fresh supply through J, both begin..
The inlet port I and the exhaust port E are both open at once during part
of the operation. To prevent, as far as possible, the fresh gas from,
escaping directly to the exhaust, the baffle G is fixed on the piston. It is
only by skillful proportioning of port areas, piston speed, and pressure in
D that large loss from this cause is avoided. * The burned gases in the
cylinder, it is sometimes claimed, form a barrier between the fresh enter
ing gas and the exhaust port.
291. PV Diagram. This is shown for the working side (space F) in
Fig. 120 and for the pumping side (space D) in Fig. 121. The exhaust
port is uncovered at tf, Fig. 120, and the pres
sure rapidly falls. At a, the inlet port opens,
the fresh supply of gas holding up the pres
sure. From a out to the end of the diagram,
and back to 6, both ports are open. At & the
inlet port closes, and at c the exhaust port,
when compres
sion begins. The
pump diagram of
FIG. 120. Art 291. Twostroke Fig. 121 COrre
y cle  spends with the
negative loop deal of Fig. 118. Aside from FIG. 13L Art. 291. Twostroke
the slight difference at dabc, Fig. 120, the Cycle Pump Diagram.
* Two cycle gas engines should never be governed by varying the quantity of
mixture drawn in (Art. 348) because of the disturbing effect which such variations
would have on these factors.
THE OTTO CYCLE
175
diagrams for the twocycle and fourcycle engines are precisely the same;
and in actual indicator cards, the difference is yery slight.
292. Ideal Diagram. The perfect PV
diagram for either engine would be that of
Fig. 122, ebfd, in which expansion and com
pression are adiabatic, combustion instan
taneous, and exhaust and suction unre
FIG. 122. Arts. 292, 293, 29i, Stricte(1 I so that the area of the negative
295, 314, 329, 1329a, 329Z>, loop dg becomes zero, and eb and fd are
331. Prob. 15. Idealized . * , ,**
Gas Engine Diagram. lmes of constant volume. From inspection
of the diagram we find
293. Work Done. The work area under Jfis
under ed is
TT _ p I?"
' '  
; that
; the net work of the cycle is
This may be written in terms of two pressures and two volumes only,
for P e V e = PtVjfVJv and P f V f = P d F 6 "F^, giving
a P* W V^  P Vf VJ*
294.
4. Relations of Curves. Expressing ^ = ^Y and ^ = (" Y, and
I/ v^i/ fd \y*j
remembering that F 5 = V* 7,= V d , we have ^ = ^ and ^ = ?f . This
/ *, d ** *d
permits of rapidly plotting one of the curves when the other is given.
We also find ^ and = 
* *d * Li
176
APPLIED THERMODYNAMICS
295. Efficiency. In Fig. 122, heat is absorbed along 06, equal to
l(T b Tg); this is derived from the combustion of the gas. Heat
is rejected along fd, =l(T f T a ). Using the difference of the two
quantities as an expression for the work done, we obtain for the
efficiency
T t  T e  T f
The efficiency thus depends solely upon the extent of compression
TF ) > while ^ ~r=the clearance of the engine,
V d / V d~ V e
'and since
&
1
LOO
.80
.60
.40
,20
i
\
\
X
^x,
^
^.
""
JO .20 .30 .40 .50 .60 .70 .80 .90 LOO
Clearance
PIG. 122a. Art. 295. Relation between Efficiency and Clearance in the Ideal Cycle.
the efficiency may be expressed in terms of the clearance only. (See
Fig. 122a.)
295 a. The Sargent Cycle. Let the engine draw in its charge at atmos
pheric pressure, along ad, Fig. 122 c. The inlet valve closes at d and the
charge expands somewhat, along dc. It is then compressed along cde,
ignited along e& ; and expanded along bg. The exhaust valve opens at g,
the pressure falls to that of the atmosphere along gh, and the cylinder
contents are expelled along ha. The work area is debfgh^ there is 110
negative loop work area dhc. The entropy diagram shows the cycle to
THE SARGENT AND THE FRITH CYCLES 177
be more efficient than the Otto cycle debf between the same temperature
limits ; the superior Otto cycle ebgc has wider temperature limits. The
gain by the Sargent cycle is analogous to that in a steam engine by an
increased ratio of expansion (Art. 411), and involves a reduction in capac
ity in proportion to the size of cylinder. The efficiency is
debgh __ mebn mdhgn
mebn mebn
_ T h T d
~~ y T b T< T,T e
295 o. The Frith Regenerative Cycle (Jowr. A. S. M. ., XXXII, 7). In Fig.
122 d, abed is an ordinary Otto cycle. Suppose that during expansion some of the
fluid passes through a regenerator, giving up heat, following some such path as ae.
Then let the regenerator in turn impart this heat to the working substance during
or just before combustion, as along di in the entropy diagram.
If the regenerator were perfect, and the transfers as described could occur, the
heat absorbed from external sources would be jiah and the work would be daec.
The quotient of the latter by the former, if the path through the regenerator were
ac (limiting case), would be unity. But this would involve a contravention of the
second law, since heat would have to pass from the regenerator (at c) to a sub
stance hotter than itself (at d). If, however, we make the temperature range T d T e
very small, a large proportion of the heat transferred to the regenerator may again
be absorbed along da, and as the output of the engine approaches zero, its efficiency
approaches 100 per cent.
If, as in Fig. 122 5, the expansion curve strikes the point c, we may assume
that of all the heat (fcaty delivered to the regenerator, only that portion (Ikah),
the temperature of which exceeds T& can be redelivered to the fluid along da.
The efficiency is then
dac _ fdah fcah
fdah Ikah fdah Ikah
n 2/
 T d ) s(T a  T e ) _ n 1
T a T d   r (7 T a  T d )
where s = I ~ is the specific heat along the path akc, the equation of which is
pv n '= const. Since ' P a Va n = PcV c n t
while PdVd v =P G Vc v ,
PC , PC
178
APPLIED THERMODYNAMICS
FIG. 1226 Art. 2955.
T
FIG. 122e. Art
T
/
FIG. 122d. Art. 295b.
Let P c = 14.7, P d = 147, P tf = 294.
Then
ny 0.561
n  1 0.963
log 0.10
= 0.582.
1.963,
JY
IV
ATKINSON ENGINE
179
Now if T c = 300 P. = 760 abs,, T d = 1470 abs., and if T a = 3000 abs., the
efficiency becomes
1530  (0.582 x 2240) = 230 =
1530  (0.582 x 1530) 640 " *
while that of the Otto cycle is
T d  T c _ 1470  760 = Q 4g
T d 1470
For a discussion of limiting values, see the author's paper in Polytechnic Engineer^ 1914.
296. Carnot Cycle and Otto Cycle; the Atkinson Engine. Let nbcd,
Fig. 123, represent a Carnot cycle drawn to pv coordinates, and bfde, the
corresponding Otto cycle between the
same temperature limits, T and t. For the
Carnot cycle, the efficiency is (T t) = T 7 ;
for the Otto, it is, as has been shown,
(T e T d ) H T e . It is one of the disad
vantages of the Otto cycle, as shown in
Art. 294, that the range of temperatures
during expansion is the same as that dur
ing compression. In the ingenious Atkin
son engine (13), the fluid was contained in
the space between two pistons, which space
was varied during the phases of the cycle. This permitted of expansion
independent of compression ; in the ideal case, expansion continued down,
to the temperature of the atmosphere, giving such a diagram as ebcd, Fig.
123. The entropy diagrams for the Carnot, Otto, and Atkinson cycles are
correspondingly lettered in Fig. 124. For
the Atkinson cycle, in the ideal case, we
have iii Fig. 124 the elementary strip
vicxy, which may stand for dH, and the
isothermal dc at the temperature t. Let
the variable temperature along eb be T x ,
having for its limits T b and T^ Then, for
the area ebcd, we have
FIG. 124. Arts 296, 297, 305, 307,
3296. Efficiencies of Gas Engine
Cycles.
The efficiency is obtained by dividing by I (T b T e ) and is equal to
f /TT
^ v i J /
FIG. 123. Art. 2% Carnot, Otto,
and Atkinson Cj eles.
C^dff fr b dT x (
= I Tjr^* V = ' L ~m~ ( 2 * "
+J TC ^ x *J T e L s
297. Application to a Special Case. Let T e = 1060, T
whence, from Art, 294, T/ = 1688. We then have the following ideal efficiencies:
180
APPLIED THERMODYNAMICS
Carnot,
Atkinson, 1
Otto,
Tt_ 3440 520
T ~ 3440
.. n . 620,.
= 0.85.
3440
A _
= 0.74.
T e t_ 1060  520
T e ~~ 1060
= 0.51.
The Atkinson engine can scarcely be regarded as a practicable type ; the
Otto cycle is that upon which most gas engine efficiencies must be based;
and they depend solely on the ratio of temperatures or pressures during
compression.
298. Lenoir Cycle. This is shown in Fig. 125. The fluid is drawn
into the cylinder along Ad and exploded along df. Expansion then
occurs, giving the path/, when the exhaust valve opens, the pressure
INSTANT VOLUME
FIG. 125. Arts 298, 301, 302.
Lenoir Cycle.
FIG. 12(5. Art 298. Entropy
Diagram, Lenoir Cycle.
falls, g7ij until it reaches that of the atmosphere, and the gases are finally
expelled on the return stroke, liA. It is a twocycle engine. The net
entropy diagram appears in Fig. 126.
The efficiency is
Heat absorbed  heat rejected _ ?(2>  !T d )  l(T 9  T h )  k(T h  T d )
Heat absorbed "~ ( 7/ T d )
299. Brayton Cycle. This is shown in Fig. 127. A separate
pump is employed. The substance is drawn in along Ad, compressed
along dn, and forced into a reservoir along n. The engine begins
to take a charge from the reservoir at B, which is slowly fed in and
ignited as it enters, so that combustion proceeds at the same rate as
the piston movement, giving the constant pressure line 1. Expan
sion then occurs along lg, the exhaust valve opens at g, and the
charge is expelled along Ji A. The net cycle is dnbgh^ the net ideal
entropy diagram is as in Fig. 128. This is also a twocycle
BRAYTON CYCLE
181
FIG. 127. Arts, 2<>9, P>02. Bray ton
Cycle.
FIG. 128 Art. 2VI9 Bray ton Cycle,
Entropy Diagram.
engine. The " constant pressure " cycle which it uses was suggested
in 1865 by Wilcox. In 1873, when first introduced in the United
States, it developed an efficiency of 2.7 Ib. of (petroleum) oil per
brake hp.hr.
The efficiency is (Fig. 127)
If expansion is complete, the cycle becoming dnli, Pigs. 127, 128, then
T g = T h = T t} and the efficiency is
/in rrj fTi fTi __ fji
HH r r* ^r* r r* '
a result identical with that in Art. 295 ; the efficiency (with complete ex
pansion) depends solely upon the extent of compression.
300. Comparisons with the Otto Cycle. It is proposed to compare the capacities
and efficiencies of engines working in the Otto,* Brayton, and Lenoir cycles; the
engines being of the same size, and working "between the same limits of temperature.
For convenience, pure air will be regarded as the working substance. In each case
let the stroke be 2 ft., the piston area 1 sq. ft., the external atmosphere at 17 C.,
the maximum temperature attained, 1537" 1 C. In the Lenoir engine, let ignition
occur at half stroke; in the Brayton, let compression begin at half stroke and con
tinue until the pressure is the same as the maximum pressure attained in the Lenoir
cycle, and let expansion also begin at half stroke. These are to be compared with
an Otto engine, in which the pump compresses 1 cu. ft. of free air to iO Ib. net
pressure. This quantity of free air, 1 cu. ft., is then supplied to each of the three
engines.
301. Lenoir Engine. The expenditure of heat (in work units) along df, Fig.
125, is Jl(T  0> in which T = 1537, t  17, J is the mechanical equivalent of a
Centigrade heat unit, and / is the specific heat of 1 cu. ft. of free air,
* The " Otto cycle " in this discussion is a modified form (as suggested by Clerk)
in which the strokes are of unequal length.
182 APPLIED THERMODYNAMICS
heated at constant volume 1 (J. Now J ' 778 x 3 = HOO.i, and *// ]d a;>p
mately 0.1689 x 0.075 x 1400.4 = 17.72. The expendituie of heat is then
17.72(1537  17) = 26,900 ft.lb.
The pressure at /is
UJ 1587 + 273 == Q1A lb< absolufce .
17 + 27*3
and the pressure at g is
91.4 (i)v = 34.25 Ib. absolute.
The work done under fg is then
= 8190 ft.lb.
The negative woik under fid is 14 7 x 144 x 1 = 2107 ft.lb., and the net work is
8190  2107 = 6083 ft.lb. The efficiency is then 6083  26,900 = 226.
302. Brayton Engine. We first find (Fig. 127)
v!
Tn = T d (^\ v = (273 + 17) (~~}^ = 489 absolute or 216 C.
Proceeding in the same way as with the Lenoir engine, we find the heat expendi
ture to be
Jk(Ti  T n ) = 2375 x 0.075 x 1400.4(1537  216) = 33,000 ft.lb.
The pressure at n is by assumption equal to p f in the case of the Lenoir engine;
the pressure at g in the Brayton type then equals that at g in the Leuoir. The
work under Ig is the same as that under fg in Fig. 125. The work under nb is
found by first ascertaining the volume at n. This is
UTj^LO =0.272.
9.14/
The work under nb is then 91.4 x 144 x (1  0.272) = 9650 ft.lb., and the gross
work is 9650 + 8190 = 17,840 ft.lb. Deducting the negative work under hd,
2107 ft.lb., and that under dn,
i44 *_ x = 3650 ft.lb.,
the net work area is 12,083 ft.lb., and the efficiency, 12,083  33,000 = 0.366'.
303. Clerk's Otto Engine. In Fig. 129, a separate pump takes in a charge
along AB, and compresses it along BC, afterward forcing it into a receiver along
CD at 40 Ib. gauge pressure. Gas flows from
the receiver into the engine along DC, is ex
ploded along CEj expands to F, and is expelled
along GA. The net cycle is BCEFQ. The
volume at C is
~ y = 0.393 cu. ft.
FIG. 129. Arts. 303, 305, Clerk's
Otto Cycle.
CLERK'S GAS ENGINE 183
The temperature at C is
0.393) (278 + 17)
14.7 x 1
The pressure at E is then
(1537 + 273)54.7 = 2311
Io3 + 273
The pressure at F is
231 (^f^V = 23.64 Ib. absolute.
The work under EF is
_ 27:3 = 133 o a
that under 5(9 is 2107 ft.lb., and that under EC is
ltf / (54.7 x 0.393) (14. 7 x
V 1.402  1.0
The net work is 15,600  2107  2430 = 11,063 ft.lb. The heat expenditure in
this case is Jl(T E  T c ) = 17.72 x (1537  153) = 24,500 ft.lb., and the efficiency
is 11,063  24,500 = 0.453; considerably greater than that of either the Lenoir or
the Brayton engine (14). If we express the cyclic area as 100, then that of the
Lenoir engine is 52 and that of the Brayton engine is 104. (See Art. 295a.)
304. Trial Results. These comparisons correspond with the consumption of
gas found in actual practice with the three types of engine. The three efficiencies
are 0.226, 0.366, and 0.453. Taking 4 cu. ft. of free gas as ideally capable of giv
ing one horse power per hour, the gas consumption per hp.lir. in the three cases
would be respectively 4  0.226 = 17.7, 4  0.3C6 = 10.9, and 4  0.453 = 8.84 cu. ft.
Actual tests gave for the Lenoir and Hugon engines 90 cu. ft. ; for the Brayton,
50 ; and for the modified Otto, 21. The possibility of a great increase in economy
by the use of an engine of a form somewhat similar to that of the Brayton will be
discussed later.
305. Complete Pressure Cycle. The cycle of Art. 303 merits detailed exami
nation. In Fig. 129, the heat absorbed is l(T E  7 j that rejected is
the efficiency is
The entropy diagram may be drawn as ebmnd, Fig. 124, showing this cycle to be
more efficient than the equalleugthstroke Otto cycle, but less efficient than the
Atkinson. With complete expansion down to the lower pressure limit, the cycle
becomes BCEFH, Fig. 129, or ebo<U Fig. 121; the strokes are still of unequal
length, and the efficiency is (Fig. 129)
184 APPLIED THERMODYNAMICS
If the strokes be made of equal length, with incomplete expansion, T G =T i the
cycle becomes the ordinary Otto, and the efficiency is
1 T F Tn = TcTn
r r r n / 7 T "
1 E  1 c JC
306. Oil Engines : The Diesel Cycle. Oil engines may operate in either
the twostroke or the fourstroke cycle, usually the latter; and combus
tion may occur at constant volume (Otto), constant pressure (Brayton), or
constant temperature (Diesel). Diesel, in 1893 (15), first proposed what
has proved to be from a thermal standpoint the most economical heat
engine. It is a fourcycle engine, approaching more closely than the
Otto to the Carnot cycle, and theoretically applicable to solid, liquid, or
gaseous fuels, although actually used only
with oil. The first engine, tested by Schroter
in 1897, gave indicated thermal efficiencies
ranging from 0.34 to 0.39 (16). The ideal
ized cycle is shown in Fig. 130. The opera
tions are adiabatic compression, isothermal
~ v expansion, adiabatic expansion, and dis
FIG. 130. Arts. 306, 307. Diesel _/ ' , . 1 ^ ' . .
Cycle, charge at constant volume. Pure air is com
pressed to a high pressure and temperature,
and a spray of oil is then gradually injected by means of external air
pressure. The temperature of the cylinder is so high as at once to ignite
the oil, the supply of which is so adjusted as to produce combustion
practically at constant temperature. Adiabatic expansion occurs after
the supply of fuel is discontinued. A considerable excess of air is used.
The pressure along the combustion line is from 30 to 40 atmospheres, that
at which the oil is delivered is 50 atmospheres, and the temperature
at the end of compression approaches 1000 P. The engine is
started by compressed air; two or more cylinders are used. There is
no uncertainty as to the time of ignition; it begins immediately
upon the entrance of the oil into the cylinder. To avoid preignition
in the supply tank, the highpressure air used to inject the oil must
be cooled. The cylinder is waterjacketed. Figure 131 shows a three
cylinder engine of this type; Fig. 132, its actual indicator diagram,
reversed.
The Diesel engine has recently attracted renewed interest, especially
in small units: although it has boen built in sizes up to 2000 hp. It
has been applied in marine service, and has successfully utilized by
product tar oil.
THE DIESEL ENGINE
FIG. 131. Art. 306. Diesel Engine. (American Diesel Engine Company.)
FIG. 1J2. Art. OOC. Indicator Diagram, Diesel Engine.
(16 X 24 In. engine, 100 r.p.m. Spring 400.)
186 APPLIED THERMODYNAMICS
307. Efficiency. The heat absorbed along J, Fig. 130, is
The heat rejected along/c? is l(T f T^. We may write the efficiency
as
i
i
But 2>= r rancl Z^; whence
y
For the heat rejected tilong/d we may therefore write
*rfY r 'Y~ 1 il
i,/ 1  1 i ,
y LVT' a / J
and for the efficiency,
This increases as T a increases and as ~~ decreases. The last conclu
Ka
sion is of prime importance, indicating that the efficiency should in
crease at light loads. This may be apprehended from the entropy
diagram, abfd, Fig. 124. As the width of the cycle decreases (If
moving toward ad), the efficiency increases,
307 &. Diesel Cycle with Pressure Constant. In common present practice,
the engine is supplied with fuel at such a rate that the pressure, rather
than the temperature, is kept constant during combustion. This gives a
much greater work area, in a cylinder of given size, than is possible with
isothermal combustion. The cycle is in this case as shown in Fig. 132 a,
combining features of those of Otto and Brayton. The entropy diagram
shows that the efficiency exceeds that of the Otto cycle ebfd between the
THE DIESEL ENGINE
187
same limits ; but it is less than that of the Diesel cycle with isothermal
combustion. The definite expression for efficiency is
r,) T f T d
mabn
Inspection of the diagram shows that the efficiency decreases as the load
increases.
(For a description of the Junkers engine, see the papers by Junge, in
Power, Oct. 22, 29, Nov. 5, 1912 )
P T
N
m
FIG. 132a. Art. 307&. Constantpressure Diesel Cycle.
3070. Entropy Diagram, Diesel Engine. In constructing the entropy diagram
from an actual Diesel indicator card a difficulty arises similar to one met with in
steam engine cards; the quantity of substance m the cylinder is not constant (Art. 454 ) .
This has been discussed by Eddy (17), Frith
(18), and Reeve (19). The illustrative dia
gram, constructed as in Art. 347, is sugges
tive. Figure 133 shows such a diagram for
an engine tested by Denton (20). The
initially hot cylinder causes a rapid ab
sorption of heat from the walls during the
early part of compression along db. Later,
along be, heat is transferred in the opposite
direction. Combustion occurs along cd, the
temperature and quantity of heat increas
ing rapidly. During expansion, along de,
the temperature falls with increasing
rapidity, the path becoming practically
adiabatic during release, along ef. The TV diagram of Fig; 133 indicates that no
further rise of temperature would accompany increased compression; the actual
path at y has already become practically isothermal.
308. Comparison of Cycles. Figure 134 shows all of the cycles that
have been discussed, on a single pair of diagrams. The lettering cor
responds with that in Pigs. 122128, 130. The cycles are,
FIG. 133. Art. ^07. Diesel Engine
Diagrams.
188
APPLIED THERMODYNAMICS
Garnet, abed, Lenoir, d/o0^o>#Mb Diesel, ddbf,
Otto, ebfd, Brayton, diibgli, dnU, Atkinson, ebcd,
Complete pressure, debgh, debi.
FIG. 134. Art. 308, Probs. 7, U5. Comparison of Gas Engine Cycles.
3080. The Humphrey Internal Combustion Pump. In Kg. 134a,
C is a chamber supplied with water through the check valves V from
the storage tank .ET, and connected by the discharge pipe D with the
delivery tank F. Suppose the lower part of C, with the pipe D and
the tank F, to be filled with water, and a combustible charge of gas
to be present in the upper part of C, the valves I and E being closed.
The gas charge is exploded, and expansion forces the water down
in C and up in F. The movement does not stop when the pressure
of gas in C falls to that equivalent to the difference in head between
F and C ; on the contrary, the kinetic energy of the moving water
carries it past the normal level in F, and the gases in C fall below
that pressure due to head. This causes the opening of E and F,
an inflow of water from ET to C, and an escape of burnt gas from C
through E. The water rises in C. Meanwhile, a partial return flow
from F aids to fill C, the kinetic energy of the moving water having
been exhausted, and the stream having come to rest with an abnor
mally high level in F. Water continues to enter C until (1) the valves
V are closed, (2) the level of E is reached, when that valve closes by
the impact of water; and (3) the small amount of burnt gas now trapped
in the space Ci is compressed to a pressure higher than that correspond
ing with the difference of heads between F and Ci. As soon as the
returning flow of water has this time been brought to rest, the excess
pressure in C\ starts it again in the opposite direction from Ci toward
F. When the pressure in Ci has by this means fallen to about that of
the atmosphere, a fresh charge is drawn in through J. Frictional
losses prevent the water, this time, from rising as high in F as on its
first outflow; but nevertheless it does rise sufficiently high to acquire
THE HUMPHREY INTERNAL COMBUSTION PUMP 189
a static head, which produces the final return flow which finally com
presses the fresh charge.
The water here takes the place of a piston (as in the hydraulic
piston compressor, Art. 240). The only moving parts are the valves.
Gas ft
FIG. 134a. Art. 308a. Humphrey Pump.
The action is unaccompanied by any great rise of temperature of the
metal, since nearly all parts are periodically swept by cold water. The
pump as described works on the
fourcycle principle, the operations
being (Fig. 1346):
a. Ignition (a&) and expansion
b. Expulsion of charge (cd, de),
suction of water, com
pression of residual
charge (ef) ;
c. Intake (feg, gh) ;
d. Compression (ha).
Disregarding the two loops ehg,
dcm, the cycle is bounded by two
polytropics, one line of constant volume and one of constant pressure.
Between the temperature limits T b and T h it gives more work than
the Otto cycle habj, and if the curves be and ah were adiabatic would
necessarily have a higher efficiency than the Otto cycle. The actual
paths are not adiabatic: during expansion (as well as during ignition)
some of the heat must be given up to the water; while the heat generated
by compression is similarly (in part) transferred to the water along ha.
With the adiabatic assumption adopted for the purpose of classification,
the cycle is that described in Art. 305 and shown in Fig. 134 as debi.
The strokes are of unequal length. (See Power, Dec. 1, 1914.)
The gases are so cool toward the end of expansion that a fresh
FIG. 1346. Art. 308a. Cycle of
Humphrey Pump.
190 APPLIED THERMODYNAMICS
charge may be safely introduced at that point, by outside compression
on the twocycle principle (Art. 289). The pump may be adapted
for high heads by the addition of the hydraulic intensifies It has
been built in sizes up to 40,000,000 gal. per twentyfour hours, and
has developed a thermal efficiency (to water) under test of about
22 per cent. (See American Machinist, Jan. 5, 1911.)
PRACTICAL MODIFICATIONS OF THE OTTO CYCLE.
309. Importance of Proper Mixture. The working substance used in gas
engines is a mixture of gas, oil vapor or oil, and air. Such mixtures will not
ignite if too weak or too strong Even when so proportioned as to permit of
ignition, any variation from the correct ratio has a detrimental effect; if
too little air is present, the gas will not burn completely, the exhaust will be dart
colored and odorous, and unburned gas may explode in the exhaust pipe when
it meets more air. If too much air is admitted,
the products of combustion will be unnecessarily
diluted and the rise of temperature daring
ignition will be decreased, causing a loss of work
area on the PV diagram. Figure 185 shows the
effect on rise of temperature and pressure of
varying the proportions of air and gas, assuming
the variations to remain within, the limits of
~~ possible ignition. Fail Lire to ignite may occur
Bto. 133. Art 309. Effect T ^^ of the ^ of eMM( rf ^ ag
Mixture otrengLn. A ,.,,., n 7 ,
well as when the air supply is deficient. Rapidity
of flame propagation is essential fur efficttnry, and this is only possible with a
proper mixture. The gas may in some ca*es bum so slowly as to leave the cyl
inder partially unconsumed In an engine of the t\pe shown hi Fig. 119, this
may result in a spread of flame through /, B, and C back to D, with dangerous
consequences.
310. Methods of Mixing. The constituents of the mixture must be intimately
mingled in a finely divided state, and the governing of the engine should prefei 
ably be accomplished by a method which keeps the proportions at those of highest
efficiency. Variations of pressure in gas supply mains mav interpose serious dif
ficulty in this respect. Fluctuations in the lights which may be supplied from the
same mains are also excessive as the engine load changes. Both difficulties are
sometimes obviated in. small units by the use of a rubber supply receiver. Varia
tions in the speed of the engine often change the proportions of the mixture.
"When the air is drawn from out of doors, as with automobile engines, variations
in the temperature oE the air affect the mixture composition. In simple types of
engine, the relative openings of the automatic gas and air inlet valves are fixed
when the engine is installed, and are not changed unless the quality or pressure
of the gas changes, when a new adjustment is made by the aid of the indicator or
by observation of the exhaust. Mechanically operated mixing valves, usually of
the "butterfly" type, are used on highspeed engines; these are positive in their
ALLOWABLE COMPRESSION 191
action. The use of separate pumps for supplying air and gas permits of proportion
ing in the ratio of the pump displacements, the volume delivered being constant,
regardless of the pressure or temperature. Many adjustable mixing valves and
carbureters are made, in which the mixture strength may be regulated at will.
These are necessary where irregularities of pressure or temperature occur, but
require close attention for economical results. In the usual type of carbureter or
vaporizer, used with gasoline, a constant level of liquid is maintained either by an
overflow pipe or by a float. The suction of the engine piston draws air through a
nozzle, and the fuel is drawn into and vaporized by the rapidly moving air current.
Kerosene cannot be vaporized without heating it: the kerosene carbureter may be
jacketed by the engine exhaust, or the liquid may be itself spurted directly into
the cylinder at the proper moment, air only being present in the cylinder during
compression. The presence of burned gas in the clearance space of the cylinder
affects the mixture, retarding the flame propagation. The effect of the mixture
strength on allowable compression pressures remains to be considered.
311. Actual Gas Engine Diagram. A typical indicator diagram from
a good Otto cycle engine is shown in Fig. 136. The various lines differ
somewhat from those established in Art. 28S. These differences we now
discuss. Figure 137 shows the portion bcde of the diagram in Fig. 133
to an enlarged vertical scale, thus representing the action more clearly.
The line/0 is that of atmospheric pressure, omitted in Fig. 136. TVe will
begin our study of the actual cycle with the compression line.
FIG. 136. Arts. 311, 342, 345. FIG. 137. Arts. 311, 3^H, 328.  Eii
Otto Engine Indicator Diagram. larged Portion of Indicator Diagram.
312. Limitations of Compression. It has been shown that a high degree
of compression is theoretically essential to economy. In practice, com
pression must be limited to pressures (and corresponding temperatures)
at which the gases will not ignite of themselves ; else combustion will
occur before the piston reaches the end of the stroke, and a backward
impulse will be given. Gases differ widely as to the temperatures at
which they will ignite; hydrogen, for example, inflames so readily that
Lucke (21) estimates that the allowable final pressure must be reduced
one atmosphere for each 5 per cent of hydrogen present in a mixed gas.
The following are the average final gauge compression pressures
recommended by Lucke (22) : for gasoline, in automobile engines,
45 to 95 lb., in ordinary engines, 60 to 85 Ib. ; for kerosene, SO to 85 lb.;
for natural gas, 75 to ISO lb. ; for coal gas or carbureted water gas,
192 APPLIED THERMODYNAMICS
60 to 100 II. ; for producer gas, 100 to 160 Ib. ; and for blast furnace
ffas, 120 to 190 Ib. The range of compression depends also upon the
pressure existing in the cylinder at the beginning of compression ; for
twocycle engines, this varies from 18 to 21 Ib., and for fourcycle
engines, from 12 to 14 Ib., both absolute.
The precompression temperature also limits the allowable range below the
point of self ignition. This temperature is not that of the entering gases, but it
is that of the cylinder contents at the moment when compression begins ; it is
determined by the amount of heat given to the incoming gases by the hot cylin
der walls, and this depends largely upon the thoroughness of the water jacketing
and the speed of the engine. This accounts for the rather wide ranges of allow
able compression pressures above given. Usual precompression temperatures are
from 140 to 300 F. " Scavenging" the cylinder \uth cold air, the injection of
water, or the circulation of water in tubes in the clearance space, may reduce this.
Usual practice is to thoroughly jacket all exposed sm faces, including pistons
and valve faces, and to avoid pockets where exhaust gases may collect. The
primary object of jacketing, however, is to keep the cylinder cool, both for
mechanical reasons (e g., for lubrication) and to avoid uncontrollable explosions at
the moment when the gas reaches the cylinder.
313. Practical Advantages of Compression. Compression pressures have
steadily increased since 1881, and engine efficiencies have increased correspond
ingly, although the latter gain has been in part due to other causes. Improved
methods of ignition have permitted of this increased compression. Besides the
therm odynarnic advantage already discussed, compression increases the engine
capacity. In a noncompressive engine, no considerable range of expansion could
be secured without allowing the final pressure to fall too low to give a large work
area; in the compressive engine, wide expansion limits may be obtained along
with a fairly high terminal pressure. Compression reduces the exposed cylinder
surface in proportion to the weight of gas present at maximum temperature, and
so decreases the loss of heat to the walls. The decreased proportion of clearance
space following the use of compression also reduces the proportion of spent gases
to be mixed with the incoming charge.
314. Pressure Rise during Combustion. In Art. 292, the pressure P b after
combustion was assumed. "VVhile, for reasons which will appear, any computation
of the. rise of pressure by ordinary methods is unreliable, the method should be
described. Let H denote the amount of heat liberated by combustion, per pound
of fuel. Then, Fig. 122, H= l(T b  IT.), T b  T e = and T b = + TV But
= S * + i. Tim ft !>. = . But * whence
11 e
P.
IT.
COMPUTED MAXIMUM TEMPERATURE
193
Then
315. Computed Maximum Temperature. Dealing now with the constant
volume ignition line of the ideal diagram, let the gas be one pound of pure
carbon monoxide, mixed with just the amount of air necessary for com
bustion (2.48 lb.), the temperature at the end of compression being 1000
absolute, and the pressure 200 lb. absolute. Since the heating value of 1
lb. of CO is 4315 B. t. u., while the specific heat at constant volume of
C0 2 is 0.1692, that of N being 0.1727, we have
rise in temperature = 
4315
 = 7265F.
(1.57 x 0.1692}+ (1.91 x 0.1727)
The temperature after complete ignition is then 8265 absolute. The
pressure is 200 x ^ = 1653 lb. If the volume increases during igni
1000
tion, the pressure decreases. Suppose the volume to be doubled, the rise
of temperature being, nevertheless, as computed : then the maximum pres
sure attained is 826.5 lb.
Compression Ratio ( \* "Fijr.122 )
FIG. 137a. Art. 316. Rise of Pressure in Practice.
316. Actual Maxima. No such temperature as 8265 absolute is
attained. In actual practice, the temperature after ignition is usually
194 APPLIED THERMODYNAMICS
about 3500 absolute, and the pressure under 400 Ib. The rise of either
is less than half of the rise theoretically computed, for the actual air
supply, with the actual gas delivered. The discrepancy is least for
oil fuels and (mixtures being of proper strength) is greatest for fuels
of high heat value. It is difficult to measure the maximum temperature,
on account of its extremely brief duration. It is more usual to ni3asure
the pressure and compute the temperature. This is best dons by
a graphical method, as with the indicator. Fig. 137a gives the results
of a tabulation by Poole of pressure rises obtained in usual practice.
317. Explanation of Discrepancy. There are several reasons for the disagree
ment between computed and observed results. Charles' law does not hold rigidly
at high temperatures; the specific heats of gases are known to increase with the
temperature (Meyer found in one case the theoretical maximum temperature to
be reduced from 4250 E. to 3330 F. by taking account of the increases iu specific
heats as determined by Mallard and Le Chatelier); combustion is actually not
instantaneous throughout the mass of gas and some increase of volume always
occurs ; and the temperature is lowered by the cooling effect of the cylinder walls.
Still another reason for the discrepancy is suggested in Art. 318.
318. Dissociation. Just as a certain maximum temperature must be attained
to permit of combustion, so a certain maximum temperature must not be exceeded
if combustion is to continue. If this latter temperature is exceeded, a suppression
of combustion ensues. Mallard and Le Chatelier found this "dissociation " effect
to begin at about 3200 F. with carbon monoxide and at about 4500 F. with steam.
Deville, however, found dissociative effects with steam at 1800 F., and with car
bon dioxide at still lower temperatures. The effect of dissociation is to produce,
at each temperature within the critical range for the gas in question, a stable
ratio of combined to elementary gases, e.g. of steam to oxygen and hydrogen,
which cannot widely vary. No exact relation between specific temperatures and
such stable ratio has yet been determined. It has been found, however, that the
maximum temperature actually attained by the combustion of hydrogen in oxygen
is from 3500 to 3800 C, although the theoretical temperature is about 9000 C.
At constant pressure (the preceding figures refer to combustion at constant vol
ume), the actual and theoretical figures are 2500 and 6000 C. respectively. For
hydrogen burning in a,ir, the figures are 1830 to 2000, and 3800 C. Dissociation
here steps in to limit the complete utilization of the heat in the fuel. In gas en
gine practice, the temperatures are so low that dissociation, cannot account for all
of the discrepancy between observed and computed values ; but it probably playa
a part. (See Art. 1276.)
319. Rate of Flame Propagation. This has been mentioned as a factor influ
encing the maximum temperature and pressure attained. The speed at which
flame travels in an inflammable mixture, if at rest, seldom exceeds 65 ft. per sec
ond. If under pressure or agitation, pulsations may be produced, giving rise to
"explosion waves," in which the velocity is increased and excessive variations in
pressure occur, as combustion is more or less localized (23). Clerk (24), experi
RATE OP FLAME PROPAGATION
195
meriting on mixtures of coal gas with air, found maximum pressure to be obtained
in minimum time \\hen the proportion of air to gas by volume was 5 or 6 to 1 :
for pure hydrogen and air, the best mixture was 5 to 2. The Massachusetts Insti
tute of Technology experiments, made with carbureted water gas, showed the best
mixture to be 5 to 1 ; with 86 gasoline, the quickest inflammation was obtained
^lien 0.0217 parts of gasoline were mixed with 1 part of air; with 76 gasoline,
when 0.0203 to 0278 parts were used.* Grover found the best mixture for coal
gas to be 7 to 1 ; for acetylene, 7 or 8 to 1, acetylene giving higher pressures than
coal gas. Vt'ith coal gas, the weakest i^nitible mixture was 15 to 1, the theoreti
cally perfect mixture being 5.7 to 1. The limit of weakness with acetylene was 18
to 1. Both Grover and Lucke (2G) have investigated the effect of the presence of
"neutrals" (carbon dioxide and nitrogen, derived either fiom the air, the incom
ing gases, or from residual burnt gas) on the rapidity of piopagation. The re
tJ> 5 5.5
PARTS AIR PER ONE PART GAS
6.6
FIG. l.TS. Art 319 Effect of Presence of Neutrals.
(From Button's " The Gas Engine, 11 by permisbion of Joku Wiley L Sons, Publishers )
suits of Lucke's study of water gas are shown in Fig. 13$. The ordinates show
the maximum pressures obtained with various propoitions of air and gas. These
are highest, for all percentages of neutral, at a ratio of air to gas of 5 to 1 ; but
they decrease as the proportion of nentnil increases. The experiments indicate
that the speed of flame travel varies widely with the nature of the mixture and tlie
conditions of pressure to which it is subjected. If the mixture is too weak or too
strong, it will not Inflame at alL (See Art. 105a.)
320. Piston Speed. The actual shape of the ideally vertical ignition line will
depend largely upon the speed of flame propagation as compared with the speed
of the piston. Figure 139, after Lucke, illustrates this. The three diagrams were
taken from the same engine under exactly the same conditions, excepting that the
speeds in the three cases were 150, 500, and 750 r. p. m. Similar effects may be
obtained by varying the mixture (and consequently the flame speed) while keep
ing the piston speed constant. High compression causes quick ignition. Throt
* The theoretical ratio of air to C 6 H 14 is 47 to 1.
196
APPLIED THERMODYNAMICS
tlinrg of the incoming charge increases the percentage of neutral from the burnt
gases and retards ignition.
150 r. p. m.
500 i. p. m
750 r. p. m.
FIG. 139. Art. 320. Ignition Line as affected by Piston Speed.
(From Lucke's "Gas Engine Design.")
321, Point of Ignition. The spreading of flame is at first slow. Ignition is,
therefore, made to occur prior to the end of the stroke, giving a practically verti
cal line at the end, where inflammation is well under way. Figure 140, from
Poole (27), shows the effects of change in the point of ignition. In (a) and (b),
ignition was so early as to produce a negative loop on the diagram. This was cor
rected in (c), but (d) represents a still better diagram. In () and (/), ignition
was so late that the comparatively high piston speed kept the pressure down, and
the work area was small. It is evident that too early a point of ignition causes a
backward impulse on the piston, tending to stop the engine. Even though the
inertia of the fly wheel carries the piston past its " dead point," a large amount of
power is wasted. The same loss of power follows accidental preignition, whether
due to excessive compression, contact with hot burnt gases, leakage past piston
rings, or other causes. Failure to ignite causes loss of capacity and irregularity
IGNITION
197
IGNITION 25% EARLY
IGNITION 20 ft EARLY
IGNITION 10 ^ LAT
FIG. 140. ArL. C.I. 7i.c of Ii^...L_.
(From Poole'a " The Gas Engine," by permission of the Hill Publishing Company.)
366
fife
355
350
345
340
2.500
335
I
a
117666
320 ^K^ 10,500
w
10 11 12 13 14 15 16 17 18
Ignition Advance, Per Cent
FIG. 140a. Art. 321. Mixture Strength and Ignition Point.
198 APPLIED THERMODYNAMICS
Oi apeed, but theoretically at least does not affect economy. For reasons already
suggested, light loads (where governing is effected by throttling the supply) and
weak mixtures call for early ^qnltwn Fig. 140a, based on tests of a natural gas
engine reported by Poole, shows the effect of a simultaneous varying of mixture
strength and ignition point. The splitting of each curve at its lefthand end is
due to the use of two mixture strengths at 10 per cent ignition advance.
322. Methods of Ignition. An early method for igniting the gas was to use
an external flame enclosed in a rotating chamber which at proper intervals opened
communication between the flame and the gas. This arrangement was applicable
to slow speeds only, and some gas always escaped. In early Otto engines, the
external flame with a sliding valve was used at speeds as high as 100 r. p. m. (28).
The insertion periodically of a heated plate, once practiced, was too uncertain.
The use of an internal flame, as in the Brayton engine, was limited in its applica
tion and introduced an element of danger. Selfignition by the catalytic action
of compressed gas upon spongy platinum was not sufficiently positive and reliable.
The use of an incandescent wire, electrically heated and mechanically brought
into contact with the gas, was a forerunner of modern electrical methods. The
"hot tube "method is still in frequent use, particularly in England. This in
volves the use of an externally heated refractory tube, which is exposed to the gas
either intermittently by means of a timing valve, or continuously, ignition being
then controlled by adjusting the position of the external flame. In the Hornsby
Akroyd and Diesel engines, ignition is selfinduced by compression alone; but
external heating is necessary to start these engines.
What is called "automatic ignition" is illustrated in Fig 151. Here the external
vaporizer is constantly hot, because unjacketed. The liquid fuel is sprayed into the
vaporizer chamber. Pure air only is taken in by the engine during its suction
stroke. Compression of this air into the vaporizer during the stroke next succeeding
brings about proper conditions for selfignition.
323. Electrical Methods. The two modern electrical methods are
the (t make and break " and " jump spark." In the former, an electric
, current, generated from batteries or a small dynamo, is passed through
two separable contacts located in the cylinder and connected in series
with a spark coil. At the proper instant, the contacts are separated
and a spark passes between them. In the jump spark system, an
induction coil is used and the igniter points are stationary and from
0.03 to 0.05 in. apart. A series of sparks is thrown between them when
the primary circuit is closed, just before the end of the compression
stroke. Occasionally there are used more than one set of igniter points.
324. Clearance Space. The combustion chamber formed in the clearance
space must be of proper size to produce the desired final pressure. A common
ratio to piston displacement is 30 per cent. Hutton has shown (29) that the limits
for best results may range easily from 8.7 to 56 per cent (Arts. 295, 332).
IGNITION AND EXPANSION
199
FIG.
141. Art. 323. After
Burning.
'325. Expansion Curve. Slow inflammation has been shown to result in u
decreased maximum pressure after ignition. Inflammation occurring during expan
sion as the result of slow spreading of the flame is callod "after burning. 1 ' Ideally,
the expansion curve should be adiabatio; actually it falls m many cases above the
air adiabatic, py 1402 = constant, although it is known that during expansion from 80
to 40 per cent of the total heat in the gas is being
earned away ly the jacket water. Figure 141 repre
sents an extreme case; afterburning has made the
expansion line almost horizontal, and some uuburnt
gas is being discharged to the exhaust. Those who
hold to the dissociation theory would explain this
line on the ground that the gases dissociated during combustion are gi adually
combining as the temperature falls ; but actually, the temperature is not falling,
and the effect which we call after binning is most pronounced with weak mix
tures and at such low temperatures as do not permit of any considerable
amount of dissociation. Practically, dissociation has the same effect as an
increasing specific heat at high temperature. It affects the ignition line to
some extent; but the shape of the expansion line is to a far greater de
gree determined by the slow inflammation of the gases. The eifect of
the transfer of heat between the fluid and the cylinder walls is dis
cussed in Art. 347. The actual exponent of the expansion
curve varies from 1.25 in large engines to 1.38 in good small
engines, occasionally, however, rising as high as
1.55. The compression curve has
usually a somewhat
higher exponent. The
adiabatic exponent for a
FIG. 142. Art. 325. Explosion Waves. mixture of hydrocarbon
gases is lower than that
for air or a perfect gas; and in many cases the actual adiabatic, plotted for the
gases used, would be above the determined expansion line, as should normally be
expected, in spite of after burning. The presence of explosion waves (Art, 319)
may modify the shape of the expansion curve, as in Fig. 142. The equivalent
curve may be plotted as a mean through the oscillations. Care must be taken
not to confuse these vibrations with those due to the inertia of the indicating
instrument.
326 The Exhaust Line. This is
shown to ati enlarged vertical scale
as 6c, Fig. 137. "Low q>ring" dia
grams of this form are extremely u^e
F I0 . 143. Art. '^Delayed Exhaust Valve f^ ^ ^^ ^ ^ ^ ^
" lost motion " becomes present in the
valveactuating gear, an 4 the tendency of this is to vary the instant of opening
or closing the inlet or the exhaust valve. The effect of delayed opening of the
latter is shown in Fig. 143; that of an inadequate exhaust passage, in Fig. 144.
An early opening^of the exhaust valve may cause loss also, as in Fig. 145. There
200
APPLIED THERMODYNAMICS
FIG. 144. Ait SCO Thiottlod Exhaust Passages.
is always a loss of this kind, more or less pronounced: the expansion ratio is
never quite equal to the compression ratio The exhaust valve begins to open
when the expansion stroke is
only from 80 to 93 per cent
completed In multiple cylinder
engines having common exhaust
and suction mains, early exhaust
from one cylinder may produce
a rise of pressure during the
latter part of the exhaust stroke
of another. Obstructions to suction and discharge movements of gas are com
monly classed together as " fluid friction. " This may in small engines amount to
as much as 30 per cent of the
power developed. In good
engines of large or moderate
size, it should not exceed 6
per cent. It increases, pro
portionately, at light loads;
and possibly absolutely as
well if governing is effected
by throttling the charge FIQ. lj Art. 326 Exhaust Valve Opening too Early,
327. Scavenging. To avoid the presence of burnt gases in the clear
ance space, and their subsequent mingling with the fresh, charge, " scav
enging," or sweeping out these gases from the cylinder, is sometimes prac
ticed. This may be accomplished by means of a separate air pump, or by
adding two idle strokes to the four strokes of the Otto cycle. In the
Crossley engines, the air admission valve was opened before the gas valve,
and before the termination of the exhaust stroke. By using a long ex
haust pipe, the gases were discharged in a rather violent puff, which pro
duced a partial vacuum in the cylinder. This in turn caused a rush of
air into the clearance space, which swept out the burnt gases by the time
the piston had reached the end of its stroke. Scavenging decreases the
danger of missing ignitions with weak gas, tends to prevent preignition,
and appears to have reduced the consumption of fuel.
328. The Suction Stroke. This also is shown in Fig. 137, line cd. The effect
of late opening of the valve is shown in Fig. 146 ; that oi an obstructed passage
or of throttling the supply, in Fig'.
147. If the opening is too eaily,
exhaust gases will enter the supply
pipe. If closure is too early, the
gas will expand during the re
mainder of the suction stroke, but
the net work lost is negligible; if
too late, some gas will be discharged
back to the supply pipe during the
beginning of the compression stroke,
FIG. 146. Art. 328. Delayed Opening of
Suction Valve.
DIAGRAM FACTOR
201
ACTUAL
FIG 147. Art. 328. Throttled Suction.
as in Fig. 148. Excessive obstruc
tion in the suction passages de
creases the capacity of the engine,
in a way already suggested in the
study of air compressors (Art. 224).
329. Diagram Factor. The
discussion of Art. 309 to Art.
328 serves to show why the
work area of any actual dia
gram must always be less than
that of the ideal diagram for
the same cylinder, as given in
Fig. 122. The ratio of the
two is called the diagram
factor. The area of the ideal card would constantly increase as
compression increased ; that of the actual card soon reaches a limit
in this respect; and, consequently, in general, the diagram factor
decreases as compression increases. Variations in excellence of
design are also responsible for variations of diagram factor.
FIG. 148. Art. 328. Late Closing ot
Suction Valve.
Gasolene Vapor
Kerosene Spray
Natural and Dlmnmating Gases
Mond Producer Gas ^Jp"^
 Suction Anthracite Producer Gag
10
75 85 100 115 ISO 145 160
Absolute Pressure at the End of Compression, Lbs.per Sa.In,
FIG. 148a. Art. 329. Maximum Mean Effective Pressures Realized in Practice.
202
APPLIED THERMODYNAMICS
In the best recorded tests, its value has ranged from 0.38 to 0.59; in
ordinary practice, the values given by Lucke (30) are as follows: for
kerosene, if previously vaporized and compressed, 30 to 0.40, if injected
on a hot tube, 20; for gasoline, 0.25 to 50; for producer gas, 0.40 to
0.56; for coal gas, 0.45; for carbureted water gas, 0.45; for blast furnace
gas, 0.30 to 0.48; for natural gas, 0.40 to 0.52. These figures are for four
cycle engines. For twocycle engines, usual values are about 20 per cent
less. Figure 149 shows on the PV and entropy planes an actual indica
tor diagram with the corresponding ideal cycle.
Some of the highest mean effective pressures obtained in practice
with various fuels, tabulated by Poole, have been charted in Fig. 148a,
ACTUAL DIAGRAM
IDEAL DIAGRAM
FIG 149. Art. 329. Actual and Ideal Gas Engine Diagrams.
MODIFIED ANALYSIS
329 a. Specific Heats Variable. Suppose k = c 4 bt, I a + bt, M=Jcl
= c a. For a differential adiabatic expansion
Idt = pdv,
Also, from pv =
adt . ,,, ndv
f oat = H .
pdv + vdp = Edt ?? = ; whence
v p t
dv
(1)
p
MODIFIED ANALYSIS 203
(a 4 H) log e v + <i log fi p \bt= constant,
c log e u + a log c /> + it = constant,
 log fl v + log, 2? + = constant.
a a
r M
2iv eft = constant,
where e is the Napierian logarithmic base.
Between given limits, the approximate value of n may be obtained as
follows: from Equation (1),
log.g + (a + JB) log, & = & ft  *,) (2)
If we assume an equation in the form p^vf =^ 2 v 2 n to be possible, then
log, "== ^ 71 log
9?2 ^l
Substituting in Equation (2),
^i

(3)
a al SJ
The external work done during the expansion is
J/ 6
Idt = I (CL J oi) dt'=' ct \t% ti) ^2 ~ ^i)^ or
x 2
n1 "'
where n has the value given in Equation (3).
We may find a simple expression for n by combining these equations :
204 APPLIED THERMODYNAMICS
The efficiency of the Otto cycle debf, Fig. 122, may now be written
C\a+bt)dt
in which  (t,  t d + t e  %) = log, f?*?*} = log, (&A a relation obtained
by dividing the equation of the path bf by that of the path ed.
Following the method of Art. 169, the gain of entropy between the
states a and b is, for example,
a log. k + &(*.  * a ) + o log. 4 + 6ft  0,
If we apply an equation in this general form to each of the constant
volume paths eb, df, Fig. 122, we find
a log & + 6&0 = ^ log
C e C
log/^ = fe
V^// a
as already obtained.
o W
329 b. Application of the Equations. The expression pv a e*= con
stant is exceedingly cumbersome in application excepting as t is employed
independently. If t is to be assumed, however, we may write
log a p + log e v H = log, constant ,
a a
Gog. R + log. t log.p) 4 = log, constant,
a a
^log. p +(log. JJ + log. t) +  = log. constant.
a & a
MODIFIED ANALYSIS 205
Consider one pound of air at the absolute pressure of 100 Ib. per
square inch and a volume of 1 cu. ft. Let = 0.23327+0.00002652,
Z=0.1620 + 0000265*. We find
__
tl ~~R
ptle^=100X 144X 1X2.7183 ** 15030.
ac 0.1620 0.23327 c 0.23327 ,.. . _ , A _
=  01620  =  4il ; a~oi62 L44; lo &*37
Let < 2 =200. Then ^ 2 = 0.0327, Iog e f 2 = 5.3, (log. B+logA) 13.32,
T i oge p s =log e 1503013.320.0327= 9.61 13.35= 3.74, Iogp 3
= 8.48, log p 2 = 3.685, p 2 =4845 Ib. per square foot =33.63 Ib. per square
inch. Also
Rk_ 53.36X200 . 01
V%   7n* ,  ^ &.].
p 2 4845
(0 233^7\
' 1g ^ ) we should have had
U.lO^ /
pi \tj \.70J '
log p 2 = 2+(3.27x0.131) = 1.571,
p 2 =37.23 Ib.. per square inch,
and v 2 =  = ' Qx/1 . .= 1.99. Proceeding in this way, we plot the two
PZ &* & X
curves as required. The y curve is the steeper of the two, and for
expansion to a given lower temperature reaches a point of considerably
less volume.
By Equation (3) ; for the upper of the two curves, between
P! = 100 ; *i=270, ri=l, and p 2
, AA 0.0000265X170
,3 log 11.14
206 APPLIED THERMODYNAMICS
the curve being somewhat less steep than the y curve. This value of n
(1.43) will be found to fit the whole expansion with reasonable accuracy.
Also, by Equation (4),
_, , 5336^778 _,
n i H /o nooo f >fi ^ "" 9
0163 + f
a fairly close check value. If we take p at 50 lb. p?r s uare inch, and
ti at 135 absolute, instead of the conditions given, we have,
iwe a " = 50 x 144 x 1 X 2.7183 (>2L '= 7360.
If we let f s = 100, ^ = 0.01635, Iog e i 2 = 4.6 ;  (log. 22 + log. f a ) = 12.3,
^^log.p^log, 736012.3 0.01635 = 3.42, logj>,=7.75, log^ 2 =3.37,
0.0000205 x 35
0.162X2.3 log 2.2
53.36 778
0.182 + x 283
. .
=l42.
The value of n is thus about the same for this curve as for that formerly
considered, and (approximately), in Fig. 122,
hjL
te U'
If this relation were exact, the efficiency of an Otto cycle would be
expressed by the same formula as that which holds when the specific
heats are constant. In Fig. 124 ; the efficiency of the strip cycle qvwp is
= l t a ,
and if 2. =  = j2 = ^ etc., the efficiency of the whole cycle
*d * q t p V
1 ^ = 1 tf^t^t.t,
*. *6 *. t,
For a path of constant volume, in Equation (5), = 1, = $i, and
^a Pa, t a
the gain of entropy is
GAS ENGINE DESIGN 207
In the case under consideration, t b 270, t a = 135, a = 0.162,
b = 0.0000265, so that Equation (6) gives for the path eb,
0.162 x 2.3 log ^4 + (0.0000265 x 13,)) = 01123 + 0.0036 = 0.1158.
If m Fig. 122 the temperature at d is 100, we may write
0.1158 = 0.162 x 2.3 log ^ + 0.0000265 ($, 100)
100
= 0.372 log t,  0.744 + 0.0000265 t f  0.00265,
log *, + 0.0000712^ = 2.32,
from which t f is, nearly, log 1 2.32, and ^ = 200, about. In expanding
from 270 to 200, the volume increased from 1.0 to 2.21 ; in expand
ing from 135 to 100, it increased from 1 to 2 28. We have computed the
change of entropy from p = 50, v = 1, t = 135, to p = 100, r = 1.0, t = 270,
as 0.1158. This must equal the change from p = \\\ 3  = 1(5.85, = 100,
= 2.28, to ^ = 33.6, <y = 2.28, * = ? Now for ^ = 33.6, = 2.21, it was
found that t = 200, Adiabatic expansion from this point to the greater
volume 2.28 means that t f must be slightly less than 200; but a very
slight change in temperature produces a large change in volume since the
isothermals and the adiabatics nearly coincide.
GAS ENGINE DESIGN
330. Capacity. The work done per stroke may readily be computed for the
ideal cycle, as in Art 293. This may be multiplied by the diagram factor to
determine the probable performance of an actual engine. To develop a given
power, the number of cycles per minute must be established. Ordinary piston
speeds are from 450 to 1000 ft. per minute, usually lying between 550 and 800 ft.,
the larger engines having the higher speeds. The stroke ranges from 1.0 to 2.0
times the diameter, the ratio increasing, generally, with the size of the engine.
A gas engine has no overload capacity, strictly speaking, since all of the factors
entering into the determination of its capacity are intimately related to its effi
ciency. It can be given a margin of capacity by making it larger than the
computations indicate as necessary, but this or any other method involves a con
siderable sacrifice of the economy at normal load.
331. Mean Effective Pressure. Since in an engine of given size the extreme
volume range of the cycle is fired, the mean net ordinate of the work area measures
the capacity. The quotient of the cycle area by the volume range gives what is called
the mean effective pressure (m. e. p.). In Fig. 122, it is ebfd (V d  7). We
yl
then write m. e. p. = W  ( V 4  7 e ); but from Art. 295, W = Q[I  (fr) * ] 5
208
APPLIED THERMODYNAMICS
being the gross quantity of heat absorbed iu the cycle. Then, in proper units,
without allowance for diagram factor,
332. Illustrative Problem To determine the cylinder dimensions of a fourcycle^
twocylinder, doubleacting engine of 500 Tip.) using producer gas (assumed to contain
CO, 394; N, 60; If, 06; parh in 100 by weight) (Art. 285), at 150 r. p. m. and a
piston speed of 825ft. per minute.
We assume (Fig. 150), P L = 12, P 2 = 144.7, I\ = 200 F., and diagram factor
= 0.48 (Arts 312, 329).
V /P\y /1447\ 718
Since P l 7^ = P 2 IV, L = ( y~ 1 = f ' I = 5.9. Let the piston displace
I'z vPi/ \ 12 /
ment V l  7 3 = D. Then 7 9 = 0.2045 D and V l = 1.2045 D. The clearance is
=0.2045 (Art. 324)*. Also T* = ^ = 659 ' 6 ^ ^^ = 1357
absolute. The heat evolved per pound of the mixed gas (taking the calorific
value of hydrogen burned to steam as 53,400) is (0.394 x 4315) + (0.006 x 53,400)
= 2021 B. t. u. The products of com p
bustion consist of $ x 0.394 = 3,
0.619 Ib. of C0 2 (specific heat = 0.1G92),
0.006 x 9 = 0.054 Ib. of H 2 (steam,
specific heat 0.37), and H (0619 
0.394) = 0.751 Ib. of N" accompanying
the oxygen introduced to burn the
CO, with (0.054 0.006)H =0.1007 Ib.
of N" accompanying the oxygen in
troduced to burn the H; and 0,60 Ib.
of K originally in the gas, making a
total of 1.5117 Ib. of N (specific lieat
0. 1727) . To raise the temperature of
these constituents 1 F. at constant
FIG. 150. Arts. 332335. Design of Gas
Engine.
volume requires (0.619 x 0.1692) + (0.054x0.37) + (1.5117 x 0.1727) = 0.3849
B. t, u. Adding the heat required for the clearance gases always present, this may
be taken as 0.3849 X 1.2045 = 464 B. t u. The rise in temperature T 3  T 2 is
then 2021 f 0.464 = 4370, and T* = 4370 + 1357 = 5727 absolute. Then
P 3
144.7 :
,5727
1357
= 613,
and
> P*  ir 6 13 _ KO q
] F;" 12 144.7" 509 
* While the use of a " blanket " diagram factor as in this illustration may be justi
fied, in any actual design the clearance at least must be ascertained from the actual
exponent of the compression curve. The design as a whole, moreover, would better
be based on special assumptions as in Problem 15, (i), page 227.
GAS ENGINE DESIGN 209
The work per cycle is
y i
= 144 x 0.48 D [ (613 x0 ' 2045 ) " ( 5Q 9 x 1.2045) (144.7 x 0.2045) + (12 x 1.2045)1
L 0.402 J
= 8410 D foot pounds.
In a twocylinder, fourcycle, doubleacting engine, all of the strokes are work
ing strokes ; the footpounds of work per stroke necessary to develop 500 hp. are
 ^ = 55,000. The necessary piston displacement per stroke, D, is
55,000 + 8410 = 6.52 cu. ft, The stroke is 825 s (2 X 150) = 2.75 ft. or 38 in. The
piston area is then 6.52 + 2.75 = 2.37 sq. ft. or 342 sq. in. The area of the water
cooled tail rod may be about 33 sq. in., so that the cylinder area should be 342
+ 33 = 375 sq. in. and its diameter consequently 21.8 in.
333. Modified Design. In an actual design for the assumed conditions, over
load capacity was secured by assuming a load of 600 hp. to be carried with 20 per
cent excess air in the mixture. (At theoretical air supply, the power developed
should then somewhat exceed 600 hp.) The air supply per pound of gas is now
[(0.394 x Jf) + (0.006 x 8)] VJfx 1.2 = 1.422 Ib.
Of this amount, 0.23 x 1.422 = 0.327 Ik is oxygen. The products of combustion
are f f x 0.394 = 0.619 Ib. C0 21 0.006 x 9 = 0.054 Ib. H 2 O, (1.422  0.327) + 0.60
= 1.693 Ib. N, and 0.327  (if x 0.394)  (8 x 0.006) =0.054 Ib. of excess oxygen ; a
total of 2.422 Ib. The rise in temperature r 3  T 2 is
_ 2021*1.2045
(0.619 X 0.1692)'+ (0 054 X 37) + (1,693 X 0.1727) + (0.054 X 0.1551)
Then !T 3 3950 + 1357  5307 absolute,
" *
P  P .
Pz  P
' 4 ~ l Pi " 1447
and the work per cycle is
i AA v n AC n ["(569 X O.C045) (47.2JX 1 2045) f 144.7 X 0.2045yH(12 X 1.2045)1
144XU.4S^ 04Q2 J
=7630 D fooirpounds.
600 X 33000
The piston displacement per stroke is 2 v 150 x 7630 ~ 8 ' 6 ^ CU " **"' tbe ^tinder
area is (8.65 i 2.75)144 + 33 = 486 sq. in., and its diameter $4.9 in. The cylinders
were actually made 23J by 33 in., the gas composition being independently assumed.
334. Estimate of Efficiency. To determine the probable efficiency of the engine
under consideration : each pound of working substance is supplied with 1.422 Ib.
of air. Multiplying the weights of the constituents by their respective specific
volumes, we obtain as the volume of mixture per pound of gas, 31.33 cu. ft. at
14.7 Ib. pressure and 32 F., as follows :
210 APPLIED THERMODYNAMICS
CO, 394 x 12 75 = 5.01
H, 006 x 178 t>3 = 1.07
N. 0.600 x 12.75 = 7.65
Air, 1.422 x 12.387 =1760
31.33
At the state 1, Fig. 150, 7^ = 659 6, P l = 12, whence
v = PI P,.r, l= 659.6 x 147 x 31.83 51
1 P,T 12 x 491.6
The piston displaces 8.65 X 300 = 2595 cu, ft. of this mixture per minute. The heat
taken in per minute is then 2021 X (2595 s 51.2) = 102,400 B. t. u. The work done
fiOO V ^l^ftOfl
per minute is  ^  = 25,500 B t. u. The efficiency is then 25,500 f 102,400
= 0.249. An actual test of the engine gave 0.282, with a load somewhat under
1 ^ t\7 fi f\Q fi
600 hp. The Otto cycle efficiency is  1357 = 0.516.*
335. Automobile Engine. To ascertain, the probable capacity and economy of a
fourcylinder^ fourcycle, singleacting gasoline engine with cylinders 4 by 5 in. 3 at
J.500 r. p. m.
In Fig. 150, assume P 3 = 12, P 2 = 84.7, 7\ = 70 F., diagram factor, 0.375
(Arts. 312, 329). Assume the heating value of gasoline at 19,000 B. t. u , and its
composition as C Q H^: its vapor density as 3.05 (air = 1.). Let the theoretically
necessary quantity ot air be supplied.
The engine will give two cycles per revolution. Its active piston displacement
is then ' 7854 x W* x 5 x 3000 = 145.5 cu. ft. per minute, which may be repre
1728
seated as V^  F 2 , Fig. 150. We now find
s = " = 0,2495: F 2 = 0.2495 F I; 0.7505 7i = 145. 5; 7 1 =
V 1 V84.7/
Clearance = = 0.384 (Art. 324); r a = , = 936 absolute.
145.0 1 x iy^
To burn one pound of gasoline there are required 3.53 Ib. of oxygen, or 15.3 Ib.
of air. For one cubic foot of gasoline, we must supply 3.05 x 15.3 = 46.6 cu. ft.
of air. The 145.5 cu. ft. of mixture displaced per minute must then consist of
* The actual efficiency will always be less than the product of the Otto cycle
efficiency by the diagram factor. Thus, let the actual cycle be described as 1234,
Fig. 160, and let the corresponding ideal cycle be 123'4 ; . The efficiencies are,
respectively,
1234
The quotient 1234 f 123'4' = the diagram factor Then write
1( ? T3} * diagram factor x Jf= JSL^
CURRENT GAS ENGINE FORMS 211
145.5 ^ 47.6 = 3.06 cu. ft. of gasoline and 142.44 cu. ft. of air, at 70 F. and 12 Ib.
pressure. The specific volume of air at this state is 52<Q ' 6 x 14 '' 7 x 12  3S7 _ 16.33
491.6 x 12
cu. ft. ; that of gasolene is 16.38 * 3.03 = 3.37 cu. ft. The weight of gasoline
used per minute is then 3.06  5.37 = 0.571 II. The heat used per minute is
0.571 x 19,000 = 10,840 B. t. u. The combustion reaction may be written
86 + 304 = 264 + 126
W= 3.06 lb.C0 2 per lb.C 6 H 14
V/ = 135 Ib. II 2 per Ib. C 6 H 14
11 x W = 1182 Ib. N per Ih. C fl H 14
16.23 = 1. + 15.3, approximately.
The heat required to raise the temperature of the products of combustion 1 F. is
[(3.06 X 0.1692) + (1.35 X 0.37) + (11.82 X 0.1727)] 0571 = 1.746 B. t. u. per
minute. Adding for clearance gas, this becomes 1,746 X 1.334 = 2.327 B. t. u.
The rise in temperature T*  T is then 10,840 = 2 327 = 4660, T*  4660  936
= 5596 absolute, P 3 = 84.7^  508, P 4 = 12  72 0, and the wr* per m^n
ute K 0.375X144[^ 508X48 ' 5
1 200 000
footbounds. This is equivalent to '^ * = 1540 B. t u. per minute or to
77o
1 20' A OCO
33 000 = 86 * l TSe ' power ' Tli e e ff lcienc y is 1540 * 10,840 = 0.142. In an auto
mobile running at 50 miles per hour, this would correspond to 50 s (0.571 X 60)
= 1.46 miles run per pound of gasohne. In practice, the air supply is usually incor
rect, and the power and economy less than those computed.
It is obvious that with a given fuel, the diagram factor and other data of
assumption are virtually fixed. An approximation of the power of the engine may
then be made, based on the piston displacement only. This justifies in some
measure the various rules proposed for rating automobile engines (30 a). One
d*n
of these rules is, brake hp. == , where n is the number of fourcycle cylinders of
2.5
diameter d inches, running at a piston speed of 1000 ft. per minute,
CURRENT GAS ENGINE FOKVIS
336. Otto Cycle Oil Engines. This class includes, among many others, the
Mietz and Weiss, twocycle, and the Daimler, Priebtman, and HornsbyAkroyd,
fourcycle. In the lastnamed, shown in Fig. 151, kerosene cil is injected by a
small pump into the vaporizer. Air is drawn into the cylinder during the suction
stroke, and compressed into the vaporizer on the compression stroke, where the
simultaneous presence of a critical mixture and a high temperature produces the
explosion. External heat must be applied for starting. The point of ignition is
determined by the amount of compression; and this may be varied by adjusting
212
APPLIED THERMODYNAMICS
the length of the connecting rod on the valve gear. The engine is governed by
partially throttling the charge of oil, thus weakening the mixture and the force of
FIG. 151. Arts 322, 336, Kerosene Engine with Vaporizer.
(From " The Gaa Engine," by Cecil P. Poole, with the permission of the Hill Publishing Company )
the explosion. The oil consumption may be reduced to less than 1 Ib. per brake hp,
per hour.
In the Priestman engine, an earlier type, air under pressure sprayed the oil
into a vaporizer kept hot by the exhaust gases. The method of governing was to
reduce the quantity of chaige without changing its proportions. A hand pump
and external heat for the vaporizer were necessary in starting. An indicated
thermal efficiency of 0.1 Go has been obtained. The Daimler (German) engine
uses hottube ignition without a timing valve, the hot tube serving as a vaporizer.
Extraordinarily high speeds are attained.
337. Modern Gas Engines : the Otto. The presentday small Otto engine is ordi
narily singlecylinder and singleacting, governing on the "hit or miss" principle
(Art. 343). It is used with all kinds of gas and with gasoline. Ignition is elec
trical, the cylinder water jacketed, the jackets cast separately from the cylinder.
The Foos engine, a simple, compact form, often made portable, is similar in princi
ple, lu the CrossleyOtto, a leading British type, hottube ignition is used, and
the large units have two horizontal opposed singleacting cylinders. In the
Andrews form, tandem cylinders are used, the two pistons being connected by
external side rods.
TYPES OF GAS ENGINE
213
338. The Westinghouse Engine. This has recently been developed in very
large units. Figure 132 shows the \\oiking side of a twocylinder, tandem,
double acting engine, representing the mlt valves on top of the cylinders.
FIG. 152. Arts. 338, 330. Westinghouse Gas Engine. Twocylinder Tandem, Fourcycle.
Smaller engines are often built vertical, with one, two, or three singleacting
cylinders. All of these engines are fourcycle, with electric ignition, governing
by varying the quantity and proportions of the admitted mixture. Sections of
the cylinder of the Riverside horizontal, tandem, doubleacting engine are shown in
Fig. 15;?. It has an extremely massive frame. The AllisChalmers engine is built
in laige units along similar general lines. Thirtysix of the latter engines of
4000 hp. capacity each on blast furnace gas are now (1009) being constructed.
They weigh, each, about 1,500,000 lb., and run at83J r. p. m. The cylinders are
44 by 54 in. Nearly all are to be directconnected to electric generators.
339. Twocycle Engines. In these* the explosions are twice as frequent as
with the fourcycle engine, and cooling is consequently more difficult. With an
equal number of cylinders, single or doubleacting, the twocycle engine of course
gives better regulation. The first important twocycle engine was introduced by
Clerk in 18SO. The principle was the same as that of the engine shown in Fig. 11>.
The Oechelhaueser engine has two singleacting pistons in one cylinder, which are
connected with cranks at ISO , so that they alternately approach toward and
lecede from each other. The engine frame is excessively long. Changes in the
quantity of fuel supplied control the speed. The Eoerting engine, a doubleacting
horizontal form, has two pumps, one for air and one for gas. A ' scavenging "
charge of air is admitted just prior to the entrance of the gas, sweeping out the
burnt gases and acting as a cushion between the incoming charge and the exhaust
ports. The engine is built in large units, with electrical ignition and compressed
air starting gear. The speed is conti oiled by changing the mixture propoitions.
340. Special Engines. For motor bicycles, a single aircooled cylinder is often
used, with gasoline fuel. Occasionally, tv\o cyhndeis are employed. The engine
214
APPLIED THERMODYNAMICS
TYPES OF GAS ENGINE 215
is fourcycle and runs at high speed. Starting is effected by foot power, which
can be employed whenever desired. Ignition is electiical and adjustable. The
speed is controlled by throttling. Extended surface aircooled cylinders have also
been used on automobiles, a fan being employed to circulate the air, but the limit
of size appears to be about 7 hp. per cylinder. Most automobiles have water
cooled cylinders, usually four in number, fourc\cle, singleacting, running at
about 1000 to 1200 r. p. m., normally. Governing is by throttling and by chang
ing the point of ignition. The cylinders are usually vertical, the jacket water
being circulated by a centrifugal pump, and being used repeatedly. Both hottube
and electiical methods of ignition have been employed, but the former is now
almost wholly obsolete. The number of cylinders varies from one to six ; occa
sionally they are arranged horizontally, duplex, or opposed. Twocycle engines
have been introduced. The fuel in this country is usually gasoline. For launch
engines, the twocycle piinciple is popular, the crank case forming the pump
chamber, and governing being accomplished by throttling. Kerosene or gasoline
are the fuels.
341. Alcohol Engines. These are used on automobiles in France. A special
carburetor is employed. The cylinder and piston an angement is sometimes that
of the Oechelhaueser engine (Art. 330). The speed is controlled by varying the
point of ignition. In launch applications, the alcohol is condensed, on account of
its high cost, and in some cases is not burned, but serves merely as a working fluid
in a " steam " cylinder, being alternately vaporized by an externally applied gaso
line flame and condensed in a surface condenser. The low value of the latent
heat of vaporization (Art. 360) of alcohol permits of * getting up steam " more
rapidly than is possible in an ordinary steam engine.
342. Basis of Efficiency. The performances of gas engines may be compared
by the cubic feet of gas, or pounds of liquid fuel, or pounds of coal gasified in the
producer, per horse power hour ; but since none of these figures affords any really
definite basis, on account of variations in heating value, it is usual to express the
results of trials in heat units consumed per horse power per minute. Since one horse
power equals 33,000  778 = '2A'2 B. t. u. per minute, this constant divided by the
heat unit consumption gives the indicated thermal efficiency. In making tests, the
overall efficiency of a producer plant may be ascertained by weighing the coal.
When liquid fuel is used, the engine efficiency can readily be determined separately.
To do this with gas involves the measurement of the gas, al\\ ays a matter of some
difficulty with any but small engines. The measurement of power by the indicator
is also inaccurate, possibly to as great an extent as o per cent, which may be reduced to
2 per cent, according to Hopkinson, by employing mirror indicatoi s. This error has
resulted in the custom of expressing performance in heat units consumed per brake
horse power per hour or per kw.hr., where the engines are directly connected to
generators. There is some question as to the proper method of considering the
negative loop, bcde, of Fig. 136. By some, its area is deducted from the gross work
area, and the difference used in computing the indicated horse power. By others,
the gross work area of Fig. 136 is alone considered, and the "fluid friction " losses
216 APPLIED THERMODYNAMICS
producing the negative loop aie then clashed with engine friction as reducing the
"mechanical efficiency." Yaiious codes for testing gas engines aie in use (31).
343. Typical Figures Small oil or gasoline engines may easily show 10 per
cent brake efficiency. Alcohol engines of small size consume less than 2 pt. per
brake hp.hr. at full load (>2). A welladjusted Otto engine has given an indicated
thennal efficiency of 010 \\itli gasoline and 023 with kerosene (33). Ordinary
producer gas engines of average size under test conditions have repeatedly shown
indicated thermal efficiencies of 2.5 to 2,9 per cent. A Cockerill engine gave 30 per
cent. Hubert (31) tested at Seramg an engine shov\ ing neaily 32 per cent indicated
thermal efficiency. Mathob (3o) reports a test of an Ehihardt and Lehiner double
acting, fomcycle 000 hp engine at Ilemitz which reached nearly 38 per cent. A
blast furnace gas engine gave at full load 25.4 per cent. Expressed in pounds of
coal, one plant with a low load factor gave a kilowatthour per 1 8 Ib. In another
case, 1.59 Ib. was reached, and in another, 2 97 Ib of wood per kw.hr. It is common
to hear of guarantees of 1 Ib of coal per brake hp.hr., or of 11,000 B. t. u. in gas.
A recent test of a Crossley engine is reported to have shown the result 1.13 Ib. ot
coal per kw.hr. Under ordinary running conditions, 1.5 to 2.0 Ib ^ith varying
load may easily be realized. These latter figures are of course for coal burned iu
the producer. They repiesent the joint efficiency of the engine and the pioducer.
The best results have been obtained m Germany. For the engine alone, Schroter
is reported to have obtained on a Guldner engine an indicated thermal efficiency of
0.<t27 at full load with illuminating gas (30).
The efficiency cannot exceed that of the ideal Otto cycle. In one test of an
Otto cycle engine an indicated thermal efficiency of 0.37 was obtained, while the
ideal Otto efficiency was only 0.41. The engine was thus within 10 per cent of
perfection for its cycle *
The Diesel engine has given from 0.32 to 0.412 indicated thermal efficiency.
Its cycle, as has been shown, peimits of higher efficiency than that of Otto.
Plant Efficiency. Frames ba\e been given on coal consumption. Over
all efficiencies from fuel to indicated u oik have ranged from 0.14 upward. At the
Maschinenfabrik Wiuterthur, a consumption of 0.7 Ib. of coal (13,850 B. t. u.) per
brake hp.hr. at full load has been reported (37). This is closely paralleled by the
285 indicated plant efficiency on the Guldner engine mentioned in Art. 343 when
opeiated with a suction producer on anthiacite coal. At the Royal Foundry,
Wurtemburg (38), 0.78 Ib. ol anthracite weie burned per 1 hp.hr., and at the
Imperial Post Office, Hamburg, O.U3 Ib. of coke. In the best engines, variations of
efficiency with reasonable changes of load below the normal have been greatly
reduced, largely by impi oved methods of governing.
345. Mechanical Efficiency. The ratio of work at the brake to net indicated
work ranges about the same for gas as for steam engines having the same arrange
ment of cylinders. When mechanical efficiency is understood in this sense, its
* At the present tune, any reported efficiency much above 30 per cent should be
regarded as needing authoritative confirmation,
GAS ENGINE TRIALS 217
value is nearly constant for a given engine at all loads, decreasing to a slight
extent only as the load is reduced In the other sense, suggested in Art. 342, i.e.
the mechanical efficiency being the ratio of work at the brake to gross indicated
work (no deduction being made for the negative loop area of Fig 136), its value
falls off sharply as the load decreases, on account of the increased proportion of
"fluid friction." Lucke gives the following as average values for the mechanical
efficiency in the latter sense:
ENGINE
MECHANIC \L
AT FULL
EFFICIENCY
LOVD
F<nn>~ryalf
7V/ ncyvlc
Lar^e, 500 Ihp. and over, .....
SI to SO
63 to 0.70
Medium, 25 to 500 Ihp , .....
O.VO to O.S1
U.D i to 06
Small, 4 to 25 Ihp.,
0.74 to 0.80
0.00 to 70
The friction losses for a singleacting engine are of course relative!} 71 greater
than those for an ordinary doubleacting steam engine.
346. Heat Balance. The principal losses in the gas engine are due to
the cooling action of the jacket water (a necessary evil in present prac
tice) and to the heat carried away in the exhaust. The arithmetical
means of nine trials collated by the writer give the foil owing percentages
representing the disposition of the total heat supplied: to the jacket,
40.52; to the exhaust, 33.15; work, 21.87; unaccounted for, 6.23.
Hutton (40) tabulates a large number of trials, from which similar
arithmetical averages are derived as follows: to the jacket, 37.96; to the
exhaust, 29.84; work, 22.24; unaccounted for, 8.6. In general, the
larger engines show a greater proportion of heat converted to work, an
increased loss to the exhaust and a decreased loss to the jacket. In
working up a "heat balance," the loss to the exhaust is measured by a
calorimeter, which cools the gases below 100 F. The heat charged to
the engine should therefore be based on the " high" heat value of the
fuel (Arts. 561, 561a). The * k work " item in the above heat balance is
indicated work, not brake work.
Unlike the jacket water heat (Art. 352), the heat carried off in the exhaust gases
is at fairly high temperature. There would be a decided gain if this heat could be
even partly utilized. Suppose the engine to have consumed, per hp., 10,000 B. t. u.
per hour, of which 30 per cent, or 3000 B. t. u., passes off at the exhaust. At 80
per cent efficiency of utilization, 2400 B. t. u. could then be recovered. In form
ing steam at 100 Ib. absolute pressure from feed water at 212 F., 1006.8 B. t. u.
are needed per pound of steam. Each horse power of the gas engine would then
give as a waste gas byproduct 2400 1006.3 = 2.39 Ib. of steam. Or if the steam
plant had an efficiency of 10 per cent, 240 B. t. u. could be obtained in work from
the steam engine for each horse power of the gas engine. This is 240  2545 = Q\
per cent of the work given by the gas engine. A much higher gain would be
possible if the steam generated by the exhaust gases were used for heating rather
than for power production.
218
APPLIED THERMODYNAMICS
347. Entropy Diagram. When the PV diagram is given, points may be trans
Vb Pb
f erred to the entropy plane by the formula n 6 7? a = fc log e  + / log e (Art.
y a * o
169). The state a may be taken at 32 F. and atmospheric pressure; then the
entropy at any other state b depends simply upon V d and P&. To find V a , we
must know the equation of the gas. According to Richmond, (41) the mean
value of k may be taken at 0.246 on the compression curve and at 0.26 on the ex
pansion curve, while the mean values of I corresponding are 0.17G and 0.189. The
values of R are then 778(0.240  0.176) = 54.46 and 778(0.260  0.189) = 55.24.
The characteristic equations are, then, PV = 54.46 T along the compression curve;
and PV = 55.24 T along the expansion curve. The formula gives changes of en
tropy per pound of substance. The indicator diagram does not ordinal ily depict
the behavior of one pound; but if the weight of substance used per cycle be
known, the volumes taken from the PV diagram may be converted to specific
volumes for substitution in the formula.
It is sometimes desirable to study the TFielations throughout the cycle. In
Fig. 154, let ABCD be the PV diagram. Let EF be any line of constant volume
intersecting this diagram at G, H. By Charles' law, T : T B : : P G  P H . The
Pqr T
FIG. 154. Art. 347. Gas Engme TV Diagram.
ordinates JG, JH may therefore serve to represent temperatures as well as pres
sures, to some scale as yet undetermined. If the ordinate JG represent tempera
turf, then the line OG is a line of constant pressure. Let the pressure along this
line on a TV diagram be the same as that along IG on a PV diagram. Then
(again by Charles' law) the line OH is a line of constant press ui e on the TV plane,
corresponding to the line KH on the PV plane. Similarly, OL corresponds to
MJT and OQ to RB. Pioject the points 5, T, R, B, where MN and RB intersect
the PV diagram, until they intersect OL, OQ. Then points Z7, Q, W, X are
GOVERNING
219
points on the corresponding TV diagram. The scale of T is determined from
the characteristic equation; the value of R may be taken at a mean between
the two given. A tiansfer may now be made to the NT plane by the aid of the
equation n^n a l log e f + (k  Z)log. ^ (Art. 169), in which T a = 491.6,
* ra
.54.46 x 491.6
2116.8
= 12.64.
Figure 155, from Reeve (42), is from a similar fourcycle engine. The enor
mous area BA CD represents heat lost to the water jacket. The inner dead center
of the engine is at x ; thereafter, for a short
period, heat is evidently abstracted from the
fluid, being afterward restored, just as in the
case of a steam engine (Art. 431), because
during expansion the temperatm e of the gases
falls below that of the cylinder walls. This agrees
with the usual notion that most of the heat is
discharged to the jacket early in the expansion
stroke. It would probably be a fair assumption
to consider this loss to occur during ^gn^t^on t as
far as its effect on the diagram is concerned.
Reeve gives several instances in which the
expansive path resembles xBzD; other investi
gators find a constant loss of heat during expan
sion. Figure 156 gives the PV and NT dia
grams for a HornsbyAkroyd engine; the expan
sion line be here actually rises above the iso
thermal, indicative of excessive after burning.
FIG. 155. Art. 347. Gas Engine
348. Methods of Governing. The Entropy Diagram,
power exerted by an Otto cycle engine
may be varied in accordance with the external load by various
methods; in order that efficiency may be maintained, the governing
should not lower the ratio of pressures during compression. To ensure
N
FIG. 156. Art. 347. Diagrams for HornsbyAkroyd Engine.
220 APPLIED THERMODYNAMICS
this, variation of the clearance, by mechanical means or water
pockets and outside compression have been proposed, but no practicably
efficient means have yet been developed. The speed of an engine may
be changed by varying the point of ignition, a most wasteful method,
because the reduction in power thus effected is unaccompanied by any
change whatever in fuel consumption. Equally wasteful is the use of
excessively small ports for inlet or exhaust, causing an increased nega
tive loop area and a consequent reduction in power when the speed
tends to increase. In engines where the combustion is gradual, as in
the Brayton or Diesel, the point of cutoff of the charge may be changed,
giving the same sort of control as in a steam engine,
Three methods of governing Otto cycle engines are in more or less
common use. In the "Jiitormiss" plan, the engine omits drawing in its
charge as the external load decreases. One or more idle strokes ensue.
No loss of economy results (at least from a theoretical standpoint), but the
speed of the engine is apt to vary on account of the increased irregularity
of the already occasional impulses. Governing by changing the proportions
of the mixture (the total amount being kept constant) should apparently
not affect the compression; actually, however, the compression must be
fixed at a sufficiently low point to
avoid danger of preignition to the
strongest probable mixture, and
thus at other proportions the de
gree of compression will be less
than that of highest efficiency. A
change in the quantity of the mix
ture, without change in its propor
tions, by throttling the suction or
by entirely closing the inlet valve
Art. 348 Effect of lOirottlin/ tOWard the end f the suction
stroke, results in a decided change
of compression pressure, the superimposed cards being similar to those
shown in Pig. 157. In theory, at least, the range of compression pressures
would not be affected; but the variation in proportion of clearance gas
present requires injurious limitations of final compression pressure, just
as when governing is effected by variations in mixture strength. Besides,
the rapidity of flame propagation is strongly influenced by variations
in the pressure at the end of compression.
349. Defects of Gas Engine Governing. The hitormiss system may
be regarded as entirely inapplicable to large engines. The other
practicable methods sacrifice the efficiency. Further than this, the
DETAILS 221
governing influence is exerted during; the suction stroke, one full revolu
tion (in fourcycle engines) previous to the working stroke, which should
be made equal in effort to the external load. If the load changes during
the intervening revolution, the control will be inadequate. Gas engines
tend therefore to irregularity in speed and low efficiency under variable
or light loads. The first disadvantage is overcome by increasing the
number of cylinders, the weight of the fly wheel, etc., all of which
entails additional cost. The second disadvantage has not yet been
overcome. Tn most large power plant engines, both the quantity and
strength of the mixture are varied by the governor.
350. Construction Details. The irregular impulses characteristic of the gas
engine and the high initial pressures attained require excessively heavy and
strong frames. For anything like good regulation, the fly wheels must also be
exceptionally heavy. For small engines, the bed casting is usually a single heavy
piece. The type of frame usually employed on large engines is illustrated in Fig.
152. It is in contact with the foundation for its entire length, and in many cases
is tied together by rods at the top extending from cylinder to cylinder.
Each working end of the cylinder of a fourcycle engine must have two valves,
one for admission and one for exhaust. In many cases, three valves are used,
the air and gas being admitted separately. The valves are poppet, of the plain
disk or mushroom type, with beveled seats; in large engines, they are sometimes
of the doublebeat type, shown in Fig. 153. Sliding valves cannot be employed
at the high temperkture of the gas cylinder.* Exhaust opening must always be
under positive control; the inlet valves may be automatic if the speed is low, but
are generally mechanically operated on large engines. Alljshould be finally seated
by spring action, so as to avoid shocks. In horizontal fourcycle engines, a earn
shaft is driven from an eccentric at half the speed of the engine. Cams or eccen
trics on this shaft operate each of the controlling valves by means of adjustable
oscillating levers, a supplementary spring being empolyed to accelerate the closing
of the valves. In order that air or gas may pass at constant speed through the
ports, the cam curve must be carefully proportioned with reference to the varia
tion in conditions in the cylinder (43). Hutton (44) advises proportioning of
ports such that the mean velocity may not exceed 60 ft. per second for automatic
inlet valves, 90 ft. for mechanically operated valves, and 75 ft. for exhaust valves,
on small engines.
351. Starting Gear. No gas engine is selfstarting. Small engines are often
started by turning the fly wheel by hand, or by the aid of a bar or gearing. An
auxiliary hand air pump may also be employed to begin the movement. A small
electnc motor is sometimes used to drive a gearfaced fly wheel with which the
motor pinion meshes. In all cases, the engine starts against its friction load only,
and it is usual to provide a method for keeping the exhaust valve open during part
of the compression stroke so as to decrease the resistance. In multiplecylinder
engines, as in automobiles, the ignition is checked just prior to stopping. A com
pressed but unexploded charge will then often be available for restarting. In the
* The sleeve valve, analogous to the piston valve commonly used on locomotives^
has been successfully developed for automobile work
222 APPLIED THERMODYNAMICS
Clerk engine, a supply of unexploded mixture was taken during compression from
the cylinder to a strong storage tank, from which it could be subsequently drawn
Gasoline railway motor cars are often started by means of a smokeless powder
cartridge exploded in the cylinder Modern lar^e enpines are started by com
pressed air, furnished by a directdriven or independent pump, and stored in small
tanks. Kecent automobile practice has developed two new starting methods:
(a) By acetylene generated from calcium carbide and watei under pressure, and
(6) by an electric motor, operated from a storage battery which is charged while
the engine is running The same batteiy lights the car.
352. Jackets. The use of waterspray injection during expansion has been
abandoned, and air cooling is practicable only in small sizes (say, for diameters
less than 5inch). The cylinder, piston, piston rod, and valves must usually be
thoroughly waterjacketed * Positive circulation must be provided, and the water
cannot be used over again unless artificially cooled. At a heat, consumption of 200
B t u. per minute per Ihp v with a 40 per cent loss to the jacket, the theoretical
consumption of water heated from 80 to 160 F is exactly 1 Ib. per Ihp per minute.
This is greater than the water consumption of a noncondensing steam plant, but
much less than that of a condensing plant The discharge water fiom large engines
is usually kept below 130 F. In smaller units, it may leave the jackets at as high
a temperature as 160 F. The usual nss of temperature of water while passing
through the jackets is from 50 to 10 j F. The circulation may be produced either
by gravity or by pumping.
353. Possibilities of Gas Power. The gas engine, at a comparatively early
stage in its development, has surpassed the best steam engines in thermal effi
ciency. Mechanically, it is less perfect than the latter ; and commercially it is
regarded as handicapped by the greater lehabihtv, moie geneial field of applica
tion, and much lower cost (excepting, possibly, in the Idrgest sizes j) of the steam
engine. The use of producer gas f 01 power eliminates the coal .smoke nuisance ,
the standby losses of producers are low ; and gas may be stored, in small quanti
ties at least The small gas engine is quite economical and may be kept so. The
small steam engine is usually wasteful. The Otto cycle engine regulates badly, a
disadvantage which can be overcome at excessive cost; it is not self starting ; the
cylinder must be cooled. Kveu if the mechanical necessity for jacketing could be
overcome, the same loss would be experienced, the heat being then earned off in
the exhaust. The ratio of expansion is too low, cau&ing excessive waste of heat
at the exhaust, which, however, it may prove possible to reclaim. The heat in the
jacket water is large in quantity and losv in temperature, so that the proV
lem of utilization is confronted with the second law of thermodynamics.
Methods of reversing have not yet been worked out, and no important marine
applications of gas power have been made, although small producer plants have
been installed for ferryboat service with clutch reversal, and compressed and
* The piston need not be cooled in singleacting fourcycle engines.
f Piston speeds of large gas engines may exceed those of steam engines. Unless
special care is exercised in the design of ports, the efficiency will fall off rapidly with
increasing speed. Gas engines have been built in units up to 8000 hp :2000 hp. from
each of the four twintandem doubleacting cylinders.
GAS POWER 223
stored gas has been used for driving river steamers in France, England, and
Germany.
The proposed combinations of steam and gas plants, the gas plant to take the
uniform load and the steam units to care for fluctuations, really beg the whole
question of comparative desirability. The bad k  characteristic " curve low effi
ciency at light loads and absence of bona fide overload capacity "will always bar
the gas engine from some services, even where the storage battery is used as an
auxiliary. Many manufactui ing plants nuist have steam in any case for process
work. In such, it will be difficult for the gas engine to gain a foothold. For the
utilization of blast furnace waste, even aside from any question of commercial
power distribution, the gas engine has become of prime economic importance.
[A topical list of research problems in gas power engineering, the solution of
which is to be desired, is contained m the Report of the Gas Power Research Com
mittee of the American Society of Mechanical Engineers (1910).]
[See the Resume of Producer Gas Investigations, by Fernald and Smith, Bulletin
No. 13 of the United States Bureau of Mines, 1911.]
(1) Button, The Gas Engine, 190S, 545; Clerk, Theory of the Gas Engine, 1903,
75. (2) Hutton, The Gas Engine, 1908, 158. (3) Clerk, The Gas Engine, 1890,
119121. (4) Ibid., 129. (5) Ibid, 133. (6) Ibid., 137. (7) Ibid, 198. (8)
Engineering News, October 4, 1906, 357. (9) Lucke and Woodward, Tests of
Alcohol Fuel, 1907. (10) Junge, Power, December, 1907. (10 a) For a fuller exposi
tion of the limits of producer efficiency with either steam or waste gas as a diluent,
see the author's paper, Trans. Am. Inst. Chem. Engrs T Vol. II. (11) Trans. A. S.
M. E., XXVIII, 6, 1052. (12) A test efficiency of 657 was obtained by Parker,
Holmes, and Campbell. United States Geological Survey, Professional Paper No. 48.
(13) Ewing, The Steam Engine, 1906, 418. (14) Clerk, The Theory of the Gas Engine,
1903. (15) Theorie und Construction eines rationdlen Warmemotors. (16) Zeuner,
Technical Thermodynamics (Klein), 1907, I, 439, (17) Trans. A. S. M. E., XXI,
275. (18) Ibid., 286. (19) Op. ciL, XXIV, 171. (20) Op. cit., XXI, 276. (21)
Gas Engine Design, 1897, 33. (22) Op. at., p 34 et seq. (23) See Lucke, Trans.
A.S.M. E., XXX, 4, 418. (24) The Gas Engine, 1890, p 95 et seq. (25) A. L.
Westcott, Some Gas Engine Calculations based on Fuel ami Exhaust Gases; Power,
April 13, 1909, p. 693. (26) Hutton, The Gas Engine, 1908, pp. 507, 522. (27)
The Gas Engine, 1908. (28) Clerk, op. cit , p. 216. (29) Op. *., p. 291. (30)
Op. cit., p. 38. The corresponding usual mean effective pressures are given on p. 36.
(30 a) See the author's papers, Commercial Ratings for Internal Combustion Engines,
in Machinery, April, 1910, and Design Constants for Small Gasolene Engines, with
Special Reference to the Automobile, Journal A. S. M. E., September, 1911. (31)
Zeits. Ver. Deutsch. Ing., November 24, 1906; Power, February, 1907. (32) The
Electrical World, December 7, 1907, p. 1132. (33) Trans. A. S. M. E., XXIV, 1065.
(34) Bui. Soc. de V Industrie Mineral, Ser. Ill, XIV, 1461. (35) Trans. A. S. M. E.,
XXVIII, 6, 1041. (36) Quoted by Mathot, supra. (37) Also from Mathot. (38)
Mathot, supra. (40) Op. cit., pp. 342343. (41) Trans. A. S. M. E., XIX, 491.
(42) Ibid., XXIV, 171. (43) Lucke, Gas Engine Design, 1905. (44) Op. tit.,
483.
224 APPLIED THERMODYNAMICS
SYNOPSIS OF CHAPTER XI
The Producer
The importance of the gas engine is largely due to the producer process for making
cheap gas.
In the gas engine, combustion occurs in the cylinder , and the highest temperature
attained by the substance determines the cyclic efficiency.
Fuels are natural gas, carbureted and uncarburetcd water gas, coal gas, coke oven
gas, producer gas, blast furnace gas ; gasoline, kerosene, fuel oil, distillate,
alcohol, coal tars.
The gas producer is a lined cylindrical shell in which the fixed carbon is converted
into carbon monoxide, while the hydrocarbons are distilled off, the necessary heat
being supplied by the fixed carbon burning to CO.
The maximum theoretical efficiency of the producer making power gas is less than that,
of the steam boiler. Either steam or exhaust gas from the engine* must be intro
duced to attain maximum efficiency. The reactions are complicated and partly
reversible.
The mean composition of producer gas, by volume, is CO, 10.2 ; C0 2l 05, H, 12.4 ;
CH 4 , C 2 H 4 , 3.1; N, 55.8.
The "figure of merit ^ is the heating value of the gas per pound of carbon contained.
Gas En (tine
The Otto cycle is bounded by two udialH.it ics and two lines of constant volume; the
engine may operate in either thefours'rftkc eyrie or the twostroke cycle.
In the twostroke cycle, the inlet and exhaust ports are loth open at once.
In the Otto cycle, 5> = t and ^ = If
J ' P e P d T e T d
Efficiency  Tf ~ T * = 1  f ?*\ IT = 7& " T '= 1  / r\ V; it depends solely on the
extent of compression.
The Sargent and Frith cycles.
Efficiency of Atkinson engine (isothermal rejection of heat) = l " log e ~J
10 J. e J. e
higher thaii that of the Otto cycle.
Lenoir cycle: constant pressure rejection of heat, efficiency = 1  ^  y h ~" 
Tj T d T f T^"
Brayton cycle : combustion at constant pressure; efficiency = I fr g ~~ ,,  I?"" ~ >
2/( A Jn) Tb T n
T T 1
or, with complete expansion, " "" <l
TH
A special comparison shows the Clerk Otto engine to give a much higlier efficiency than
the Brayton or Lenoir engine, but that the Brayton engine gives slightly the largest
work area.
The Clerk Otto (complete pressure) cycle gives an efficiency of 1
'II
'II rri g rn
JL e JLo J. e
intermediate between that of the ordinary Otto and the Atkinson.
SYNOPSIS 225
The Diesel cycle: isothermal combustion; efficiency = 1  L \ a/  =!; increases
as ratio of expansion decreases. . yRT a log e
Va
The Diesel cycle . constant pressure combustion.
The Humphrey internal combustion pump.
Modifications in Practice
The PV diagram of an actual Otto cycle engine is influenced by
(a) proportions of the nurture, \tkich must not be too weak or too strong, and
must be controllable ,
(&) maximum allowable temperature after compression to avoid preignition ; the
range of compression, \\hich determines the efficiency, depends upon this as
well as upon the precoinpiesaion pressure and temperature;
(c) the rise of pressure and temperature during combustion; always less than
those theoretically computed, on account of (1) divergences from Charles'
law, (2) the variable specific heats of gases, (0) slow combustion, (4) disso
ciation ;
(<Z) the shape of the expansion curve, usually above the adiabatic, on account of
after burning, in spite of loss of heat to the cylinder wall;
(e) the forms of the suction and exhaust lines, which may be affected by badly
proportioned ports aud passages and by improper valve action.
Dissociation prevents the combustion reaction of more than a certain proportion of
the elementary gases at each temperature within the critical limits.
The point of ignition must somewhat precede the end of the stroke, particularly with
weak mixtures.
Methods of ignition are by hot tube, jump spark, and make and break.
Cylinder clearance ranges from 8.7 to 56 per cent. It is determined by the compression
pressure range.
Scavenging is the expulsion of the burnt gases in the clearance space prior to the
suction stroke.
The diagram factor is the ratio of the area of the indicator diagram to that of the ideal
cycle.
Analysis with specific heats variable.
4'(
Mean effective pressure  , r
Gas Engine Design
In designing an engine for a given power, the gas composition, rotative
speed and piston speed are assumed. The probable efficiency may be
estimated in advance. Overload capacity must be secured by assum
ing a higher capacity than that normally needed ; the engine will do
no more work than that for which it is designed.
226 APPLIED THERMODYNAMICS
Current Forms
Otto cycle oil engines include the Mietz and TVeiss, twocycle, and the Daimler, Priest
man, and HornsbyAkroyd, fourcycle.
Modem forms of the Otto got* engine include the Otto, Foos, CrossleyOtto, and
Andrews.
The TTestinghouse, Riverside, and AllisChalmers engines are built in the largest sizes.
Twocycle gas engines include the Oechelhaueser and Koertmg.
Special engines are "built for motor bicycles, automobiles, and launches, and for burn
ing alcohol.
The basis of efficiency is the heat unit consumption per horse power per minute
The mechanical efficiency may be computed from either gross or net indicated work.
Recorded efficiencies of gas engines range up to 42. 7 per cent; plant efficiencies to 0.7
Ib. coal per brake hp.hr.
The mechanical efficiency increases with the size of the engine, and is greater with the
fourstroke cycle.
About 38 per cent of the heat supplied is carried oS by the jacket water, and about
S3 per cent by the exhaust (jases^ in ordinary practice.
The entropy diagram may be constructed by transfer from the PFor TV diagrams.
Governing is effected
(a) by the hitormiss method; economical, but unsatisfactory for speed regulation,
V) by throttling, 1 both witehil.
(c) by changing mixture proportions, J
In all cases, the governing effort is exerted too early in the cycle.
Gas engines must have heavy frames and fly wheels; exhaust valves (and inlet valves
at high speed) must be mechanically operated by carefully designed cams; pro
vision must be made for starting; cylinders and other exposed parts are jacketed.
About 1 Ib. of jacket water is required per Ibp. minute.
Gas engine advantages: high thermal efficiency; elimination of coal smoke nuisance ;
standby losses are low ; gas may be stored ; economical in small units ; desirable
for utilizing blast furnace gas.
Disadvantages : mechanically still evolving ; of unproven reliability ; less general field
of application , generally higher first cost ; poor regulation ; not selfstarting ;
cylinder must be cooled ; low ratio of expansion ; nonreversible ; no overload
capacity ; no available byproduct heat for process work in manufacturing plants,
PROBLEMS
1. Compute the volume of air ideally necessary for the complete combustion of
1 cu. ft. of gasoline vapor, C fa Hii.
2* Find the maximum theoretical efficiency, using pure air only, of a power gas
producer fed with a fuel consisting of 70 per cent of fixed carbon and 30 per cent of
volatile hydrocarbons.
3. In Problem 2, what is the theoretical efficiency if 20 per cent of the oxygen
necessary for gasifying the fixed carbon is furnished by steam ?
4. In Problem 3, if the hydrocarbons (assumed to pass off unchanged) are half
pure hydrogen and half marsh gas, compute the producer gas composition by volume,
PROBLEMS 227
using specific volumes as follows, nitrogen, 12.75; hydrogen, 17R.83; carbon mon
oxide, 12.75; marsh gas, 22.3.
5. A producer gasifying pure carbon is supphed with the theoretically necessary
amount of oxygen from the atmospheie and from the gas engine exhaust. The latter
consists of 28.4 per cent of CO., and 71.6 per cent of X, by weight, and is admitted to
the extent of 1 Ib. per pound of pure carbon gasified. Find the rise in temperature,
the composition of the produced gas, and the efficiency of the process. The heat of
decomposition of CO., to CO may be taken at 10,050 B. t. u. per pound of carbon.
6. rind the figures of merit in Piobleins 4 and 5. (Take the heating value of H
at 53,400, of CH 4 , at 22,500.)
7. In Fig. 134, let ^ = 4, P d = 30 (Ibs. per sq. m ), P a = P ff =P d +W, T 6 = 3000,
* e
T d = 1000 (absolute). Find the efficiency and area of each of the ten cycles, for 1 Ib.
of air, without using efficiency formulas.
8 In Problem 7, show graphically by the XT diagram that the Carnot cycle is
the most efficient.
9 What is the maximum theoretical efficiency of an Otto fourcycle engine in
which the fuel used is producer gas ? (See Art. 312.)
10. What maximum temperature should theoretically be attained in an Otto en
gine using gasoline, with a temperature after compression of 780 F. ? (The heat liber
ated by the gasoline, available for inci easing the temperature, may be taken at 19,000
B. t. u per pound.)
11. Find the mean effective pressure and the work done in an Otto cycle between
volume limits of 0.5 and 2.0 cu. ft. and pressure limits of 14.7 and 200 Ib. per square
inch absolute.
12. An Otto engine is supplied with pure CO, with pure air in just the theoretical
amount for perfect combustion. Assume that the dissociation effect is indicated by the
formula * (1.00 a) (COOO 7") = 300, in which a is the proportion of gas that will
combine at the temperature T F. If the temperature after compression is 800 F.,
what is the maximum temperature attained during combustion, and what proportion
of the gas will burn during expansion and exhaust, if the combustion line is one of con
stant volume ? The value of I for CO is 0.1758.
13. An Otto engine has a stroke of 24 in., a connecting rod 00 in. long, and a pis
ton speed of 400 ft. per minute. The clearance is 20 per cent of the piston displace
ment, and the volume of the gas, on account of the speed of the piston as compared
with that of the flame, is doubled during ignition. Plot its path on the PV diagram
and plot the modified path when the piston speed is increased to 800 ft, per minute,
assuming the flame to travel at uniform speed and the pressure to increase directly as
the spread of the flame. The pressure range during ignition is from 100 to 200 Ib.
14. The engine in Problem 11 is fourcycle, twocylinder, doubleacting, and makes
100 r. p. m. with a diagram factor of 0.40. Find its capacity.
15. Starting at P d = 14.7, F</ = 43.45, T<j = 32 F. (Fig. 122), plot (a) the ideal
Otto cycle for 1 Ib. of CO with the necessary air, and (b) the probable actual cycle
* This is assumed merely for illustrative purposes. It has no foundation and is
irrational at limiting values.
228
APPLIED THERMODYNAMICS
modified as described in Arts 309328, and find the diagram factor. Clearance is 25
per cent of the piston displacement in both cases.
16. Find the cylinder dimensions in Art. 332 if the gas composition be as given in
Art. 285. (Take the average heating value of C II 4 and C^ at 22,500 B t. u. per pound,
and assume that the gas contains the same amount of each of these constituents )
17. Find the clearance, cylinder dimensions, and probable efficiency in Art. 332 if
the engine is twocycle.
18. Find the size of cylinders of a fourcylinder, fourcycle, singleacting gasoline
engine to develop 30 blip, at 1200 r. p. in , the cylinder diameter being equal to the
stroke. Estimate its thermal efficiency, the theoretically necessary quantity of air
being supplied.
19. An automobile consumes 1 gal. of gasoline per 9 miles run at 50 miles per
hour, the horse power developed being 23. Find the heat unit consumption per Ihp.
per minute and the thermal efficiency , assuming gasoline to weigh 7 Ib. per gallon
SO. A twocycle enyine gives an indicator diagram in which the positive work
area is 1000 ft.lb., the negative work area 00 ft Ib. The work at the brake is 700
ft,lb. Give two values for the mechanical efficiency
21. The engine in Probtem 17 dischaiges 30 per cent of the heat it receives to the
jacket. Find the water consumption in pounds per minute, if its initial temperature
is 72" F.
22. In Art 344, what was the producer efficiency in the case of the Guldner en
\
0.20 0.40 0.00 u.SO 100
FIG. 158. Prob. 23. Indicator Diagram for Transfer.
PROBLEMS 229
gine, assuming its mechanical efficiency to have been 0.85? If the coal contained
13,800 B. t. u. per pound, what was the cual consumption per brake hp lir. ?
23 Given tbe indicator diagram of Fi. 158, plot accurately the TV diagram, the
engine using 0.0402 Ib. of substance per cycle. Draw the compressive path on the NT
diagram by both of the methods of Art. 347.
24. The engine in Problem 17 governs by throttling its charge. To what percent
age of the piston displacement should the clearance be decreased in ordei that the pres
sure after compression may be unchanued when the precompression pressure drops to
10 Ib. absolute ? What would be the object (if huch a change in clearance ?
25. In the Diesel engine, Problem 7, by what, percentages will the efficiency and
capacity be affected, theoretically, if the supply of fuel, is cut off 30 per cent earlier in
T r T*
the stroke ? (i.e , cutoff occurs when the volume is u ~~ * + F, Fig. 134.)
2
26. Under the conditions of Art. 835, develop a relation between piston displace
ment in cubic inches per minute, and Ihp., lor four cylinder fourcycle single acting
gasolene engines Also find the relation between cylinder volume and Ihp. if endues
run at 1500 r. p. m., and the relation between cylinder diameter and Ihp. if bore = stroke,
at 1500 r. p. in.
27. In an Otto engine, the range of pressures during compression is from 13 to
130 Ib,, the compression curve pa 1 * = /. Find the percentage of clearance.
28 The clearance space of a 7 by 12 in. Otto engine is iound to hold Ib. of water
at 70 F. Find the ideal efficiency of the engine. (See Art. 295.)
29. An engine uses 220 cu. ft. of gas, containing 800 B. t. u. per cubic foot, in 39
minutes, while developing 12.8 hp. Find its thermal efficiency.
30. In the formula, brake hp. =  (Art. 335), if the mechanical efficiency is
80, what mean effective pressure is assumed in the cylinder ?
31. A sixcylinder fourcycle engine, singleacting, with cylinders 10 by 24 in.,
develops 500 hp. at 200 r. p. m. What is the mean effective pressure ?
32. An engine uses 1.62 Ib. of gasolene (210! K> B. t. u. per pound) per Blip hr.
What is its efficiency from fuel to shaft ? If it is a 2cycle engine with a pressure of
00 Ib. gage at the end of compression, estimate the ideal efficiency.
33. Derive an expression for the mean effective pressure in Ait. 293.
CHAPTER XII
THEORY OF VAPORS
354. Boiling of Water. If we apply heat to a vessel of water open
to the atmosphere, an increase of temperature and a slight increase
of volume may be observed. The increase of temperature is a gain
of internal energy; the slight increase of volume against the constant
resisting pressure of the atmosphere represents the performance of
external work, the amount of which may be readily computed. After
this operation has continued for some time, a temperature of 212 F.
is attained, and steam begins to form. The water now gradually
disappears. The steam occupies a much larger space than the water
from which it was formed ; a considerable amount of external work is
done in thus augmenting the volume against atmospheric pressure ;
and the common temperature of the steam and the water remains con
stant at 212 F. during evaporation.
355. Evaporation under Pressure. The same operation may be
performed in a closed vessel, in which a pressure either greater or less
than that of the atmosphere may be maintained. The water will now
boil at some other temperature than 212 F. ; at a lower temperature,
if the pressure is less than atmospheric^ and at a higher temperature^ if
greater. The latter is the condition in an ordinary steam boiler. If
the water be heated until it is all boiled into steam, it will then be
possible to indefinitely increase the temperature of the steam, a result
not possible as long as any liquid is present. The temperature at
which boiling occurs may range from 32 F. for a pressure of
0.089 Ib. per square inch, absolute, to 428 F. for a pressure
of 336 Ib. ; but for each pressure there is a fixed temperature of
ebullition.*
* A striking illustration is in the case of air, which has a boiling point of 314 B 1 .
at atmospheric pressure. As we see "liquid air," it is always boiling. If we
attempted to confine it, the pressure which it would exert would "be that corresponding
with the room temperature, several thousand pounds per square inch.
Hydrogen has an atmospheric boiling point of 423 2T.
230
SATURATED AND SUPERHEATED VAPOR 231
356. Saturated Vapor. Any vapor in contact with its liquid and
in thermal equilibrium (i.e. 7 not constrained to receive or reject heat)
is called a saturated vapor. It is at the minimum temperature (that
of the liquid) which is possible at the existing pressure. Its density
is consequently the maximum possible at that pressure. Should it
be deprived of heat, it cannot fall in temperature until after it has
been first completely liquefied. If its pressure is fixed, its temperature
and density are also fixed. Saturated vapor is then briefly definable
as vapor at the minimum temperature or maximum density possible
under the imposed pressure.
357. Superheated Vapor. A saturated vapor subjected to ad
ditional heat at constant pressure, if in the presence of its liquid,
cannot rise in temperature ; the only result is that more of the liquid
is evaporated. When all of the liquid has been evaporated, or if the
vapor is conducted to a separate vessel where it may be heated while
not in contact with the liquid, its temperature may be made to rise,
and it becomes a superheated vapor. It may be now regarded as an
imperfect gas; as its temperature increases, it constantly becomes
more nearly perfect. Its temperature is always greater, and its
density less, than those properties of saturated vapor at the same
pressure ; either temperature or density may, however, be varied at
will, excluding this limit, the pressure remaining constant. At
constant pressure, the temperature of steam separated from water
increases as heat is supplied.
The characteristic equation, PV = R T, of a perfect gas is inapplicable to steam.
(See Art. 390.) The relation of pressuie, volume, and temperature is given by
various empirical formulas, including those of Joule (1), Rankine (^), Him (3),
Racknel (4), Clausius (5), Zeuner (6), and Knoblauch Linde and Jakob (7).
These are in some cases applicable to either saturated or superheated steam.
SATURATED STEAM
358. Thermodynamics of Vapors. The remainder of this text is
chiefly concerned with the phenomena of vapors and their application
in vapor engines and refrigerating machines. The behavior of vapors
during heat changes is more complex than that of perfect gases.
The temperature of boiling is different for different vapors, even at
the same pressure ; but the following laws hold for all other vapors
as well as for that of water ;
232 APPLIED THERMODYNAMICS
(1) The temperatures of the liquid and of the vapor in contact with
it are the same ;
(2) The temperature of a specific saturated vapor at a specified pres
 sure is always the same ;
(3) The temperature and the density of a vapor remain constant
during its formation from liquid at constant pressure ;
(4) Increase of pressure increases the temperature and the density of
the vapor ; *
(5) Decrease of pressure lowers the temperature and the density ;
(6) The temperature can beincreased and the density can be decreased
at will, at constant pressure, when the vapor is not in contact
with its liquid ;
(7) If the pressure upon a saturated vapor be increased without allow
ing its temperature to rise, the vapor must condense ; it cannot
exist at the increased pressure as vapor (Art. 356). If the
pressure is lowered while the temperature remains constant, the
vapor becomes superheated.
359. Effects of Heat in the Formation of Steam. Starting with
a pound of water at 32 F., as a convenient reference point, the heat
expended during the formation of saturated steam at any temperature
and pressure is utilized in the following ways :
(1) h units in the elevation of the temperature of the water. If the
specific heat of water be unity, and t be the boiling point,
h = t 32 ; actually, h always slightly exceeds this, but the
excess is ordinarily small. fJ
* Since mercury boils, at atmospheric pressure, at 675 F., common thermometers
cannot be used for measuring temperatures higher than this ; but by filling the space in
the thermometric tube above the mercury with gas at high pressure, the boiling point
of _the mercury may be so elevated as to permit of its use for measuring flue gas
temperatures exceeding 800 F.
t According to Barnes 1 experiments (8), the specific heat of water decreases from
1.0094 at 32 F. to 91)735 at 100 3 P., and then steadily increases to 1.0476 at 428 F.
J In precise physical experimentation, it is necessary to distinguish between the
value of h measured above 32 F. and (ttinrispheric pressure, and that measured above
32 F. and the corresponding pressure nf the saturated vapor. This distinction is of no
consequence in ordinary engineering work.
FORMATION OF STEAM 233
(2) P^ ' v ) units in the expansion of the water (external work), p
( To
being the pressure per square foot and v and T^the initial and
final specific volumes of the water respectively. This quantity
is included in item , as above defined ; it is so small as to be
usually negligible, and the total heat required to bring the
water up to the boiling point is regarded as an internal energy
change.
(3) e = ^^ }  units to perform the external work of increasing
7 ( 8
the volume at the boiling point from that of the water to that of
the steam, HHbeing the specific volume of the steam.
(4) r units to perform the disgregation work of this change of state
(Art. 15) ; items (3) and (4) being often classed together as L.
The total heat expended per pound is then
The values of these quantities vary widely with different vapors, even when
at the same temperature and pressure; in general, as the pressure increases, h
increases and L decreases. Watt was led to believe (erroneously) that the sum of
h and L for steam was a constant; a result once described as expressing ^ Watt's
Law." This sum is now known to slowly increase with increase of pressure.
360. Properties of Saturated Steam. It has been found experimentally
that as p, the pressure, increases, t } 7i, e, and H increase, while r and L
decrease. These various quantities are tabulated in what is known as a
steam table.*
* Begnaalt's experiments were the foundation of the steam tables of Rankine (9),
Zeuner (10), and Porter (11). The last named have been regarded as extremely accu
rate, and were adopted as standard for use in reporting trials of steam boilers and
pumping engines by the American Society of Mechanical Engineers. They do not
give all of the thermal properties, however, and have therefore been unsatiKtactory lor
some purposes. The tables of DwelshaueversDery (12) were based on Zeuner's ;
Duel's tables, originally published in Weisbach's Jtfeefaz/ucs (13), on Rankine's,
Peabody's tables are computed directly from Regnaulfs work ( 14). The principal
differences in these tables were due to Rome uncertainty as to the specific volume of
steam (15). The precise work of Holborn and Henning (16) on the pressuretempera
ture relation and the adaptation by Davis (17) of recent experiments on the specific
heat of superheated steam to the determination of the total heat of saturated steam
(Art. 388) have suggested the possibility of steam tables of greater accuracy. The
most recent and satisfactory of these is that of Marks and Davis (1R), values from
which are adopted in the remainder of the present text. (See pp. 247, 248.)
234 APPLIED THERMODYNAMICS
Our original knowledge of these values was derived from the com
prehensive experiments of Regnault, whose empirical formula for the
total heat of saturated steam was ff = 1081.94 + 0.305*. The recent
investigations of Davis (17) show, however, that a more accurate ex
pression is
ff = 1150.3 + 0.3745(*212)0.00055(212)2 (Art. 388).
(The total heat at 212 F, is represented by the value 1150 3.) Barnes'
and other determinations of the specific heat of water permit of the com
putation of h; and L =H h. The value of e may be directly calculated
if the volume W is known, and r=Le. The value of r has a straight
line relation, approximately, with the temperature. This may be
expressed by the formula r = 1061.3 0.79 t F. The method of deriv
ing the steam volume, always tabulated with these other thermal
properties, will be considered later. When saturated steam is con
densed, all of the heat quantities mentioned are emitted in the reverse
order, so to speak. Regnault's experiments were in fact made, not
by measuring the heat absorbed during evaporation, but that emitted
during condensation. Items h and r are both internal energy effects;
they are sometimes grouped together and indicated by the symbol E]
whence H=E + e. The change of a liquid to its vapor furnishes the
best possible example of what is meant by disgregation work. If there
is any difficulty in conceiving what such work is, one has but to com
pare the numerical values of L and r for a given pressure. What
becomes of the difference between L and e? The quantity L is often
called the latent heat, or, more correctly, the latent heat of evapora
tion. The " heat in the water " referred to in the steam tables is h\
the " heat in the steam " is #", also called the total heat.
361. Factor of Evaporation. In order to compare the total expen
ditures of heat for producing saturated steam under unlike condi
tions, we must know the temperature T, other than 32 F. (Art.
359), at which the water is received, and the pressure p at which
steam is formed ; for as T increases, h decreases ; and as p increases,
S increases. This is of much importance in comparing the results
of steam boiler trials. At 14.7 Ib. (atmospheric) pressure, for ex
ample, with water initially at the boiling point, 212 F., A = and
H~ L*= 970. 4 (from the table, p. 247). These are the conditions
adopted as standard, and with which actual evaporative performances
PRESSURETEMPERATURE 235
are compared. Evaporation under these conditions is described as
being
From (a feed water temperature of) and at (a pressure correspond
ing to the temperature of) 212 F.
Thus, for p = 200, we find L = 843.2 and h = 354.9 ; and if the tem
perature of the water is initially 190 F., corresponding to the heat
contents of 157,9 B. t. u.,
H= L + (354.9  157.9) = 843.2 + 197 = 1040.2.
The ratio of the total heat actually utilized for evaporation to that
necessary " from and at 212 F/' is called the factor of evaporation.
In this instance, it has the value 1040.2 r 970.4 = 1.07. Generally,
if L> h refer to the assigned pressure, and A is the heat correspond
ing to the assigned temperature of the feed water, then the factor of
evaporation is
F = \L+ (h A )]* 970.4.
362. Pressuretemperature Relation. Regnault gave, as the result of his ex
haustive experiments, thirteen temperatui es corresponding to known pressures
at saturation. These range from  32 C. to 220 C. He expressed the relation
by four formulas (Art. 19); and no less than fifty formulas have since been.
devised, representing more or less accurately the same experiments. The deter
minations made by Holborn and Kenning (16) agree closely with those of Reg
nault; as do those by Wiebe (19) and Thiesen and Scheel (20) at temperatures
below the atmospheric boiling point.
The steam table shows that, beginning at 32 F. ; the pressure rises with the
temperature, at first slowly and afterward much more rapidly. The fact that
slight increases of temperature accompany large increases of pressure in the working
part of the range seems fatal to the development of the engine using saturated
steam, the high temperature of heat absorption shown by Caraot to be essential
to efficiency being unattainable without the use of pressures mechanically objection
able.
A recent formula for the relation between pressure and temperature is (Power>
March 8, 1910)
6 
in which t is the Fahrenheit temperature and p the pressure in pounds per square
inch. This has an accuracy within 1 or so for usual ranges.
Marks gives (Jour. A.S.M. E., XXXIII, 5) the equation,
log p 10.515354  0.00405096 T+ 0.00000 1392964 T 2 ,
T being absolute and p in pounds per square inch. This has an established accuracy
within i of 1 per cent for the whole range of possible temperatures.
236 APPLIED THERMODYNAMICS
363. Pressure and Volume. Fairbairn and Tate ascertained experimentally
in 1860 the relation between pressure and volume at a few points; some experi
ments were made by Hira; and BatteUi has reported results which have been
examined by Tumhrz (21) who gives
BT
where p is in pounds per square inch, c = 0.256, 5 = 0.5962 and T is in degrees
absolute.
More recent experiments by Knoblauch, Linde, and Klebe (1905) (22) give
the formula
j0 5962 rp(l+0.0014 p) ( 150 ' 3 ff' 00 0.0833] ,
in which p is in pounds per square inch, *> in cubic feet per pound, and T in degrees
absolute.
Goodenough's modified form of this equation is more convenient:
in which =0.5963, log w = 13.67938, n=5, c=0.088, a=00006.
A simple empirical formula is that of Rankine, pptt = constant, or that of Zeuner,
ppri.owe constant. These forms of expression must not be confused with the
PV n = c equation for various polytropic paths. An indirect method of determin
ing the volume of saturated steam is to observe the value of some thermal piop
erty, like the latent heat, per pound and per cubic foot, at the same pressure.
The incompleteness of experimental determinations, with, the diffi
culty in all cases of ensuring experimental accuracy, have led to the use of
analytical metliods (Art. 368) for computing the specific volume. The
values obtained agree closely with those of Knoblauch, Linde, and Klebe.
364. Wet Steam. Even when saturated steam is separated from
the mass of water from which it has been produced, it nearly always
contains traces of water in suspension. The presence of this water
produces what is described as wet steam, the wetness being an indi
cation of incomplete evaporation. Superheated steam, of course,
cannot be wet. Wet steam is still saturated steam (Art. 356) ; the
temperature and density of the steam are not affected by the pres
ence of water.
The suspended water must be at the same temperature as the
steam; it therefore contains, per pound, adopting the symbols of
Art. 359, h units of heat. In the total mixture of steam and water,
then, the proportion of steam being x, we write for L, xL ; for r, xr ;
for e, xe i for j, xr + h ; while, h remaining unchanged, J3T= Ji + xL.
FORMATION OF STEAM
237
o
FNJ K>l) Arts. :*M, 3f>6, 371). Paths
of Steam Formation.
The factor of evaporation (Art. 361), wetness considered, must be
correspondingly reduced ; it is F= [sL + (Ji  7/ )] H 070.4.
Tlie specific volume of wet steam is TF, r = V+x( TF F)=^^+ T",
where #= TF T. For dry steam, .r= 1, and TF; r = V+ ( W V) = TF
The error involved in assuming W n = ./ TFis usually inconsiderable,
since the value of T r is comparatively small.
365. Limits of Existence of Saturated Steam. In Fig. 160, let
ordinates represent temperatures, and abscissas, volumes. Then db
is a line representing possible condi
tions of water as to these two proper
ties, which may be readily plotted if
the specific volumes at various tem
peratures are known; aud cd is a
similar line for steam, plotted from the
values of TFand t in the steam table.
The lines db and cd show a tendency
to meet (Art. 370). The curve cd is
called the curve of saturation, or of con
stant steam weight; it represents all possible conditions of constant
weight of steam, remaining 1 saturated. It is not a path, although
the line db is (Art. 3G3). States along db are those of liquid; the
area lade includes all wet saturated states ; along rfc, the steam is
dry and saturated; to the right of dc^ areas include superheated
states.
366. Path during Evaporation. Starting at 32, the path of the
substance during heating and evaporation at constant pressure would
be any of a series of lines aef, old, etc. The curve db is sometimes
called the locus of boiling points. If superheating at constant pres
sure occur after evaporation, then (assuming Charles' law to hold)
the paths will continue as fg* ij, straight lines converging at 0.
For a saturated vapor, wet or dry, the isothermal can only be a straight
line of constant pressure,
367. Entropy Diagram. Figure 161 reproduces Fig. 160 on the
entropy plane. . The line ab represents the heating of the water at
constant pressure. Since the specific heat is slightly variable, the
238
APPLIED THERMODYNAMICS
increase of entropy must be computed for small differences of tem
perature. The more complete steam tables give the entropy at various
boiling points, measured above 32. Let evaporation occur when the
g M
FIG. 161. Arts. 367, 3. M73, 370, 370, 386, 426 The Steam Dome.
temperature is T b . The increase of entropy from the point b (since
the temperature is constant during the formation of steam at constant
pressure) is simply L s (2^ + 459. G), which is laid off as be. Other
points being similarly obtained, the saturation curve cd is drawn.
The paths from liquid at 82 to dry saturated steam are ale, a VN,
aUS, etc.
The factor of evaporation may be readily illustrated. Let the area
eUSf represent L^ the heat necessary to evaporate one pound from and
at 212 P. The area gjbcJi represents the heat necessary to evaporate one
pound at a pressure b from a feedwater temperature j. The factor of
evaporation is gjbch** eUSf. For wet steam at the pressure b, it is, for
example, gjbik 5 eUSf.
368. Specific Volumes* Analytical Method. This was developed by
Clapeyron in 1834, In Fig. 102, let abed represent a Carnot cycle in
which steam is the working substance and the range of temperatures is
dT. Let the substance be liquid along da and dry saturated vapor along be.
VOLUME OF VAPOR
239
The heat area alfe is L\ the work area abed is (L + T)dT. In Fig. 163,
let abed represent the corresponding work area on the pv diagram. Since
the range of temperatures is only dT, the range of pressures may be
FIGS. 162 and 1(>3 Arts. otiS, 400, ^Ou. nj
\ oiuuieb
taken as c/P; whence the area abed in Fig. 1C3 is dP( W F), where W
is the volume along be, and Fthat along ad. This area must by the first
law of thermodynamics equal (778 L = T)dT\ whence
78 L d'.
Thus, if we know the specific volume of the liquid, and the latent heat
of vaporization, at a given temperature, we have only to determine the
dT
differential coefficient in order to compute the specific volume of the
vapor. The value of this coefficient may be approximately estimated from
the steam table; or may be accurately ascertained when any correct formula
for relation between P and T is given. The advantage of this indirect
method for ascertaining specific volumes arises from the accuracy of
experimental determinations of T, L, and P.
369. Entropy Lines. In Fig. 161, let ab be the water line, cd
the saturation curve ; then since the horizontal distance between
these lines at any absolute temperature T is equal to is2 7 , we
deduce that, for steam only partially dry, the gain of heat in passing
from the water line toward cd being xL instead of i, the gain of
entropy is xL * T instead of L + T. If on be and ad we lay off bi
and al = x be and x ad, respectively, we have two points on the
constant dryness curve e7, along which the proportion of dryness is x.
Additional points will fully determine the curve. The additional
curves zn, pq, etc., are similarly plotted for various values of 2:, all
of the horizontal intercepts between ab and cd being divided in the
same proportions by any one of these curves.
210 APPLIED THERMODYNAMICS
370. Constant Heat Curves. Let the total heat at o be H. To
find the state at the temperature be, at which the total heat may also
equal IT, we remember that for wet steam H= li I xL, whence
x = (ff h) * L = bj> f be. Additional points thus determined for
this and other assigned values of H give the constant total heat
curves op, mr, etc. The total heat of saturated vapor is not, however,
a cardinal property (Art. 10). The state points on this diagram
determine the heat contents only on the assumption that heat has
been absorbed at constant pressure ; along such paths as abc, aUS,
aVN, etc.
371. Negative Specific Heat. If steam passes from o to r, Tig. 161,
heat is absorbed (area sort) while the temperature decreases. Since the satu
ration curve slopes constantly downward toward the light, the specific heat
of steam kept saturated is therefore negative. The specific heat of a vapor
can be positive only when the saturation curve slopes downward to the left,
like CM, as in the case, for example, of the vapor of ether (Fig. 315). The
conclusion that the specific heat of saturated steam is negative was
reached independently by Kankine aud Clausins in 1850. It was experi
mentally verified by Him in 1862 aud by Cazin in 1866 (24). The
physical significance is simply that when the temperature of dry saturated
steam is increased adiabatically, it becomes superheated ; heat must be
abstracted to keep it saturated. On^the other hand, when dry saturated
steam expands, the temperature falling; it tends to condense, and
lieat must be supplied to keep it dry. If steam at c, Fig. 161, having
been formed at constant pressure, works along the saturation curve to 2?,
its heat contents are not the same as if it had been formed along aVN,
but are greater, beiug greater also than the " heat contents " at c.
372. Liquefaction during Expansion. If saturated steam expand adia
batically from c, Fig. 161, it will at v have become 10 per cent wet. If
its temperature increase adiabatically from y, it will at c have become
dry. If the adiabatic path then continue, the steam will become superheated.
Generally speaking, liquefaction accompanies expansion and drying or
superheating occurs during compression. If the steam is very wet to begin
with, say at the state #, compression may, however, cause liquefaction, and
expansion may lead to drying. Water expanding adiabatieally (path bz)
becomes partially vaporized. Vapors may be divided into two classes,
depending upon whether they liquefy or dry during adiabatic expansion
under ordinary conditions of initial dryness. At usual stages of dryness
and temperature, steam liquefies during expansion, while ether becomes
dryer, or superheated.
INTERNAL ENERGY OF VAPOR
241
373. Inversion. Figure 161 shows that when x is about 0.5 the constant dry
ness lines change their direction of curvature, so that it is possible for a single
adiabatic like DE to twice cut the same dryness curve ; x may therefore have the
same value at the beginning and end of expansion, as at D and E. Further, it
may be possible to draw an adiabatic which is tangent to the dryness curve at A.
Adiabatic expansion below A tends to liquefy the steam ; above A, it tends to dry
it. During expansion along the dryness curve below A, the specific heat is nega
tice; above .4, it is positive. By finding other points like A, as F 9 (7, on similar
constant dryness curves, a hue BA may be drawn, which is called the zero line or
line of inversion. During expansion along the dryness lines, the specific heat
becomes zero at their intersection with AB, where they become tangent to the
adiabatics. If the line AB be projected so as to meet the extended saturation
curve dc, the point of intersection is the tempeiature of imersion. There is no
temperature of inversion for dry steam (Art. 379), the saturation curve reaching
an upper limit before attaining a vertical direction.
374. Internal Energy. In Fig. 164, let 2 he the state point of a wet vapor.
Lay off 2 4 vertically, equal to (TL)(L r). Then 1 2 4 3 (3 4 being drawn
hoiizontally and 1 3 vertically) is equal to
This quantity is equal to the external work of
vaporization = xe, which is accordingly repre
sented by the area 1243. The irregular
area 651347 then represents the addition
of internal energy, 6518 having been ex
pended in heating the water, and 8 3 4 7=xr
being the disgregation work of vaporization.
FIG. 161. Art 374. Internal Energy
and External Work.
375. External Work. Let Jl/jV, Fig. 165, be any path in the saturated region.
The heat absorbed is mMNn. Construct J/cfa, Nfed, as in Art. 374. The inter
nal energy has increased from Oabcm to Odefn, the
amount of increase being adefnmcb. This is greater
than the amount of heat absorbed, by dei^fcba iNf,
which difference consequently measures the external
work done upon the substance. Along some such curve
as XY 9 it will be found that external work has been
done by the substance.
o
FIG. 165. Art. 375. In
ternal Energy of Steam.
376. The Entropy Diagram as a Steam Table. In
Fig. 161, let the state point be H. We have T= HI,
from which P may be found. HJ is made equal to (T L)(L r), whence
Oa VKJI E and VH.TK xe. Also x = VH s FTV", the entropy measured from
the water line is VH 9 the momentary specific heat of the water along the dif
ferential path jL is g}LH^Tj\ xL = PVHI, xr  KJIP, A = OaVP, and
H = Oa VHI. The specific volume is still to be considered.
242
APPLIED THERMODYNAMICS
FIG
166. Art. 377 Constant
Tolume Lines.
377. Constant Volume Lines. In Fig. 166, let JA be the water
line, JBGf the saturation curve, and let vertical distances below ON
represent specific volumes. Let xs equal the volume of boiling water,
sensibly constant, and of comparatively
small numerical value, giving the line ss.
From any point B on the saturation
curve, draw BD vertically, making QD
represent by its length the specific volume
at B. Draw BA horizontally, and AH
vertically, and connect the points J^andD.
Then ED shows the relation of volume of
vapor and entropy of vapor, along AB,
the t\vo increasing in arithmetical ratio.
Find the similar lines of relation KL and
HJFioT the temperature lines JTand YGr.
Draw the constant volume line TD, and
find the points on the entropy plane
w, v, JB, corresponding to t, u, D. The line of constant volume wB
may then be drawn, with similar lines for other specific volumes, qz,
etc. The plotting of such lines on the entropy plane permits of the
use of this diagram for obtaining
specific volumes (see Fig. 175).
378. Transfer of Vapor States. In
Fig. 167, we have a single represen
tation of the four coordinate planes
pt, tn, m\ and pv. Let ss be the line
of water volumes, db and ef the satura
tion curve, Od the pressuretempera
ture curve (Art. 362), and Op the
water line. To transfer points a, 5 on
the saturation curve from the pv to the
tn plane, we have only to draw a (7,
Cfe, bd, and df. To transfer points
like i, Z, representing wet states, we
first find the vn lines qh and rg as in Art. 377, and then project
(7, jk> Im, and mn (25).
FIG. 167 Art. 378 Transfer of
Vapor States.
CRITICAL TEMPERATURE 243
Consider any point t on the pv plane. By drawing tu and uv we
find the vertical location of this point in the tn plane. Draw w A and
#2?, making zB equal to the specific volume of vapor at x (equal to
EF on the pv plane). Draw AS and project t to c. Projecting this
last point upward, we have D as the required point on the entropy
plane.
379. Critical Temperature. The water curve and the curve of saturation
in Figs. 160 and 161 show a tendency to meet at their upper extremities.
Assuming that they meet, what are the physical conditions at the critical
temperature existing at the point of intersection ? It is evident that here
L = 0, T = 0, and e = 0. The substance would pass immediately from the
liquid to the superheated condition ; there would be no intermediate state
of saturation. "No external work would be done during evaporation, and,
conversely, no expenditure of external work could cause liquefaction. A
vapor cannot be liquefied, when above its critical temperature, by any
pressure whatsoever. The density of the liquid is here the same as that
of the vapor : the two states cannot be distinguished. The pressure re
quired to liquefy a vapor increases as the critical temperature is approached
(moving upward) (Arts. 358, 360) ; that necessary at the critical temperature
is called the critical pressure. It is the vapor pressure corresponding to the
temperature at that point. The volume at the intersection of the saturation
curve and the liquid line is called the critical volume. The " specific heat
of the liquid 5 ' at the critical temperature is infinity.
The critical temperature of carbon dioxide is 88.5 F. This substance is
sometimes used as the working fluid in refrigerating machines, particularly on
shipboard. It cannot be used in the tropics, however, since the available supplies
of cooling water have there a temperature of more than 885 F., making it im
possible to liquefy the vapor. The carbon dioxide contained in the microscopic
cells of certain minerals, particularly the topaz, has been found to be in the critical
condition, a line of demarcation being evident, when cooling was produced, and
disappearing with violent frothing when the temperature again rose. Here the
substance is under critical pressure; it necessarily condenses with lowering of
temperature, but cannot remain condensed at temperatures above 88.5 F. Ave
narius has conducted experiments on a large scale with ether, carbon disulphide,
chloride of carbon, and acetone, noting a peculiar coloration at the critical point (26).
For steam, Regnault's formula for H (Art 360), if we accept the approximation
h = /  3*2, would give L = H  h = 1118.94  0.695 1, which becomes zero when
t = 1603 F. Davis* formula (Art 360) (likewise not intended to apply to temper
atures above about 400 F.) makes L  when t  1709 F. The critical tempera
ture for steam has been experimentally ascertained to be actually much lower, the
best value being about 689 F. (27). Many of the important vapors have been
studied in **"' direction by Andrews.
214
APPLIED THERMODYNAMICS
380. Physical States. We may now distinguish between the gaseous
conditions, including the states of saturated vapor, superheated van) or, and
true gas. A saturated vapor, which may be either dry or icet, is a gaseous
substance at its maximum, density for the given temperature or pressure ;
and below the critical temperature. A superheated vapor is a gaseous sub
stance at other than maximum density whose temperature is either less
than, or does not greatly exceed, the enticed temperature At higher tempera
tures, the substance becomes a true gas. All imperfect gases may be regarded
as superheated vapors.
Air, one of the most nearly perfect gases, shows some deviations from Boyle's law
at pressures not exceeding 2500 Ib. per square inch. Other substances show far more
marked deviations. In Fig. 168, QP is an equilateral hyperbola. The isothermals
for air at vaiious temperatures centi
grade are shown above. The lower
curves are isothermals for carbon di
oxide, as determined by Andrews (28).
They depart widely from the perfect
gas isothermal, PQ. The dotted lines
show the liquid curve and the satura
tion curve, running together at , at the
critical temperature. There is an evi
dent increase in the irregularity of the
curves as they approach the ei itical tem
perature (from above) and pass below
it. The cuive for 21.5 C. is paiticu
larly interesting. From I to c it is a
liquid curve, the volume remaining
practically constant at constant temperature in spite of enormous changes of "pres
sure. From b to d it is a nearly straight horizontal line, like that of any vapor
between the liquid and the dry saturated states; T\hile fiom d to e it approaches
the perfect gas form, the equilateral hyperbola. All of the isothermals change
their direction abruptly whenever they ap T
proach either of the limit curves ctf or ag.
381. Other Paths of Steam Formation.
The discussion has been limited to the
formation of steam at constant pressure,
the method of practice. Steam might con
ceivably be formed along any arbitrary
path, as for instance in a closed vessel at
constant volume, the pressure steadily in
creasing. Since the change of internal
energy of a substance depends upon its
initial and final states only, and not on the intervening path, a change of path
affects the external work only. " For formation at constant volume, the total heat
equals E> no external work being done. Jf in Fig. 169 water at c could be com
so
S 85 '
Soft
2*'
I 7fi
$70
W
60
55
50
FIG. 168. Art 380. Critical Temperature.
FIG. 109. Art. 381. Evaporation at
Constant Volume.
VAPOR ISODYNAMIC 245
pletely evaporated along en at constant volume, the area acnd would represent the
addition of internal energy and the total heat received. If the process be at con
t>tatit pressure, along cbn, the area acbnd lepresents the total heat received and the
area cbn represents the external work done.
382. Vapor Isodynamic. A saturated vapor contains heat above 32 F. equal
to li f r f e ; or, at some other state, to \ f r L f e r If the two states are isody
narnic (Art. 83), h + r = 7^ f r 1? a condition which is impossible if at both states
the steam be dry. If the steam be wet at both states, h + xr = 7^ 4 a^. Let y>,
p r v be given ; and let it be required to find v r the notation being as in Art. 304.
"We have x 1 =  xr ~~ \ all of these quantities being known or readily ascertain
able. Then
i = ^ + ^ (W,  V^x^ + l\ ^V. + Z^h + zr h,).
r i
If x = 1.0, the steam being diy at one state, x l = * "^ r "" ' and
Substitution of numerical values then shows that if p exceed pi, v is less than vi;
i.e. the curve slopes upward to the left on the pv diagram and x is less than
x r The curve is less " steep" than the satuiatiou curve. Steam cannot be worked
isodynaimcally and remain dry; each isodynamic curve meets the saturation curve
at a single point.
382ft. Sublimation. It has been pointed out that a vapor cannot exist at a
temperature below that which "corresponds" to its pressure. It is likewise true
that a substance cannot exist in the liquid form at a temperature above that which
" corresponds ' ' to its pressure. When a substance is melted in air, it usually becomes
a liquid; and if a further addition of heat occurs it will at some higher temperature
become a vapor. If, however, the saturation pressure at the melting temperature
exceeds the pressure of the atmosphere, then at atmospheric pressure the saturation
temperature is less than the melting temperature, and the substance cannot become
a liquid, because we should then have a liquid at a higher temperature than that
which corresponds to its pressure. Sublimation (Art. 17), the direct passage from
the solid to vaporous condition, occurs because the atmospheric boiling point is
below the atmospheric melting point.
Water at 32 has a saturation pressure of 0.08&6 Ib. per square inch. If the
moisture in the air has a lower partial pressure than this, ice cannot be melted,
but will sublime, because water as a liquid cannot exist at 32 at a less pressure
than 0.0886.
THERMODYNAMICS OF GAS AND VAPOR MIXTURES
3825. Gas Mixture. (See Art. 52 b.) When two gases, weighing ?i and w Ib.
respectively, together occupy the same space at the conditions p, v, /, we may write
the characteristic equations, using subscripts to represent the different gases,
conforming to Dalton's law,
246
APPLIED THERMODYNAMICS
WEIGHTS OF AIR, VAPOR OF WATER, AND SATURATED MIXTURES OF AIR AND VAPOR
AT DIFFERENT TEMPERATURES, UNDER THE ORDINARY ATMOSPHERIC PRESSURE
OF 29 921 INCHES OF MERCURY.
Temperature
Fahrenheit
MIXTURES OP AIR SATURATED WITH VAPOR
Elaatio Force of the Air
in the Mixture of Air
and Vapor in ins
of Mercury
Weight of Cubic Foot of the Mixture
of Air and Vapor
Weight of the Air
in Pounds
Weight of the Vapor
in Pounds
29 877
.0863
000079
12
29 849
.0840
000130
22
29 803
.0821
000202
32
29 740
.0802
000304
42
29 654
.0784
.000440
52
29 533
.0766
000627
62
29 365
0747
.000881
72
29 136
.0727
.001221
82
28 829
.0706
.001667
92
28 420
,0684
.002250
102
27 885
.0659
.002997
112
27 190
.0631
003946
122
26 300
.0599
005142
132
25 169
.0564
.006639
142
23.756
.0524
.008475
152
21 991
.0477
.010716
162
19 822
.0423
.013415
172
17.163
.0360
.016682
182
13 961
0288
020536
192
10 093
.0205
.025142
202
5 471
.0109
030545
212
000
0000
036820
These yield as the equation of the mixture,
where R (R\w\{Ry.w^^(wi\w^^ For pure dry air, containing by weight 0.77
nitrogen to 0.23 oxygen, the value of R should then be
(48.2X0,23) +(54.9X0 77)=53 2.
382c. Air and Steam. We are apt to think of the minimum boiling point of water
(except in a vacuum) as 212 F. But water will boil at temperatures as low as
32 F. under a definite low partial pressure for each temperature. Thus at 40 F.,
if an adequate amount of moisture is exposed to the normal atmosphere it will
THERMODYMAMICS OF GAS AND VAPOR MIXTURES 247
be vaporized until the mixture of air and steam contains the latter at a partial
pressure of 0.1217 Ib. per square inch, the partial pressure of the air then being
only 14.6970.1217 = 145753 Ib. per square inch. Such air is saturated. If there
is a scant supply of moisture, the partial pressure of vapor will be less than that
corresponding with its temperature, and such vapor as is evaporated will be super
heated The weight of moisture in a cubic foot of saturated air is the tabular
density of the vapor at its temperature. What is commonly called the absolute
humidity of air may be expressed either in terms of the weight of vapor per cubic
foot of mixture or of the partial vapor pressure.
The weight of gas or superheated vapor in any assigned space at any stated
temperature is directly proportional to the partial pressure thereof. The relative
humidity of moist air may therefore be expressed either as or as , where w and
W PI
W are respectively the weights of water vapor in a cubic foot of moist air, unsatu
rated and saturated, and p*., P* are the corresponding partial pressures. The value
of R in the characteristic equation is obtained, for moist air at a relative humidity
below 1.0, by the method of the first paragraph, using for the water vapor Rz =85.8.
If the air temperature is 92 F., and a wickcovered ("wet bulb") thermometer
reads 82, the partial pressure of the vapor is that corresponding with saturation
at 82, that is, 0.539 Ib. per square inch; for the air about the wetbulb thermometer
is saturated, evaporation from the moist wick causing the cooling. Saturated air
at 92 would have a partial vapor pressure of 0.741 Ib. per square inch. The air in
539
question has therefore a relative humidity of o~74i~^'^" ^ e va ^ ue f ^ * or ^^
air is not 53.2, but
a subordinate relation being
(14.6970.539)144
 53.2X552    069 '
If the respective specific heats are fci and kz t then the specific heat of the mix
ture is
which for our conditions, with fa =0.2375, fe =0.4805, gives A; =0.248.
382^. Thennodynamic Equations. When dealing with mixtures of wet vapors,
or of wet vapors and air, the ordinary equations for expansion do not in general
apply. This is the more unfortunate in that any general analysis of the subject
must include consideration of expansion paths which will partially liquefy
one or more of the constituents of even a wholly superheated mixture. The
internal energies of the constituents and their entropies are dependent upon
and may be computed from their thermal conditions alone, however; mixing
248 APPLIED THERMODYNAMICS
does not affect the energy, and adiabatic expansion does not affect the entropy;
so that it is by no means impracticable to study the phenomena accompanying
(a) the operation of mixing and (&) the expansion or compression of the mixture.
3820. Wet Vapor and Gas. As a simple case, consider a mixture of wet steam
and air. the condition of a supersaturated atmosphere. Let such a mixture be
at the state p, v, t' } the steam state being w*, p*, x, and that of the air wi, pi. Then
P=pi+Pz, and v = u> 2 X2V2=  , where v 2 is the specific volume of the dry steam.
PL
The internal energy of the mixture is
E
where I is the specific heat of air at constant volume and A 2 and r 2 are tabular thermal
properties at the pressure pa The entropy of the mixture is
l Iog fl +$ 0
where k is the specific heat of air at constant pressure, v is the volume of w\ Ib.
of air under standard conditions and n^ and n$ are the entropies of steam at the
pressure #.
In an isothermal change of such mixture, EI remains constant and (the dryness
of the steam changing to xs) E 2 increases by w&sfa #2). The air conforms to
its usual characteristic equation , piViRt lf In reaching the expanded volume zfy
the external work done by the air is then
The steam remaining wet expands at constant pressure, and does the external work
s v) } so that the whole amount of external work done is
W = piv log fl +#2(03  v) .
The heat absorbed may be expressed as the sum of the external work done and
the internal energy gained; or as
H=piv log, ^Hp2(Ps v) +Wzr 2 (xs 3%) =piv log* ^+w 2 I z (or 3 a^),
where k is the latent heat of vaporization corresponding with the pressure p.
Alternatively, the heat 'absorbed is equal to the product of the temperature by the
increase of entropy; or
H=t I Wi(kl) log* j
as before; ^er being the entropy of vaporization at the pressure pa Let it be
noted also that vs=w<tx*vs=~l, so that
=, Pz r denoting the partial pressure of air in the mixture after expansion.
The mixing of air with saturated steam produces a total pressure which is higher
than the saturation pressure of steam at the given temperature. Such a mixture
THERMODYNAMICS OF GAS AND VAPOR MIXTURES 249
may therefore be regarded as the reverse of superheated vapor, in which latter
the pressure is less than that corresponding with the temperature.
In adiabatic expansion, let the final condition be x 3 , U, p z . The entropy remaining
constant,
wik loge y + wi (k I) log e , + Ws(n w ' \x 3 ne n w xne) 0,
where n w is entropy of liquid and primes refer to final conditions. The paHial
pressure of the vapor is tabular for Z 3 . If v$ is the specific volume of steam for .
then
where p 3 " and p z f are the partial pressures of air and steam, respectively. The
external work is written as the loss of internal energy, or, as
382 /. High Pressure Steam and Air. The pressure attained by mixing cannot
exceed the initial pressure of the more compressed constituent. Assume 1 Ib. of
steam, 0.85 dry, at an absolute pressure of 200 Ib., to be mixed with 2 Ib. of air
at 220 Ib. pressure and 400 F. The respective volumes are
,=0.85X2 29 = 1.945; *&
and the volume of mixture will be, under the usual condition of practice,
1.945+2.9=4.845.
The internal energy before (and after) mixing is
(2X0.1689X860)+354.9+(0.85+759.5)=1288 B. t. u.
This we put equal to (2X0. 1689X0+^+^^; 3a= =  ; and (assuming
values of t) we find by trial and error,
(1285)
ft
<=314(+460), fo=2S4, 7 2
pi = 118.2, p 200.5.
Mixing has caused an increase in dryness of steam, a considerable reduction of tem
perature, and a final pressure between the two original pressures.
The entropy of the mixture is now
2  (o.!689X2.3 log ^ + ^0.0686X2.3 log g
+0.456+(0.908X1.1617) =1.438
250 APPLIED THERMODYNAMICS
Let isothermal expansion increase the dryness to 0.95. The volume then
becomes 0.95X5.33 = 5.08 =z> 8 . The external work done is
=i= j (l44XHS 2X4.845X2.3 log ff) +144X82.3(5084.845) [ =8.45 B. t. u.
778 \
The internal energy increases by 042 X759.5 = 31 9 B. t. u , and the heat absorbed
should then be 31.9+8.45=40.35 B. t. u. The entropy in the expanded condition
is
j (0.1
2 0.1689X2.3 log + o 0686X2.3 log
+0456+(0.95X11617)=1.49,
and the check value for heat absorbed is (460 +314) X (1.491. 438)  40.3 B. t. u.
The partial pressures after expansion are
Air, 393' =PI= 118.2 = us; and steam, 82.3, as before.
In the usual expression for external work,
pvpy pv ~py+w
W ~ nl > n ~ W
the equivalent value of n is
1441 (200 5X4.845)(195.3X5.08)} + (8.45X778)
845X778
Consider next the adidbatic expansion from the same initial condition to a
temperature * 3 = 50(+460); when v 3 ' = 1702, p 3 '=0178, n*' 0.0361, rc e '=2.0865,
v z = 1702%. Then
1.438 =2 j ^0.1689 X2.3 log ) + ^0.0686 X2.3 log ^^) j +0.0361 +2.0865^,
and a* =0.47, v 3 = 802.
The internal energy in the expanded condition is
18.08 +(0.47X1007.3) +2(0.1689X510) =665 B. t. u.,
and the external work done is 1288 665=623 B. t u. The steam expanding
alone from its original condition would have had a final dryness of 0.65, and would
have afforded external work amounting to
354.9+(0.85X759.5)18 08 (0.65X1007.3)= 323 B. t. u.
The air expanding alone to 50 , according to the law piv i y =p 1 'v ] .' v would have
given
.1^220X2 90085X80^ .
THERMODYNAMICS OF GAS AND VAPOR MIXTURES 251
The total work obtainable without mixture, down to the temperature 3 =50
would then have been 262+323 = 585 B. t. u.
The equivalent value of n for the expansion of the mixture is
144) (200.5 X4.845)  (0 263 X802) + (665 X778)
665X778
Since y for steam initially 0.91 dry is 1.126, and y for air is 1.402, the value
of n might perhaps have been expected to be about
(2X1.402) +1.126
3 *
382^. Superheated Steam and Air. If the steam is superheated, its initial
volume is (from the Tumlirz equation, Art. 363),
where B=* 0.5962, c= 0.256. The internal energy of superheated steam may be
written as that at saturation (hz+xzr z ) plus that of superheating,
where k s is the specific heat of the superheated steam, y, = 1.298, and t s is the satu
ration temperature for the partial pressure pt. The entropy of the steam is
Its behavior during expansion may be investigated by the relations previously
given.
382/j. Mixture of Two Vapors. Let two wet vapors at the respective conditions
w*j Pa, 2, s, ht, k, r z , and w 2 , p2, ts, 12* hj, *2r r 2, be so mixed that the volume
of the aggregate is v =02+v 4 . The internal energy of the mixture is
the numerical value of which may be computed for the conditions existing prior
to mixing. After mixing, the temperature t being attained, the internal energy
is the same as before, and the drynesses are
**
where v ' is the tabular volume at the temperature t. The known internal energy
may then be written as a function of tabular properties at the temperature t, and the
252 APPLIED THERMODYNAMICS
value of t found by trial and error The equation for adiabatic expansion entirely
in the saturated field to the state fe is w'2(Hw5+2rc e )+W2fn w +X2He) = w*(n v ,' +X2 f n & f )
+W2(n w '+x 2 'ne'), primes denoting final conditions. Thus, let 1 Ib of steam at
107 Ib. pressure, 90 dry, be mixed with 2 Ib. of carbon tetrachloride at the same
pressure, 95 dry. The tables give t 2 = 320, h 2 61 2, r 2 = 58.47, v =0.415, n w =
0.1003, n e = 0855; .=333, ^=303 4, r 2 =802 5, r =4 155, 7^ = 0,4807, n e = l 1158.
Then r 2 = 090X4.155*3 75, v 2 =0 95X2X0.415 = 789, z>= 3 75+0.789=4.539.
The internal energy is
303.4+(090XS02.5)+2{61.2 + (0.95X58.47)} =1258 B. t. u.
Since * 2 '>1 for values of t between 320 and 333 the carbon tetrachloride
is superheated after mixture occurs. We must then express the energy as
~t 2 ') j
=1258,
k
in which k =0.056, # = 1.3,  = 0.043, and t 2 ' is the saturation temperature corre
sponding with the partial pressure of the carbon tetrachloride. Assuming that this
vapor when superheated conforms with the usual characteristic equation for gases,
and putting 5 = 100, P2'=jg^f= 0,0307 2. Assuming values of t, the trial
and error method gives a resulting mixture temperature close to 319, at which
p 2 '=239 I t 2 '=200, and
7814.0+2(34 59+72.64+0.043X119) 1255(1258) B.t.u.
4.980
The entropy computed as before mixing Is
0.481+(0.9Xl1152)+2(0.1003+0.95X0.0855)
after mixing, it is
0.4627+ (HJ2L1492) +2 (o. 1846 +0.056X2.3 log gg) =1.89.
Mixing has again lowered the temperature. Let adiabatic expansion proceed
until the temperature is 212. The tetrachloride will stall be superheated, and
0.3118+1.4447jr 2 r +2
20 X 672
For every assumed value of t 2 ', the whole volume of mixture is 7 r v', say:
144p2
Then 2'=% where 0/=26.79, the volume of saturated steam at 212. At
THERMODYNAMICS OF GAS AXD VAPOR MIXTURES 253
t 2 ' = 106^ p 2 ' =4.37,0' =21^4, z 2 '=^ =0.798, n w '+ n</ =0.1865; and the en
tropy is
0.3118+1.15+2(01865 +0.0094) = 1.89.
The internal energy is now
180 0+(0.798X897.6)+2(14.92+81.76+O.Q43+106) =1098 B. t. u.,
and the external work done during expansion is 12581098 = 160 B. t. u. If the
two vapors had expanded from their original condition to 212 separately, the
external work done would have been, very nearly, 126 B. t. u.
382 1. Technical Application of Mixtures in Heat Engines The preceding
illustration shows that the expanded mixture, although at 212 F., has a pressure
4.37 Ib. per sq. in. greater than that of the atmosphere. A mixture at an absolute
pressure of 1 Ib. (about the lowest commercially attainable) might similarly exist
at a temperature considerably lower than the 102 F. which is characteristic of
steam alone. A lowering of the temperature of heatrejection is thus the feature
which makes the use of a fluid mixture of practical interest. This is the more
important, since from a powerproducing standpoint the most fruitful part of the
cyclic temperature range is the lower part. The operation of mixing itself reduces
the initial temperature, but it in no way impairs the stock of internal energy of the
constituents
If one of the constituents is at the lower temperature of the cycle a superheated
vapor, it cannet be condensed at that temperature: but since cooling water con
ditions permit of normal condensing temperature around 65, the use of a mixture,
even one of air and steam, may permit the attainment of that temperature without
the necessity for an impracticably high vacuum.
The total heat of saturated steam increases less than J B. t. u. per degree of
temperature; that of superheated steam increases from 0.5 to 0.6 B. t.u. It follows
that at the same temperature superheated steam "contains" more heat than
saturated steam. The internal energy of saturated steam increases about 0.2 B. t. u.
per degree of temperature; that of superheated steam, about 0.4 to 0.45 B. t. u.
The total internal energy at a given temperature is thus also greater with super
heated than with saturated steam. The less the internal energy at the end of the
expansion, the greater is the amount of external work performed during expansion
for given initial conditions. The analyses show that in general the effect of mixing
air or vapor with steam is to decrease the dryness of the steam after expansion,
and thus to decrease its final stock of internal energy and to increase the external
work performed. Saturated steam expands (i e., increases in volume) more rapidly
than air, as its temperature is lowered. Similarly, for a given rate of increase in
volume, the temperature of air falls more rapidly than that of steam. TVhen the
two fluids are mixed, a condition of uniform temperature must prevail. This
necessitates a transfer of heat from the steam to the air, decreasing the entropy
of the former and increasing that of the latter. The decrease in entropy of the
steam is responsible for its decreased dryness at the end of expansion.
254 APPLIED THERMODYNAMICS
SUPERHEATED STEAM
383. Properties : Specific Heat. In comparatively recent years, superheated
steam has become of engineering importance in application to reciprocating en
gines and turbines and in locomotive practice.
Since superheated steam exists at a temperature exceeding that of saturation,
it is important to know the specific heat for the range of superheating. The first
determination was by Regnault (1S62), who obtained as mean values k = 0.4805,
I = 0.346, y = 1.39. Fenner found I to be variable, ranging from 0.341 to 0.351.
Hirn, at a later date, concluded that its value must vary with the temperature.
Weyrauch (29), who devoted himself to this subject from 1876 to 1904, finally
concluded that the value of k increased both with the pressure and with the
amount of superheating (range of temperature above saturation), basing this con
clusion on his own observations as collated with those of Regnault, Hirn, Zeuner,
Mallard and Le Chatelier, Sarrau and Teille, and Langen. Rankine presented a
demonstration (now admitted to be fallacious) that the total heat of superheated
steam was independent of the pressure. At very high temperatures, the values
obtained by Mallard and Le Chatelier in 1883 have been generally accepted by
metallurgists, but they do not apply at temperatures attained in power engineer
ing. A list by Dodge (30) of nineteen experimental studies on the subject shows
a fairly close agreement with Regnault's value for k at atmospheric pressure and
approximately 212 F. Most experimenters have agreed that the value increases
with the pressure, but the law of variation with the temperature has been in
doubt. Holborn's results (31) as expressed by Kutzbach (32) would, if the em
pirical formula held, make k increase with the temperature up to a certain limit,
and then decrease, apparently to zero.
384. Knoblauch and Jakob Experiments. These determinations (33)
have attracted much attention. They were made by electrically super
heating the steam and measuring the input of electrical energy, which
was afterward computed in terms of its heat equivalent. These experi
menters found that k increased with the pressure, and (in general)
decreased with the temperature up to a certain point, afterward increas
ing (a result the reverse in this respect of that reported by Holborn).
Figure 170 shows the results graphically. Greene (34) has used these
in plotting the lines of entropy of superheat, as described in Art. 398.
The Knoblauch and Jakob values are more widely used than any others
experimentally obtained. They are closely confirmed by the equation
derived by Goodenough (Principles of Thermodynamics, 1911) from
fundamental analysis :
SUPERHEATED STEAM
255
where k is the true or instantaneous value of the specific heat at the
constant pressure p (Ibs. per sq, in.) and at the temperature T abso
FiG. 170.
340 280 320 360 400 440 480 520 SCO 600 G40 680 720
TEMPERATURE DECREES FAHRENHEIT
Arts. 384, 421. Specific Heat of Superheated Steam. Knoblauch and
Jakob Results.
lute, and log (7 = 14.42408. Values given by this equation should
correspond with those of the curves, Fig. 170. The values in Fig. 171
are for mean specific heat at the pressure p from saturation to the
temperature T, for which Goodenough's equation is
256
APPLIED THERMODYNAMICS
Amp(n+l) (l+^p hyTn
To being the saturation temperature,
a=0.367, 6=00001. log m = 13.67938,
n = o 3 A =7YT> 1S {Am(n\l) } =11.566.
385. Thomas' Experiments. In these, the electrical method of heating
and a careful system of radiation corrections were employed (35). The
conclusion reached was that 7c increases with increase of pressure and
decreases with increase of temperature. The variations are greatest near
the saturation curve. The values given included pressures from 7 to 500 Ib.
FIG. 171. Aits. 385, 388,
$, 417, Prob 42. Specific Heat of Superheated Steam.
Thomas' Experiments.
per square inch absolute, and superheating ranging up to 270 F. The
entropy lines and total heat lines are charted in Thomas' report. Within
rather narrow limits, the agreement is close between these and the Knob
lauch and Jakob experiments. The reasons for disagreement outside
these limits have been scrutinized by Heck (36), who has presented a
table of the properties of superheated steam, based on. these and other data.
The steam tables of Marks and Davis (see footnote, p. 202) contain
a complete set of values for superheated states. Figure 171 shows
the Thomas results graphically.
386. Total Heat. As superheated steam is almost invariably formed
at constant pressure, the path of formation resembles dbcW, Fig. 161, ab
SPECIFIC HEAT 257
being the water line and cd the saturation curve. Its total heat is then
H c ik(T t), where T, t refer to the temperatures at W and c. If we
take Begnault's value for H c , 1081.94 + 0.305* (Art. SCO), then, using
7c = 0.4805, we find the total heat of superheated steam to be 108] .94
0.1755 1 f 0.4805 T. A purely empirical formula, m which P is the pres
sure in pounds per square foot, is ff= OASOo(T 10. 37 jP ^; f 857.2.
For accurate calculations, the total heat must be obtained by using correct
mean values for & during successive short intervals of temperature between
t and T.
387. Variations of k. Dodge (37) has pointed out a satisfactory method
for computing the law of variation of the specific heat. Steam is passed
through a small orifice so as to produce a constant reduction in a constant
pressure. It is superheated on both sides the orifice ; but, the heat coii
tents remaining constant during the throttling operation, the temperature
changes. Let the initial pressure be ^>, the final pressure j^ Let one
observation give for an initial temperature t, a final temperature t x ; and
let a second observation give for an initial temperature T, a final tempera
ture 2i. Let the corresponding total heat contents be 7d, 7^, H, JI^ Then
h H= Je p (t T) and 7^ H : = fc, (^ T 2 ). But k = 7^ H= H^ whence
TP f m
h H= hi H^ and * = ^ ^  If ive know the mean value of k for any
K D t jt
*\
given range of tem2}erature, we may then ascertain the mean value for a
series of ranges at various pressures.
388. Davis' Computation of H. The customary method of deter
mining k has been by measuring the amount of heat necessarily added
to saturated steam in order to produce an observed increase of tem
perature. Unfortunately, the value of H for saturated steam has
not been known with satisfactory accuracy ; it is therefore inade
quate to measure the total heat in superheated steam for comparison
with that in saturated steam at the same pressure. Davis has sho\N ri
(17) that since slight errors in the yalue of H lead to large errors
in that of &, the reverse computation using known values of k to
determine H must be extremely accurate ; so far so, that while
additional determinations of the specific heat are in themselves to be
desired, such determinations cannot be expected to seriously modify
values of ^BTas now computed.
The basis of the computation is, as in Art. 387, the expansion of
superheated steam through a nonconducting nozzle, with reduction
258 APPLIED THERMODYNAMICS
of temperature. Assume, for example, that steam at 38 Ib. pres
sure and 300 F. expands to atmospheric pressure, the temperature
becoming 286 F. The total heat before throttling we may call
H c = H b + kyT c 2&), i n which H b is the total heat of saturated
steam at 38 Ib. pressure, T e = 300 F., and T b is the temperature of
saturated steam at 38 Ib. pressure, or 264.2 F. After throttling,
similarly, H d = 2Zi + * a (Ztf 2^), in which H e is the total heat of
saturated steam at atmospheric pressure, T & is its temperature
(212 FO, and T d is 286 F. Now JZ d = H e , and H e = 1150.4 ; while
from Fig. 171 we find * x = 0.57 and * a = 0.52 ; whence
S b =  0.57(800  264.2) + 1150.4 + 0.52(286  212) = 1168.47.
The formula given by Davis as a result of the study of various
throttling experiments may be found in Art. 360. The total heat
of saturated steam at some one pressure (e.g. atmospheric) must be
known.
A simple formula (that of Smith), which expresses the Davis results with an
accuracy of 1 per cent, between 70 and 500, was given in Power, February 8, 1910.
t being the Fahrenheit temperature.
389. Factor of Evaporation. The computation of factors of evapora
tion must often include the effect of superheat. The total heat of super
heated steam which we may call H t may be obtained by one of the
methods described in Art. 386. If ?IQ is the heat in the water as sup
plied, the heat expended is H t Ji^ and the factor of evaporation is
(H 9 o)* 970.4.
390. Characteristic Equation. Zeuner derives as a working formula,
agreeing with Hirn's experiments on specific volume (38),
PF= 0.64901 T 22.5819 P 03 *,
in which P is in pounds per square inch, V in cubic feet per pound, and
T in degrees absolute Fahrenheit. This applies closely to saturated as
well as to superheated steam, if dry. Using the same notation, Tumlirz
gives (39) from Battelli's experiments,
PV= 0.594 T 0.00178 P.
The formulas of Knoblauch, Linde and Jakob, and of Goodenough, both
given in Art. 363, may also be applied to superheated steam, if not too
PATHS OF VAPORS 259
highly superheated. At very high temperatures , steam behaves like a
perfect gas, following closely the law PV=RT. Since the values of R
for gases are inversely proportional to their densities, we find R for steam
to be 85.8.
391. Adiabatic Equation. Using the value just obtained for 72, and Regnault's
constant value 0.4805 for k, we find y 1.208. The equation of the adiabatic
would then be ^?i 1298 = c. This, like the characteristic equation, does not hold
for wide state ranges; a more satisfactory equation remains to be developed
(Art. 397). The exponential form of expression gives merely an approximation
to the actual curve.
PATHS OF VAPORS
392. Vapor Adiabatics. It is obvious from Art. 372 that during
adiabatic expansion of a saturated vapor, the condition of dryness
must change. We now compute the equa
tion of the adiabatic for any vapor. In
Fig. 172, consider expansion from J to c.
Draw the isothermals T, t. We have
FIG. 172. Art. 392. Equa ing the variable temperature along da. But
tion of Vapor Adiabatic. ^ = ^ ^ . f the specific heat of the liquid be
constant and equal to <?, ^=6 j log t! I ^, the desired equation.
t t JL
If the vapor be only X dry at J, then
393. Applications. This equation may of course be used to derive the results
shown graphically in Art. 373. For example, for steam initially dry, we may
make X = 1, and it will be always found that x e is less than 1. To show that
water expanding adiabatically partially vaporizes, we mate X 0. To determine
the condition under which the dryness may be the same after expansion as before
it, we make x = X.
394. Approximate Formulas. Rankine found that the adiabatic might be
represented approximately by the expression,
PP"^ = constant;
which holds fairly well for limited ranges of pressure when the initial dryness is
1.0, but which gives a curve lying decidedly outside the true adiabatic for any con
siderable pressure change. The error is reduced as the dryness decreases, down to
a certain limit. Zeuner found that an exponential equation might be written in
260
APPLIED THERMODYNAMICS
the form P V n = constant, if the value of were made to depend upon the initial
dryness. He represented this by
n = 1.035 + 0.100 X,
for values of X ranging from 0.70 to 1.00, and found it to lead to sufficiently accu
rate results for all usual expansions. For a compression from an initial dry ness r,
n = 1.034 + 0.11 x. "Where the steam is initially dry, n = 1.135 for expansion and
1,144 for compression. There is seldom any good reason for the use of exponential
formulas for steam adiabatics. The relation between the true adiabatic and that
described by the exponential equation is shown by the curves of Fig. 173, after
o & 10
FIG. 173. Arts. 394, 395. Adiabatic and Saturation Corves.
Heck (40). In each of these five sets of curves, the solid line represents the
adiabatic, while the shortdotted lines are plotted from Zeuner's equation, and the
longdotted lines represent the constant dryness curves. In I and II, the two
adiabatics apparently exactly coincide, the values of x being 1.00 and 0.7o. In
IH, IV, and V, there is an increasing divergence, for x = 0.50, 0.25 and 0. Case
V is for the liquid, to which no such formula as those discussed could be expected
to apply.
395. Adiabatics and Constant Dryness Curves. The constant dryness curves
I and II in Fig. 173 fall above the adiabatic, indicating that heat is absojbed during
expansion along the constant dryness line. Since the temperature falls during
expansion, the specific heat along these constant diyness curves, within the limits
shown, must necessarily be negative, a result otherwise derived in Art. 373, The
points of tangency of these curves with the corresponding adiabatics give the
points of inversion, at which the specific heat changes sign.
STEAM ADIABATICS
261
396. External Work. The work during adiabatic expansion from
PVto pV) assuming pv n = PF", is represented by the formula
PVpv
711 '
More accurately, remembering that the work done equals the loss of
internal energy, we find its value to be H h f XR xr, in which
H and h denote the initial and final heats of the liquid,
397. Superheated Adiabatic. Three cases are suggested hi Fig. 174, paths //,
jk t de, the initially superheated vapor being either dry, ^wet, or superheated at the
/
J ij
/
w
&
f
1^
/ I
/
\
k
N
FIG. 174. Art. 397. Steam Adiabatics.
end of expansion. If k be the mean value of the specific heat of superheated
steam for the range of temperatures in each case, then
for>, c log. + ^
2 T
for jk,c log, +
398. Entropy Lines for Superheat. Many problems in superheated
steam are conveniently solved by the use of a carefully plotted entropy
diagram, as shown in Fig. 175.* The plotting of the curves within the
saturated limits has already been explained. At the upper righthand
corner of the diagram there appear constant pressure lines and constant
total heat curves. The former may be plotted when we know the mean
specific heat fc at a stated pressure between the temperatures T and t : the
T
entropy gained being Tc log e . The lines of total heat are determined
* This diagram, is based on saturated steam tables embodying Regnault's results, and
on Thomas' values for k ; it does not agree with the tables given on pages 247, 248. The
same remark applies to Figs. 159 and 177.
262
APPLIED THERMODYNAMICS
0.5 00 017 0>9 1.0 11 12 13 14 1.5 16 17 1.8 1
Fio. 175. Arts. 377, 398, 401, 411, 417, 516, Problems. Temperatureentropy
for Steam.
ENTROPY LINES FOR SUPERHEAT
263
by the following method: For saturated steam at 103.38 Ib. pressure,
#=1182.6, T= 330 F. As an approximation, the total heat of 1200
B. t. u. will require (1200  1182. 6 js 0.4805 = 36.1 F. of superheating.
For this amount of superheating at 100 Ib. pressure, the mean specific
heat is, according to Thomas (Fig. 171), 0.604; whence the rise in tem
perature is 17.4 r 0.604 = 28.7 F. For this range (second approxima
tion), the mean sp3cific heat is 0.612, whence the actual rise of temperature
is 17.4 4 0.612 = 28.4 F. No further approximation is necessary ; the
amount of superheating at 1200 B. t. u. total heat may be taken as 28 F.,
which is laid off
yertically from the
point where the satu
ration curve crosses
the line of 330 F.,
giving one point on
the 1200 B. t. TL total
heat curve.
A few examples
in the application of
the chart suggest
themselves. Assume
steam to be formed
at 103.38 Ib. pres
sure ; required the
necessary amount of
superheat to be im
parted such that the
steam shall be just
dry after adiabatic
expansion to atmos
pheric pressure. Let
rs, Fig. 176, be the
line of atmospheric pressure. Draw st vertically, intersecting di\ then
t is the required initial condition. Along the adiabatic ts, the heat contents
decrease from 1300 B. t u. to 1150.4 B. t. u., a loss of 149.6 B. t. u.
To find the condition of a mixture of unequal weights of water and super
heated steam after the establishment of thermal equilibrium, the whole
operation being conducted at constant pressure : let the water, amounting
to 10 Ib., be at r, Fig. 176. Its heat contents are 1800 B, t. u. Let one
pound of steam be at t, having the heat contents 1300 B. t. u. The heat
gained by the water must equal that lost by the steam ; the final heat con
tents will then be 3100 B. t. u., or 282 B. t u. per pound, and the state
FIG. 17G. Arts. 398, 399, 401. Entropy Diagram, Superheated
Steam.
1460
1440
1480
1420
141
1400
1380
1380
1370
1360
1350
1340
1330
1320
131
1COO
1200
1280
127
1260
x25
124
121
1200
1190
1180
i fc a " g * i i * l ".
EtfTHOPV
THE MOLLIER HEAT CHART
265
be /, where the temperature is 312 F. ; the steam "will have been
completely liquefied.
We may find, from the chart, the total heat in steam (wet, dry, or
superheated) at any temperature, the quality and heat contents after
adiabatic expansion from any initial to any final state, and the specific
volume of saturated steani at any temperature and dryness.
399. The Mollier Heat Chart. This is a variant on the temperature
entropy diagram, in a form rather more convenient for some purposes. It
has been developed by Thomas (41) to cover his experiments in the
superheated region, as m Fig. 177. In this diagram, the vertical coordi
nate is entropy ; and the horizontal, total heat. The constant heat lines
are thus vertical, while adiabatics are horizontal. The saturation curve
is inclined upward to the right, and is concave toward the left. Lines of
constant pressure are nearly continuous through the saturated and super
heated regions. The quality lines follow the curvature of the saturation
line. The temperature lines in the superheated region are almost vertical.
It should be remembered that the " total heat" thus used as a coordinate
is nevertheless not a cardinal property. The " total heat '' at t, Fig. 176,
for exam pie, is that quantity of heat which would have been imparted had
water at 32 F. been converted into superheated steam at constant pressure.
It will be noted that within the portion of saturated field which is
shown, the total heat at a given pressure is directly proportional to the
total entropy. This would be exactly true if the water line in Fig. 175
PRESSURE. POUNDS PER SQUARE INCH
1550
80 100 & UO MO 180 SOO *2D iltt 2<KI 2W 300 SSO MO SCO 3*0 400 4204401004
SATURATED TEAM TEMPERATURE DEGREES P.
FIG. 185, Art, 399, PruWw*, Total Heavpressure Diagram,
266 APPLIED THERMODYNAMICS
were a straight line and if at the same time the specific heat of water
could be constant. An empirical equation might be written in the form
where n s , H and P are the total entropy, total heat and pressure of
a wet vapor.
The socalled total heatpressure diagram (Fig 185) is a diagram in which the
coordinates are total heat above 32 F. and saturation temperature; it usually includes
curves of (a) constant volume, (b) constant dryness, and (c) in the superheated field,
constant temperature. Vertical lines show the loss or gain of heat corresponding
to stated changes of volume or quality at constant pressure. Horizontal lines show
the change in pressure, volume, and quality of steam resulting from throttling
(Art. 387). This diagram is a useful supplement to that of Mollier.
Heck has developed a pressuretemperature diagram for both saturated and
superheated fields, on which curves of constant entropy and constant total heat
(throttling curves) are drawn. By transfer from these, there is derived a new
diagram of total heat on pressure, on which are shown the isothermals of superheat.
A study of the shape of these isothermals illustrates the variations in the specific
heat of superheated steam.
VAPORS IN GENERAL
400. Analytical Method: Mathematical Thermodynamics. An expression
for the volume of any saturated vapor was derived in Art. 368:
Where the specific volume is known by experiment, this equation may be used for
computing the latent heat. A general method of deriving this and certain related
expressions is now to be described. Let a mixture of x Ib. of dry vapor with
(1  x) Ib. of liquid receive heat, dQ. Then
dQ = kxdT + c (1  x)dT + Ldx,
in which k is the "specific heat" of the continually dry vapor, L the latent heat
of evaporation, and c the specific heat of the liquid. If P,V are the pressure and
volume, and E the internal energy, in footpounds, of the mixtuie, then
dQ = PdV + dE = IxdT + c (1  x) dT 4 Ldx, whence
/ 78
dE = 778 [kx + c (1  a;)] dT + 778 Ldx  PdV.
Now V = (/) T, x] whence d V = f dT + 1? dx, whence
bjT Sx
dE = 778 [for + c (1  or)] dT + 778 Ldx  P* dTP ^dx
= J778 [for + C (l _ *)] _ pZJ dT + ( 77SL P^\ dx.
Moreover, E = (/) T, x, whence
VAPORS IN GENERAL 267
giving
(all properties excepting V and x being functions of T only).
The volume, V, may be written xu f r, where u is the volume of the liquid and
X T r
w the increase of volume during vaporization. This gives 8 J r = wSx or = u.
ox
Also, since F= (/) T, or, 1 = JJ5L, and equation (A) becomes
Now if the heat is absorbed along any reversible path, = dN, or
dN _ kzdT + cQ  x*)dT + Ldx = kx + c(l  s)
/'
6V
+*. (
which may be combined with (B), giving
778 = u = F  0, as in Art 369. (D)
401. Computation of Properties. Equation (D), as thus derived, or as obtained
in Art. 369, may be used to compute either the latent heat or the rdume of any
vapor when the other of these properties and the relation of temperature and pres
sure is known. The specific heat of the saturated vapor may be obtained from
(C) ; the temperature of inversion is reached when the specific heat changes sign.
For steam, if L  1113.94  0.695 1 (Art. 379), where t is in degrees P., or
1113.94  0.695(2 T  459.6) where T is the absolute temperature: ~ T =  0.695.
Also c = 1 ; whence, from equation (C), k = 0.305  , which equals zero when
T= 1433 absolute.* At 212 *\k= 0.303  =  1.135. This may be roughly
Oil.Q
* This would be the temperature of inversion of dry steam if the formula for L held :
but L becomes zero at 689 F. (Art. 379), and the saturation carve 'for steam slopes downward
toward the right throughout its entire extent. For the dry vapors of chloroform and ben
zine, there exist known temperatures of inversion.
268 APPLIED THERMODYNAMICS
checked fiom Fig. 175. In Fig. 176, consider the path ,s^ from 212 F, to 157 F.,
and fiom n = 1 735 to n = 1.835 (Fig. 175). The average height of the area ctibe
representing the heat absorbed is 459.6 + 212 * ln/ = 644.1 ; whence, the area is
fiU 1(1 835  1.735) = 04.41 B. t. u., and the mean specific heat between s and b is
61.11  (212  157) = 1.176. The properties of the volatile vapors used in refriger
ation are to some extent known only by computations of this sort. When once
the pressuretemperature i elation and the characteristic equation are ascertained by
experiment, the other propeities follow.
402. Engineering Vapors'. The properties of the vapors of steam, carbon
dioxide, ammonia, sulphur dioxide, ether, alcohol, acetone, carbon disulphide, carbon
tetrachlonde, and chloroform have all been more or less thoroughly studied. The
firnt five are of considerable importance. For ether, alcohol, chloroform, carbon disul
phide, carbon tetrachloride, and acetone. Zeuner has tabulated the pressure, tempera
tui e. volume, total heat, latent heat, heat of the liquid, and internal and external
woik of vaporization, in both French and English units (42), on the basis of
Regnault's experiments. The properties of these substances as given in Peabody's
"Steam Tables" (1890) are reproduced from Zeuner, excepting that the values
 273.7 and 426.7 aie used instead of  273.0 and 424.0 for the location of the
absolute zero centigrade and the centigiade mechanical equivalent of heat,
respectively. Peabody's tables for these vapors are in Fiench units only. Wood
has derived expressions for the properties of these six vapois, but has not tabulated
their values (40). Rankine (44) has tabulated the pressure, latent heat, and density
of ether, per cubic foot, in English units, fiom Regnault's data. Forcrzr&n/i dioxide,
the experimental results of Andrews, Cailletet and Hautefeuille, Cailletet and
Mathias (45), and, finally, Ainagat (46), have been collated by Mollier, whose
table (47) of the properties of this vapor has been reproduced and extended, in
French and English units, by Zeuner (48). The vapor tables appended to Chapter
XVIII, it will be noted, are based on those of Zeuner. The entropy diagrams for am
monia, ether, and carbon dioxide, Figs. 314316, have the same foundation
The present writer (in Vapor* for Heat Engines, D. Van Nostrand Co., 1911)
has computed the entropies and prepared temperatureentropy diagrams for alcohol,
acetone, chloroform, carbon chloride and carbon disulphide.
403. Ammonia. Anhydrous ammonia, largely used in refrigerating
machines, was first studied by Regnault, who obtained the relation
Q IAO
= S.40<9
t
in which p is in pounds per square foot and t is the absolute temperature.
A " characteristic equation " between p, v, and t was derived by Ledoux
(49) and employed by Zeuner to permit of the computation of V> L, e, r
and the specific heat of the liquid (the last having recently been deter
mined experimentally (50)). The results thus derived were tabulated by
Zeuner (51) for temperatures below 32 P. ; 'while for higher temperatures
he uses the experimental values of Dietrici (52). Peabody's table (53),
also derived from Ledonx, uses his values for temperatures exceeding
32 F. 5 Zeuner regards Ledgux's values in this region as unreliable.
VAPORS IN GENERAL 269
Peabody's table is in French units ; Zeuner's is in both French arid Eng
lish units. The latent heat of evaporation has been experimentally de
termined by Regnault (54) and Ton Strombeck Coo). The specific volume
of the vapor at 26.4 F. and atmospheric pressure is 17.51 cu. ft. ; that of
the liquid is 0.025; whence from equation (D), Art. 400,
= 778 " ^ dT
= 433.2 (17 51 _ oog) /2196 x 2.3026 X 14 7 X 144\
778 ^ " V 433.2x433.2 /
dP
the value of being obtained by differentiating Regnault's equation,
above given. From a study of Regnault's experiments, Wood has derived
the characteristic equation,
PF == oi _ 16920
T ~~
which is the basis of his table of the properties of ammonia vapor (56).
Wood's table agrees quite closely with Zeuner's, as to the relation between
pressure and temperature ; but his value of L is much less variable. For
temperatures below C., the specific volumes given by Wood are rather
less than those by Zeuner; for higher temperatures, the volumes vary
less. Zeuner's table must be regarded as probably more reliable. The
specific heat (0.508) and the density (0.597, when air = l.Q) of the super
heated vapor have been determined by experiment.
404. Sulphur Dioxide. The specific heat of the superheated vapor is given by
Regnau.lt as 0.15438 (57). The. specific volume, as compared with that of air, is
2.23 (58). The specific volume of the liquid is 0.0007 (oO) ; its specific heat is
approximately 0.4. A characteristic equation for the saturated vapor has been
derived from Regnault's experiments :
P F = 26.4 !T  184 P  22 ;
in which Pis in pounds per square foot> Tin cubic feet per pound, and T in abso
lute degrees. The relation between pressure and temperature has been studied by
Reguault, Sajotschewski, Blumcke, and Miller. Regnault's observations were
made between  40 and 149 F. ; Miller's, between 68 and 211 F. ; a table repre
senting the combined results has been given by Miller (00). lu the usual form
of the general equation,
log p = a bd* ce *,
the values given by Peabody for pleasures in pounds per square inch are (61)
a = 3.9527847, log b = 0.4792425, log d = 1.9984994, log c = 1J659562, logc =
1.99293890, n = 18.4 f Fahrenheit temi>erature. The specific volumes, determined
by the characteristic equation and the pressuretemperature formula, permit of the
computation of the latent heat from equation (D), Art, 400. An empirical formula
270 APPLIED THERMODYNAMICS
for this property is L = 176 0.27(  32), in which t is the Fahrenheit tempera
ture. The experimental icsults of Cailletet and Mathias, and of Mathias alone (62) ,
have led to the tables of Zeuner (63). Peabody, following Ledoux's analysis, has
also tabulated the properties in French units. Wood (61) has independently com
puted the properties in both French and English units. Comparing Wood's, Zeu
ner's, and Peabody's tables, Zeunei's values for L and V are both less than those of
Peabody. At F., he makes L less than does Wood, departing even more widely
than the latter from. Jacobus' experimental results (65) ; at 30 F., his value of L is
greater than Wood's, and at 104 F., it is again less. The tabulated values of the
specific volumes differ correspondingly. Zeuner's table may be regarded as sus
tained by the experiments of Cailletet and Mathias, but the lack of concordance
with the experimental results of Jacobus remains to be explained
405. Steam at Low Temperatures. Ordinary tables do not give the properties
of water vapor for temperatures lower than those corresponding to the absolute
pressures reached in steam engineering. Zeuner has, however, tabulated them for
temperatures down to 4 F. (66).
40 5. Vapors for Heat Engines. Engines have been built using,
instead of steam, the vapors of alcohol, gasolene, ammonia, ether,
sulphur dioxide and carbon dioxide, with good results as to thermal
efficiency, if not with commercial success. In a simple condensing
engine, with a rather low expansive ratio, a considerable saving may
be effected with some of these vapors, as compared with steam; and
the cost of the fluid is not a vital matter, since it may be used over and
over again. Strangely enough, in the case of none of the vapors is a
very low discharge temperature practically desirable, under usual
simple condensing engine conditions. This statement applies even
T t
to steam. The Carnot criterion  , does not exactly apply, sinca
it refers to potential efficiency only: but the use of a substitute vapor
might perhaps be justified on one of the two grounds, (a) an increased
upper temperature without excessive pressures or (6) a decreased
lower temperature at a reasonable vacuum, say of 1 Ib. absolute.
To meet both requirements the vapor would have to give a pt curve
crossing that of steam. It is probable that carbon tetrachloride is
uch a vapor, bearing such a relation to steam as alcohol does to it.
ITo great gain is possible in respect to the lower temperature limit,
since this limit is in any case established by the cooling water. The
criterion given in Art. 630 measures the relative efficiencies of fluids
working in the Clausius cycle. On this basis steam surpasses all other
common vapors in potential thermal efficiency.
The lower " heat content " per pound of the more volatile and
heavy vapors leads to a greatly reduced nozzle velocity with adiabatic
flow, and this suggests the possibility of developing a turbine expanding
in one operation without excessive peripheral speeds (see Chapter XIV).
STEAM PLANT CYCLE
271
The greater density of the volatile sapors also leads to the con
clusion that the output from a cylinder of given size might in the cases
of some of them be about twice what it is from a steam cylinder.
On the whole, the use of a special vapor seems to be more promising,
technically and commercially, than the binary vapor principle (Art.
4S3). For a fuller discussion of this subject, reference may be made
to the work referred to in Art. 402.
STEAM CYCLES
406. The Carnot Cycle for Steam. This is shown in Figs. 163,
179. The efficiency of the cycle abed may be rend from the entropy
Tt
diagram as
T
The external
work done per pound of steam
T t
is L  ; or if the steam at I
Tt
is wet, it is xL
T
If the
1
I
FIG. 179. Art 40t> Carnot Cycle for Steam.
fluid at the beginning of the
cycle (point a) is wet steam
instead of water, the dryness
being x^ then the work per
pound of steam is L(x # )
m *
. i In the cycle first discussed, in order that the final adiabatic
compression may bring the substance back to its initially dry state at
a, such compression must begin at d, where the dryness is md s mn.
The Carnot cycle is impracticable
with steam; the substance at d is
mostly liquid, and cannot be raised
in temperature by compression.
What is actually done is to allow
condensation along cd to be com
pleted, and then to warm the liquid
or its equivalent along ma by trans
mission of heat from an external
source. This, of course, lowers
the efficiency.
407. The Steam Power Plant. The cycle is then not completed in
the cylinder of the engine. In Fig. 180, let the substance at d be
o
FIG. 180. Arts. 407, 408, 410 T 412, 413.
The Steam Power Plant.
272
APPLIED THERMODYNAMICS
cold water, either that resulting from the action of the condenser
on the fluid which luis passed through the engine, or an external
supply. This water is now delivered by the feed pump to the boiler,
iu which its temperature und pressure become those along al. The
work done by the feed pump per pound of fluid is that of raising
unit weight of the liquid against a head equivalent to the pressure;
or, what is the same thing, the product of the specific volume of the
water by the range in pressure, in pounds per square foot. From
a to b the substance is in the boiler, being changed from water to
steam. Along fit, it is expanding in the cylinder; along ed it is
being liquefied in the condenser or being discharged to the atmos
phere. In the former case, the resulting liquid reaches the feed
pump at <Z. In the latter, a fresh supply of liquid is taken in at d,
but this may be thermally equivalent to the liquid resulting from
atmospheric exhaust along cd. (See footnote, Art. 502.) The four
organs, feed pump, boiler, cylinder,
and condenser, are those essential in
a steam power plant. The cycle rep
resents the changes undergone by
the fluid in its passage through them.
408. Clausius Cycle. The cycle
of Fig. ISO, worked without adialatic
fiompresxion, is known as that of
Chutius. Its entropy diagram is
shown as dele in Fig. 181, that of
the corresponding Carnot cycle being
dhbc. The Carnot efficiency is obviously greater than that of the
Clausius cycle. For wet steam the corresponding cycles are deM
and dhkl.
FIG. 181. Arts 40&41.1. Rteain
Cydes.
409. Efficiency.
dele
In Fig. 181, cycle dele, the efficiency is
_ ft, Ji a + L b xjj f
But x c =
% if the specific heat of the
RANKIXE CYCLE 273
liquid be unity. Then letting 7, L refer to the state J, and t, I to
the state <?, the efficiency is
Tt+ L
which is determined s0ZeZ# by tJie temperature limits Tand t. For
steam initially wet, the efficiency is
T
410. Work Area. In Figs. 180, 181, we have
W= W ab + W bc  W cd  W da
ignoring the small amount of work done by the feed pump in forcing
the liquid into the boiler. But p b (v b a ) = e b and j^Oy i\i) J'Sj
(Art. 359), whence
W=h e + L b hx t Lsi
a result identical with the numerator of the first expression in Art.
409.
411. Rankine Cycle. The cycle delgq, Fig. 181, af>gq<J, Fig. 180.
is known as that of Rankine (67). It differs from that of Clansuis
merely in that expansion is incomplete, the "toe"" gey, Fig. ISO,
being cut off by the limiting cylinder volume line gq. This is the
ideal cycle nearest which actual steam engines work. The line yy in
Fig. 181 is plotted as a line of constant volume (Art. 877). The
efficiency is obviously less than that of the Clausius cycle ; it is
elgqd __ W ab +W^~ W qd (Fig. 180)
"
 O] + (** + ?b  K 
The values of h^ X T r t , x q , depend upon the limiting volume v g = v r
and may be most readily ascertained by inspecting Fig, 175. The
computation of these properties resolves itself into the problem : given
274 APPLIED THERMODYNAMICS
the initial state, to find the temperature after adidbatic expansion to a
given volume. We have
v g  v r = x g (v s  fl r ), n g = w 6 ,
^
9 n s n r n s ?? r L s + T r
whence
in which v ff , T e , LI, are given, v r =0.017, and v sj L s are functions of
T T , the value of which is to be ascertained. The greater the ratio
of expansion, ^sr*, Fig. 181, with given cyclic limits, the greater
is the efficiency.
412. NoBexpansive Cycle. This appears as debt, Fig. 181 ; and'a&ed, Fig. 180.
No expansion occurs; work is done only as steam is evaporated or condensed.
The efficiency is (Fig. 181)
del* = W*  W ed (Fig. 180) = p b (v b  r a )  p t (r  v d ) t
h e h d +L b h t  h d + 5
This is the least efficient of the cycles considered.
413. Pambonr Cycle. The cycle debf. Fig. 181, represents the operation of a
plant in which the steam remains dry throughout expansion. It is called the
Pamhtur cycle. Expansion may be incomplete, giving such a diagram as debuq*
Let abed in Fig. 180 represent debfiu. Fig. 181. The efficiency is
external work done _ _ external work done _
gross heat absorbed ~" heat rejected + external work done
_ TFqft + TtV  W ed _
in which the saturation curve If may be represented by the formula pv& = con
stant (Art. 363). A second method for computing the efficiency is as follows:
& T L
the area debf= \ ~=dT, in which T and t are the temperatures along eb and df
jt y
respectively, and L =(J)T= 1433  0.695 T (Art. 379). This gives
debf= 1433 log.  0.695(T  *)*
and the efficiency is
1433 log fl   0.695( T  1)
debf __ debf _
debf+idfv imiogf I_ M g 6(T _ t)+L/
SUPERHEATED CYCLES
275
The two computations will not precisely agree, because the exponent $ does not
exactly represent the saturation curve, nor does the formula for L in terms of T
hold rigorously.
Of the whole amount of heat supplied, the portion Kbfv was added
during expattswi, as by a steam jacket (Art. 439). To ascertain this
amount, we have
heat added by jacket
= whole heat supplied heat present at beginning of expansion
= 1433 log,^ O.G95 (!T /) + Zy h, + U d  L
L
The efficiency is apparently less than that of the Clausius cycle (Pig.
181). In practice, however, steam jacketing increases the efficiency of
engines, for reasons which will appear (Art. 439).
414. Cycles with Superheat. As in Art. 397, three cases are pos
sible. Figure 182 shows the Clausius cycles debzw, debgf, debzAf,
in which the steam is respectively wet, dry, and superheated at the
end of expansion. To appreciate
the gain in efficiency due to super
heat, compare the first of these
cycles, not with the dry steam
Clausius cycle dele, but with the
superior Oarnot cycle dhbe. If the
path of superheating were b C, the
efficiency would be unchanged;
the actual path is Jj?, and the work
area bxO is gained at 100 per cent
efficiency. The cycle dhbxw is
thus more efficient than the Car
not cycle dhbc, and the cycle
debxw is more efficient than the Clausius cycle debc. It is not more
efficient than a Carnot cycle through its own temperature limits,
The cycle debyf shows a further gain in efficiency, the work area
added at 100 per cent effectiveness being byE. The cycle debzAf
shows a still greater addition of this desirable work area, but a loss of
area AfB now appears. Maximum efficiency appears to be secured
with such a cycle as the second of those considered, in which the
steam is about dry at the end of expansion. The Carnot formula
FIG 182. Art. 414. Cycles with
Superheat.
276 APPLIED THERMODYNAMICS
suggests the desirability of a high upper temperature, and superheating
leads to this ; "but when superheating is carried so far as to appreciably
raise the temperature of heat emission, as in the cycle debzAf, the
efficiency begins to fall.
415. Efficiencies. The work areas of the three cycles discussed
may be thus expressed :
in which Jc v Jc# k# k# refer to the mean specific heats over the re
spective pressure and temperature ranges. The efficiencies are
obtained by dividing these expressions by the gross amounts of heat
absorbed. The equations given in Art. 397 permit of computation
of such quantities as are not assumed.
416. Itemized External Work. The pressure and temperature at the
beginning of expansion being given, the volume may be computed and
the external work during the reception of heat expressed in terms of
P and F. The temperature or pressure at the end of expansion being
given, the volume may be computed and the negative external work
during the rejection of heat calculated in similar terms. The whole
work of the cycle, less the algebraic sum of these two work quantities
(the feed pump work being ignored), equals the work under the
adiabatic, which may be approximately cheeked from the formula
pypv^ ^ suitable value being used for n (Art. 394). A second
n 1
approximation may be made by taking the adiabatic work as equivalent
to the decrease in internal energy, which at any superheated state has
the value h + r +  (T f), T being the actual temperature, and A, r,
t referring to the condition of saturated steam at the stated pressure.
The most simple method of obtaining the total work of the cycle is to
COMPARISONS
277
read from Fig. 177 the " total heat " values at the beginning and end
of expansion. (See the author's " Vapors for Heat Engines/' D.
Van Nostrand Co., 1912.)
417. Comparison of Cycles. In Fig. 183, we have the following
cycles:
\
tpawq
FIG. 183. Arts. 417, 441, 442. Seventeen Steam Cycles.
Clausius,
Rankine,
Nonexpansive,
with dry steam, dele (the corresponding Carnot
cycle being dhbe) ;
with wet steam, dekl ;
with dry steam, debgq ;
with wet steam, dekJq;
with dry steam, debt ;
with wet steam, dekK 9
Pambour, complete expansion, debf;
incomplete expansion, debuqi
Superheated to a;, complete expansion, debxw ;
incomplete expansion, debxLuq\
no expansion, debxNp;
Superheated toy, complete expansion, debyfi
incomplete expansion, debyMuqi
BO expansion, debyRs;
Superheated to z, complete expansion, debzAfi
incomplete expansion, debzTuq ;
no expansion, debt Vw.
278
APPLIED THERMODYNAMICS
The lines tl, pNx, sRy, icTz, quT, are lines of constant volume,
Superheating without expansion would be unwise on either technical
or practical grounds ; superheating with incomplete expansion is the
condition of "universal practice in reciprocating engines. The
seventeen cycles are drawn to PJ 7 " coordinates in Fig. 184.
x y z
Iff J
FIG 184. Arts 417, 420, 424, 517 Seventeen Steam Cycles.
ILLUSTRATIVE PROBLEM
To compare the efficiencies, and the cyclic areas as related to the maximum volume at
tained: let the maximum pressure be 110 lh.,the minimum pressure 2 Ib , and consider
the Clausiua cycle (a) with steam initially dry, () with steam initially 90 per cent
dry ; the Rankine with initially dry steam and a maximum volume of 13 cu. ft ,
the same Kankine with steam initially 90 per cent dry; the nonexpansive
with steam dry and 00 per cent dry ; the Pambour (a) with complete expansion
and (6) with a maximum volume of 13 cu ft. ; and the nine types of superheated
cycle, the steam being; (a) 06 per cent dry, (ft) dry, (c) 40 F. superheated, at the
end of complete expansion ; and expansion being (a) complete, (/>) limited to a
maximum volume of 13 cu. ft., (c) eliminated.
L Cla usius cycle. The gross heat absorbed is h lta  7< a f 140 = 324  6  9^0 + 86"6
= 1098. S.
The</rytt&MJattheend of expansion is dc * df, Fig. 183, ~(n e n d + n ab ') n d/
= (0.5072  0.174!) 4 1.0075)  1.74;U = O.SOJ*
The teat rejected along cd is x<.L f = 0.80S X 1021 = 8194.
1098.2"
The uork done is 1008.2  819.4 = 273.8 B. t. u. The efficiency is ^ = 0354.
The efficiency of the corresponding Carnot cycle is
TWT* 353.1 ,120.15
= 0.88,
n.
!T 14() 353.1+ 459.0 '
Clawdwt cycle with tret steam. The gross teat absorbed is h l4D h Si + x*L J40
=324.6  94.0 + (0.00 x 8C7.G) = 1015.44*
The dryrwif at the end of expansion is dl 5 df (n  nj f n&) * n^
= (0.5072  0.174D + 0.90 x 1.0b73) * 1.7431 = 0.741.
COMPARISONS 279
The heat rejected along Iff is XiL f 0.741 x 1021 = 756.
The work done is 1013.44:  73fj = 359.44 B> * *
The efficiency is
(It is in all cases somewhat less than that of the initially dry steam cycle.)
til. Rankine cycle* dry steam. The grouts Iwtt absorbed, as in T, is 10QS.2.
The work along rte, Fig. 181, is 14 1 x liJS x 0.017 = ;A75. ~> footpounds (Art. 407);
along eb is 144 x 140 x (Fi 0.017) = 64,300 footpounds ;
"
is A c + r 6 Ji z ay^ = 103.76 B. t. u.
(Prom Fig. 175, f,=247 P., whence ,=947.4, F a = 11.52, *,=
0.8950
= [0.50722.3 (log T n  log 491.6) + 1.0075] TV
1433  0.093 T ff
For !T ff = 247 F. = 700.6 absolute, this equation gives x ff = 0.905 ; a suffi
cient check, considering that Fi. 173 in based on a different set of values
than those used in the steam talle. Then It 2 = 213., I* r g = 871.6.
The work along qd is P d ( F f  T d ) = 144 x 2 x (13  0.017)= 3740 foot
pounds.
The whole work of the cycle is 64anft ~ : ^ 8 ' 5 " 374 + 100.76 = .
^7$
The efficiency is
IV ^an^tfne cyr/e, ?e </eai. The ^ro.w Aca^ afoorbed is as in IT, 1015 J4*
The negatire work along </<? and ^ is, as iu III, 338.5 f 3740 = 4078.5 foot
pounds*
The work along ek Is 14i x 140 X 0.90(T" 6  0.017)= 57J70 footpowids.
The work along kJ is A f Xtf* A x jrjr r = 99.8 B. t, u.
(From Fig. 175, t x = 242 F., whence A x = 210.3, r r = 875.3, V r = 15.78,
IS 0.017
35.78 0.017 ^
The taAo/c M7orJt of the cycle is 5787 "I 4 078>5 + 99.8 = ^5.1 B. t.
The efficiency is
V. Non*xpQn*ive cycle, dry steam. The gross heat absorbed, as in I, is
The wrb <dong d*> s in III, is 33B J footpounds;
along eb? as in IFI, iw $4,300 footpoundt ;
along td is^(F fc  T*) = 144 x 2 x (3.21&  0.017)= 9$2 footpound*.
The wA^ tcorjfe efttie cycle is
 338.5  922 = 63,039.5 footpounds = 81.0$ B.
is
280 APPLIED THERMODYNAMICS
VI. Nonexpansive cycle t wet steam. The gross heat absorbed, as in II, is 1015.44+
The work along de, ek, as in IV, is  338.5 + 57,870 = 57,531.5 footpounds*
The work along Kd is
J>(r* 0.017)= 144 x 2 x 0.90 x (3.219  0.17)= 829.8 footpounds.
The whole work of the cycle is
57,531.5  829.8 = 56,701.7 footpounds = 73 B. t. u.
The efficiency is  = 0.072*.
VII. Pambour cycle, complete expansion. The heat rejected is L f 102LO.
The work along de, eb, as in in, is  338.5 f 64300 = 63,961.5 footpounds.
The work along bfis
= ^ 800 foot _ pounds .
The work along fd is P d ( r,  Vd) = 2 x 144 (173.5  0.017) = 49,900 foot
pound*.
The whole work of the cycle is 63,961.5 1 236,800  49,900 = 250J61.5 foot
pounds.
(Otherwise 1433 log,  0.695 (2^ /)= 312 B. t. u. = 42, 000 footpounds
(Art. 413).) '
Using a mean of the two values for the whole work, the gross Jieat absorbed
is ?iMp + 1021 = 1340 B. t. u. and the efficiency is ^ 2464 ^ = M8.
The heat supplied by the jacket is 1340  1098.2 = S46.S B. t. u.
VIII. Pambour cycle, incomplete expansion (debuq). In this case, we cannot
directly find the heat refected, nor can we obtain the work area by inte
gration.* From Fig. 175 (or from the steam table), we find T u =253.8 F.,
P M = 31.84. The heat area under bu is then, very nearly,
T + r (n u  712 ' 6 + 812 ' 7 (1.6953  1.5747) = 9S B. t. u.
2 2i
The whole heat absorbed is then 1098.2 f 92 = 1190 S B. t. u.
The work along de, eb y as in VII, is 6^96 1.5 footpounds.
The work along bu is 144 x 16[(140 x 3.219)  (31.84 x 13)] = 85,800 foot
pounds.
The work along qd, as in III, is 37 40 footpounds.
The whole work of the cycle is
63,961.5 + 85,800  3740 = 146,021,5 footpounds = 188.2 B. t. u.
The efficiency is = 0.1585.
* A satisfactory solution may be had by obtaining the area of the cycle in two parts, a
horizontal line being drawn through u to de. The upper part may then be treated as a com
pleteexpansion Fambour cycle and the lower as a nonexpansive cycle. The gross heat
absorbed IB equal to the work of the upper cycle plus the latent heat of vaporization at the
division temperature plus the difference of the heats of liquid at the division temperature
and the lowest temperature.
A somewhat similar treatment leads to a general solution for any Rankine cycle : in
which, if the temperature at the end of expansion be given, the use of charts becomes
unnecessary.
COMPARISONS 281
IX. Superheated cycle, steam 0.96 dry at the end of expansion ; complete expansion;
cycle debxw. We have n v ,=n d +x 1D n^ / = 0.1749 + (0.96 x 1.7431) = 1.8449.
The state x(n x = n tt ) may now be found either from Fig. 175 or from the
superheated steam table. Using the last, we find 7*, = 081.1 F., .7*= 1481.8,
V x = 5.96. The whole heat absorbed, measured above T d , is then
1481.8  94.0 = 1387.8.
The heat rejected is x v L f = 0.96 x 1021 = 981.
The external work done is 1387.8 981 = 4063, and the efficiency is
SB
(The efficiency of the Carnot cycle within the same temperature limits is
931. 1  126.15 ^p^v
931.1 + 459.6 " *'
X. !T&e same superheated cycle, with incomplete expansion.
The whole heat absorbed, as before, is 1387.8,
The work done along de, eb, as in HI, is 63,961.5 footpounds.
The work done along bx is
P b (V,  T 5 ) = 144 x 140(5.96  3.219)= 55,000 footpounds.
The w?0r& cfone atony a; is
x 5J>51.1 x 13 = 81 j 00 footpounds.
(V L = 13, P*F s i* = p z ?yj, p, = 140^y ** = 51.1 ; a procedure
which is, however, only approximately correct (Art. 391).)
The work along gd, as in III, is 3740 footpounds.
The whole work of the cycle is
63,961.5 + 55,000 + 81,500  3740 = 196,721.5 footpounds = 2SS.5 B. t. u.
The efficiency is
XI. T?ie *ame superheated cycle, worked nonezpansively. The (7r(w fta/ alwrbed
is
...
The j<?rA: a/on^ <?, eb 9 bx, as in X, is 118,961.3 footpounds.
The worfc along pd is 2 x 144 X (5.96  0.017)= 1716 footpounds.
The whole work of the cycle is
118,961.5  1716 = 111 ',246.5 footpounds = 150.6 B. t. u.
The efficiency is ^ = OJ086.
XIL Superheated cycle, steam dry at Ihe end of expansion, complete expansion ; cycle
debyf.
We have n f = n/= 1.018. This makes the temperature at y above the
range of our table. Figure 171 shows, however, that at high tempera
tures the variations in the mean value of k are less marked. We may
perhaps then extrapolate values in the superheated steam table, giving
r r = 1120.1 F., H 9  1578.5, T r r = 6.81. The whole heat absorbed, above
T* is then 157&5  94.0 = U79J. The heat refected is L/st 1Q&1.
282 APPLIED THERMODYNAMICS
The external work done is 1479.5  1021 = 458.6 S. t. u., and the efficiency
XIIL Superheated cycle as above^ but with incomplete expansion. The gross heat
absorbed is 1470.5.
The work done along de, eb, as in III, is 63,961. o fontpounds f
The work done ahmy ly is 144 x 140 X (6.81  3 219) = 72,200 footpounds.
(6 81\ 1>23 ^
Hi J =60.3poun d.% approximately, s
The we* done along yX is lf I 140 * ^T.f ' 3 X 13) ) = *V< >*
v o.ijyo /
pounds, alf*o approximately.
The ipori ^/ir a/o/zy ^/, as in III, is 3740 footpounds.
The 7r&0/e ttorl' of the cycle is
63,961.5 + 72,200 + 81,100  3740 = 213,521.5 footpounds = 875 B. t. u.
The efficiency is "j ,  ^?.^5r.
XTV, Superheated cycle as above, but without expansion. The #ros$ Aeaf absorbed
The warX: a?on^ /^, eb y by, as in XIII, is 136,161.5 footpounds.
The zconfc atofl' *rf is 2 x 144 x (6.81  0.017) = 1952 footpounds.
The tota/ wor/fc' is 136,1615  1952 = 134,209.5 footpounds = 172.7 B. t. u.
The efficiency is 2j=jL. = 0.117.

XV. Superheated cyde^ steam superheated 4&* F* at the end of expansion; expan
sion complete ; cycle debzAf. TV T e have n A = n x = 1.9486. A rather
doubtful extrapolation now makes T s = 1202.1 F., #, = 1613.4, V*
= 7.18. The irhule heat absorbed is 1613.4  94.0 = 1519.4 The heat re
jected is H A = 1133.2. The total work is 1519.4  1133.2 = 386.2 B. t. u.,
SS6 ^
and the efficiency is ' '" = 0355.
lolU.4
XVL The same superheated cycle, with incomplete expansion. The pressure at T is
140 (TQ) 65.8 pounds. The work along zT (approximately) is
144 ((140 x 7.18) (65.8 x!3)\ = 7Sj900 foot _ pmnds . TheoZ, work is
\ o.2yo /
63,961.5 + [144 x 140 x (7.18  3.219)] + 73,900  3740 = 213,921.5 foot
pounds = 875.$ B. t. u., and the efficiency is
1519.4
XYII. The same superheated cycle without expansion. The total work is 63,961.5 +
[144 x 140 x (7.18  3.219)]  [2 x 144 x (7.18  0.017)] =141,701.5 foot
pounds = 1833 B. t. u. T aud the efficiency is 0.1803.
418. Discussion of Results. The saturated steam cycles rank in
order of efficiency as follows: Carnot, 0.28; Clausius, with, dry steam,
COMPARISONS
283
0.254; with wet steam, 0.254 (a greater percentage of initial wetness
would have perceptibly reduced the efficiency); Pambour, with com
plete expansion, 0.238 ; with incomplete expansion, 0.1585 ; Rankine,
with dry steam, 0.1704 ; with wet steam, 0.1667; nonexpansive, with
dry steam 0.074; with wet steam, 0.0722. The economical impor
tance of using initially dry steam and as much expansion as possible
is evident. The Pambour type of cycle has nothing to commend it,
the average temperature at which heat is received being lowered.
The Rankine cycle is necessarily one of low efficiency at low expan
sion, the nonexpansive cycle showing the maximum waste.
Comparing the superheated cycles, we have the following
efficiencies :
CYCLE
COMPLETE EXPANSION
INCOMPLETE EXPANSION
No EXPANSION
debxw
0.293
0.183
0.1086
debyf
debzAf
0.31
0.255
0.187
0.182
0.117
0.1203
The approximations used in solution* will not invalidate the
conclusions (a) that superheating gives highest efficiency when it is
carried to such an extent that the steam is about dry at the end of
complete expansion; (J) that incomplete expansion seriously re
duces the efficiency ; (V) "that in a nonexpansive cycle the effi
ciency increases indefinitely with the amount of superheating. As
a general conclusion* the economical development of the steam en
gine seems to be most easily possible by the use of a superheated
cycle of the finallydrysteam type, with as much expansion as pos
sible. We shall discuss in Chapter XIII what practical modifica
tions, if any, must be applied to this conclusion.
The limiting volumes of the various cycles are
F c for the Garnet, I, = 139.3. V w for IX = 166.5.
V l for H = 128.2. V x for XI = 5.96.
F* for V = 3.219.
V k for VI = 2.9.
F^for VII, XII =: 173, 5.
* See footnote, Problem 53, page 296.
A for XV = 186.1.
; for XVII = 7.18.
284
APPLIED THERMODYNAMICS
The capacity of an engine of given dimensions is proportional to
cyclic area ^ w hich. quotient has the following values* :
maximum volume
Car/not, temperature range x entropy range
31
= 226.95(1.5747  0.1749)= 317.5 : quotient = ^i^ = 2.29,
L 278.85130.3 = 2.0
II. 259.445128.2 = 2.
III. 187.2913 = 14.4.
IV. 169.1 + 13 =13.0.
V. 81.05* 8.219 = 25.1.
VL 73.052.9=25.1.
VII. E18f 173.5 =1.84.
VIII. 188.213 = 14.5.
IX, 400.8^166.5 = 2.445.
139.3
X. 253.5 i 13 = 19.45.
XL 150.6 f 5.96:= 25.3.
XII. 458.5 r 173.5 = 2.65.
XIII. 27513 = 21.1.
XIV. 172,7 6.81 =25.4.
XV. 386,2186.1 = 2.075.
XVI. 275.313 = 21.1.
XVII. 182,257.18 = 25.5.
Here we find a variation much greater than is the case with the
efficiencies ; but the values may be considered in three groups, the
first including the five nonexpansive cycles, giving maximum
capacity (and minimum efficiency); the second including the six
cycles with incomplete expansion, in which the capacity varies from
13 to 21.1 and the efficiency from 0.1585 to 0.187; and the third
including six cycles of maximum efficiency hut of minimum capacity,
ranging from 1,84 to 2.65. In this group, fortunately, the cycle of
maximum efficiency (XII) is also that of maximum capacity.
* The assumption of a constant limiting volume line Tuq, Pig. 183, is scarcely
fair to the superheated steam cycles. In practice, either the ratio of expansion or the
amount of constant volume pressuredrop at the end of expansion is assumed. As the
firKt increases and the second decreases, the economy increases and the capacity figure
decreases. The following table suggests that with either an equal pressure drop or an
equal expansion ratio the efficiencies of the superheated cycles would compare still
more favorably with that of the Rankine :
CYCLES WITH INCOMPLETE EXPANSION
ClCLB
RATIO OP EXPAXSIOX
PEESSETEE DROP
Rankine
^i
r* = 13  3.219 = 4.04
P ff  P 9  26.3
Superheat I
VL
r, = 18 * 5.06 =2.185
PL P q = 49.1
Superheat II
VM^
V, = IS ^ 0.81 = 1.91
Pjf P 9 = 68.3
Superheat III
TV
F^=137.18 =1.815
P T  P f = 63.3
THE STEAM TABLES 285
Practically, high efficiency means fuel saving and high capacity
means economy in the first cost of the engine. The general incom
patibility of the two affords a fundamental commercial problem in
steam engine design, it being the function of the engineer to estab
lish a compromise.
419. The Ideal Steam Engine. No engine using saturated steam can develop
an efficiency greater than that of the Clausius cycle, the attainable temperature
limits m present practice being between 100 and 400 Q F., or, for noncondensing
engines, between 212 F, and 400 F. The steam engine is inherently a wasteful
machine ; the wastes of practice, not thus far considered in dealing with the ideal
cycle, are treated with in the succeeding chapter,
THE STEAM TABLES
420. Saturated Steam. The table on pages 247, 248 is abridged from Marks'
and Davis' Tables and Diagrams (18). In computing these, the absolute zero
was taken at 459.64 F. ; the values of h and n w were obtained from the expei i
ments of Barnes and Dietrici (68) on the specific heat of water; the mechanical
equivalent of heat was taken at 777.52 ; the pressuretemperature relation as found
by Holborn and Henning (Art. 360); the thermal unit is the "mean B. t, u."(se
footnote, Art. 23) ; the value of H is as in Art. 388 ; and the specific volumes
were computed as in Art. 368. The symbols have the following significance :
P = pressure in pounds per square inch, absolute ;
T temperature Fahrenheit ;
V = volume of one pound, cubic feet ;
h = heat in the liquid above 32 P., B. t. u. ;
H= total heat above 32 F., B. t. u.;
L = heat of vaporization = ZT A, B. t. ti. ;
r = disgregation work of vaporization = L e (Art. 359), B. t. u.;
n^ = entropy of the liquid at the boiling point, above 32 F* ;
n, = entropy of vaporization = ;
n, = total entropy of the dry vapor = n f n+
421. Superheated Steam. The computations of Art. 417 may suggest the
amount of labor involved in solving problems involving superheated steam. This
is' largely due to the fact that the specific heat of superheated steam is variable.
Figure 177, representing Thomas' experiments, may be employed for calculations
which do not include volumes; and volumes may be in some cases dealt with by
the Linde formula (Art. 3fl#). The most convenient procedure is to use a table,
such as that of Heck (71) T or of Marks and Davis, in the work already referred to.
On the following page is an extract from the latter table. The values of naed
are the result of a harmonization of the determinations of Knoblauch and Jakob
(Art 384) and Holborn and Henning (&9) and other data (70). They differ
somewhat from &OSB given in, Fig. 170. The total heat values are obtained by
286
APPLIED THERMODYNAMICS
adding the values of k(Tt) over successive short intervals of temperature to
the total heat at saturation ; the entropy is computed iu a corresponding manner.
The specific volumes are from the Linde formula.
PROPERTIES OF SUPERHEATED STEAM
hirmiiBAT, *F
40
90
200
300
400
500
600
Absolute Pre&Rnre
Lbfl. per Square Inch
1
' * = 141.7
V = 357.8
' # = 1122.6
11)1.7
387.9
1145.3
301.7
4:037
1195.6
401.7
513.4
1241.5
501.7
573.1
1287.6
601.7
632.7
1334.1
701.7
692.4
1381.0
n = 2.0060
2.0434
2.1145
2.1701
2.2218
2.2679
2.4100
' t = 166 1
216.1
326.1
426,1
526.1
626.1
726.1
F= 186.1
201.2
234.2
264.1
293.9
323.8
353.6
9
1 IT =1133.2
1156.1
12064
1252.4
1298.6
1345.2
1392.2
n = 1.9486
1.0836
2.0529
2.1071
2.1586
2.2044
2.2459
(t = 280.1
330.1
440.1
540.1
640.1
740.1
840.1
35
#=1179.6
18.61
1203.4
21.32
1255.6
23.77
1302.8
26.20
1350.1
28.61
1397.5
31.01
1445.4
n = 1.7402
1.7712
1.8330
1.8827
1.9277
1.9688
2.0078
' t = 367.8
417.8
527.8
627.8
727.8
827.8
927.8
100
# = 1208 4
5.07
1234.6
5.80
1289.4
6,44
1337.8
7.07
1385.9
7.69
1434.1
8.31
1482.5
n = 1.6294
1.6600
1.7188
1.7656
1.8079
1.8468
1.8829
' t = 393.1
443.1
553.1
653.1
733.1
853.1
953.1
140
F=3.44
' # = 1213.8
3.70
1242.8
4,24
1298.2
4.71
1346.9
5.16
1395,4
5.61
1443.8
6.06
1492.4
n = 1.6031
1.6338
1.6916
1.7376
1.7792
1.8177
1.8533
' 1 = 398 5
448.5
558.5
658.5
758.5
858.5
958.5
150
1 #=1217.3
3.46
1244.4
3.97
1300.0
4.41
1348.8
4.84
1397.4
5.25
1445.9
5.67
1494.6
n = 1.5978
1.6286
1.6862
1.7320
1.7735
1.8118
1,8474
t = temperature Fahrenheit ; V = specific volume ; H =s total heat above 32 P. ;
n = entropy above 32 F.
(Condensed from Steam Tables and Diagram, by Marks and Davis, with the per
mission of the publishers, Messrs. Longmans, Green, & Co.)
THEORY OF VAPORS
287
PKOPERTIES OF DRY SATURATED STEAM
(Condensed from Steam Tables and Dirrr/ntmn, by Marks and Duvis, with the permit
sion of the publishers, Messrs Longmans, Green, & Co.)
JP
T
r
h
L
H
r
"a
"
n t
1
101.83
333.0
698
1034
1104.4
072.9
1327
1.8427
1.0754
2
126.15
173.5
94.0
1U21.0
1115.0
030.7
0.1740
1.74,31
1.0180
a
141.52
118.5
109.4
1012.3
1121.0
040.4
02008
1.0840
1.8848
4
153.01
90.5
120.9
1003.7
11"J<J.3
038.0
!>108
1.641(5
1.8614
5
162.28
73.33
130.1
1000.3
1130.5
932.4
0.2:J48
1.6084
1.8432
6
170.00
61 80
137.9
905.8
1133.7
027.0
0.2471
1.5814
1,8285
7
170.85
53.50
144.7
991.8
1 ];](>. 5
02.4
0.2570
1.6582
1.8161
8
182.86
47.27
150.8
988.2
1130.0
018 2
0.2073
1.5380
1.8053
9
188.27
42.30
156.2
985,0
1141.1
OH.4
2750
1.&WW
1.7958
10
193.22
38.38
161.1
982.0
1143.1
010.9
2832
1.6042
1.7874
11
197,75
35.10
165.7
970.2
1144.0
007.8
0.2002
1.4805
1.7797
12
201.96
32.36
160.9
9706
1140 5
904.8
0.2007
1.4700
1.7727
18
205.87
30.03
173.8
074.2
1148.0
90:!.
0.3025
1.4080
1.7064
14
209.55
28.02
177.5
971.0
1140.4
81W.3
0.3081
1.4523
1.7604
15
213.0
26.27
181.0
909.7
1150.7
81*0.8
0.3133
1.4410
1.7549
16
216.3
24.79
184.4
007.0
1152.0
804.4
3183
1.4311
1.7494
17
219.4
23.38
187.5
905.6
1163.1
802.1
3220
1.4215
1.7444
18
222.4
22.16
100.5
903.7
1154.2
880.9
0.3273
1.4127
1.7400
19
225.2
21.07
193.4
961.8
1155.2
887.8
3316
1.4045
1.7360
20
228.0
20.08
190.1
960.0
1156.2
885.8
0.3355
1.3005
1.7320
21
230.6
19.18
198.8
958.3
1157.1
883.0
0.3303
1.3887
1.7280
22
233.1
18.37
201.3
950.7
1158.0
882.0
0.3430
1.3811
1.7241
23
235.5
17.62
203.8
0551
115H.8
880.2
0,3405
1.3730
1.7204
24
237.8
16.93
206.1
953.5
1100.0
878.5
O.:)409
1.0070
1.7169
25
240.1
16.30
208.4
952.0
1100.4
870.8
0.8532
1.3004
1.7136
26
242.2
15.72
210.6
950.6
1161.2
876.1
0.3504
1.3642
1.7106
27
244.4
15.18
212.7
949.2
1101.9
873.6
0.3504
1.848$
1.7077
28
246.4
14.67
214.8
JH7.8
1162.0
872,0
0.3023
1.3425
1.7048
29
248,4
14,19
216.8
946.4
1KS3.2
870.5
o.30:>2
1.8807
1.7019
80
250.3
13.74
218.8
945.1
1103.9
809.0
0.3080
1.8311
1.6991
81
252.2
13.32
220.7
943.8
1164.5
867.0
0.3707
1.3357
1.6KH
82
254.1
12.93
2^22.6
942.5
11(15.1
800.2
0.978$
1.8315
1.6938
88
255.8
12.57
224.4
941,3
1105.7
804.8
0.3759
1.3153
1.6V14
84
257.6
12.22
226.2
940.1
netu
803.4
0.3784
1.3107
1.6891
85
259.3
11.89
227.9
938.9
1106,8
8C2.1
0.3808
1.30BO
1.0868
86
261.0
11,58
229.6
937.7
1167.3
860.8
0.&&2
1.3014
1.U846
87
262.6
11.29
28tS
930.6
1167.8
869.5
0.3865
1.2960
1JW34
88
264.2
11.01
232.9
955.5
1108.4
858.3
0.3877
1.2&25
1.0803
89
265.8
10.74
234.5
934.4
1108.9
867.1
0.3SiJ
1.288i
1.6781
40
267.3
10.49
236.1
m$
IMHU
&66.0
0.3920
1.2841
1.6761
41
268.7
10.26
237.6
932.2
116&.8
864.7
0.3941
1.2800
1 6741
42
270.2
10,02
*89.i
&U.2
1170.8
$5#.6
0.3962
1.2769
1,6721
48
271.7
9.80
240.5
930.2
1170.7
852.4
0.3982
, 1.2720
1.6702
44
273.1
9,69
242.0
929.3
1171/2
51.3
0.4002
1.2081
1.0683
45
274.5
9.39
243.4
028.2
1171.6
860.3
0.4021
1.2644
1.6665
46
275.8
9.20
244.8
927.2
1178.0
649. 2
0.4040
1.2607
1.6647
47
277.2
9.02
246.1
9U
H75U
848.1
0.4059
12671
1.6630
48
278.6
8.S4
247.6
926.3
ira.s
847.1
0.4077
1.25S6
1.6613
49
S79.8
8.07
248.8
924.4
117.^
&4<U
0.40%
1.2502
1.6697
50
281.0
8.61
fctfU
02S6
1175U
&45.0
0,411$
1.2468
1.65&1
i
288
APPLIED THERMODYNAMICS
PROPEKTIES OF DRY SATURATED STEAM
(Condensed from Steam Tables and Diagrams, by Marks and Davis, with the permis
sion of the publishers, Me&srs. Longmans, Green, & Co.)
p
T
r i h L n
r
n u
e i n s
51
282.3
8.35
i
231 4 ! 922.0
11740
8440
0.4130
1 2435 i 1 6505
52
283.5
8.20
2526 921.7
1174.3
843.1 0.4147
1 2402
1 0549
58
284.7
8.05
253.0 920.8
1174.7
8421
0.4164
1.2370
16534
54
285.9
7.01
5*55.1 911>.0
11750
841.1
0.4180
1.2330
1 6519
55
287.1
7,78
256.3
9U.0
1175.4
840.2
0.4196
1.2309
16505
56
288.2
7.65
257.5
0182
1175.7
839.3
0.4212
1.2278
1.6400
57
289.4
7.02
258.7
917.4
1170.0
838.3
0.4227
1.2248
1 6475
5S
200.5
7.40
250.8
910.5
1176.4
837.4
0.4242
1.2218
16460
59
291.6
7.28
261.0
013.7
1176.7
836.5
4257
1.2189
16446
60
21*2.7
7.17
262 1
9140
1177.0
8356
0.4272
1.2100
1.6432
61
293.8
7.06
203.2
914.1
1177.3
834.8
0.4287
1.2132
1.6419
62
204.1)
6.05
2(54 3
913.3
1177.6
833.0
0.4302
1.2104
1.6406
63
295.9
6.85
205.4
912.5
1177,9
833.1
0.4316
1.2077
1.6393
64
297.0
0.75
2<XU
911.8
1178 2
832.2
0.4330
1.2050
16380
65
2080
6.65
207.5
911.0
1178.5
831.4
0.4344
1.2034
16368
66
290.0
656
268.5
910.2
11788
830.5
0.4368
1.2007
16355
67
300.0
6.47
200.6
000.5
1170.0
829.7
0.4371
1.1972
16343
68
301.0
638
270.6
008.7
1170.3
828.9
0.4385
1.1946
1.6331
69
302.0 6.29
271.0
908.0
11796
828.1
0.4398
1.1921
1.6319
70
302.9
6.20 j 272.6
907.2
1179.8
827.3
0.4411
1.1896
1.6307
71
303.9
6.12
2736
906.5
1180.1
82.5
0.4424
1.1872
1 6296
72
304.8
6.04
274.5
905.8
11804
825.8
04437
1.1848
16285
78
305.8
5,90
275.5
9051
1180.6
825.0
0.4440
1.1825
1.6274
74
30.7
5.89
270.5
904.4
1180.0
824.2
0.44G2
1.1801
16263
75
307.6
581
277.4
903.7
1181.1
823.5
0.4474
1.1778
1.6262
80 i 312.0
5.47
282.0
900.3
1182.3
819.8
0.4535
1.1665
1.6200
85
310.3
516
286. 3
897.1
1183.4
816.3
0.4590
1.1561
1 6151
90
320.3
4.89
290.5
893.9
1184.4
818.0
0.4644
1.1461
1.6105
95
324.1
4.65
294.5
890.9
1185.4
809.7
0.4604
1.1367
16061
100
327.8
4.429
298.3
888.0
1186.3
800.6
0.4743
1.1277
1.6020
105
331.4
4.230
302.0
885.2
1187.2
803.6
04780
1.1191
1.5980
110
334.8
4.047
305.5
8825
1188.0
800.7
04834
1.1108
1.5942
115
338.1
3.880
309.0
879.8
1188.8
797.9
04877
1.1030
15907
120
3413
3.720
312.3
877.2
1189.6
795.2
0.4919
1.0964
1.5873
125
344.4
3.583
315.5
874.7
1190.3
792.6
0.4959
1.0880
1.5839
180
347.4
3.452
318.6
872.3
i 1191.0
790.0
0.4998
1.0809
15807
140
353.1
3.219
324.6
8076
1192.2
785.0
0.5072
10675
15747
150
358.5
3.012
330.2
863.2
1193.4
7804
0.5142
1.0550
15692
160
303.C
2.834
336.6
858.8
1194.5
775.8
0.5208
1.0431
1.5639
170
368.5
2.675
340.7
854.7
1195.4
771.5
0.5269
1.0321
15590
180
373.1
2.533
345.6
850.8
1196.4
767.4
0.5328
10215
15543
190
377.6
2,406
350.4
846.9
1197.3
763.4
0.5384
1.0114
16498
200
381.9
2.290
354.9
843.2
1198.1
769.5
0.6437
10019
1.5456
210
386.0
2,187
3592
839.6
1198.8
766.8
0.5488
0.9928
1.6416
220
389.9
2.091
363.4
836.2
1109.6
752.3
0.5538
0.9841
15379
280
393.8
2.004
367.5
832.8
1200,2
748.8
0.6686
0.9758
1.5344
240
397.4
1.924
371.4
829.5
1200.9
7454
0.5633
0.9676
15309
250
401.1
1.850
375.2
82J.3
1201.5
742.0
05676
0.9600
1.6276
THEORY OF VAPORS 289
(1) PhiL Trans., 1851, CXLIV, 360. (2) Phil. T/V/TW., 1854, 330 ; 1862, 579.
(3) Theorie Mecanique de la Chaleur, 2d ed., I, 195. (4) Wood, Th*rmoi~?ynrunf t
1905, 390. (5) Wiedemann, Ann. Her Phys. und Chem., 1880, Vol. IX. ff5) Technical
Thermodynamics (Klein), 1907, II, 215. (7) Mitteilungcn Wter ForirtuntwirMteu.
etc., 21 ; 33. (8) Peabody, Steam Tables, 1908, 9 ; Marks and Davis, Tables awl
Diagrams, 1909, 88; Phil. Trans., 199 A (1902), 1492(33. (0) The AV^wi Buying
1897, 001. (10) Op. rft., II, App. XXX. (11) The EicharOs Strim, Etujhie Indica
tor, by Charles T. Porter. (12) Trans. A. S. .If. E., XL (13) Dubols ed , II. 11, 1H84.
(14) Peabody, op. ctt. (15) Trans A S. M. E., XII, 590. (10) Ann for Ffty*rt\ t,
26,1908,833. (17) Trans. A. S. M. E.. XXX, 14191432. (18) Tables and &WJMM*
of The Thermal Properties of Saturated and Superheated Steam, Itttf. (1'J) Zrttx.
fur Instrumentenkunde, XIII, 329, (20) Wissenschnftliche Ahhandlungpii, III, 71.
(21) Sitzungsberichte K. A. W. in Wien, Math.natur *Kla$se, CVII, II, Oct, 1809.
(22) Loc. tit., note (7), (24) Comptes Rendus, LXII, 56; Bull, de la Soc. Industr.
de Mulhouse, CXXXIII, 129. (25) Boulvin's method: see Berry, The Tempera
ture Entropy Diagram, 1906, 34. (26) Zeuner, op. cd., II, 207208 (27) Nichols
and Franklin, Elements of Physics, I, 194. (28) Phil. Trans., 1869, II, 575. (29)
Zeits. Ver. Deutsch Ing., 1904, 24. (30) Trans. A. S. M. E., XXVIII, 8, 1264.
(81) Ann. der Phys., Leipzig, 1905, IV, XVIII, 739. (32) Zeits. Ver. Deutsch.
Ing., Oct. 19, 1907. (33) MitteiL uber Forschungsarb., XXXVI, 109. (34) Tnttt*.
A. fl. M. E., XXVIII, 10, KJ95. (35) Trans. A. S. M. E., XXIX, 0, 033. (30) Hid.,
XXX, 5, 533. (37) Ibid., XXX, 9, 1227. (38) Op. cit., II, 239. (39) Pea
body, Op. cit., 111. (40) The Steam Engine, 1905, flg. (41) Trantt. A. S.
M. E., XXIX, 6. (42) Op. tit., II, Apps. XXXIV, XXXV, XL, XLIV, XLU,
XXXVIII. (43) Op. cit., 407 et. seq. (44) Qp. c#.,600. (45) Cvmpte* Rrwlu*,
Oil, 1886, 1202. (40) Ibid., CXIV, 1892, 1093 ; CXIII, 1891. (47) Zetts.JVr die
gesamte KatieIndustrie, 1895, 6685. (48) Op. cit., II, App. L. (49) JfcirAjn** a
froid, Paris, 1878. (60) Elleau and Ennis, Jour. Frank. Inst., Mar., Apr., 1K98 ;
Dietrici, Zeits. KalteInd., 1904. (51) Op. cit., II, App. XL VI. (52) Zrit*.fur die
gesamte KalteIndustrie, 1904. The heavy line across the table on pa^e 422 indicates
a break in continuity between the two sources of data. The same break is resj>onsible
for the notable irregularity in the saturation and constant dry ness carves on the ammonia
entropy diagram, Fig. 316. (53) Tables of the Properties of Saturated Steam and other
Vapors, 1890. (54) See Jacobus, Trans. A. S. M. E., XH. (55) Jour. Fmnl\ /**.,
Dec., 1890. (56) Op. cit., 466. (57) Mem. de rinstttiit de France, XXI, XXVI.
(58) Landolt and Bdrnstein, Physitetbachechemische Tabfllen; Gmeliii ; Peabody,
Thermodynamics, 118. (59) Andreeff, Ann. Chem. Phartn., 1859. (tiO) Trans.
A. $. M. E., XXV, 176. (61) Tables, etc., 1890. (02) Comptes Sendw, CXIX,
1894, 404407. (63) Op. tit, App. XLVIII. (04) Op. cit., 48. (to) Trans.
A. 8. M. E. t XEL (66) Op. cit., IE, App. XXXII. (07) Trans. A. S. 3f. E.. XXI,
3, 406. (08) WieA. Annallen,, (4), XVI, 1905, 5S)3620. (09) WM. AnnaUen, (4),
XVIII, 1905, 73&~756; (4), XXIII, 1907, 809845. (70) Marks and Davis, op. cit., 5.
(71) Trans. A. S. M. E., May, 1908.
SYNOPSIS OP CHAPTER XII
The temperature remains constant during evaporation j that of the liquid is the same
as that of the vapor; increase of pressure raises the toiling point, and wre wr<i;
it also increases tM density. There is a definite boiling point for each pr**ure.
Saturated vapor is vapor at minimum temperature and maximum density for the given
pressure.
Superheated vapor is *n imperfect gas, produced by adding heat to a dry saturated vapor.
290 APPLIED THERMODYNAMICS
Saturated Steam
FC W V")
The principal effects of heat are, h = t 32, e = s ^ p ^
( to
Asp increases, i, h, e and H increase, and r and X decrease.
&= Mm + 0.3745(2 212) 0.00065 ($
o/ evaporation, = X + ~ ^'
The pressure increases more rapidly than the temperature.
Characteristic equation for steam, JH? = o!T j)(l + Z>p) rjg
Saturated steam may he dry or wef . 3Tor wet steam,
and the /actor of evaporation is t 7  Tne volume is TF=F+a:(TFo 7).
The zoa^r Zine shows the volume of water at various temperatures j the saturation curve
shows the relation between volume and temperature of saturated steam. Approxi
mately, pv$ = constant. The isothermal is a line of constant pressure.
The path during evaporation is (a) along the water line (&) across to the saturation
curve at constant pressure and temperature. If superheating occurs, the path pro
ceeds at constant pressure and increasing temperature to the right of the satura
tion curve.
T
On the entropy diagram, the equation of the water line is n = clog, . The distance
between the water line and the saturation curve is JV r =^ Constant dry ness
curves divide this distance in equal proportions. Lines of constant total heat may
be drawn. The specific heat of steam kept dry is negative. The dryness changes
during adiabatic expansion. The temperature of inversion is that temperature at
Trtiich the specific heat of dry steam is zero. The change of internal energy and
the external work along any path of saturated steam may be represented on the
entropy diagram.
W= F .
Constant volume lines may be plotted on the entropy diagram, permitting of the trans
fer of any point or path from the PFto the T2? plane. The temperature after
expansion at oontant entropy to a limiting volume can best be obtained from the
entropy diagram,
The critical temperature is that temperature at which the latent heat becomes zero
(68SP F.}.
Saturated vapor (dry or wet), superheated vapor, gas ; physical states in relation to the
critical temperature ; shape of isothermals.
The i&odynamic path for saturated steam touches the saturation curve at one point
only.
SYNOPSIS 291
Sublimation occurs if the saturation pressure at the melting temperature exceed*
that of the surrounding medium.
Gax and Tapor ^fixtures
Value of E for gas mixtures : mixture of air and steam ; absolute and relative
humidities ; wet and diy bulb thermometers ; in mixtures, mixing does not
affect the internal energy and adiabatic expansion, ih without influence on the
aggregate entropy.
Mixture and expansion of (a) wet vapor and jras, (6; hi^hpressure steam and air,
(c) superheated steam and air, (d\ two vapors. Equivalent values of n. In the
heat engine, mixtures may lower the temperature of heat rejection*
Superheated Steam
The specific heat has been in doubt. Its value increases with the pressure, and varies
with the temperature.
j5T=jff, + %,(:ro. r = ^ rL r HiJ3i=*C^afi) + ^(2lr.).
Kpi T t
Factor of evaporation = Saa + *?' f) ~ h PF= 0.64901 T 22.5819 J
y iU.4
PF= 0.694 T  0.00178 P. J? = 85.8. y = 1.298.
Paths of Vapors
Adiabatic equation : = doge +  Approximately, PF n =constank Values of n.
t t T
External work along an adiabatic = h
Continuously superheated adiabattc, e
Adtabatfc crossing the saturation curve :
m
Method of drawing constant pressure lines on the entropy diagram : n = Aplog. 
Method of drawing lines of constant total heat.
Use of the entropy diagram for graphically solving problems: dryness after expansion j
work done during expansion ; mixing ; heat contents.
The Mollier coordinates, total heat and entropy. The total heat^pressvre diagrams.
Vapors in General
&* ** '
When the pressuretemperature relation and the characteristic equation are given, we
may compute L for various temperatures, and the specific heat of the vapor.
*=0.608,
292 APPLIED THERMODYNAMICS
vapor density =0.597 (air = l), specific volume of lic][uid= 0.025, its specific heat
= 1.02. Sulphur dioxide: =0.15438, vapor density = 2. 23, specific volume of
liquid = 0.0007, its specific heat = 0.4. PY = 26.4T1S4P 23 . Pressuretem
perature relation. L = 176 0.27(232). Engine capacity and economy is
influenced by the vapor employed.
Steam Cycles
Efficiency = work done r gross heat absorbed.
The Carnot cycle is impracticable , the steam power plant operates in the Clausius cycle.
Efficiency of Glauslus cycle

J.
Sankine cycle (incomplete expansion) determination of efficiency, with steam
initially wet or dry.
tfbnopanszoe cycle: efficiency = (frlX 3 **  017 ).
1483 log. 0.695(r0
Pambour cycle : steam dry during expansion ; efficiency = 
computation of heat supplied by jacket.
Superheated cycle : efficiency is increased if the final dryness is properly adjusted and
the ratio of expansion is not too low.
Numerical comparison of seventeen cycles for efficiency and capacity : steam should
be initially dry. The ratio of expansion should be large for efficiency and small
for capacity.
The Steam Tables
Computation is from p (or ) to t (or j>), H, h, L, 3?, F, e, r, n u , ^ n,.
at
The superheated tables give /*, F, H, f, for various superheats at various pressures ; all
values depending on H^t, w, and kp.
PROBLEMS
NOTE. Problems not marked T are to be solved without the use of the steam
table. In all cases where possible, computed results should be checked step by step
with those read from the three charts, Figs. 175, 177, 185.
Tl. The weight per cubic foot of water at 32 P. being 62.42, and at 250.3 F ,
58.84, compute in heat units the external work done in heating one pound of water at
pressure from 32 to 250.3. (The pressure is that of saturated steam at a tempeiature
of 250,3.) (J.ns., 0.0055 B. t. u.)
T la. 10 Ib. of water at 212 are mixed with 20 Ib. at 170.06. What is the
total heat per pound, above 32 F., of the resulting mixture?
2. Forp^lOO, =327.8, FW.429, compute h (approximately), fl", X, e, r, in
the order given. Why do not the results agree with those in the table?
., ^=295.8, J5T=1186.3, ^890.5, e = 81.7, r=SOS.8.)
PROBLEMS
T 2a. Water at 90 F. is fed to a boiler in which the pressure is 105 Ib. per sq.
in. absolute. How much heat must be supplied to evaporate one pound ?
T 3. Find the factor of evaporation for dry steam at 95 Ib. pressure, the feed
water temperature being 153 F. (Ans^ 1.097.)
273*^ 396945
T 4. Given the formula, log p = c ^ ^j, T being the absolute tempera
ture and p the pressure per square foot, find the value of ~ f or p = 100 Ib, per square
inch, t = 327.8 F. Check roughly by observing nearest differences in the steam table.
T 5. What increase in steam pressure accompanies an increase in temperature
from 353.1 F. to 393.8 F? Compare the percentages of increase of absolute pressure
and absolute temperature.
T 6. Find the values of the constants in the KanMne and Zeuner equations (Art.
363), at 100 Ib. pressure.
T 7. From Art. 363, find the volume of dry steam at 240.1 F. hi four ways.
Compare with the value given in the steam table and explain the disagreement.
8. At 100 Ib. pressure, the latent heat per pound is 888.0 j per cubic foot, it is
200.3. Find the specific volume. (Ans., 4.433.)
9. For the conditions given in Problem 2, W being the volume of dry steam, find
the five required thermal properties of steam 95 per cent dry. Find its volume.
T 9a. How much heat is consumed in evaporating 20 Ib. of water at 90 F. into
steam 96 per cent dry at 100 Ib. absolute pressure per sq. in. ?
T 96. What is the volume occupied by the mixture produced in Problem 9a ?
T 9c. Five pounds of a mixture of steam and water at 200 Ib. pressure have a
volume of 3 cu. ft. How much heat must be added to increase the volume to 6 cu. ft.
at the same pressure ?
T 9d. A boiler contains 2000 Ib. of water and 130 Ib. of dry steam, at 100 Ib.
presssure. What is the temperature ? What are the cubic contents of the boiler ?
T 9e. Water amounting to 100 Ib. per min. is to be heated from 65 to 200 by
passing through a coil surrounded by steam 90 per cent dry, kept at 100 Ib. pressure.
What is the TninimiTm weight of steam required per hour ?
T 9f. Water amounting to 100 Ib. per min. is to be heated from 55 to 200 by
blowing into it a jet of steam at 100 Ib. pressure, 90 per cent dry. What is the
minimum weight of steam required per hour f
T10. State the condition of steam (wet, dry, or superheated) when (a)p=100,
<=327.8; (&)p=95, 0=4.0; (c) jp= 80, 2=360.
II. Determine the path on the entropy diagram for heating from 200 to 240 F.
a fluid the specific heat of which is LOOfoft, in which t is the Fahrenheit temperature
and a =0.0044.
T 12. Find the increases in entropy during evaporation to dry steam at the f o 1 
lowing temperatures : 228% 261, 386 F.
T 13. Compute from Art. 368 the specific volume of dry steam at 327.8 F. What
is its volume if 4 per cent wet 1 (See Problem 4.)
Tl3a. Steam at 100 Ibs. pressure 2 per cent wet, is blown into a tank having a
capacity of 175 cu. ft. The weight of steam condensed in the tank, after the flow is
discontinued, is 60 Ib. What weight of steam was condensed during admission ?
294 APPLIED THERMODYNAMICS
T 14. Find the entropy, measured from 32 F., of steam at 327.8 F., 65 per cent
dry, (a) by direct computation, (5; from the steam table. Explain any discrepancy,
T15. Dry steam at 100 IK pressure is compressed without change of internal
energy until its pressure is 200 IK rind its dryness after compression.
T 16. Find the diyness of steam at 300 F. if the total heat is 800 B. t. u.
T ita. One pound of steam at 200 Ib. pleasure occupies 1 cu. ft. "What per cent
of moisture is present in the steitm ?
T 17. Pind the entropy of steam at 130 Ib. pressure when the total heat is 840 B. t. u.
T 18. One pound of steam at 327.8 E., having a total heat of 800 B. t. u., expands
adiabatically to 1 Ib. pressure, rind its diyness, entiupy, and total heat after expan
sion. What weight of steam wab condensed during expansion ?
18 a. Three pounds of water at 760 absolute expand adiabatically to 660 absolute.
What weight of steam is pretext at the end of expansion ? (Use Pig. 175.)
19. Transfer a wet steam adiabatic from the TJUfto the PV plane, by the graphi
cal method.
20. Transfer a constant dryness line in the same manner.
21. Sketch on the T^anrl PV planes the saturation curve and the water line in
the region of the critical temperature.
T22, At what stage of dryness, at 300 F., is the internal energy of steam equal
to that of dry steam at 228 F. ?
T23. At what specific volume, at 300 F., is the internal energy of steam equal
to that of dry steam at 228 F?
T 23 a. A boiler contains 4000 Ib. of water and 400 Ib. of steam, at 200 Ib. absolute
pressure. If the boiler should explode, its contents cooling to 60 F. and completely
liquefying, in 1 sec., how much energy would be liberated ? What horse power
would be developed during the second following the explosion ?
724. Compute from the Thomas experiments the total heat in steam at 100 Ib.
pressure and 440 F.
T 25. Find the factor of evaporation for steam at 100 Ib. pressure and 500 F. from
feed water at 153 F.
T26. In Problem 18, find the volume after expansion, and compare with the vol
ume that would have been obtained by the use of Zeuner's exponent (Art. 394).
Which result is to be preferred?
T 27. Using the Knoblauch and Jakob values for the specific heat, and determin
ing the initial properties in at least five steps, compute the initial entropy and total
heat and the condition of steam after adiabatic expansion from P=100, T=7QQ F.
to p = 13. Find its volume from the formula in Art. 390. Compare with the volume
given by the equation PV 1 aWw^oi^w. (Assume that the superheated table shows
the steam to be superheated about 55 F. at the end of expansion.)
T27a. Steam at 100 Ib. pressure, 95 per cent dry, passes through a superheater
in which its temperature increases to 450 F. Find the heat added per Ib. and the
increase of volume,
T2S. Compute the dryness of steam after adiabatic expansion from P=140 r
T 753.1 F, t to t = 153 F. Find the change in volume during expansion.
^29. Find the external work done in Problems 27 and 28, along the expansive
paths.
PROBLEMS 295
T29a. Three pounds of steam, initially dry, expand adiabatically from 100 Ibs. to
1 Ib. pressure. Find the initial and final volumes and the external work done.
T 30. At what temperature is the total heat in steam at 100 Ib. pressure 1200 B. t. u. ?
31. Find the efficiency of the Carnot cycle between 341.3 F. and 101 83 F.
T 32. Find the efficiency of the Clausius cycle, using initially dry steam between
the same temperature limits.
T 33. In Problem 32, find the efficiency if the steam is initially 60 per cent dry.
T 34. In Problem 32, find the efficiency if expansion terminates when the volume
is 12 cu. ft. (Rankine cycle).
T 35. In Problem 32, find the efficiency if there is no expansion.
T36. Find the efficiency of the Pambour cycle between the temperature limits
given in Problem 31. How much heat is supplied by the jacket ?
T 37. Find the efficiency of this Pambour cycle if expansion terminates when the
volume is 12 cu. ft.
T 38. Steam initially at 140 Ib. pressure and 443.1 F. is worked (a) in the Clau
sius cycle, (5) in the Rankine cycle, with the same ratio of expansion as in Problem
37. Find the efficiency in each case, the lower temperature being 101.83 F. Find the
efficiency of the Rankine cycle in which the maximum volume is 5 cu. ft. (See foot
nqte, Case VIII, Art. 417.)
T 39. At what per cent of dryness is the volume of steam at 100 Ib. pressure
3 cu. ft. ?
7*40. Steam at 100 Ib. pressure is superheated so that adiabatic expansion to
261 F. will make it just dry. Find its condition if adiabatic expansion is then carried
on to 213 F. Find the external work done during the whole expansion,
T 41. Steam passes adiabatically through an orifice, the pressure falling from 140
to 100 Ib. When the inlet temperature of the steam is 500 F. 7 its outlet temperature
is 494 F. ; and when the inlet temperature is 000 F., the outlet temperature is 505 F.
The mean value of the specific heat at 140 Ib. pressure between 600 F. and 600 F. is
0.498. Find the mean value at 100 Ib. pressure between 505* F. and 404 F. How
does this value agree with that found by Knoblauch and Jacob ?
T 42. Find from Problem 41 and Fig. 171 the total beat in saturated steam at 140
Ib. pressure, in two ways, that at 100 Ib. pressure being 1186 3.
T 43. Plot on a total heatpressure diagram the saturation curve, the constant
dryness curve for x = 0.8$, the constant temperature curve for T= 500 F^ and a
constant volume curve for V = 13, passing through both the wet and the superheated
regions. Use a vertical pressure scale of 1 in. = 20 Ib., and a horizontal heat scale of
1 in. = 20 B. t. n,
44 Compute the temperature of inversion of ammonia, given the equation,
L = 666.6  0,613 T F M the specific heat of the liquid being 1.0, What is the result
if L = 656.5  0.01S r 0.00021& f* (Art 401) t
45. Compute the pressure of the saturated vapor of sulphur dioxide at 60 F (ArL
404). (Compare Table, page 424,)
T 48*. Compare the capacities of the cycles in Problems 8137, as in Art. 418.
47. Sketch the water line, the saturation curve, an adiabatic lor saturated, steam,
and a constant dryness line on the PT plane.
296 APPLIED THERMODYNAMICS
7 T 48. A 10gal. vessel contains 0.1 Ib. of water and 0.7 Ib. of dry steam. What
is the pressure ?
T 49. A cylinder contains 0.25 Ib. of wet steam at 58 Ib. pressure, the volume of
the cylinder being 1.3 cu. ft. What is the quality of the steam ?
T 50. What is the internal energy of the substance in the cylinder in Problem 49 ?
T&I. Steam at 140 Ib. pressure, superheated 400 F., expands adiabatically until
its pressure is 5 Ib. Find its final quality and the ratio of expansion.
T 52. The same steam expands adiabatically until its dryness is 98. Find its
pressure.
T 53. * The same steam expands adiabatically until its specific volume is 50. Find
its pressure and quality.
T 54. Steam at 200 Ib. pressure, 94 per cent dry, is throttled as in Art. 387. At
what pressure must the throttle valve be set to discharge dry saturated steam ?
T 55. Steam is throttled from 200 Ib. pressure to 15 Ib. pressure, its temperature
becoming 235.5 F. What was its initial quality ? (Use Fig. 175.)
56. Represent on the entropy diagram the factor of evaporation of superheated
steam.
57. Check by accurate computations all the values given in the saturated steam
table for t = 180 F., using 459.64 F. for the absolute zero, 14.696 Ib. per square
inch for the standard atmosphere, 777.52 for the mechanical equivalent of heat, and
0.017 as the specific volume of water. Use Thiesen's formula for the pressure :
(t 4 459.6) log ~L = 5.409 ( 212) 8.71 x 10w[(689 O 4  477*];
t being the Fahrenheit temperature and p the pressure in pounds per square inch. Use
the Knoblauch, Linde and Klebe formula for the volume and the Davis formula for
the total heat. Compute the entropy and beat of the liquid in eight steps, using the
following values for the specific heat of the liquid :
at 40, 1.0045; at 120, 0.9974 ;
at 60, 0.9991; at 140, 0.9987 ;
at 80, 0. 997 ; at 160, 1.0002 ;
at 100, 0.99675 ; at 180, 1.002e.
Explain the reasons for any discrepancies.
* This is typical of a class of problems the solution of which is difficult or impos
sible without plotting the properties on charts like those of Figs. 175, 177, 185. Prob
lem 53 may be solved by a careful inspection of the total heatpressure and Mollier
diagrams, with reasonable accuracy. The approximate analytical solution will be
found an interesting exercise. We have no direct formula for relation between V
and T, although one may be derived by combining the equations of Bankine or
Zeuner (Art. 363) with that in Problem 4. The following expression is reasonably
accurate between 200 and 400 F., where a is in cu. ft. per Ib. and t is the Fahrenheit
temperature :
(0.005 1 +0.505) 8 0**=477.
For temperatures between 200 and 260 F., an approximate equation is
PROBLEMS 297
T58. Check the properties given m the superheated steam table for P^ 25 with
200 of superheat, UMIU; Knoblauch values for the specific heat, in at least three steps,
and using the Knoblauch, Lmde and Klebe formula for the volume. Explain any
discrepancies.
59. Represent on the entropy diagram the temperature of inversion of a dry
vapor.
60. Sketch the Molher Diagram (Art, 399) from T=0 to JBT=r400, n = to 7i = 0.5.
CHAPTER
THE STEAM ENGINE
PBACTICAL MODIFICATIONS OF THE RANKINE
422. The Steam Engine. Figure 186 shows the working parts.
The piston P moves in the cylinder A, communicating its motion
through the piston rod R, crosshead (7, and connecting rod M to the
disk crank D on the shaft S, and thus to the belt wheel W. The
guides on which the crosshead moves are indicated by 6r, H", the
frame which supports the working parts by J. Journal bearings
at B and support the shaft. The function of the mechanism is to
transform the toandfro rectilinear motion of the piston to a rotatory
movement at the crank. Without entering into details at this point,
it may be noted that the valve V, which alternately admits of the
passage of steam through either of the ports JT, Y", is actuated by a
valve rod I traveling from a rocker J", which derives its motion from
the eccentric rod N and the eccentric E. In the end view, L is the
opening for the admission of steam to the steam chest JI", Q is a sim
ilar opening for the exit of the steam (shown also in the plan), and
the valve.
423. The Cycle. With the piston in the position shown, and
moving to the left, steam is passing from the steam chest through Y
into the cylinder, while another mass of steam, which has expended
its energy, is passing from the other side of the piston through the
port JTand the opening Q to the atmosphere or the condenser.
When the piston shall have reached its extreme lefthand position,
the valve will have moved to the right, the port Y will have been
cut off from communication with 2> and the steam on the right of
the piston will be passing through Yto Q. At the same time the
port X will be cut off from Q and placed in communication with E
The piston then makes a stroke to the right, while the valve moves
to the left. The engine shown is thus
298
THE STEAM ENGINE
299
300 APPLIED THERMODYNAMICS
If the valve moved instantaneously from one position to the other
precisely at the end of the stroke, the PV diagram representing
the changes in the fluid on either side of the piston would resemble
efcd, Fig. 184. Along eb, the steam \vould be passing from the
steam chest to the cylinder, the pressure being practically constant
because of the comparatively enormous storage space in the boiler,
while the piston moved outward, doing work. At 5, the supply of
steam would cease, while communication would be immediately
opened with the atmosphere or the condenser, causing the fall of
pressure along It. The piston would then make its return stroke,
the steam passing out of the cylinder at practically constant pressure
along id, and at d the position of the valve would again be changed,
closing the exhaust and opening the supply and giving the instan
taneous rise of pressure indicated by de.
424. Expansion. This has been shown to be an inefficient cycle
(Art. 41 7j, and it would be impossible, for mechanical reasons, to
more than approximate it in practice. The inlet port is nearly
always closed prior to the end of the stroke, producing such a diagram
as debgq, Fig. 184, in
_B which the supply of
steam to the cylin
der is less than the
whole volume of the
piston displacement,
and the work area
under bg is obtained
without the supply of
_ v heat, but solely in
FIG. 187. Arts. 424, 42o, 427, 430, 431, 436, 441, 445, 446, consequence of the
448, 449, 450, 451,452, 454. Indicator Diagram and . ,. r
RanJdnc Cycle. expansive action of
the steam. Appar
ently, then, the actual steam engine cycle is that of RanMne * (Art.
411) . But if we apply an indicator (Art. 484) to the cylinder, an instru
* It need scarcely be said that the association of the steam engine indicator dia
gram and its varying quantity of steam with the ideal Bankine cycle is open to
objection (Art. 454). Yet there are advantages on the ground of simplicity in this
method of approaching the subject.
WIREDRAWING 301
ment for graphically recording the changes of pressure and volume
during the stroke of the piston, we obtain some such diagram as
abodes, Fig. 187, which may be instructively compared with the cor
responding Rankine cycle, ABGDE. The remaining study of the
steam engine deals principally with the reasons for the differences
between these two cycles.
425. Wiredrawing. The first difference to be considered is that along the
lines 6, AB. An important reason for the difference in volumes at ft and B will
be discussed (Art. 430) ; we may at present note that the pressures at a and b aie
less than those at A and B, and that the pressure at b is less than that at a. This
is due to the frictional resistance of steam pipes, valves, and ports', which caufes
the steam to enter the cylinder at a pressure somewhat less than that in the boiler ;
and produces a further drop of pressure while the steam enters. The action of
the steam in thus expanding with considerable velocity through constricted pas
sages is described as "wiredrawing." The average pressure along ab will not
exceed 0.9 of the boiler pressure; It may be much less than this. A loss of \voik
area ensues. The greater part of the loss of pressure occurs in the ports and pas
sages of the cylinder and steam chest. The friction of a suitably designed &team
pipe is small. The pressuredrop due to wiredrawing or "throttling," as it is
sometimes called, is greatly aggravated when the steam is initially wet; Clark
found that it might be even tripled. Wet steam may be produced as a result of
priming or frothing in the boiler, or of condensation in the steam pipes. Its evil
effect in this as in other respects is to be prevented by the use of a steam separator
near the engine; this automatically separates the steam and entrained moisture,
and the water is then trapped away.
426. Thermodynamics of Throttling. Wiredrawing is a non~rever$
ible process, in that expansion proceeds, not against a sensibly equivalent
external pressure, but against a lower and comparatively nonresistant
pressure. If the operation be conducted with sufficient rapidity, and
if the resisting pressure be negligible, the external work done should be
zero, and the initial heat contents should be equal to the final heat
contents; i.e., the steam expands adiabatically (though not isentropic
ally) along a line of constant total heat like nir, Fig. 161. The steam
is thus dried by throttling; but since the temperature has been reduced,
the heat has lost availability. Figure 188 represents the case in which the
steam remains superheated throughout the throttling process. A is the
initial state, DA aixd EC Enee of constant pressure, AB an adiabatic,
A.F a line of constant total heat, and C the final state. The areas
SHJDAG and SHECK, and, consequently, the areas JDABEH and
GBCK, are equal; the temperature at C is less than that at A. (See
the superheated steam tables : at p~140 ; H = 1298.2 when 553.1 F.;
302
APPLIED THERMODYNAMICS
at p100, H = 1298.2 when t is about 548 F.) The effect of wire
drawing is generally to lower the temperature, while leaving the
total quantity of heat unchanged.
FIG. 188. Art. 426. Throttling
of Superheated Steam.
inn
FIG. 189. Arts. 426, 445, 453.
Converted Indicator Dia
gram and Rankine Cycle.
427. Regulation by Throttling. On some of the cheaper types of steam
engine, the speed is controlled by varying the extent of opening of the admis
sion pipe, thus producing a wiredrawing effect throughout the stroke. It is
obvious that such a method of regulation cannot be other than wasteful; a better
method is, as in good practice, to vary the point of cutoff, &, Fig. 187. (See
Art. 507.)
428. Expansion Curve. The widest divergence between the theo
retical and actual diagrams appears along the expansion lines 6c, BC,
Fig. 1ST. In neither shape nor position do the two lines coincide.
Early progress in the development of the steam engine resulted in the
separation of the three elements, boiler, cylinder, and condenser. In
spite of 'this separation, the cylinder remains, to a certain extent, a
condenser as well as a boiler, alternately condensing and evaporating
large proportions of the steam supplied, and producing erratic effects
not only along the expansion line, but at other portions of the diagram
as well.
429. Importance of Cylinder Condensation. The theoretical analysis of the Ran
kine cycle (Art. 411) gives efficiencies considerably greater than those actually attained
in practice. The principal reason for this was pointed out by Clark's experiments on
locomotives in 1855 (1); and still more comprehensively by Isherwood, in his
classic series of engine trials made on a vessel of the United States Navy (2). The
further studies of Loring and Emery and of Ledoux (3), and, most of all, those
conducted under the direction of Him (4), served to point out the vital importance
of the question of heat transfers within the cylinder. Recent accurate measure
ments of the fluctuations in temperature of the cylinder walls by Hall, Callendar
and Nicholson (5) and at the Massachusetts Institute of Technology (6) have
furnished quantitative data.
CYLINDER CONDENSATION 303
430. Initial Condensation. When hot steam enters the cylinder at
or near the beginning of the stroke, it meets the relatively cold surface
of the piston and cylinder head, and partial liquefaction immediately
occurs. By the time the point of cutoff is reached the steam may
contain from 25 to 70 per cent of water. The actual weight of steam
supplied by the boiler is, therefore, not determined by the volume at
b, Fig. 1ST; it is practically from 33 to 233 per cent greater than the
amount thus determined. If ABCDE, Fig. 1ST, represents the ideal
cycle, then b will be found at a point where V b =from 0.30 V B to 0.75 V B
(Art. 436).
Behavior during Expansion. The admission valve closes at
6, and the steam is permitted to expand. Condensation may continue
for a time, the chilling wall surface increasing ; but as expansion pro
ceeds the pressure of the steam falls until its temperature becomes less
than that of the cylinder walls, when an opposite transfer of heat begins.
The walls now give up heat to the steam, drying it, i.e., evaporating a
portion of the commingled water. The behavior is complicated, how
ever, by the liquefaction which necessarily accompanies expansion,
even if adiabatic (Art. 372). The reevaporation of the water during
expansion is effected by a withdrawal of heat from the walls; these
are consequently cooled, resulting in the resumption of proper conditions
for a repetition of the whole destructive process during the next succeed
ing stroke. Reevaporation is an absorption of heat by the fluid. For
maximum efficiency, all heat should be absorbed at maximum tempera
ture, as in the Camot cycle. The later in the stroke that reevaporation
occurs, the lower is the temperature of reabsorption of this heat, and
the greater is the loss of efficiency.
Data on Condensation. Even if the cylinder walls were per
fectly insulated from the atmosphere, these internal transfers would
take place. The Callendar and Nicholson experiments showed that the
temperature of the ianer surface of the cylinder walls followed the
fluctuations of steam temperature, but that the former changes were
much less extreme and lagged behind in point of time. Clayton has
demonstrated (7) that the expansion curve may be represented (in
noncondensing ttnjacketed cylinders) by the equation
* constant, n*0.&c 0.465,
where x is the proportion, of dryness at cutoff: the value of n being
independent of the initial pressure or ratio of expansion. The initial
304 APPLIED THERMODYNAMICS
wetness is thus the important factor in determining the rate of reevapora
tion during expansion. With steam very dry at cutoff (due to jacket
ing or superheat) heat may be lost throughout expansion. In ordinary
cases, the condensation which may occur after cutoff, during the early
part of expansion, can continue for a very brief period only: the prob
ability is that in most instances such apparent condensation has been
in reality nothing but leakage (Art. 452), and that condensation prac
tically ends at cutoff.
432. Continuity of Action. When unity of weight of steam condenses, it gives
up the latent heat L] when afterward reevaporated, it reabsorbs the latent heat
Li; meanwhile, it has cooled, losing the heat h hi. The net result is an increase of
heat in the walls of LLi+hh^HHi, and the walls would continually become
hotter, were it not for the fact that heat is being lost by radiation to the external
atmosphere and that more water is reevaporated than was initially condensed; so
much more, in fact, that the dryness at the end of expansion zs usually greater than
it would have been, had expansion been adiabatic, from the same condition of initial
The outer portion of the cylinder walls remains at practically uniform tem
perature, steadily and irreversibly losing heat to the atmosphere. The inner portion
has been experimentally shown to fluctuate in temperature in accordance with the
changes of temperature of the steam in contact with it. The depth of this " peri
odic " portion is small, and decreases as the time of contact during the cycle decreases,
e.g., in high speed engines*
433. Influences Affecting Condensation. Four main factors are
related to the phenomena of cylinder condensation: they are (a) the
temperature range y (6) the size of the engine, (c) its speed and (most
important), (d) the ratio of volumes during expansion. Of extreme
importance, as affecting condensation during expansion, is the condi
tion of the steam at the beginning of expansion.
The greater the range of pressures (and temperatures) in the engine, the more
marked are the alternations in temperature of the walls, and the greater is the dif
ference in temperature between steam and walls at the moment when steam is
admitted to the cylinder. A wide range of working temperatures, although practi
cally as well as theoretically desirable, has thus the disadvantage of lending itself
to excessive losses.
434. Speed. At infinite speed, there would be no time for the transfer of heat,
however great the difference of temperature. Willans has shown the percentage
of water present at cutoff to decrease from 20.2 to 5.0 as the speed increased from
122 to 401 r. p. m., the steam consumption per Ihphr. concurrently decreasing
from 27.0 to 24.2 Ib. (8). In another test by Willans, the speed ranged from 131
to 405 r. p. m., the moisture at cutoff from 29.7 to 11.7, and the steam consumption
from 23.7 to 20 3; and in stifl another, the three sets of figures were 116 to 401,
20.9 to 8.9, and 20.0 to 17.3. In all cases, for the type of engine under <5onsideca
EXPANSION AND CYLINDER CONDENSATION 305
tion, increase of speed decreased the proportion of moisture and increased the
economy: but it should not be inferred from this that high speeds arc necessarily
or generally associated with highest efficiency.
435. Size. The volume of a cylinder is sD*L+4 and its exposed wall surface
is (3cZ)L)h(xD 2 ^2), if D denotes the diameter and L the exposed length. Tie
volume increases more rapidly than the wall surface, as the diameter is increased
for a constant length. Since the lengths of cylinders never exceed a certain hn.it,
it may be said, generally, that small engines show greater amounts of condensation,
and lower efficiencies, than large engines.
436. Ratio of Expansion. This may be defined as Fd*7, Fig. 187 (Art. 450).
The greater the ratio of expansion, the greater is the initial condensation. This
would be true even if expansion were adiabatic; with early cutoff, moreover, the
time during which the metal is exposed to high temperature steam is reduced, and
its mean temperature is consequently less. Its activity as an agent for cooling
the steam during expansion is thus increased. Again, the volume of steam during
admission is more reduced by early cutoff than is the exposed cooling; surface, since
the latter includes the two constant quantities, the surfaces of the piston and of the
cy Under head (clearance ignored Art. 450). The following Bhows the results of
several experiments:
OBSERVERS
BATIO or
PEE CENT. OF WATEB
ITEA.W CuNMrMrrmv.
EXPANSION
AT Cl TOFF
Poi"KI*s PEC laVIIB
L&ie
High
Low
High
Zo/r ! ///i/1
Loring and Emery
Willans (9)
4.2
4.0
16.8
8.0
8.9
25.0
21.2 ! 5.1
20.7 ' 2JU
Barrus (10) gives the following as average results from a large number of
of Corliss engines at normal speed :
CUTOFT, PBB CEMTT.
OF STBOKB
PBBOEWTAGE or
GOXWBXSATCOX
CtTT*>rr T Fm* CENT.
OP STUCK*
PlnCETTAl/E >F (
COMPSSPITIOH '
2.5
62
25.0
24
5.0
54
SO.O
20
10.0
44
40.0
16 :
15.0
36
45.0
15
20.0
28
j
In these three sets of experiments, it was found that the propor
tion of water steadily decreased as the ratio of expansion decreased.
The steam consumption, however, decreased to a certain mfriininm
figure, and then increased (a feature not shown by the tabulation)
see Fig. 189a. The beneficial effect of a decrease in condensation
306
APPLIED THERMODYNAMICS
was here, as in general practice, offset at a certain stage by the thenno
dynamic loss due to relatively incomplete expansion, discussed in
Art. 418. The proper balancing of
these two factors, to secure best
efficiency, is the problem of the
engine designer. It must be solved
by recourse to theory, experiment,
and the study of standard practice.
In American stationary engines, the
ratio of expansion in simple cylinders
is usually from 4 to 5.
RATIO OF EXPANSION
FIG. I89a. Art 436. Effect of Ratio
of Expansion on Initial Conden
sation and Efficiency.
437. Quantitative Effect. Empirical formulas for cylinder condensation have
been presented by Marks and Heck, among others. Marks (11) gives a curve
of condensation, showing the proportion of steam condensed for various ratios of
expansion, all other factors being eliminated. A more satisfactory relation is
established by Heck (12), whose formula for nonjacketed engines is
0.27
in which M is the proportion of steam condensed at cutoff, N is the speed of the
engine (r. p. m.)> is the quotient of the exposed surface of the cylinder in square
feet by its volume in cubic feet
12 /2Z)

+4 ) where D and L are in inches, p is the
D\L
TABLE: VALUES FOR T
Pa
Const.
Po
Const.
PO
Const.
Po
Const.
170
45
262
115
348
185
409
1
175
50
269}
120
353
190
413
2
179
55
277
125
358
195
416}
3
183
60
284
130
362}
200
420
4
186
65
291
135
367
210
427
6
191
70
297}
140
371}
220
434
8
196
75
304
145
376
230
441
10
200
80
310
150
380}
240
447}
15
210
85
316
155
385
250
454
20
220
90
321}
160
389
260
460}
25
229
95
327
165
393
270
467
30
238
100
332}
170
397
280
473
35
246
105
338
175
401
290
479
40
254
110
343
180
405
300
485
(T in the formula is equal to the difference in constants corresponding with the
highest and lowest absolute pressures in the cylinder.)
STEAM JACKETS 307
absolute pressure per square inch at cutoff, e. is the reciprocal of the ratio of expan
sion, and T is a function of the pressure range in the cylinder, which may be obtained
from the table on p. 306. Heck estimates that the steam consumption of an
engine may be computed from its indicator diagram (Art. 500) within 10 per cent
by the application of this formula. If the steam as delivered from the boiler is
wet, some modification is necessary.
438. Reduction of Condensation. Aside from careful attention to
the factors already mentioned, the principal methods of minimizing
cylinder condensation are by (a; the use of steamjackets, (b) super
heating the steam, and (r) the employment of multiple expansion.
439. The Steam Jacket. Transfers of heat between steam and
cylinder walls would be eliminated if the walls could be kept at the
momentary temperature of the steam. Initial condensation is elimi
nated if the walls are kept at the temperature of steam during admis
sion : it is mitigated if the walls are kept from being cooled by the
lowpressure steam during the latter part of expansion and exhaust.
The steam jacket, invented by Watt, is a hollow casing enclosing the
cylinder walls, within which steam is kept at high pressure. Jackets
have often been mechanically imperfect, and particular difficulty has
been experienced in keeping them drained of the condensed water.
In a few cases, the steam has passed through the jacket on its way to
the cylinder; a bad arrangement, as the cylinder steam was thus made
wet. It is usual practice, with simple engines, and at the highpressure
cylindeis of compounds, to admit steam to the jacket at full boiler
pressure; and in some cases the pressure and temperature in the jacket
have exceeded those in the cylinder. Hotair jackets have been used, in
which flue gas from the boiler, or highly heated air, was passed about
the body of the cylinder.
440. Arguments for and against Jackets. The exposed heated
surface of the cylinder is increased and its mean temperature is raised;
the amount of heat lost to the atmosphere is thus increased. The jacket
is at one serious disadvantage : its heat must be transmitted through the
entire thickness of the walls; while the internal teat transfers are
effected by direct contact between the steam and the inner " skin "
of the walls.
Unjacketed cylinder walls act like heat sponges. The function of
the jacket is preventive, rather than remedial, opposing the formation
of moisture early in the stroke, liquefaction being transferred from the
cylinder to the jacket, where its influence is less harmful. The walls
are kept hot at all times, instead of being periodically heated and cooled
308
APPLIED THERMODYNAMICS
by the action of the cylinder steam. The steam in the jacket does
not expand; its temperature is at all times the maximum temperature
attained in the cycle. The mean temperature of the walls is thus
raised.
i
441. Results of Jacketing. In the ideal case, the action of the jacket may be
regarded as shown by the difference of the areas dekl and debf, Fig. 183 The total
heat supplied, without the jacket, is Ideb2, but cylinder condensation makes the
steam wet at cutoff, giving the work area dekl only. The additional heat 2&/3,
supplied by the jacket, gives the additional work area kbfl, manifestly at high
efficiency. In this country, jackets have been generally employed on wellknown
engines of high efficiency, particularly on slow speed pumping engines; but their
use is not common with standard designs. Slow speed and extreme expansion,
which suggest jackets, lead to excessive bulk and first cost of the engine. With
normal speeds and expansive ratios, the engine is cheaper and the necessity for
the jacket is less. The use of the jacket is to be determined from considerations
of capital charge, cost of fuel and load factor, as well as of thermodynamic efficiency.
These commercial factors account for the far more general use of the jacket in Europe
than in the United States.
From 7 to 12 per cent of the whole amount of steam supplied to the engine
may be condensed in the jacket. The power of the engine is almost invariably
increased by a greater percentage than that of increase
of steam consumption. The cylinder saves more than
the jacket spends, although in some cases the amount
of steam saved has been small. The range of net
saving may be from 2 or 3 up to 15 per cent. The
increased power of the engine is represented by the
^ , ^j ^j^ i i t difference between the areas abodes and aXYdes,
4 Vis 7 6 J M aJ w il f Fig. 187. The latter area approaches much more
3 P^T^mrr.: dosely the ideaj fflrea ABCDEf Jacketing pays best
when the conditions are such as to naturally induce
excessive initial condensation. The diagram of Fig.
190, after Donkin (14), shows the variation in value
of a steam jacket at varying ratios of expansion in the same engine run at constant
speed and initial pressure. With the jacket, the best ratio of expansion was about
10, giving 25 Ib. of steam per hp.hr: without the jacket, the lowest steam consump
tion (of 39 Ib. per hp.hr) was reached at an expansion ratio of 4.
442. Use of Superheated Steam. The thermodynamic advantage of
superheating, though small, is not to be ignored, some heat being taken
in at a temperature higher than the mean temperature of heat absorp
tion; the practical advantages are more important. Adequate super
heat fills the " heat sponge " formed by the walls, without letting the
steam become wet in consequence. If superheating is slight, the steam,
during admission, may be brought down to the saturated condition,
and may even become wet at cutoff, following such a path as debxbkl,
Fig. 183. With a greater amount of superheat, the steam may remain
Vis
POINT OF CUTOFF
FIG. 190. Art. 441. Effect
of Jackets at Various Ex
pansion Ratios.
SUPERHEAT
309
dry or even superheated at cutoff, giving the paths debzijf, deblzA.
The minimum amount of superheat ordinarily necessary to give dry
ness at cutoff seems to be about 100 F.; it may he much greater.
Ripper finds (15) that about 7.5 F. of superheat are necessary for each
1 per cent of wetness at cutoff to be expected when working with
saturated steam. We thus obtain Fig. 191, in which the increased work
areas acbd, cefb, eghf are obtained by superheating along jk, kl f Im,
each path representing 7o of superheat. Taking the pressure along aj
as 120 lb. ; and that along hb as 1 lb., the absolute temperatures are 800 S)~
.UK! 561.43, respectively, and since the latent heat at 120 lb. is 87T.1 1
U. t. u., the work gained by each of the areas in
question is aceg
ms
2iS
000
Slik
dbJK
800.9 ' *
If we take the specific heat of superheated
steam, roughly, at 0.48, the heat used in secur
ing this additional work area is 0.48 x 75 = 36
P> t. u. The efficiency of superheating is then
1^.1536 = 0.73, while that of the nonsuper FIG. 101. Art. 442. Snper
heated cycle as a whole, even if operated at Car heat for overcoming Initial
nnt efficiency, cannot exceed 239. 47 =800.9= 0.30. Condensation.
Great care should be taken to avoid loss of heat in pipes between the super
heater and the cylinder; without thorough insulation the fall of tem
perature here may be so great as to
considerably increase the amount of
superheating necessary to secure the
desired result in the cylinder.
443. Experimental Results with Super
heat The AJaace teats of 1892 showed, with
from 60 to 80 of superheat, mi average net
saving of 12 per cent, baaed on fuel, even when
the coal consumed in the separately fired
superheaters was considered; and when the
superheaters were fired by waste heat from
the boilers, the average saving was 20 per
cent. WiUflns found a considerable saving
by superheat, even when cutn>ff was at half
stroke, a ratio of expansion certainly not unduly favorable to superheating. As
with jackets, the advantage of superheat is greatest in engines of low speeds and
high expansive ratios. Striking results have been obtained by the use of high
superheats, ranging from 200 to 300 F. above the temperature of saturation.
The mechanical design, of the engine must then be considerably modified. Vaughan
INDICATED HORSE POWER
FIG. 193. Art. 443, Prob. 7. Steam
Economy in Relation to Superheat.
310 APPLIED THERMODYNAMICS
(16) has reported remarkably large savings due to superheating in locomotive
practice. Figure 193 shows the decreased steam consumption due to various
degrees of superheat in a small high speed engine.
444, Actual Expansion Curve. In Fig. 187, bY represents the
curve of constant dryness, bC the adiabatic. The actual expansion
curve in an un jacketed cylinder using saturated steam will then be
some such line as be, the entropy increasing in the ratio xz+xy and
the fraction of dryness in the ratio xz+xw. Expressed exponentially,
the value of n for such expansion curve depends on the initial dryness
(Art. 4316); it is usually between 0.8 and 1.2, and averages about
1,00, when the equation of the curve is PV=pv. This should not
be confused with the perfect gas isothermal; that the equation has
the same form is accidental. The curve PV =pv is an equilateral
hyperbola, commonly called the hyperbolic line.
The actual expansion path be will then appear on the entropy dia
gram, Fig, 189, as be, bc f , usually more like the former. The point b
(cutoff) specifies a lower pressure and temperature than does B in the
ideal diagram, and lies to the left of B on account of initial condensa
tion. If expansion is then along bc 3 the walls are giving up, to the
steam, heat represented by the area mbcn. This is much less than
the area mbBAf, which represents roughly the loss of heat to the walls
by initial condensation.
445. Work done during Expansion: Engine Capacity. From Art.
y
95, this is, for a hyperbolic curve, BC, Fig. 187, P B V B log, ^
Assume no clearance, and admission and exhaust to occur without
change of pressure; the cycle is then precisely that represented by
ABODE, excepting that the expansive path is hyperbolic. Then the
work done during admission is P B V B ] the negative work during exhaust
is Pj)V c ; and the net work of the cycle is
The mean effective pressure or average ordinate of the work area ia
obtained by dividing this by V c , giving
p a
MEAN EFFECTIVE PRESSURE 311
Y
or, letting = =r, it is
Pg(l + log.r)
Letting m stand for this mean effective pressure, in pounds per square
inch, A for the piston area in square inches, L for the length of the stroke
in feet, and N for the revolutions per minute, the total average pressure
on the piston (ignoring the rod) is mA pounds, the distance through
which it is exerted per minute is in a doubleacting engine 2 LN feet,
and the work per minute is 2 mALN footpounds, or 2 mALX 4 33,000
horse power. This is for an ideal diagram, which is always larger than
the actual diagram abcdes; the ratio of the latter to the former gives the
diagram factor, by which the computed value of m must be multiplied
to give actual results.
Diagram factors for various types of engine, as given by Seaton, are as follows:
Expansion engine, with special valve gear, or with a separate cutoff valve,
cylinder jacketed . . . 0.90;
Expansion engine having large ports and good ordinary valves, cylinders jacketed
. . , 0.86 to 0.88;
Expansion engines with ordinary valves and gear as in general practice, and
unjacketed . . . 0.77 to 0.81;
Compound engines, with expansion valve on high pressure cylinder, cylinders
jacketed, with large ports, etc. . . . 0.86 to 0.88; (see Art. 466),
Compound engines with ordinary slide valves, cylinders jacketed, good ports,
etc. . . . 0.77 to 0.81;
Compound engines with early cutoff in both cylinders, without jackets or
separate expansion valves . . . 0.67 to 0.77;
Fastrunning engines of the type and design usually fitted in warships . . . 0.57
to 0,77.
The extreme range of values of the diagram factor is probably between 0.50 and
0.90. Regulation by throttling gives values 0.10 to 0.25 lower than regulation
by cutoff control. Jackets raise the value by 0.05 to 0.15. Extremely early
cutoff in simple unjacketed engines (less than 1) or high speed (above 225 r. p. m.)
may decrease it by 0.025 to 0.125. Features of valve and port design may cause
a variation of 0.025 to 0175.
Piston speeds of large engines at around 100 r. p. m. now range from 720 ft.
per minute upward. The power output of an engine of given size is almost directly
proportional to the piston speed. Rotative speeds (r. p. m.) depend largely on the
type of valve gear, and are limited by the strength of the flywheel. Releasing
gear engines do not ordinarily run at over 100 r. p. m. (Art. 507): nor do fourvalve
engines often exceed 240 r. p. m. The smaller engines are apt to have the higher
rotative speeds and the larger ratios of cylinder diameter to stroke. Long strokes
favor small clearances, with many types of valve* Engines of high rotative speed
will generally have short strokes. Speeds of stationary reciprocating engines seldom
exceed 325 r. p. m.
312 APPLIED THERMODYNAMICS
446, Capacity from Clayton's Formula. If the expansion curve can be repre
sented by the equation pv n = const., in which n^l, the mean effective pressure
(clearance ignored) is, with the notation of Art. 445,
nPs ~ Ps

lilt f Z~T X if f TT .
r(n 1) r n (n I)
The best present basis for design is to find n as suggested in Arts. 4316, 437,
to assume a moderate amount of hyperbolic compression (see Art 451) and to
allow for clearance. This is in fact the only suitable method for use where there
is high superheat: in which case n> LO.
Thus, let the pressure limits be 120 and 16 Ib. absolute, the apparent ratio of
expansion 4, clearance 4 per cent, compression to 32 Ib. absolute, n = 1.15. The
approximate equation above gives
1.15X120 120 _ ,
m 060 16 ~4ii* X 0.15
More exactly, calling the clearance volume 0.04, the length of the diagram is 1.0,
the volume at cutoff is 0.29, and the maximum volume attained is 1.04. The
mean effective pressure is
f 6Xl ^16 (1.040.08)
 (30 X0.04 log 2) = 54.5 Ib. per square inch,
(0 29\ 1  15
r^j J  27.6 Ib. and the volume at
the beginning of compression being 0.04X11=0.08.
Any diagram factor employed with this method will vary only slightly from 1.0,
depending principally upon the type of valve and gear. Unfortunately, we do not
as yet possess an adequate amount of information as to values of n in condensing
and jacketed engines
447. Capacity K$. Economy. If we ignore the influence of con
densation, the Clausius cycle (Art. 409), objectionable as it is with
regard to capacity (Art. 418), would be the cycle of maximum effi
ciency ; practically, when we contemplate the excessive condensation
that would accompany anything like complete expansion, the cycle of
Rankine is superior. This statement does not apply to the steam tur
bine (Chapter XIV). The steam engine may be given an enormous
range of capacity by varying the ratio of expansion ; but when this
falls above or below the proper limits, economy is seriously sacrificed.
In purchasing engines, the ratio of expansion at normal load should
be set fairly high, else the overload capacity will be reduced. In
marine service, economy of fuel is of especial importance, in order to
save storage space. Here expansive ratios may therefore range
CLEARANCE AXD COMPRESSION 313
higher than is common in stationary practice, where economy in first
cost is a relatively more important factor.
448. The Exhaust Line : Back Pressure. Considering now the line de of Fig.
187, \\e find a noticeable U)hS of \\ork area as compared with that in the ideal
catse. (Line J)E represents the pressure existing outside the cylinder.) This is
due to several causes. The f notional resistance of the ports and exhaust pipes
(greatly increased by the prepuce of water) produces a wiredrawing effect, mak
ing the pressure in the cylinder higher than that of the atmosphere or of the con
denser. The presence of air in the exhaust passages of a condensing engine may
elevate the pressure above that corresponding to the temperature of the steam,
and fto cause undesirable resistance to the backward movement of the piston.
This air may be present as the re>ult of leakage, under poor operating conditions;
more or less air is always bi ought in the cycle with the boiler feed and condenser
water. The effect of these causes is to increase the pressure during release, even
in good engines, from 1 to 3.0 Ib. above that ideally obtainable.
Hee'vaporation may be incomplete at the end of expansion; it then proceeds
during exhaust, sometimes, in flagrant cases, being still incomplete at the end of
exhaust. The moisture then present greatly increases initial condensation. The
evaporation of any water during the exhaust stroke seriously cools the cylinder
walls. In general good practice the steam is about dry during exhaust; or at least
during the latter portion of the exhaust.
449. Effect of Altitude. The possible capacity of a noncondensing engine is
obviously increased at low barometric pressures, on account of the lowering of the
line DE, Fig. 187. "With condensing engines, the absolute pressure attained along
DE depends upon the proportion of cooling water supplied and the effectiveness
of the condensing apparatus. It is practically independent of the barometric pres
sure, excepting at very high vacua; consequently, the capacity of the engine is
unchanged by variations in the latter. A slightly decreased amount of power,
however, will suffice to drive the air pump which delivers the products of conden
sation against any lessened atmospheric pressure.
450. Clearance. The line e*a does not at any point come in contact with the
ideal line EA, Fig. 187. In all engines, there ia necessarily a small space left
tatween the piston and the inside of the cylinder heat! at the end of the stroke.
This space, with the port spaces back to the contact surfaces of the inlet valves, is
filled with steam throughout the cycle. The distance t* in the diagram represents the
volume of these " clearance " spaces. In Fi;. 195, the apparent ratio of ex
pansion is ^ . If the zero volume line OP be found, the real ratio of expansion,
ab
FD
clearance volume included, IB , The proportion of clearance (always ex
Ab
Aa
pressed in terms of the piston displacement) is . The clearance in actual engines
314 APPLIED THERMODYNAMICS
varies from 2 to 10 percent of the piston displacement, the necessary amount
depending largely on the type of valve gear. In such an engine as that of
Fig. 186, it is necessarily large, on account
of the long ports. In these flat slide valve
engines it averages 5 to 10 per cent* with
rotary (Corliss) valves, 3 to 8 per cent; with
single piston valves, 8 to 15 per cent. These
figures are for valves placed on the side (bar
rel) of the cylinder. When valves are placed
on heads, the clearance may be reduced 2 to
6 per cent. In the unidirectional flow
(Stumpf) engine (Art. 507), it is only about
2 per cent. It is proportionately greater in
v small engines than in those of large size.
FIG. 195. Arts. 450, 451. Real and Ap Tt ma y be accurately estimated by placing
parent Expansion. the piston at the end of the stroke and fill
ing the clearance spaces with a weighed or
measured amount of water. All waste spaces, back to the contact surfaces of the
valves, count as clearance.
451. Compression. A large amount of steam is employed to fill the clearance
space at the beginning of each stroke. This can be avoided by closing the exhaust
valve prior to the end of the stroke, as at e, Fig. 187, the piston then compressing
the clearance steam along es, so that the pressure is raised nearly or quite to that
of the entering steam. This compression serves to prevent any sudden reversal of
thrust at the end of the exhaust stroke. If compression is so complete as to raise
the pressure of the clearance steam to that carried in the supply pipe, no loss of steam
will be experienced in filling clearance spaces. The work expended in compression,
eahg t Fig. 195, will be largely recovered during the next forward stroke by the expan
sion of the clearance steam: the clearance will thus have had httle effect on the
efficiency; the loss of capacity efa will be just balanced by the saving of steam,
for the amount of steam necessary to fill the clearance space would have expanded
along ae, if no other steam had been present.
Complete compression would, however, raise the temperature of the com
pressed steam so much above that of the cylinder walls that serious condensation
would occur. This might be counteracted by jacketing, but in practice it is cus
tomary to terminate compression at some pressure lower than that of the entei ing
steam. A certain amount of unresisted expansion then takes place during the
entrance of the steam, giving a wiredrawn admission line. If the pressure at s,
Fig. 187, is fixed, it is, of course, easy to determine the point e at which the
exhaust valve must close. Considered as a method of warming the cylinder walls
so as to prevent initial condensation, compression is " theoretically less desirable
than jacketing, for in the former case the heat of the steam, once transformed to
work, with accompanying heavy losses, is again transformed into heat, while in
the latter, heat is directly applied." For mechanical reasons, some compression is
usually considered necessary. It makes the engine smoothrunning and probably
iecreases condensation if properly limited. Compression must not be regarded as
bringing about any nearer approach to the Carnot cycle. It is applied to a very
3mall portion only of the working substance, the major portion of which is
jxternally heated during its passage through the steam plant.
VALVE ACTION: LEAKAGE 315
452. Valve Action: Leakage. We have now considered most of the differences
between the actual and ideal diagrams of Fig. 187. The rounding of the corners
at b, and along cdu, is due to sluggish valve action; valves must be opened slightly
before the full effect of their opening as realized, and they cannot close instantaneously.
The round corner at e is due to the slow closing of the exhaust valve. The inclined
line sa shows the admission of steam, the shaded work area being lost by the slow
movement of the valve. In most cases, admission is made to occur slightly prior
to the end of the stroke, in order to avoid this very effect. If admission is too early,
a n3gative lost work loop, mno, may be formed. Important aberrations in the
diagram, and modifications of the phenomena of cylinder condensation; may result
from leakage past valves or pistons. In an engine like that of Fig. 186, steam
may escape directly from the steam chest to the exhaust port. Valves are more
apt to leak than pistons, A valve may be tight when stationary, but leak when
moving; it may be tight when cold and leak when hot. Unbalanced slide valves,
poppet and Corliss valves tend to wear tight; piston valves and balanced slide
valves become leaky with wear. Leakage is increased when the steam is wet.
Jacketing the cylinder decreases leakage. The steam valve may allow steam to
enter the cylinder after the point of cutoff has been passed. Fortunately, as the
difference in pressure between steam chest and cylinder increases, the overlap of
the valve also increases. Leakage past the exhaust valve is particularly apt to
occur just after admission, because then (unless there is considerable compression)
the exhaust valve has only just closed.
The indicator diagram cannot be depended on to detect leakage, excepting as
the curves are transferred to logarithmic coordinates (7). Such steam valve leakage
as has just been described produces the same apparent effect as reevaporation
occurring shortly after cutoff. Leakage from the cylinder to the exhaust, occurring
during this period, produces the effect which was formerly regarded as due to cylinder
condensation immediately following cutoff. In engines known to have tight
exhaust valves, this latter effect is not found. *
An engine may be blocked and examined for leakage (Trans. A.S.M. E., XXIV,
719) but it is difficult to ascertain the actual amount under running conditions.
In one test of a small engine, leakage was found to be 300 Ib. per hour. Tests have
shown that with sin pie flat slide or piston valves the steam consumption ir creases
about 15 per cent in from 1 to 5 years, on account of leakage alone, A large
number of tests made on all types of engine gave steam consumptions averaging
5 per cent higher where leakage was apparent than where valves and pistons were
known to be tight.
THE STEAM ENGUTE CYCLE ON THE ENTROPY DIAGRAM
453. Cylinder Feed and Cushion Steam. Fig. 189 has been left incomplete, for
reasons which are now to be considered. It is convenient to regard the working
fluid in the cylinder as made up of two masses, the " cushion steam/' which
alone nils the compression space at the end of each stroke, and is constantly present,
and the " cylinder feed," which enters at the beginning of each stroke, and leaves
before the completion of the next succeeding stroke. In testing steam engines by
weighing the discharged and condensed steam, the cylinder feed is alone measured ;
it alone is chargeable as heat consumption ; but for an accurate conception of the
cyclical relations in the cylinder, the influence of the cushion steam must be con
sidered. '
316
APPLIED THERMODYNAMICS
In Fig. 196, let abcde be the PV diagram of the mixtme of cushion steam and
cylinder feed, and let gh he the expansion line of the cushion steam if it alone were
present. The total volume rq at any point q of the combined paths is made up
of the cushion bte<im volume co and the
cylinder feed volume, obviously equal to
og. If we wish to obtain a diagram
shoeing the behavior of the cylinder
feed alone, we must then deduct from
the volumes around alc<U the correspond
ing volumes of cushion steam. The point
p is then derived by making rp = vq vo,
and the point t by making rt = ru rs.
Proceeding thus, we obtain the diagram
nzjklm, representing the behavior of the
cylinder feed. Along nz the diagram
coincides with the OP axis, indicating
that at this stage the cylinder contains
cushion steam only.
o
FIG. 196.
Arts. 453, 457 Elimination of
Cushion Steam.
454, The Indicator Diagram. Our study of the ideal cycles in Chapter XII has
dealt with representations on a single diagram of changes occurring in a given mass
of steam at the boiler, cylinder, and condenser, the locality of changes of condition
being ignored. The energy diagram abcdes of Fig. 187 does not represent the
behavior of a definite quantity of steam working in a closed cycle. The pressure
and volume changes of a varying quantity of fluid are depicted. During expansion,
along he, the quantity remains constant; during compression along es, the quantity
is likewise constant, but diiferent. Along sab the quantity increases ; while along
cde it decreases. The quality or dryness of the steam along es or be may loe readily
determined by comparing the actual volume with the volume of the same weight
of dry steam ; but no accurate information as to quality can be obtained along the
admission and release lines sab and cde. The areas under these lines represent
work quantities, however, and it is desirable that we draw an entropy diagram
which shall represent the corresponding heat expenditures. Such a diagram will
not give the thermal history of any definite
amount of steam ; it is a mere projection of
the PV diagram on diiferent coordinates.
It tacitly assumes the indicator diagram to
represent a reversible cycle, whereas in fact
the operation of the steam engine is neither
cyclic nor reversible.
455. Boulvin's Method. In Fig. 197,
let abode be any actual indicator diagram,
YZ the pressure temperature curve of
saturated steam, and QR the curve of satu
ration, plotted for the total quantity of
FIG. 197 Art 455. Transfer from PV steam in the cylinder during expansion.
to JVT Diagram (Boulvin's Method). The water line OS and the saturation curve
CONVERTED DIAGRAMS 317
MT are now drawn for 1 Ib. of steam, to any convenient scale, on the entropy plane.
To transfer any point, like B, to the entropy diagram, we draw BD, DK, EH, KT,
BA, AT, HT, BG, and GF as in Art. 378. Then F is the required point on the
temperature entropy diagram. By transferring other points m the same way, we
obtain the area NVFU. The expansion line thus traced correctly represents the
actual hLstory of a definite quantity of fluid; other parts of the diagram are imaginary.
It is not safe to make deductions as to the condition of the substance from the NT
diagram, excepting along the expansion curve. For example, the diagram apparently
indicates that the dryness is decreasing along the exhaust line SU; although we have
seen (Art. 448) that at this stage the dryness is usually increasing (17).
456. Application in Practice. In order to thus plot the entropy diagram, it is
necessary to have an average indicator card from the engine, and to know the
quantity of steam in the cylinder. This last is determined by weighing the dis
charged condensed steam during a definite number of strokes and adding the
quantity of clearance steam, assuming this to be just dry at the beginning of com
pression, an assumption fairly well substantiated by experiment.
45705. Reeve's Method. By a procedure similar to that described in Art. 453,
an indicator diagram is derived from that originally given, representing the behav
ior of the cylinder feed alone, on the assumption that the clearance steam works
adiabatically through the point e, Fig. 196. This often gives an entropy diagram
in which the compression path passes to the left of the water line, on account of
the fact that the actual path of the cushion steam is not adiabatic, but is occasion
ally less " steep."
The Reeve diagram accurately depicts the process between the points of cut
off and release and those of compression and admission with reference to the cylinder
feed only.
4575. Preferred Method. The most satisfactory method is to make
no attempt to represent action between the points of admission and
cutoff and of Release and compression. During these two portions
of the cycle we know neither the weight nor the dryness of steam
present at any point. The method of Art. 155 should be used for the
expansion curve alone. For compression, a new curve corresponding
with RQj Fig. 197, should be drawn, representing the pv relation for
the weight of clearance steam alone. Points along the compression
curve may then be transferred to the upper righthand quadrant by
the same process as that described in Art. 455. The TN diagram
then shows the expansion and compression curves, both correctly
located with reference to the water line OS and the dry steam curve
TM } for the respective weights of steam; and the heat transfers and
dryness changes during the operations of expansion and compression
are perfectly illustrated.
458. Specimen Diagrams. Figure 199 shows the gain by high initial pressure
and reduced back pressure. The augmented work areas befc, cfho, are gained at high
efficiency; adji and adlh cost nothing. The operation of an engine at back pressure,
318
APPLIED THERMODYNAMICS
to permit of using the exhaust steam for heating purposes, results in such losses as
adji, adlk. Similar gains and losses may be shown for nonexpansive cycles. Figure
200 shows four interesting diagrams plotted from actual indicator cards from a small
FIG. 199.
Art. 458. Initial Pressure and
Back Pressure.
FIG. 200. Art. 458. Effects of Jacket
ing and Superheating.
engine operated at constant speed, initial pressure, load, and ratio of expansion
(18). Diagrams A and C were obtained with saturated steam, B and D with super
heated steam. In A and B the cylinder was un jacketed; in C and D it was jacketed.
The beneficial influence of the jackets is clearly shown, but not the expenditure of
heat in the jacket. The steam consumption in the four cases was 45.6, 28.4, 27 25
and 20.9 Ib. per Ihphr., respectively.
MULTIPLE EXPANSION
459. Desirability of Complete Expansion. It is proposed to show that a large
ratio of expansion is from every standpoint desirable, excepting as it is offset by
increased cylinder condensation ; and to suggest multiple expansion as a method
for attaining high efficiency by making such large ratio practically possible.
From Art. 446, it is obvious that the maximum work obtainable from a cylinder is
a function solely of the initial pressure, the back pressure, and the ratio of expan
sion. In a nonconducting cylinder, maximum efficiency would be realized when
the ratio of expansion became a maximum between the pressure limits. Without
expansion, increase of initial pressure very slightly, if
at all, increases the efficiency. Thus, in Fig. 201,
the cyclic work areas abed, aefg, ahij, would all be
equal if the line XY followed the law po == PV.
As the actual law (locus of points representing
steam dry at cutoff) is approximately,
\
d
the wort areas increase slightly as the pressure in
creases; but the necessary heat absorption also
increases, and there is no net gain. The thermody FrQ ^ Art 459 _ Non _
namic advantage of high initial pressure is realized only ' expaiig i ve Cycles.
wksn the ratio of expansion is large.
By condensing the steam as it flows from the engine, its pressure may be re
duced from that of the atmosphere to an absolute pressure possibly 13 Ib. lower.
The cyclic work area is thus increased ; and since the reduction of pressure is ac
MULTIPLE EXPANSION
319
companied by a reduction in temperature, the potential efficiency is increased.
Figure 202 shows, however, that the percentage gain in efficiency is smaU with no
expansion, increasing as the expansion ratio increases. Wide ratios of expansion are
from all of these standpoints essential to efficiency.
We have found, however, that wide ratios of
expansion are associated with such excessive losses
from condensation that a compromise is necessary,
and that in practice the best efficiency is secuied
with a rather limited ratio. The practical attain
ment of large expansive ratios without correspond
ing losses by condensation is possible by multiple
expansion. By allowing the steam to pass suc
cessively through two or more cylinders, a total
expansion of 15 to 33 may be secured, with condensa
tion losses such as are due to much lower ratios.
FIG. 202. Art. 439. Gain due
to Vacuum.
460. Condensation Losses in Compound Cylinders. The range of pres
sures, and consequently of temperatures, in any one cylinder, is reduced
by compounding. It may appear that the sum of the losses in the two
cylinders would be equal to the loss in a single simple cylinder. Three
considerations may serve to show why this is not the case :
(a) Steam ree'vaporated during the exhaust stroke is rendered avail
able for doing work in the succeeding cylinder, whereas in a simple
engine it merely causes a resistance to the piston;
(&) Initial condensation is decreased because of the decreased fluctua
tion in wall temperature;
(c) The range of temperature in each cylinder is half what it is in the
simple cylinder, but the whole wall surface is not doubled.
461. Classification. Engines are called simple, compound, triple, or quadruple,
according to the number of successive expansive stages, ranging from one to four.
A multipleexpansion engine may have any number of cylinders ; a triple expan
sion engine may, for example, have five cylinders, a single highpressure cylinder
discharging its steam to two succeeding cylinders, and these to two more. In a
multipleexpansion engine, the first is called highpressure cylinder and the last
the lowpressure cylinder. The second cylinder in a triple engine is called the
intermediate; in a quadruple engine, the second and third are called the first
intermediate and the second intermediate cylinders, respectively. Compound en
gines having the two cylinders placed end to end are described as tandem ; those
having the cylinders side by side are crosscompound. This last is the type most
commonly used in highgrade stationary plants in this country. The engines may
be either horizontal or vertical ; the latter is the form generally used for triples or
quadruples, and in marine service. Sometimes some of the cylinders are horizon
tal and others vertical, giving what, in the twoexpansion type, has been called the
angle compound. Compounding may be effected (as usually) by using cylinders of
various diameters and equal strokes i or of equal diameters and varying strokes,
320
APPLIED THERMODYNAMICS
or of like dimensions but unequal speeds (the cylinders driving independent
shafts), or by a combination of these methods
462. Incidental Advantages. Aside from the decreased loss through cylinder
condensation, multipleexpansion engines have the following points of superiority .
(1) The steam consumed m filling clearance spaces is less, because the high
pressure cylinder is smaller thau the cylinder of the equivalent simple engine;
(2) Compression in the highpressure cylinder may be carried to as high a
pressure as is desnable without beginning it so early as to greatly i educe the woik
area;
(3) The lowpressure cylinder need be built to withstand a fraction only of
the boiler pressure ; the other cylinders, which carry higher pressures, are com
paratively small;
(4) In most common types, the use of two or more cylinders permits of using
a greater number of less powerful impulses on the piston than is possible with a
single cylinder, thus making the rotative speed more unifonn;
(5) For the same reason, the mechanical stresses on the crank pin, shaft, etc.,
are lessened by compounding.
These advantages, with that of superior economy of steam, have led to the
general use of multiple expansion in spite of the higher initial cost which it en
tails, whei ever steam pressures exceed 100 Ib.
463. Woolf Engine. This was a form of compound engine originated by Horn
blower, an unsuccessful competitor of Watt, and revived by Woolf in 1800, after
the expiration of Watt's principal patent.
Steam passed dhectly from the high to the A 
lowpressure cylinder, entering the latter
while being exhausted from the former.
This necessitated having the pistons either
in phase or a half revolution apart, and
there was no improvement over any other
doubleacting engine with regard to uni D
formity of impulse on the piston. Figure
203 represents the ideal indicator diagrams. 1^.303. Arts. 463, 466. Woolf Engine.
A BCD is the action in the highpressure
cylinder, the fall of pressure along CD being due to the increase in volume of
the steam, now passing into the lowpressure cylinder and forcing its piston out
ward. EFGH shows the action in the lowpres
sure cylinder; steam is entering continuously
throughout the stroke along EF. By laying off
MP  LK, etc., we obtain the diagram TABRS,
representing the changes undergone by the steam
during its entire action. This last area is ob
viously equal to the sum of the areas ABCD
and EFGH. Figure 204, from Ewing (19)
shows a pair of actual diagrams from a Woolf
engine, the length of the diagrams representing
FIG. 204 Art. 463, Prob. 31. Dia
grams from Woolf Engine.
RECEIVER ENGINE
321
the stroke of the pistons and not actual steam volumes. The lowpressure dia
gram has been reversed for convenience Some expansion in the lowpressure
cylinder occurs after the closing of the high pressure exhaust valve at a. Some
loss of pressure by wiredrawmg in the passages between the two cylinders is clearly
indicated.
464# . Receiver Engine. In this more modern form the steam passes
from the highpressure cylinder to a closed chamber called the receiver,
and thence to the lowpressure cylinder. The receiver is usually an
independent vessel connected by pipes with the cylinders; in some
cases, the intervening steam pipe alone is of sufficient capacity to
constitute a receiver. Receiver engines may have the pistons coin
cident in phase, as in tandem engines, or opposite, as in opposed beam
engines, or the cranks may be at an angle of 90, as in the ordinary
crosscompound. In all cases the receiver engine has the characteristic
advantage over the Woolf type that the lowpressure cylinder need not
receive steam during the whole of the working stroke, but may have a
definite point of cutoff, and work in an expansive cycle. The dis
tribution of work between the two cylinders, as will be shown, may
be adjusted by varying the point of cutoff on the lowpressure cylinder
(Art. 467).
Receiver volumes vary from to 1 times the highpressure cylinder
volume.
4645. Reheating. A considerable gain in economy is attained by
drying or superheating the steam during its passage through the
WITH REHEATERS
\ AND JACKETS
WITHOUT REHEATERS AND JACKETS
HEHEATEfie
AND JACKETS
.WITHOUT
~ 1EATERS
JACKETS
FIGS. 215 and 216. Art. 4646. Effect of Reheaters and Jackets (25).
receiver, by means of pipe coils supplied with highpressure steam from
the boiler, and drained by a trap. The argument in favor of reheating
is the same as that for the use of superheat in any cylinder (Art. 442).
It is not surprising, therefore, that the use of reheaters is only profit
able when a considerable amount of intermediate drying is effected.
Reheating was formerly unpopular, probably because of the difficulty
322
APPLIED THERMODYNAMICS
of securing a sufficient amount of superheat with the limited amount
of coil surface when saturated steam was used in the receiver coils.
With superheated steam, this difficulty is obviated. Reheating increases
the capacity as well as the economy of the cylinders.
465. Drop. The fall of pressure occurring at the end of expansion
(cdj Fig. 196) is termed drop. Its thermodynamic disadvantage
and practical justification have been pointed out in Arts. 418, 447.
In a compound engine, some special considerations apply. If there is
no drop at highpressure release, the diagram showing the whole expan
sion is substantially the same as that for a simple cylinder. With
drop, the diagram is modified, the ratio of expansion in the highpressure
cylinder is decreased, and the ideal output is less.
The orthodox view is that there should be no drop in the high
pressure cylinder (21). The cylinders of a compound engine work
with less fluctuation of temperature than that of a simple engine, and
may therefore be permitted to use higher ratios of expansion (i.e.,
less drop) than does the latter. In the design method to follow, dimen
sions will be determined as for no drop. Changes of load from normal
may introduce varying amounts of drop in operation.
466. Combination of Actual Diagrams: Diagram Factor. Figure 210 shows the
high and lowpressure diagrams from a pTrm.11 compound*engine. These are again
H.P.
FIG. 210. Art. 466. Compound Engine
Diagrams.
FIG. 211. Art. 466. Compound Engine
Diagrams Combined.
shown in Fig. 211, in which the lengths of the diagrams are proportioned as are
the cylinder volumes, the pressure scales are made equal, and the proper amounts
of setting off for clearance (distances a and 5) are regarded. The cylinder feed
per single stroke was 0.0498 lb., the cushion steam in the highpressure cylinder
0.0074 lb., and that in the other cylinder 0,0022 lb. No single saturation curve
is possible; the Lne *s is drawn for 0.0572 lb. of steam, and SS for 0.0520 lb. As
in Art. 453, we may obtain equivalent diagrams with the cushion steam eliminated.
COMBINED DIAGRAMS
323
In Fig. 212, the single curve SS then represents saturation for 0.0498 Ib. of steam.
The areas of the diagrams are unaltered, and correctly measure the work done;
they may be transferred to the entropy plane as
in Art. 455. The moisture present at any point
during expansion is still represented by the dis
tance cd, corresponding with the distance similarly
marked in Fig. 211. The ratio of the area of the
combined actual diagrams to that of the Ran
kine cycle through the same extreme limits of
pressure and with the same ratio of expansion
is the diagram factor, the value of which may
range up to 95, being higher than in simple
engines (Art. 459).
467. Combined Diagrams. Figure 205 shows Fro. 212. Art. 466. Combined
the ideal diagrams from a tandem receiver engine. Diagrams for Cylinder Feed.
Along CD, as along CD in Fig. 203, expansion
into the lowpressure cylinder is taking place The corresponding line on the low
pressure diagram is EF. At F the supply of steam is cut off from the lowpressure
cylinder, after which hyperbolic expansion occurs along FS. Meanwhile, the
FIG. 205. Arts. 467, 475. Elimination of Drop,
Tandem Receiver Engine.
FIG. 214. Art. 468. Effect of Low
pressure Cutoff.
exhaust from the highpressure cylinder is discharged to the receiver; and since a
constant quantity of steam must now be contained in the decreasing apace between
the piston and the cylinder and receiver walls, some compression occurs, giving
the line DE. The pressure of the receiver steam remains equal to that at E
after the highpressure exhaust valve closes (at E) and while the highpressure
cylinder continues the cycle along EABC. If the pressure at C exceeds that at
E, then there will be some dropr As drawn, the diagram shows none. If cutoff
in the lowpressure cylinder occurred later in the stroke, the line DE would be
lowered, P c would exceed P s , and drop would be shown.
An incidental advantage of the receiver engine is here evident. The intro
duction of cutoff in the lowpressure cylinder raises the lower limit of tempera
324
APPLIED THERMODYNAMICS
FIG. 206. Art. 468. Effect of Changing Low
pressure Cutoff
ture in the highpressure cylinder from D in Fig. 203 to D in Fig. 205. This reduced
range of temperature decreases cylinder condensation
468. Governing Compound Engines. Fig. 214 shows that delayed
cutoff on the highpressure cylinder greatly increases the output of
the lowpressure cylinder while (the receiver pressure being raised)
scarcely affecting its own output.
In Fig. 206, is shown the result of varying lowpressure cutoff in
a tandem receiver engine with drop, the lowpressure clearance being
exaggerated for clearness.
The highpressure diagram
is fabcde, the lowpressure is
ghjkl, p f =p d = p a and p e =p.
Lowpressure cutoff occurs at
h (point e in the highpressure
diagram). If this event occur
earlier, the corresponding
point on the highpressure
diagram is made (say) n, and
compression then raises the
receiver pressure to o instead
of /. The result is that the
drop decreases to cp instead of cd (p p =p ) The admission pressure
of the lowpressure cylinder thus becomes Pxpp^po instead of p ffj
and the gain qmg due to such increased pressure more than offsets the
loss mhj due to the fact that lowpressure cutoff now occurs at p m =
p n . The same results will be found with crosscompound engines.
The total output of the engine is very little affected by changes
in lowpressure cutoff: but (contrary to the result in simple cylinders)
the output of the lowpressure cylinder varies directly as its ratio
of expansion. With delayed cutoff, the lowpressure cylinder performs
a decreased proportion of the total work.
When the load changes in a compound engine which has a fixed
point of lowpressure cutoff , equality of work distribution becomes
impossible. The output of the engine should be varied by varying
the point of highpressure cutoff. Equal distribution of the work
should then be accomplished by variation of lowpressure cutoff.
The two points of cutoff will be changed in the same direction as the
load changes. At other than normal load, there will then be some
drop. The aim in design will be, after fixing upon a suitable receiver
pressure, to select a normal corresponding point of lowpressure cutoff at
which, with the given receiver volume and cylinder ratio, drop will be
eliminated. (Arts. 475478),
DESIGN OP COMPOUND ENGINES 325
DESIGN OF COMPOUND ENGINES
469. Preliminary Diagram. We first consider the action as repre
sented in Fig. 205, which shows the combined ideal diagrams without
clearance or compression, and with
hyperbolic expansion. Losses or
gains between the cylinders are
ignored. The following notation is
adopted :
P= initial absolute pressure, Ib.
per sq. in., along a&; FIG. 205. Arts. 469, 470, 473 Pre
p 0=t receiver absolute pressure, Ib. liminary Compound Engine Diagram.
per sq. in., along dc;
p=back pressure, absolute, Ib. per sq. in,, along gf;
Pmh= mean effective pressure, Ib. per sq. in., highpressure cylinder;
effective pressure, Ib. per sq. in., lowpressure cylinder;
=# A = ratio of expansion, highpressure cylinder;
=Ri = ratio of expansion, lowpressure cylinder;
v c
~=R = whole ratio of expansion;
z> 6 r '
1 =C = ratio of cylinder volumes, or " cylinder ratio."
v c
The following relations are useful:
R=R h Ri=CR h ; C=R t j p^^Tr log c ^; ^O=D~;
) p.
470. Bases for Design. The values of P, p and R being given,
whatever fixes the pressure or volume at c determines the proportions
of the engine. We may assume either *
(a) the receiver pressure, p ;
(b) the cylinder ratio, (7=;
^c
(c) equal division of the temperature ranges; that is,
T b T c = T c T f , or r.
* Some designers of marine engines aim at equalization of maximum pressures on
the cranks. This requires careful consideration of clearance and compression.
326 APPLIED THERMODYNAMICS
and PO is the pressure corresponding with the temperature
T c  or,
(d) equal division of the work; that is, abcd=dcefg, attained when
Any one of these four assumptions may be made, but not more than
one. Having made one, the pressures and volumes at &, c, e and /
are all fixed.
471. Diagrams with Clearance. We now employ Fig. 213, in which
clearance is allowed for. The expansion curve is still assumed to be
a continuous hyperbola, and inter
cylinder losses are ignored. (These
last need not be important.)
If dn is the highpressure clearance
^hdrdc, Fig 213), the apparent ratio of
expansion in the highpressure cylinder is
*'!
FIG. 213. Arts. 471473. Design Similarly, the apparent ratio of expansion
Diagram i Compound Engine. j^ the lowpressure cylinder is
Ri' =
where dj=T ls the lowpressure clearance. Engines are usually designed by
specifying the whole apparent ratio of expansion, (dD\gf)sab. In terms of the
real ratios, this is
The mean effective pressures in the cylinders are now
P P
and Ri=C only when d* = dj.
The mean effective pressures reached in practice will differ from
these by some small amount, the ratio of probable actual to com
puted pressure being described as the diagram factor. Generally speak
DESIGN OF COMPOUND ENGINES 327
ing, the diagram factor to be used for the cylinder of a multiple expan
sion engine of n expansion stages and R ratio of expansion is the same
as that for a simple engine of expansion ratio R s when
472. Size and Horse Power. In general, diagram factors, piston
speeds and strokes are the same for all the cylinders of the engine.
Then following Art. 446,
_2fLN
EP* DO* f\r\ \Pmh" ft \pml A. i) .
OOUUU
where /= diagram factor and A^ and A l are the areas of high and low
pressure cylinders respectively, in sq. in. Letting C denote the cylinder
ratio,
, 2fLNA l
in which ^ describes what is called the "highpressure meaneffective
pressure referred to the lowpressure cylinder."
473. Division of Work: Equivalent Simple Engine. The work
will be divided between the cylinders in the same ratio as the two areas
abed, Dcefg, Fig. 213; or in the ratio,
When the assumption of equal output is made (Art. 469), the mean
effective pressures must be inversely as the cylinder areas.
The power of the compound engine is very nearly the same as that
which would be obtained from a simple cylinder of the same size as
the lowpressure cylinder of the compound, with a ratio of expansion
equal to the whole ratio of expansion of the compound. This would
bo exactly true if the diagram factor were the same for the simple as for
the compound and if the noclearance diagram, Fig. 213, were used for
finding p m . An approximate expression for the area of the low
pressure cylinder of a compound is then
hn = 2/LJVA,(P(l+logefl) ,
( R l
328 APPLIED THERMODYNAMICS
474. Cylinder Ratio. Ratio of Expansion. Noncondensing com
pound engines usually have a cylinder ratio C =3 to 4. With condensing
engines, the ratio is 1 or 5, increasing with the boiler pressure. In
triple engines, the ratios are from 1 : 2.0 : 2.0 up to 1 : 2.5 : 2.5 in sta
tionary practice. With quadruple expansion the ratios are succes
sively from 2.0 to 2.5 : 1.
Tests by Rockwood (22) of a triple engine in which the intermediate
cylinder was cut out r permitting of running the high and lowpressure
cylinders as a compound with the high cylinder ratio of 5.7 to 1, give
the surprising result that with the same initial pressure and expansive
ratio, the compound was more economical than the triple. This was
a small engine, with large drop. The pointing out of the fact that the
conditions were unduly favorable to the compound as compared with
the triple did not explain the excellent economy of the former as com
pared with all engines of its class. Somewhat later, exceptionally
good results were obtained by Barrus (23) with a compound engine
having the extraordinary cylinder ratio of 7.2 : 1.0. Thurston, mean
while, experimented in the same manner as Rockwood, determining,
in addition, the economy of the highpressure and intermediate cylinders
when run together as a compound. There were thus two compounds
of ratios 3.1 : 1 and 7.13 : 1 and a triple of ratio 1 : 3.1 : 2 3, available
for test. The results showed the 7.1 compound to be much better than
the 3.1, but less economical than the triple (24) . As the ratio of expan
sion decreased, the economy of the intermediate compound closely
approached that of the triple; and at a very low ratio it would probably
have equaled it. It is a question whether the high economy of these
" intermediate compounds " has not been due primarily to the high
ratio of expansion which accompanied the high cylinder ratio. The
best performances have been reached by compounds and triples alike
at ratios of expansion not far from 30. Ordinary compound engines
probably have the highpressure cylinders too large for best economy.
This is due to the aim toward overload capacity. As in a simple
engine, the less the total ratio of expansion, the greater is the output:
but in a compound, the lowest ratio of expansion cannot be less than
the cylinder ratio.
Values of R for multiple expansion engines range normally from
12 to 36, usually increasing with the number of expansive stages.
Superheat, adequate reheating or jacketing justify the higher values
The use of Compound (twostage) engines is common practice every
where. For stationary service, since the development of the turbine,
the triple, even, is an almost extinct type. The extra mechanical
losses necessitated by the triple arrangement often offset the slightly
DESIGN OF COMPOUND ENGINES 329
greater efficiency. The gain by the compound over the simple is so
great (where condensing operation is possible) that excepting under
peculiarly adverse conditions of fuel cost or load factor the compound
must be regarded as the standard form of the reciprocating steam
engine using saturated steam.
475, Determination of Lowpressure Cutoff. Tandem Compound. In Fig.
205, let ABCD be a portion of the indicator diagram of the highpressure cylinder
of a tandem receiver engine, release occurring at C. At this point, the whole volume
of steam consists of that m the receiver plus that in the highpressure cylinder.
Let the receiver volume be represented by the distance CX. Then the hyperbolic
curve XY may represent the expansion of the steam between the states C and D,
and by deducting the constant volumes CX, LR, MZ, etc., we obtain the curve
CCr, representing the expansion of the steam in the two cylinders. For no drop,
the pressure at the end of compression into the receiver must be equal to that at C.
We thus find the point E y and draw EF, the admission line of the lowpressure
cylinder, such that ac+ad = ae, etc ; the abscissa of cC being to that of Ed in the
same ratio as the respective cylinder volumes. By plotting ED we find the point
D at its intersection with CD. A horizontal projection from D to EF gives F. The
point F is then the required point of cutoff in the lowpressure cylinder. The
diagram EFSHI maj 7 be completed, the curve FS being hyperbolic.
476. Analytical Method. Let the volume of highpressure cylinder be taken
as unity, that of the receiver as R, that of the lowpressure cylinder as L. Let x
be the fraction of its stroke completed by the lowpressure piston at cutoff, and let
p be the pressure at release from the highpressure cylinder, equal to the receiver
pressure at the moment of admission to the lowpressure cylinder. The volume
of steam at this moment is 1+K; at lowpressure cutoff, it is 1 \RixL x If
expansion follows the law pv = PV t and P be the pressure in the lowpressure cylinder
at cutoff,
x), or P^
The remaining quantity of steam in the highpressure cylinder and receiver has
the volume 1 re+fl, which, at the end of the stroke, will have been reduced to R.
If the pressure at the end of the stroke is to be p, then
~
or
Combining the two values of P, we find
R+l
477. Crosscompound: Cranks at Right Angles. In Fig. 208, let dbC be a
portion of the highpressure diagram, release occurring at C. Communication is
now opened with the receiver. Let the receiver volume be laid off as Cd, and let
de be a hyperbolic curve. Then the curve C/, the volume of which at any pressure
is Cd less than that of de, represents the path in the highpressure cylinder. This
continues until admission to the lowpressure cylinder occurs at g. The whole
volume of steam is now made up oiL that in the two cylinders and the receiver; the
volumes in the cylinders alone are measurable out to /C. In Fig. 209, lay off hi = 1C
and jk so that jk+hi is equal to the ratio of volumes of low and highpressure
330
APPLIED T
cylinder. At the point C of the cycle, the highpressure crank is at i, the lowpres
sure "crank 90 ahead or behind it. When the highpressure crank has moved from
^ to m, the volume of steam in that cylinder is represented by the distance hn, the
lowpressure crank is at o and the volume of steam in the lowpressure cy under is
represented by pk. Lay off qr, in Fig. 208, distant from the zero volume line al
by an amount equal to hn+pk. Draw the horizontal line is. Lay off tu=hn and
tv=n,s^pk. Then u is a point on the highpressure exhaust line and v is a point
on the lowpressure admission line. Similarly, we find corresponding crank posi
tions w and x, and steam volumes hy and zk t and lay off AB = hy+zk, Ac = hy,
AD=cB=zk, determining the points c and Z? t The highpressure exhaust line
FIG. 208, Arts. 477479. Elimination of Drop, Crosscompound Engine.
guc is continued to some distance below I. For no drop, this line must terminate
at some point such that compression of steam in the highpressure cylinder and
receiver will make I the final state. At I the highpressure cylinder steam volume
is zero; all the steam is in the receiver. Let IE represent the receiver volume
and EF a hyperbolic curve. Draw IG so that at any pressure its volumes are equal
to those along EF, minus the constant volume IE. Then T, where IG intersects
guc, is the state of the highpressure cycle at which cutoff occurs in the lowpressurq
cylinder. By drawing a horizontal line through H to intersect vD, we find the point
of cutoff J on the lowpressure diagram. If we regard the initial state as that when
admission occurs to the lowpressure cylinder, then at lowpressure cutoff the
JTC*
highpressure cylinder will have completed the ^ proportion of a full stroke.
to
Modifications of this construction permit of determining the point of cutoff for no
drop in triple or quadruple engines with any phase relation of the cranks.
478. Crosscompound Engine: Analytical Method. In this case, the fraction
of the stroke completed at lowpressure cutoff is different for the two cylinders.
Let X be the proportion of the highpressure stroke occurring between admission and
cutoff in the lowpressure cylinder. Proceeding as before, the volume of the steam
at lowpressure admission is 0.5 +R, and that at lowpressure cutoff is 0.5 X+R
}xL. The volume of steam in the highpressure cylinder and the receiver at the
end of the highpressure exhaust stroke is R; the volume just after lowpressure
cutoff occurs is 0.5 X+R. The volume at the beginning of exhaust from the high
pressure cylinder is l+R. In Fig. 208, let the pressure at C and I be p; let that at
ffbeP. Then
CROSSCOMPOUND ENGINE
331
XI + ) = ^(0.5 + R) 01 P=
Let the pressure at H be Q : then
P(0.5 + R) = Q(0.5  X + tf +
.
0.5 + Jrc
0.5  X + ,tt + sL*
f (0.5  X + j
whence,
y=0.5 + ^:
(A)
In Pig. 209, we have the crank
circles corresponding to the
discussed movements. If Ow
and Ox are at right angles,
then for a highpressure pis
ton displacement Oy, we have
the corresponding lowpres
sure displacement kz. If these
displacements be taken as at
lowpressure cutoff, then
~~ A* "jk
We may also draw OwP^ PQ,
and write X = ~z In the
mm Arts. 47r i 478.Cr^k Circles and Piston ^^ O p^ Q ^ QQ =
xzjk . X, xz 2 + ~Oz'^ 5?, and
(jk  X) z + (* x  jk\ = ( 4J J whence X = Var ar 2 . Substituting this value iu
Equation (A), we find jR (ar 1) = 0.5 Var z 2 as the condition of no drop.
479. Practical Modifications. The combined diagrams obtained from actual
engines conform only approximately to those of Figs. 205 and 208. The receiver
spaces are. usually so large, in proportion to the volume of the highpressure
cylinder, that the fluctuations of pressure along the release lines are scarcely notice
able. The fall of pressure during admission to the lowpressure cylinder is, how
ever, nearly always evident. Marked irregularities arise from the angularity of the
connecting rod and from the clearance spaces. The graphical constructions may
easily be modified to take these into account. In assuming crank positions and
piston displacements to correspond, we have tacitly assumed the rod to be of
infinite length; in practice, it seldom exceeds five or six times the length of the
crank. We have assumed all expansive paths to be hyperbolic; an assumption
not strictly justified for the conditions considered.
482. Superheat and Jackets. Since multiple expansion itsalf
decreases cylinder condensation, these refinements cannot be expected
332
APPLIED THERMODYNAMICS
to lead to such large economies as in simple engines. Adequately
superheated steam has, however, given excellent results, eliminating
cylinder condensation so perfectly as to permit of wide ranges of expan
sion without loss of economy and thus making the efficiency of the
engine, within reasonable limits, almost independent of its load. ^ The
best test records have been obtained from jacketed engines, A simple
engine with highly superheated steam (see Chapter XV) will be nearly
as economical as a compound with saturated steam.
483. Binary Vapor Engine. This was originated by Du Tremblayin 1850
(26). The exhaust steam from a cylinder passed through a vessel containing
coils filled with ether. The steam being at a temperature of almost 250 F.,
vrhile the atino^ihmc boiling point of ether is 94 F., the latter was rapidly
vaporized at a considerable pressure, and was then used for performing work in
a second cylinder. Assuming the initial temperatuie of the steam to have been
320 F., and the final temperature of the ether 100 F,, the ideal efficiency should
thus be increased from
320  250 = 09 to 320  100 _.
320 + 460 ~~ ' 320 + 460
a gain of over 200 per cent. The advantage of the binary vapor principle arises
from the low boiling point of the binary fluid. This permits of a lower tempera
ture of heat emission than is possible with ^ater. Binary engines must be run
condensing. Since condensing water is generally not available at temperatures
below f>0' or TO J F., the fluid should be one which may be condensed at these tem
peratures. Etliw satisfies this requirement, and gives, at its initial temperature
of, >ay, SoO^ F., a woiking pressure not far fiom 151) Ib. On account of its high
boiling point* however, its pressure is less than that of the atmosphere at 70 F. ?
and an air pump w m cessary to discharge the condensed vapor from the condenser
just us is the case with condensing steam engines. Sulphur dioxide has a much
lower boiling point, and may be used without an air
pump: but its pressure at 250 would be excessive, and
the best results are secured by allowing the steam cylinder
to run condensing at a final temperature as low as pos
sible ; at 104 3 F., the pressure of sulphur dioxide is only
OO.o Hi. The best steam engines have about this lower
temperature limit; the maximum gain due to the use of a
binary fluid cannot exceed that corresponding to a reduc
tion of this temperature to about 60 or 70 F., the usual
temperature of the available supply of cooling water.
The steamether engines of the vessel Brestt operated
at 43.2 Ib. boiler pressure and 7.G Ib. back pressure of
ether. The cylinders were of equal size, and the mean
effective pressures were 11,6' and 7. 1 Ib. respectively. The
Fio.217. Art.483, Prob. coal colwlllll ption was brought down to 2.44 Ib. per
raBiuaiy Vapor Eu Ihl) _ br> . ft ^ favorab le result than that obtainable from
glne> good .steam engines of that time. Several attempts have
THE INDICATOR
333
been made to revive the binary vapor engine on a small scale, the most important
recent experiments are those of Josse (27), on a 200hp. engine using steam
at 160 Ib. pressure and 200 of superheat, including four cylinders. The first three
cylinders constitute an ordinary triplecondensing steam engine, a vacuum of 20
to 25 in of mercury being maintained in the lowpressure cylinder by the circula
tion of sulphur dioxide in the coils of a surface condenser. The dioxide then enters
the fourth cylinder at from 120 to 180 Ib .pressure and leaves it at about 35 Ib. pressure.
The best result obtained gave a consumption of 167 B t u, per Ihp. per minute,
a result scarcely if ever equaled by a highgrade steam engine (Art 550). The ideal
entropy cycle for this engine is shown m Fig. 217, the three steam cylinders being
treated as one. The steam diagram is abode, and the heat delivered to the sulphur
dioxide vaporizer is aerm This heated the binary liquid along M and vaporized
it along ?/, giving the work area hifg. The different liquid lines and saturation curves
of the two vapors should be noted The binary vapor principle has been suggested
as applicable to gas engines, in which the temperature of the exhaust may exceed
1000 F.
ENGINE TESTS*
484. The Indicator. Two special instruments are of prime importance in
measming the perfoimance of an engine. The first of these is the indicator, one
of the secret inventions of Watt (28), which
shows the action of the steam in the cylinder.
Some conception of the influence of this device
on progress in economical engine operation may
be formed from the typically bad and good dia
grams of Fig. 218. The indicator furnishes a
method for computing the mean effective pres
sure and the horse power of any cylinder.
Figure 219 shows one of the many common
forms. Steam is admitted from the engine cylin
der through 6 to the lower side of the movable
piston 8. The fluctuations of pressure in the
FIG. 218. Arts. 484, 486. Good
and Bad Indicator Diagrams.
cylinder cause this piston to rise or fall to an extent determined by the stiffness
of the accurately calibrated spring above it. The piston movements are trans
mitted through, the rod 10 and the parallel motion linkage shown to the pencil
23, where a perfectly vertical movement is produced, in definite proportion to
the movement of the piston 8. By means of a cord passing over the sheaves
37, 27, a toandfro movement is communicated from the crosshead of the engine
to the drum, 24. The movements of the drum, under control of the spring, 31 J
are made just proportional to those of the piston; so that the coordinates of the
diagram traced by the pencil on the paper are pressures and piston movements.
485. Special Types. Various modifications are made for special applications.
For gas engines, smaller pistons are used on account of the high pressures; springs
of various stiffnesses and pistons of various areas are employed to permit of accu
rately studying the action at different parts of the cycle, safety stops being pro
vided in connection with the lighter springs. The Mathot instrument, for
example, gives a continuous record of the ignition lines only of a series of suc
* See Trans, 4, 8* M, E., XXIV 7 713; Jour, 4.& M. J&, XXXIV, 11,
334
APPLIED THERMODYNAMICS
cessiye gas engine diagrams. "Outsidespiing" indicators are a recent type, in
which the spring is kept away from the hot steam. The Ripper meanpressure
indicator (29) is a device which shows continuously the mean effective pressure
in the cylinder. Instruments are often provided with pneumatic or electrical
operating mechanisms, permitting one observer to take exactly simultaneous dia
grams from two or more cylinders. Indicators for ammonia compressors must
luive ail internal parts of steel; special forms are also constructed for heavy hy
FIG. 219. Art. 484. Crosby Steam Engine Indicator.
draulic and ordnance pressure measurements. For very high speeds, in \\hich the
inertia of the moving parts would distort the diagram, optical indicators are used.
These comprise a small mirror which is moved about one axis by the pressure and
about another by the piston movement. The path of the beam of light is pre
served by photographing it. Indicator practice constitutes an art in itself; for
the detailed study of the subject, with the influence of drum reducing motions,
methods of calibration, adjustment, piping, etc., reference should be made to such
works as those of Carpenter (30) or Low (31). In general, the height of the dia
gram is made of a convenient dimension by varying the spring to suit the maxi
mum pressure; and accuracy depends upon a just proportion between (a) the
movements of the drum and the engine piston and (6) the movement of the indi
cator piston and the fluctuations in steam pressure.
INDICATOR DIAGRAMS
335
486. Measurement of Mean Effective Pressure. This may be accomplished
by averaging a laige number of equidistant ordinates across the diagram, or,
mechanically, by the use of the planimeter (32). In usual practice, the indicator is
either piped, with intervening valves, to both ends of the cylinder, in which case a
pair of diagrams is obtained, as in Fig. 218, one cycle after the other, representing
the action on each side of the piston ; or two diagrams are obtained by separate
indicators. In order that the diagrams may be complete, the lines ab, representing
the boiler pressuie, cd, of atmospheric pressure, and efof vacuum in the condenser,
should be drawn, together with the line of zero volume ea^ determined by measur
ing the clearance, and the hyperbolic curve (/, constructed as in Art. 92. The
saturation curve gh for the amount of steam actually in the cylinder is sometimes
added. As drawn in Fig 218, the position of the saturation curve indicates that the
steam is dry at cutoff scarcely the usual condition of things.
487. Deductions. By taking a "fullload" card, and then one with the ex
ternal load wholly removed, the engine overcoming its own frictional resistance
only, we at once find the me
chanical efficiency, the ratio of
power exerted at tie shaft to
power developed in the cylin
der; it is the quotient of the
difference of the two diagrams
by the former. By measur
ing the pressure and the vol
ume of the steam at release,
and deducting the steam pres
ent during compression, we
may in a rough way com
\
pute the steam consumption
per Ihp.hr., on the assumption
that the steam is at this point
dry; and, as in Art. 500, by
properly estimating the per
centage of wetness, we may
closely approximate the actual
steam consumption.
Some of the applications
of the indicator are suggested
by the diagrams of Fig. 220.
In a, we have admission oc
curing too early; in b, too
late. Excessively early cutoff
is shown in c ; late cutoff, with
excessive terminal drop, in d.
Figure e indicates too early
release ; the dotted curve
would give a larger wort area;
in f, release is late. The bad effect of early compression is indicated in g ; late com
pression gives a card like that of h, usually causing noisiness. Figure * shows excea
FIG. 320.
Art. 487. Indicator Diagiams and Valve
Adjustment.
336 APPLIED THERMODYNAMICS
sire throttling during admission; / indicates excessive resistance during exhaust
\\hich may be due to thiotthng or to a poor vacuum. The effect of a small supply
pipe is shown in k, in which the upper line repiesents a diagram taken with the
indicator connected to the steam chest. The abrupt rise of pressure along LC is
due to the cutting off of the flow of steam from the steam chest to the cylinder.
Figure I shows the fomi of card taken when the drum is made to derive its mo
tion from the eccentric instead of the croabhead. This is often done in order to
study more accurately the conditions near the end of the stroke when the piston
moves veiy slowly, while the eccentric moves more rapidly. Figure m is the coi
responding ordinary diagram, and the two diagrams are correspondingly letteied.
Figure is an excellent card from an air compressor ; o shows a card from an air
pump with excessive poit friction, particularly on the suction side. Figure j>
shows what is called a stroke card, the dotted line representing net pressures on
the piston, obtained by subtracting the back pressure as at cib from the initial
pressure uc, i.e. by making tic = alt. Figure q shows the effect of varying the
point of cutoff; r, that of throttling the supply. Negative loops like that of g
must be deducted from the remainder of the diagram in estimating the work done.
488 Measurement of Steam Quality. The second special instrument used in
engine testing is the steam calorimeter, so called because it determines the percent
age of dryness of steam by a series of heat measurements. Carpenter (33) classi
fies steam calorimeters as follows :
(a) Condensing
Calorimeters
' Barrel or tank
Continuous
Jet 
Surface
.Kent
External Barrus
Barrus Continuous
Hoadley
() Superheating
J Separator
^ ' \ Chemical
489. Barrel or Tank Calorimeter. The steam Is discharged directly into an
insulated tank containing cold water. Let W, w be the weights of steam and
water respectively, t, ti the initial and final temperatures of the water, correspond
ing to the heat quantities h, hi ; and let the steam pressure be P 0j corresponding
to the latent heat L Q and heat of liquid ho, the percentage of dryness being zo
The heat lost by the steam is equal to the heat gained by the water ; or, the steam and
water attaining the same final temperature,
W(x*Lo + ho  Ai) = (*!  A), whence * = M" + ^)^W%o .
The value of IT is determined by weighing the water before and after the mix
ture. The radiation corrections are large, and any slight error in the value of W
CALORIMETERS
337
greatly changes the result; this foim of calorimeter is therefore seldom used, its
average error even under the best conditions ranging from 2 to t per cent. Some
improvement is possible by causing condensation to become continuous and tak
ing the weights and temperatures at frequent intervals, as in the " Injector " or
" Jet Continuous " caloi imeter.
490. Surfacecondensing Calorimeter. The steam is in this case condensed
in a coil ; it does not mingle with the water. Let the final temperatui e of the
steam be fe, its heat contents //a ; then
 h) and x =
More accurate measurement of W is possible with this arrangement. In the
Hoadley form (34) a propeller wheel was used to agitate the u ater about the coils;
in the Kent instrument, arrangement was made for removing the coil to peimit
of more accurately determining W. In that of Barrus, the flow was continuous
and a series of observations could be made at short intervals.
491. Superheating Calorimeters. The Peabody throttling calorimeter
is shown in Fig. 221 ; steam entering at b through a partially closed valve
expands to a lower steady pressure in A and then flows into the atmos
phere. Let L Q , 7i , x be the condition at b, and assume the steam to be
superheated at A, its temperature being T, t being the
temperature corresponding to the pressure p, and the cor
responding total heat at saturation H. Then, the total heat
at I equals the total heat at A, or
(%L + AO)= #+ Tt(T f),
where 7c is the mean specific heat of superheated steam
at the pressure p between Tand ; whence
If we assume the pressure in A to be that of the atmos
phere, 27" =1150.4, and superheating is possible only when
x L Q + h exceeds 1150.4. For each initial pressure, then,
there is a corresponding minimum value of x^ beyond
which measurements are impossible; tlms, for 200 lb., FIG. 221 Art 401.
L Q = 843.2, ft = 354.9, and a*, (minimum) is 0.94. Aside
from this limitation, the throttling calorimeter is exceed
ingly accurate if the proper calibrations, corrections, and methods of
sampling are adopted. In the Barrus throttling calorimeter, the valve at
b is replaced by a diaphragm through which a fine hole is drilled, and the
range of C values is increased by mechanically separating some of the
moisture. The same advantage is realized in the Barrus superheating
calorimeter by initially and externally heating the sample of steam. The
Superheat
ing Calorimeter.
338
APPLIED THERMODYNAMICS
amount of heat thus used is applied in such a way that it may be ac
curately measured. Let it be called, say, Q per pound. Then
f)h Q  Q
492. Separating Calorimeters. The water and steam are mechanically sepa
rated and separately weighed. In Fig. 222, steam enters, through 6, the jacketed
chamber shown. The water is intercepted by the cup
14, the steam reversing its direction of flow at this
point and entering the jacket space 7, 4, whence it is
discharged through the small orifice 8. The water ac
cumulates in 3, its quantity being indicated by the
gauge glass 10. The quantity of steam flowing is de
termined by calibration for each reading of the gauge
at 9. The instrument is said to be fairly accurate un
less the percentage of moisture is very small. The
steam may be, of course, run off, condensed, and
actually weighed.
493 Chemical Calorimeter. This depends for its
action on the fact that water will dissolve certain salts
(e.g. sodium chloride) which are insoluble in. dry
steam.
494 Electric Calorimeter. The Thomas superheat
ing and throttling instrument (35) consists of a small
soapstoue cylinder in which are embedded coils of
German silver wire, constituting an electric heater.
3 . is inserfced in a brass ea8e thr U S h which fl WS
a current of steam. The electrical energy correspond
to heataugmentation to any superheated condition being known, say, as
. t.u. per pound (1 B, t. u. per minute = 17.59 watts), we have, as in Art. 491,
or Z n + 7/ + E = JT+ k(T ), whence ar = H + k ( T ~ ')* " B .
ing
E B
495. Engine Trials: Heat Measurement. "We may ascertain the heat
supplied in the steam engine cycle either by direct measurement, or by
adding the heat equivalent of the external work done to the measured amount
of heat rejected. In the former case the amount of water fed to the boiler
must be determined, by weighing, measuring, or (in approximate work) by
the use of a water meter. The heat absorbed per pound of steam is ascer
tained from its temperature, quality, and pressure, and the temperature of
the water fed to the boiler. In the latter case, the steam leaving the
engine is condensed and, in small engines, weighed; or in larger engines,
determined by metering or by passing it over a weir. This latter of the
two methods of testing has the advantage with small engines of greater
ENGINE TEST 339
accuracy and of giving accurate results in a test of shorter duration. Where
the engine is designed to operate noncondensing, the steam may be con
densed for the purposes of the test hy passing it over coils exposed to
the atmosphere, so that no vacuum is produced by the condensation. If
jackets are used, the condensed steam from them must be trapped off and
weighed. This water would ordinarily boil away when discharged at
atmospheric pressure, so that provision must be made for first cooling it.
496. Heat Balance. By measuring loth the heat supplied and that rejected, as
well as the work done, it is possible to draw up a debit and credit account show
ing the use made of the heat and the unaccounted for losses. These last are due
to the discharge of water vapor by the air pump, to radiation, and to leakage.
The weight of steam condensed may easily be four or five per cent less than that
of the water fed to the boiler. Let 71, h, be the heat contents of the steam and
the heat in the boiler feed water respectively; the heat absorbed per pound is
then H h. Let Q be the heat contents of the exhausted steam (measured
above the feed water temperature) and W the heat equivalent of the work done
per pound. Then for a perfect heat balance, H h = Q f W. In practice, W
is directly computed from the indicator diagrams ; H and Q must be corrected
for the quality of steam as determined by the calorimeter or otherwise.
The heat charged to the engine is measured from the ideal feed
water temperature corresponding with the pressure of the atmosphere
or condenser to the condition of steam at the throttle: that is, it is
(in general symbols),
R =Q(H A ), B. t. u. per Ihp. hr.,
where Q represents the steam consumption in Ib. per Ihp. hr.
Then 2545 +R is the thermal efficiency =E. Let H be the total heat
above 32 after adiabatic expansion in the Clausius cycle: then the
ideal efficiency is
and the " efficiency ratio " or relative efficiency is
E 2545
The efficiency ratio referred to the Carnot cycle is correspondingly
2545 T
^ c 'Q(Hh )(Tty
where T and t are, respectively, the absolute temperatures at the
throttle and corresponding with atmospheric or condenser pressure.
In working up a heat balance, it is convenient to measure all heat
340
APPLIED THERMODYNAMICS
quantities above 32. The gross heat charged to the engine is then
HQ, less any transmission losses between boiler and engine. If the
engine runs" condensing, and Qi Ib. of condenser water circulated
rise from *i to t 2 F., the heat rejected to the circulating water is
^i) B. t. u. There are also rejected, in the condensed steam,
~. t. u., where h 3 is the heat of liquid corresponding with the tem
perature t 3 of the condensed steam. (Note that t 3 = fe in jet condensing
engines.) Some of the heat thus rejected may, however, be returned
to the boiler, and should then be credited, the amount of credit being
the sum of the weights returned each multiplied by the respective
heat of liquid. Any steam condensed in the jackets is charged to the
engine, but the heat rejected from the jackets (usually returned to
the boiler) is then credited as Q 2 h where Qj is the weight of steam con
densed and h the heat of liquid corresponding with its pressure (usually
the throttle pressure).
497. Checks; Codes. Where engines are used to drive electrical generators
the measurement of the electrical energy gives a close check on the computation
of indicated horse power. Let G= generator output in kilowatts, E& = generator
efficiency ,E m = mechanical efficiency of the umt,#=Ihp. of engine then 1.34G =
HEffEm. In locomotive trials a similar check is obtained by comparison of the
drawbar pull and speed (36). In turbines, the indicator cannot be employed,
measurement of the mechanical power exerted at the shaft is effected by the use
of the friction brake. Standard codes for the testing of pumping engines (37), and
of steam engines generally (38;, have been developed by the American Society of
Mechanical Engineers.
3TiG. 224. Arts. 498, 499, 500. Indicator Cards from Compound Engine.
498. Example of an Engine Test* Figure 22 i, from Hall (39), gives
the indicator diagrams from a 30 and 56 by 72in. compound engine at
58 r. p. m. The piston rods were 4J and 5J in. diameter. The boiler
* Values from steam tables, used in this article, do not precisely agree with those
given on pp. 287, 288.
ENGINE TEST 341
pressure was 124.0 Ib. gauge: the pressure in the steam pipe near the
engine, 122.0 Ib. The temperature of jacket discharge was 338 F. The
conditions during the calorimetric test of the inlet steam were P = 122.08
Ib. gauge, T = 302.1 F. (Art. 491), pressure in calorimeter body (Fig. 221),
11.36 Ib. (gauge). The net weight of boiler feed water in 12 hours was
231,861.7 Ib. ; the weight of water drained from the jackets, 15,369.7 Ib.
Prom the cards, the mean effective pressures were 44.26 and 13.295
Ib. respectively; and as the average net piston areas were 697.53 and
2452.19 square inches respectively, the total piston pressures were 44.26
X 697.53=30872.7 and 13.295 x 2452.19=32601.9 Ib. respectively. These
were applied through a distance of
if X 2 x 58 = 696 feet per minute;
whence the indicated horse power was
(30872.7+32601.9) X 696 =
33000
From Art. 491> A>+^o = &+& (Tfy or in this case, 866.5 x + 322.47
= 1155.84 + 0.48* (302.1242.3) whence X Q = 0.995. The weight of
cylinder feed was 231,861.7 15,369.7 = 216,492.0 Ib. At its pressure of
136.7 Ib. absolute, =866.5, ft = 322.4. Tor the ascertained dryness, the
total heat per pound, above 32, is 322.4 + (0.995x866.5) =1184.5 B. t u.
The heat left in the steam at discharge from the condenser (at 114 F.)
was 82 B. t. u. ; the net heat absorbed per pound of cylinder feed was
then 1184.5 82.0 = 1102.5; for the total weight of cylinder feed it was
1102.5 x 216,492 = 238,682,430 B. t. u. The total heat in one pound qf
jacket steam was also 1184.5 B. t. tu This was discharged at 338 F.
(7i = 308.8), whence the heat utilized in the jackets was 1184.5 308.8
= 875.7 B. t. u. (The heat discharged from both jackets and cylinders
was transferred to the boiler feed water, the former at 338, the latter at
114 F.) The supply of heat to the jackets was then 875.7 x 15,369.7
=i 13,459,246.29 B. t. u: the total to cylinders and jackets was this quan
tity plus 238,682,430 B. t. u., or 252,141,676.29 B. t. u. Dividing this by
60 x 12 we have 350,196.77 B. t. u. supplied per minute.
499. Statement of Results. We have the following :
(a) Pounds of steam per Ihp.hr. = 231,861.7 s 12 = 1338.62 = 14.43.
(This is the most common measure of efficiency, but is wholly
unsatisfactory when superheated steam is used.)
* Value taken for the specific heat of superheated steam.
342 APPLIED THERMODYNAMICS
(6) Pounds of dry steam per Ihp.hr. = 14.43 x 995 * = 14.36.
(c) Heat consumed per Ihp. per minute = 350,196.77 * 1338.62 = 261.61
B. t. u.
($) Thermal efficiency = ^f^* 261.61 = 0.1621.
(e) Work per pound of steam=?^%^^ X 0.1621 = 176 B. t. u.
l. I
CO Camotefficiency^ =0.293.
(gf) Clausius efficiency (Art. 409), with dry steam,
810.82
^1=0.265.
351.22114+866.5
(&) Ratio of (<*)*&) = 0.1621 4 0.265 = 0.61.
500. Steam Consumption from Diagram. The inaccuracy of such estimates
will be shown. In the highpressure cards, Pig. 224, the clearance space at each
end of the cylinder was 0.932 cu. ft. The piston displacement per stroke on the side
opposite the rod was 706.86 x 72  1728 = 29.453 cu. ft.; the cylinder volume
on this side was 29.453 + 0.932 = 30.385 cu. ft. The length of the coriespond
ing card (a) is 3.79 in. ; the clearance line Ic is then drawn distant from the
admission line
3.79 x ^?i = 0.117 in.
29.453
At rf, on the release line, the volume of steam is 30.385 cu. ft., and its pressure is
31.2 Ib. absolute. From the steam table, the weight of a cubic foot of steam at
this pressure is 0.076362 Ib.; whence the weight of steam present, assumed dry, is
0.076362 x 30.385 = 2.3203 Ib. At a point just after the beginning of compres
sion, point e, the volume of steam expressed as a fraction of the stroke plus the
clearance equivalent is 0.517 * 3.907 = 0.1321, 3.907 being the length bg iu inches.
The actual volume of steam at e is then 0.1321 x 30.385 = 4.038 cu. ft., and its
pressure is 28.3 Ib. absolute, at which the specific weight is 0.069683 Ib. The
weight present at e is then 4.038 x 0.069683 = 0.2SO Ib. The net weight of steam
used per stroke is 2.3203  0.280 = 2.0403 Ib., or, per hour, 2.0403 x 58 x 60 = 7090
Ib., for this end of the cylinder only. For the other end, the weight, similarly
obtained, is 7050 Ib. ; the total weight is then 14,140 Ib. The horse power
developed being 1339, the cylinder feed per Ihp hr. from highpressure diagrairs
is 10.6 Ib., or 26 per cent less than that which the test shows. The same process
may be applied to the lowpressure diagrams. It is best to take the points d and e
just before the beginning of release and after the beginning of compression respec
* The factor 0.995 does not precisely measure the ratio of energy in the actual
steam to that in the corresponding weight of dry steam, but the correction is usually
made in this way.
MEASUREMENT OF REJECTED HEAT 343
tively. The method is widely approximate, but may give results of some value
in the absence of a standard trial (Arts 448, 4iO).
501. General Expression. In Fig. 224a, let 7=^, =D. Let the cylinder
L L
area be A sq in., the stroke S ft , the clearance d = m(Ld)=mAS: and let the
speed be n r, p. m. The horse power of the double
acting engine is
ZpnASn
n 33,000 '
for p m Ibs. mean effective pressure per square inch.
The weight of steam used per stroke, in pounds, is
w = BAS(1 +m) DASQ +m)
501. Steam
am.
M/1+ \ (JLJL\ FlG 224a ' Ali SOI. St
144 \ / \xv XVo) ' Consumption from Diagr
where v and V are the specific volumes of dry steam and x and X are the dryncss
of the actual steam, at d and e respectively Making Xx = l.Q, we find (from
the indicator diagrams alone) the weight of steam consumed per Ihp. hour to be
in pounds,
13,750(1 +m)/B D\
HP. " p m U vj"
In applying this to compound engines, p m must be taken as the total equivalent
mean effective pressure "referred to" to the cylinder of area A (Art. 472).
For the conditions of Art. 500, p*=44.26+ f^X13.295J =90.36, and the steam
rate is
/1.0317\ / 30.385 4.038 \ , n41 ,
*' U \ 90.36 / \30.3S5X13.24 30.385 X14.53/ U< * 1Dt
502. Measurement of Rejected Heat A common example is in tests in
which the steam is condensed by a jet condenser (Art. 584). In a test
cited by Ewing (40), the heat absorbed per revolution measured above the
temperature of the boiler feed was 1551 B. t u. ; that converted into work
was 225 B. t. n. The exhaust steam was mingled with the condensing
water, a combined weight of 51.108 Ib. being found per revolution. The
temperature of the entering water was 50 F., that of the discharged mix
ture was 73.4 F. ? and the cylinder feed amounted to 1.208 Ib. per revolu
tion. The temperature of the boiler feed water was 59 F. We may
compute the injection water as 51.108 1.208 = 49.9 Ib. and the heat
absorbed by it as approximately 49.9(73.4 50) = 1167 B. t. u. The
1.208 Ib. of feed were discharged at 73.4, whereas the boiler feed was at
59 ; a heat rejection of 73.4  59 = 14.4 occurred, or 14.4 x 1.208 = 17.4
344 APPLIED THERMODYNAJV1ICS
B. t. u. The total heat rejection was then 1167 + 17.4 = 1184.4 B. t. u.,
to which we must add 47 B. t. u. from the jackets, giving a total of
1231.4 B. t. u. Adding this to the work done, we have 1231.4 + 225 =
1466.4 B. t. u. accounted for of the total 1551 B. t. u. supplied; the
discrepancy is over 6 per cent.
"When surface condensers are used, the temperatures of discharge of
the condensed steam and the condenser water are different and the weight
of water is ascertained directly. In other respects the computation
would be as given.*
503. Statements of Efficiency. Engines are sometimes rated on the basis of
fuel consumption. The duty is the number of footpounds of work done in the
cylinder per 100 pounds of coal burned (sometimes and preferably the number
of footpounds of work per 1 } 000,000 B. t. u consumed at coal. The efficiency
of the plant is the quotient of the heat converted into work per pound of coal, by
the heat units contained in the pound of coal. In the test in Art. 498, the coal
consumption per Ihp.hr. was 2068.84J1338,62 = 1.54 Ib. In some cases, all state
ments are baaed on the brake horse power instead of the indicated horse power. The
ratio of the two is of course the mechanical efficiency. It may be noted that the
engine is charged with steam, not at boiler pressure, but at the pressure in the steam
pipe. The difference between the two pressures and qualities represents a loss
which may be considered as dependent upon the transmissive efficiency. The plant
efficiency is obviously the product of the efficiencies of boiler (Art. 574), transmission,
and engine.
504. Measurement of Heat Transfers: Hirn's Analysis. In the refined methods
of studying steam engine performance developed by Hirn (41), and expounded by
DwelshauversDery (42), the heat absorbed
and that rejected are both measured. Dur
ing any path of the cycle, the heat inter
change between fluid and walls is computed
from the change in internal energy, the heat
externally supplied or discharged, and the
external work done.
The internal energy of steam is, in general
symbols, h+xr. The heat received being Q,
_ __ ___ ___ and the heat lost by radiation Q', we have
. 225. Art. 504. Hirn's Analysis, the general form
where the path is, for example, from 1 to 2, and the weight of steam increases from
wi to tr*. Applying such equations to the cycle, Fig. 225, made up of the four
* It is most logical to charge the engine with the heat measured above the tem
perature of heat rejection. This, in Tig. 182, for example, makes the efficiency
d&bc dsoc
rather than ~~ *m the ordinate FJT representing the feedwater temperature,
HIRN'S ANALYSIS
345
operations 01, 12, 23, 30, we have, M Q denoting the weight of clearance steam and
M that of cylinder feed, per stroke, in pounds.
Ei = (M +M) (^
Let Q a , Q&, Q c , Qd, represent amounts of heat transferred to the walls along the
paths a, b, c, d.
Consider the path a. Let the heat supplied by the incoming steam be Q. Then
Along the path 5, Q&TPH(#Ei); along d, 
Along c , heat is carried away by the discharged steam and by the cooling water.
Let G denote the weight of cooling water per stroke, k 5 and hi its final and initial
heat contents, and h the heat contents of the discharged steam. The heat rejected
by the fluid per stroke is then G(h & h 4 )\Mk 6 . Then Q c G(h tt h^Mh^ =
W C +(E S E 2 ), and Q c ^G(k s h t )Mh 9 +W c (StS^
Values for the h and r quantities are obtained from the steam table for the pres
sures shown by the indicator diagram. The diagram gives also the work quantities
along each of the four " paths." The conditions of the test give Q 3 O t h s , h& t h*,
and M. The remaining unknown quantities are M Q and the drynesses. MQ is found
by assuming #3 = ! (see Art 500). Then the dryness at any of the remaining
points 0, 1, 2, may be found by writing
v
x = ,
WVo
where v is the volume shown by the indicator diagram, v is the specific volume
of dry steam and w is the weight of steam present, at the point in question. The
quantity w will be equal to M or (M +Mo) as the case may be.
505. Graphical Representation. In Fig. 226, from the base line xy, we may
lay off the areas oefs representing heat lost during admission, smba showing heat
gained during expansion, mhcr showing heat gained
during release, and oakr showing heat lost during
compression. If there were no radiation losses
from the walls to the atmosphere, the areas above
the line xy would just equal those below it. Any
excess in upper areas represents radiation losses.
Ignoring these losses, Him found by comparing the
work done with the value of Q Mh* G(h & A 4 )
an approximate value for the mechanical equivalent
of heat (Art. 32).
Analytically, if Q T denote the loss by radiation,
its value is the algebraic sum of Q a , Q&, Q c , Q&. If
the heat Q 3 be supplied by a steam jacket, then
Q r = Qj + sQo, 6 . c, d The heat transfer during
release, Q c , regarded by Him as in a special sense a
measure of wastefulness of the walls, may be expressed as Q T Qj~ S^ fl , b . d In
a noncondensing engine, Q r can be determined only by direct experiment.
505a. Testing of Regulation, The " regulation " of a steam engine refers to
its variations in speed. In most applications uniformity of rotation is important.
This is particularly the case when engines drive electric generators, and the momen
FIG. 226. Art. 505. Heat
Transfers.
346
APPLIED THERMODYNAMICS
tary or periodic variations in speed must be kept small regardless of fluctuations
in initial pressure, back pressure, load or ratio of expansion. This is accomplished
by using a sensitive governor and a suitably heavy flywheel. Regulation cannot
be studied by unaided observation with a revolution counter or by an ordinary
recording instrument. An accurate indicating tachometer or some special optical
device must be employed (Trans. A. S. M. E., XXIV, 742).
TYPES OF STEAM ENGINE
506.; Special Engines. "We need not consider the commercially unimportant
class of engines usmg vapors other than steam, those experimental engines built
for educational institutions which belong to no special type (43), engines of novel
and limited application like those employed on motor cars (44), nor the " fireless "
or stored hotwater steam engines occasionally employed for locomotion (45).
507. Classification of Engines. Commercially important types may be con
densing or noncondensing. They are classified as righthand or lefthand, accord
ing as the flywheel is on the right or left side of the center line of the cylinder,
as viewed from the back cylinder head. They may be simple or multipleexpan
FIG. 23$. Art 607 An^leCompouud Engine. (American Ball Engine Company.)
TYPES OF ENGINE
347
sion, with all the successive stages and cylinder arrangements made possible in
the latter case. They may be singleacting or doubleacting ; the latter is the far
more usual arrangement. They may be rotative or nonrotative. The directacting
pumping engine is an example of the latter type; the work done consists in a
rectilinear impulse at the water cylinders. In the duplex engine, simple cylinders
are used side by side. The terms horizontal, vertical, and inclined refer to the posi
tions of the center lines of the cylinders. The horizontal engine, as in Figs. 186
and 229, is mostly used in land practice ; the vertical engine is most common at
FlG. 229. Art. 607. Automatic Engine. (American Ban Engine Company.)
sea. Crosscompound vertical engines are often directconnected to electric gen
erators. Vertical engines have occasionally been built with the cylinder below
the shaft. This type, with the inclined engine, is now rarely used. Inclined
engines have been built with oscillating cylinders, the use of a crosshead and
connecting rod being avoided by mounting the cylinder on trunnions, through
which the steam was admitted and exhausted. Figure 228 shows a section of
348
APPLIED THERMODYNAMICS
the interesting anglecompound, in which a horizontal highpressure cylinder
exhausts into a vertical lou pressure cylinder. A different type of engine, but
with a similar structural arrangement, has been used in some of the largest
power stations.
Engines are locomotive, stationary, or marine. The last belong in a class by
themselves, and will not be illustrated hei e ; their capacity ranges up to that of
our laigest stationary power plants. Stationary engines are further classed as
pumping engines, mill engines, power plant engines, etc. They may be further
grouped accoiding to the method of absorbing the power, as belted, directcon
nected, rope driven, etc An engine directly driving an air compressor is shown in
Pig. 86. <k Rolling mill engines'* undergo enormous
variations in load, and must have a correspondingly
massive (tangye) frame. Power plant engines gen
erally mast be subjected to heavy load variations;
their frames are accordingly usually either tangye or
semitangye. Mill engines operate at steadier loads,
and have frequently been built with light girder
frames. Modern high steam pressures have, however,
led to the general discontinuance of this frame in
favor of the semitangye.
A slowspeed engine may run at any speed up to
125 r. p.m. From 125 to 200 r.p.m. may be re
garded as medium speed. Speeds above 200 r.p.m.
are regarded as high. Certain types of engine are
adapted only for certain speed ranges ; the ordinary
slidevalve engme, shown in Fig. ISO, may be oper
ated at almost any speed. For ]arge units, speeds
range usually from 80 to 100 r.p.m. The higher
speed engines are considered mechanically less re
liable, and their valves do not lend themselves to quite
as economical a distribution of steam. An important / /\ 8
class of mediumspeed engines has, however, been in
troduced, in which the independent valve action of
the Coiliss type has been retained, and the promptness
of cutofE only attainable by a releasing gear has been
approximated. In some cheap highspeed engines
governing is effected simply but uneconomically by
throttling the steam supply. Such engines may have
shallow continuous frames or the subbase, as in Fig.
220, which represents the large class of automatic
highspeed engines in which regulation is effected by
automatically varying the point of cutoff. Figure 230
shows three sets of indicator diagrams from a com
pound engine of this type, running noncondensing
at various loads. Some of the irregulations of these
diagrams are without doubt due to indicator inertia; but they should be care
fully compared with those showing the steam distribution with a slowspeed
TYPES OF ENGINE
349
releasing gear, in Fig. 218. All of the socalled " automatic " engines run at
medium or high rotative speeds.
The throttling engine is used only in special or unimportant applications. The
automatic type is employed where the comparatively high speed is admissible, in
units of moderate size. Better distribution is afforded by the four valve engine, in
BacKCqlinderHtoJ.
3och Cylinder Heo
Back Cyl Head
Steam pipe
feom Flanq*
^Throttle Valve
Planished 5te) Laqqinq
Heat insulating Filling
iss STeomVolve Chamber 1
^Cyl.ndtrH.od
itCtjImdtr Hcod 5tud
>^pnflod Gland Studs
Piston Rod G'and
Corliss tihomtVafvi
Eihaustthe^T , ^
^Erhoust Openmq
trhaust Pipe
FiG. 231. Art. 607. Corliss Engine Details (Murray Iron Works Company )
350
APPLIED THERMODYNAMICS
which the four events of the stroke may be independently adjusted, and this type
is often tised at moderately high speeds. Sharpness of cutoff is usually obtainable
only with a releasing geai, in which the mpchaiiihiu operating the valves is discon
nected, and the steam valve is au
tomatically and instantaneously
closed. This feature distinguishes
the Corliss type, most commonly
used, in highgrade mill and power
plant service. AVith the releasing
gear, usual speeds seldom exceed
100 r. p.m. The valve in a Cor
liss engine is cylindrical, and ex
tends across the cylinder. Some
details of the mechanism are
shown in Fig. 231. In very large
engines, the releasing principle is
sometimes retained, but "with
poppet or other forms of valve.
Figure 232 shows the parts of a
typical Corliss engine with semi'
tangye frame.
507a. The Stumpf Engine. Re
markable reductions in cylinder loss
have been effected by the unidirec
tionalflow pistonexhaust engine
of Stumpf, shown in Fig. 23 la.
The piston itself acts as an exhaust
valve by uncovering slots in the
barrel of the cylinder at i 9 o strike*
The jacketed heads form steam
chests for the poppet admission
valves. The piston is about half
as long as the cylinder. The ad
vantages of the engine are, very
slight piston leakage, no special
exhaust valve, ample exhaust ports,
low clearance (1J to 2 per cent)
and reduced cylinder condensation.
This last is due to the continuous
flow of steam from ends to center of
the cylinder, which keeps the cooled
and expanded steam from sweeping
over the heads. (The steam in an engine cylinder is by no means in a condition
of thermal equilibrium.) The condensation is so small that very large ratios of
expansion arc employed, and the simple engine with either saturated or superheated
steam seems to give an efficiency about equal to that attained by a triple expan
sion engine of the ordinary type. Compression is necessarily excessive: so much
po that when the engine is used noncondensing a special piston valve, working ID
THE STUMPF ENGINE
351
the piston, is used to prolong the exhaust period during part of the return stroke.
Some of the advantages are thereby sacrificed : this modification is not necessary on
condensing engines.
The device has been applied to locomotives on the Prussian state railways (Engi
neering Magazine, March, 1912). The cylinders are of excessive lengths: a special
valve gear, highly economical in power consumption, has been developed. The
early compression (no supplementary exhaust valve being used) requires large
clearance: but it is claimed that with a concaveended hollow piston the wall surface
of the clearance space (which influences the loss) is from 40 to 60 per cent less than
that in an ordinary locomotive cylinder. Any initial condensation is automatically
PEG. 231a. Art. 507a. The Stumpf Engine.
discharged through holes hi the Cylinder wall, so that it ceases to be a factor in
producing further condensation.
508.' The Steam Power Plant Figure 233, from Heck (4=6), is
introduced at this point to give a conception of the various elements
composing, with the engine, the complete steam plant. Fuel is burned
on the grate 1; the gases from the fire follow the path denoted by the
arrows, and pass the damper 4 to the chimney 5. Water enters, from
the pump IV, the boiler through 29, and is evaporated, the steam
passing through 8 to the engine. The exhaust steam from the engine
goes through 18 to the condenser III, to which water is brought through
21. Steam to drive the condenser pump comes from 26. Its exhaust,
with that of the feed pump 31, passes to the condenser through 27. The
condensed steam and warmed water pass out through 23, and should, if
possible, be used as a source of supply for the boiler feed. The free
exhaust pipe 19 is used in case of breakdown at the condenser.
352
APPLIED THERMODYNAMICS
509. The Locomotive,
This is an entire power plant,
made poi table. Fig me 234
shows a typical modern form.
The engine consists of t^o
horizontal double acting cyl
inders coupled to the ends of
the same axle at light an
gles. These are located tin
der the front end of the
boiler, which is of the type
described in Art. 563. A
pair of heavy frames sup
ports the boiler, the load be
ing earned on the axles by
means of an , intervening
" spring rigging." The stack
is necessarily short, so that
artificial draft is provided by
means of an expanding noz
zle in the "smoke bos,"
through which the exhaust
steam passes; live steam
may be used when necessary
to .supplement this. The
engines are noncondensing,
but superheating and heat
ing of feed water, particu
larly the former, are being
introduced extensively. The
water is carried in an aux
iliary tender, excepting in
light locomotives, in. v\ inch a
*' saddle " tank may be built
over the boilei .
The ability of a locomo
tive to start a load depends
upon the force which it can
exert at the rim of the diiv
ing wheel. If d is the cylin
der diameter in inches, L the
stroke in feet, and p the
maximum mean etfective
pressure of the steam per
square inch, the work done
per revolution by two equal
cylinders is vd*Lp. Assume
THE LOCOMOTIVE
353
this work to be trans
mitted to the point of
contact between wheel
and rail without loss,
and that the diameter
of the wheel is D feet,
then the tractive power,
the force exerted at
the rim of the wheel,
The value of p, with
such valve gears as are
employed on locomo
tives, may be taken at
80 to 85 per cent of the
boiler pressure. The
actual tractive power,
and the 'pull on the
drawbar, are reduced
by the friction of the
mechanism ; the latter
from 5 to 15 per cent.
Under ordinary con
ditions of rail, the
wheels will slip when
the tractive power ex
ceeds 0.22 to 0.25 the
total weight carried by
the driving wheels.
This fraction of the
total weight is called
the adhesion, and it is
useless to make the
tractive power greater.
In locomotives of cer
tain types, a " traction
increaser " is sometimes
used. This is a device
for shifting some of the
weight of the machine
from trailer wheels to
driving wheels. The
weight on the drivers
and the adhesion are
thereby increased. The
engineman, upon ap
354 APPLIED THERMODYNAMICS
preaching a heavy grade, may utilize a higher boiler pressure or a later cutoff
than would otherwise be useful.
510. Compounding. Mallet compounded the two cylinders as early as 1876.
The steam pipe between the cylindeis wound through the smoke box, thus becom
ing a reheating receiver. Mallet also proposed the use of a pair of tandem compound
cylinders on each side. The Baldwin type of compound has two cylinders on each
side, the high pressure being above the low pressure. Webb has used two ordinary
outside cylinders as highpressure elements, with a very large lowpressure cylinder
placed under the boiler between the wheels. In the Cole compound, two outside
lowpressure cylinders receive steam from trwo highpressure inside cylinders. The
former are connected to crank pins, as in ordinary practice, the latter drive a
forward driving axle, involving the use of a crank axle. The four crank efforts
differ in phase by 90. This causes a veiy regular rotative impulse, whence the
name balanced compound. Inside cylinders, with crank axles, are almost exclusively
used, even with simple engines, in Europe: twocylinder compounds with both
cylinders inside have been employed. The use of the crank axle has been complicated
in some locomotives with a splitting of the connecting rod from the inside cylinders
to cause it to clear the forward axle. Greater simplicity follows the standard
method of coupling the inside cylinders to the forward axle.
511. Locomotive Economy. The aim in locomotive design is not the greatest
economy of steam, but the installation of the greatest possible powerproducing
capacity in a definitely limited space. Notwithstanding this, locomotives have
shown very fair efficiencies. This is largely due to the small excess air supply
arising from the high rate of fuel consumption per square foot of grate (Art. 564).
The locomotive's normal load is what would be considered, in stationary practice,
an extreme overload. Its mechanical efficiency is therefore high. For the most
complete data on locomotive trials, the Pennsylvania Railroad Report (47) should
be consulted. The American Society of Mechanical Engineers has published a
code (48) j Reeve has worked out the heat interchange in a specimen test by Hirn's
analysis (49). (See Art. 554.)
(1) D. K. Clark, Railway Machinery. (2) Isherwood, Experimental Researches
in Steam Engineering ', 1863. (3) De la condensation de la vapeur, etc., Ann. des
mines, 1877. (4) Bull, de la Soc Indust. de Mulhouse, 1855, et seq. (5) Proc. Inst.
Civ. Eng., CXXXI. (6) Peabody, Thermodynamics, 1907, 233. (8) Min. Proc.
Inst. C. E., March, 1888; April, 1893 (9) Op. cit. (10) Engine Tests, G. H.
Barrus. (11) The Steam Engine, 1892, p. 190. (12) The Steam Engine, 1905,
109, 119, 120. (13) Proc Inst. Mech. Eng., 1889, 1892, 1895. (14) Ripper, Steam
Engine Theory and Practice, 1905, p. 167. (15) Ripper, op. tit., p. 149. (16) Trans.
A.8.M. E. f XXVIII, 10. (17) For a discussion of the interpretation of the Boulvin
diagram, see Berry, The TemperatureEntropy Diagram, 1905. (18) Proc.Inst Mech.
Eng., January, 1895, p. 132. (19) The Steam Engine, 1906. (21) Trans. A. S. M.
E., XV. (22) Ibid., XIII, 647. (23) Ibid , XIX, 189. (24) Ibid., loc. cit. (25)
Ibid., XXV, 482, 483, 490, 492. (26) Manuel du Conducteur des Machines Binaires,
Lyons, 18501851. (27) Peabody, Thermodynamics, 1907, 283. (28) Thurston,
Engine and Boiler Trials, p. 130. (29) Ripper, Steam Engine Theory and Practice,
1905, p. 412. (30) Experimental Engineering, 1907. (31) The Steam Engine Indica
SYNOPSIS 355
tor, 1898. Reference should also be made to Miller's and Hall's chapters of Prac
tical Instructions for using the Steam Engine Indicatory published by the Crosby
Steam Gage and Valve Company, 1905. (32) Low, op. at., pp. 103107; Carpen
ter, op. Git , pp. 4155, 531, 780. (33) Op. tit., p. 391. (34) Trans. A. S. M. E.,
VI, 716. (35) Ibid , XXV. (36) Ibid , 1892, also XXV, 827. (37) Ibid., XI.
(38) Ibid , XXIV, 713. (39) Op. ciL, 144. (40) The Steam Engine, p. 212. (41)
Bull. delaSoc.Ind deMulhouse, 1873. (42) Expose Succinct, etc.; Revue Unwerselle
des Mines, 1880. (43) Carpenter, Experimental Engineering, 1907, 657; Peabody,
Thermodynamics, 1907, 225. (44) Trans. A S. M. E, XXVIII, 2, 225. (45)
Zeuner, Technical Thermodijnamics (Klein), II, 449 (46) The Steam Engine, 1905,
I, 2, 3. (47) Locomotive Tests and Exhibits at the Louisiana Purchase Expositionj
1906. (48) Trans. A. S. M. E., 1892. (49) Ibtd., XXVIII, 10, 1658.
SYNOPSIS OP CHAPTER
Practical Modifications of the Rankine Cycle
With valves moving instantaneously at the ends of the stroke, the engine would
operate in the nonexpansive cycle. The introduction of cutoff makes the cycle
that of Rankine, modified as follows :
(1) Port friction reduces the pressure during admission. This causes a loss of availa
bility of the heat Regulation by throttling is wasteful.
(2) The expansion curve differs in shape and position from that in the ideal cycle.
Expansion is not adiabatic. The steam at the point of cutoff contains from 25
to 70 per cent of water on account of initial condensation. Further condensation
may occur very early in the expansion stroke, followed by reevaporation later
on, after the pressure has become sufficiently lowered. The exponent of the
expansion curve is a function of the initial dryness. The inner surfaces only of
the walls fluctuate m temperature. Condensation is influenced by
(a) the temperature range : wide limits, theoretically desirable, introduce some
practical losses ;
(6) the size of the engine : the exposed surface is proportionately greater in
small engines ,
(c) its speed : high speed gives less time for heat transfers ;
(d) the ratio of expansion : wide ratios increase condensation and decrease
efficiency, particularly because of increased initial condensation. Initial
wetness facilitates the formation of further moisture. In good design, the
ratio should be fixed to obtain reasonably complete expansion without
*^7 Is T *
excessive condensation, say at 4 or 5 to 1. M= ^ivl . Values of T.
Steam jackets provide steam insulation at constant temperature ; they oppose initial
condensation in the cylinder and are used principally with slow speeds and high
ratio of expansion. Some saving is always shown. Superheat, used under similar
conditions, increases the mean temperature of heat absorption. Each 75 of
superheat may increase the dryness at cutoff by 10 per cent. The actual expan
sion curve averages PV=pv. M.E.P.=Pj> with the RanMne
356 APPLIED THERMODYNAMICS
form of cycle. H . P . , 2 X dlagm itrtorXiiwUjr Diagnim faotor =0 .5 to
33000
0,9, With polytropic expansion, M.E.P.= ^ p D  JJ
(3) The exhaust line shows back pressure due to friction of ports, the presence of air,
and reevaporation. High altitudes increase the capacity of noncondensing
engines.
(4) Clearance varies from 2 to 15 per cent. "Real" and "apparent" ratios of
expansion.
(5) Compression "brings the piston to rest quietly ; though theoretically less desirable
than jacketing, it may reduce initial condensation.
(6) Valve action is not instantaneous, and the corners of the diagram are always
somewhat rounded. Leakage is an important cause of waste.
The Steam Engine Cycle on the Entropy Diagram.
Cushion steam, present throughout the cycle, is not included in measurements of
steam used.
Its volumes may be deducted, giving a diagram representing the behavior of the
cylinder feed alone.
The indicator diagram shows actions neither cyclic nor reversible : it depicts a
varying mass of steam.
The Boulvin diagram gives the NT history correctly along the expansion curve only.
The Reeve diagram eliminates the cushion steam J; it correctly depicts both expan
sion and compression curves, as referred to the cylinder feed.
The preferred diagram plots the expansion and compression curves separately.
Diagrams may show (a) loss by condensation, (&) gains by increased pressure and
decreased back pressure, (c) gains by superheating and jacketing.
Multiple Expansion
Increased initial pressure and decreased back pressure pay best with wide expansive
ratios.
Such ratios are possible, with multiple expansion, without excessive condensation.
Condensation is less serious because of (a) the use made of reevaporated steam,
(6) the decrease in initial condensation, and (c) the small size of the high
pressure cylinder.
Several numbers and arrangements of cylinders are possible with expansion in two,
three, or four stages.
Incidental advantages : less steam lost in clearance space ; compression begins later ;
the large cylinder is subjected to low pressure only j more uniform speed and
moderate stresses.
The Woolf engine had no receiver ; the lowpressure cylinder received steam through
out the stroke as discharged by the highpressure cylinder. The former, there
fore, worked without expansion. The piston phases coincided or differed by 180.
In the receiver engine, the pistons may have any phase relation and the lowpressure
cylinder works expansively. Early cutoS in the lowpressure cylinder increases
its proportion of the load, and is practically without effect on the total work of
the engine.
SYNOPSIS 357
The approximate point of lowpressure cutoff to eliminate drop may "be graphically
or analytically determined for tandem and crosscompound engines.
In combining diagrams, twi saturation curves are necessary, unles3 the cushion stcnm
be deducted.
The diagram factor has an approximate value the same as that in a simple engine hav
ing Wn expansions, in which n is the number of expansions in the compound
engine and c its number of expansive stages.
Cylinder ratios are 3 or 4 to 1 if noncondensing, 4 or 6 to 1 if condensing, iu com
pounds ; triples have ratios from 1 : 2.0 : 2.0 to 1 : 2.5 : 2.5. A large highpressure
cylinder gives high overload capacity.
The engine may be designed so as to equalize work areas, or by assuming the cylinder
ratio. " Equivalent simple cylinder." Values of E.
Governing should be by varying the point of cutoff in both cylinders.
Drop in any but the last cylinder is usually considered undesirable.
Exceptionally high efficiency is shown by compounds having cylinder ratios of 7 to 1.
The highpressure cylinder in ordinary compounds is too large for highest efficiency.
The binary vapor engine employs the waste heat of the exhaust to evaporate a fluid
having a lower boiling point than can be attained with steam. Additional work
may then be evolved down to a rejection temperature of 60 or 70 F. The best
result achieved is 167 B. t. u. per Ihp.minute.
Engine Tests
The indicator measures pressures and volumes in the cylinder and thus shows the
''cycle."
Its diagram gives the m. e. p. and points out errors in valve adjustment or control.
^, . , , , A ftiCio + TF) wh Who
Calorimeters : the barrel type : XQ =   ^j  J
 , . whi 4 Wh z wh
surface condensing : XQ = iZ_i __
superheating : XQ = ") . limits of capac j ty .
H+lctT^JioQ
JBarrus : XQ =   =^   ;
JL4)
separating : direct weighing of the steam and water;
chemical : insolubility of salts in dry steam ;
electrical : 1 B. t. u. per minute = 17.59 watts.
Engine trials : we may measure either the heat absorbed or the heat rejected + the work
done.
By measuring both, we obtain a heat balance.
Results usually stated : Ib. dry or actual steam per Ihp.hr.; B. t. u. per Ihp.minute ;
thermal efficiency ; work per Ib. tteam ; Carnot efficiency ; Clausius efficiency ;
efficiency ratios.
By assuming the steam dry at compression and cutoff or release, and knowing the
clearance, we may roughly estimate steam consumption from the indicator diagram.
ft.lb. of work per 100 Ib. coal (or per 1,000,000 B. t.u.) Plant efficiency
358 APPLIED THERMODYNAMICS
ffirn's analysis: E X =2M (h x +x r x Y, H X =E X +W X ; heat transfer to and from
walls may be computed from the supply of heat, the change in internal energy,
and the work done. The excess of losses over gains represents radiation.
Testing of regulation (speed control).
Types of Steam Engine
Standard engines : noncondensing or condensing, lighthand or lefthand, simple
or multiple expansion ; singleacting or doubleacting ; rotative or nonrotative ,
duplex or single ; horizontal, vertical, or inclined , locomotive, stationary (pump
ing, mill, power plant), or marine , "belted, directconnected, or ropedriven ; air
compressors ; girder, tangye, or semitangye frames ; slow, medium, or high speed ;
throttling, automatic, fourvalve, or releasing gear. The Stumpf uniflow engine.
The power plant: feedpump, boiler, engine, condenser.
The locomotive: tractive power =^rrS adhesion =0.22 to 0.25Xweight on drivers;
twocylinder and fourcylinder compounds , the balanced compound \ high econ
omy of locomotive engines.
PROBLEMS
1. Show from Art. 426 that the loss by a throttling process is equal to the prod
uct of the increase of entropy by the absolute temperature at the end of the process.
2. Ignoring ladiation, how fast are the walls gaining heat because of transfers
during expansion in an engine running at 100 r. p. m,, in which J pound of steam is
condensed per revolution at a mean pressure of 100 lb., and 0.30 pound is reevaporated
at a mean pressure of 42 lb. (Ans., 3637 B. t. u. per minute).
3 a. Plot curves representing the lesults of the tests given in Art. 434.
3 6. Represent Toy a curve the results of the Barms tests, Art. 436.
4. All other factors being the same, how much less initial condensation, at \ cut
off, should be found in an engine 30J"X48" than in one 7"x7"? (Art. 437).
5. Sketch a curve showing the variation hi engine efficiency with ratio of expan
sion.
6. Find the percentage of initial condensation at J cutoff in a noncondensing
engine using dry steam, running at 100 r. p. m. with a pressure at cutofE of 120 lb. t
the engine being 30"X48" (Art. 437).
7. In Fig. 193, assuming the initial pressure to have been 100 lb., the feedwater
temperature 90 I\, find the approximate thermal efficiencies with the various amounts
of superheat at a load of 15 hp.
8. In an ideal Clausius cycle with initially dry steam between p = 140 and p = 2
(Art. 417), by what percentage would the efficiency be increased if the initial pressure
were made 160 lb. ? By what percentage would it be decreased if the lower pressure
were made 6 lb. ?
9. Find the mean effective pressure in the ideal cycle with hyperbolic expansion
and no clearance between pressure limits of 120 and 2 lb., with a ratio of expansion
of 4. (Ans., 69.6 lb.)
10. Find the probable indicated horse power of a doubleacting engine with the
best type of valve gear, jackets, etc., operating as in Problem 9, at 100 r. p. m., the
cylinder being 30J"X4S". (Ignore the piston rod.) (Ans., 1107 hp.)
PROBLEMS 359
11. In Problem 9, what percentage of power is lost if the lower pressure is raised
to 3J Ib. ?
12. By what percentage would the capacity of an engine be increased at an altitude
of 10,000 ft. as compared with sea level, at 120 Ib. initial gauge pressure and a back
pressure 1 Ib, greater than that of the atmosphere, the ratio of expansion being 4 ?
(Atmospheric pressure decreases  Ib. per 1000 ft. of height.)
3. An engine has an apparent ratio of expansion of 4, and a clearance amounting
to 0.05 of the piston displacement, TVhat is its real ratio of expansion ? (Aiis., 3.5.)
14. In the dry steam ClausiiiR cycle of Problem 8, by what percentage are the ca
pacity and efficiency affected if expansion is hyperbolic instead of adiabatic ? Discuss
the results.
15. In an engine having a clearance volume of 1.0 and a back pressure of 2 Ib.,
the pressure at the end of compression is 40 Ib. If the compression curve is PF 1  03 =c,
what is the volume at the beginning of compression ? (Ans., 18.28.)
16. An engine works between 120 and 2 Ib. pressure, the piston displacement
being 20 cu. ft., clearance 5 per cent, and apparent ratio of expansion 4. The expan
sion curve is PV 1 02 = c, the compression curve PV 1 3 = c, and the final compression
pressure is 40 Ib. Plot the PV diagram with actual volumes of the cushion steam
eliminated.
17. In Problem 16, 1.825 Ib. of steam are present per cycle. Plot the entropy dia
gram from the indicator card by Boulvin's method.
18. In Problems 16 and 17, compute and plot the entropy diagram by Keeve's
method, assuming the steam dry at the beginning of compression. (See Art. 457.)
Discuss any differences between this diagram and that obtained in Problem 17.
19. In a nonexpansive cycle, find the theoretical changes in capacity and economy
by raising the initial pressure from 100 to 120 Ib., the back pressure being 2 Ib.
(Ans., 1.2 per cent gain in capacity : 8.5 per cent increase in efficiency.)
20. A nonexpansive engine with limiting volumes of 1 and 6 cu. ft. and an initial
pressure of 120 Ib., without compression, has its back pressure decreased from 4 to 2 Ib.
Find the changes in capacity and efficiency. The same steam is now allowed to expand
hyperbolically to a volume of 21 cu. ft. Find the effects following the reduction of
back pressure in this case. The steam is in each case dry at the point of cutoff.
(Ans., (a) 1.7 per cent increase in capacity and efficiency; (&) 3.2 per cent
increase in capacity and efficiency.
21. rind the cylinder dimensions of an automatic engine to develop 30 horse
power at 300 r. p. m., noncondensing, at J cutoff, the initial pressure being 100 Ib.
and the piston speed 300 ft. per minute. The engine is doubleacting.
22. Sketch a possible cylinder arrangement for a quadrupleexpansion engine with
seven cylinders, three of which are vertical and four horizontal, showing the receivers
and pipe connections.
23. Using the ideal combined diagram for a compound engine with a constant
receiver pressure, clearance being ignored, what must that receiver pressure be to
divide the diagram area equally, the pressure limits being 120 and 2 and the ratio of
expansion 16 ?
24. Consider a simple engine 30J"X48" and a compound engine 15 J" and
30J"X48", all cylinders having 5 per cent of clearance and no compression. What
9je the amounts of steam theoretically wasted in filling clearance spaces in the simple
360 APPLIED THERMODYNAMICS
engine and in the highpressure cylinder of the compound, the pressures being as in
Problem 23 ?
25. Take the same engines. The simple engine has a real ratio of expansion of 4;
the compound is as in Problems 23 and 24. Compression is to be carried to 40 Ib. in
the simple engine and to 60 Ib. in the compound in order to prevent waste of
steam. By what percentages are the work areas reduced in the two engines under
consideration ?
26. A crosscompound doubleacting engine operates between pressure limits of
120 and 2 Ib. at 100 r. p. m. and 800 ft. piston speed, developing 1000 hp. Find the
sizes of the cylinders under the following assumptions, there being no drop . (a) dia
gram factor 0.85, 20 expansioas, receiver pressure 24 Ib. ; (&) diagram factor O.S5,
20 expansions, work equally divided ; (c) diagram factor 0.85, ^0 expansions, cylinder
ratio 5:1; (d) diagram factor 0.83, 32 expansions, work equally divided. Find the
power developed by each cylinder in (a) and (c). Find the size of the cylinder of the
equivalent simple engine having a diagram factor of 0.80 with 20 expansions. Draw
up a tabular statement of the five designs and discuss their comparative merits.
27. lit Problem 26, Case (a), the receiver volume being equal to that of the low
pressure cylinder, find graphically and analytically the point of cutoff on the low
pressure cylinder.
28. Trace the combined diagram for one end of the cylinder from the first set of
cards in Fig. 230, assuming the clearance in each cylinder to have been 15 per cent
of the piston displacement, the cylinder ratio 3 to 1, and the pressure scales of both
cards to be the same.
29. Show on the entropy diagram the effect of reheating.
30. In Art. 483, what was the Carnot efficiency of the Josse engine ? Assuming
it to have been used in combination with a gas engine, the maximum temperature in
the latter being 3000 F., by what approximate amount might the Carnot efficiency
of the former have been increased ? (The temperature of saturated sulphur dioxide
at 35 Ib. pressure is 52 F.)
31. An indicator diagram has an area of 82,192.5 footpounds. What is the
mean effective pressure if the engine is 30"X48" ? What is the horse power of this
engine if it runs doubleacting at 100 r. p. m. ? (Ans^ 28. 1 Ib. ; 498 hp.)
32. Given points l r 2 on a hyperbolic curve, such that V* 7i = 15, P J =120,
jP 2 = 34.3, find the OPaxis.
33. An engine develops 500 hp. at full load, and 62 hp. when merely rotating its
wheel without external load. What is its mechanical efficiency * (Ans., 0.876.)
34. Steam at 100 Ib. pressure is mixed with water at 100. The weight of water
increases from 10 to 11 Ib., and its temperature rises to 197J. What was the per
centage of dryness of the steam ? ( Atis., 95 per cent.)
35. The same steam is condensed in and discharged from a coil, its temperature
becoming 210, and 10 Ib. of surrounding water rise in temperature from 100 to 204 J.
Find the quality of the steam. What would have been an easier way of determining
the quality ?
36. What is the maximum percentage of wetness that can be measured in a throt
tling calorimeter m steam at 100 Ib. pressure, if the discharge pressure is 30 Ib. ?
(Ans., 2.5 per cent.)
37. Steam at 100 Ib. pressure has added to it from an external source 30 B. t. u.
PROBLEMS 361
per pound. It is throttled to 30 Ib. pressure, its temperature becoming 270.3.
What was its diyness ? (Ans , 0.955.)
38. Under the pressure and temperature conditions of Problem 37, the added heat
is from an electric current ot 5 amperes provided for one minute, the Toltage f ailing
from 220 to 110. What was the amount of heat added and the percentage of dryness
of the steam ? (See Art. 494.) (Ans., 95.4 per cent.)
39. An engine consumes 10,000 Ib. of dry steam per hour, the moisture having
been completely eliminated by a receiver separator which at the end of one hour is
found to contain 285 Ib. of water. What was the dryness of the steam entering the
separator ? (Ans., 97.2 per cent.)
A doubleacting engine at 100 r. p. m. and a piston speed of 800 feet per minute
gives an indicator diagram in which the pressure limits are 120 and 2 Ib., the volume
limits 1 and 21 cu. ft. The apparent ratio of expansion is 4. The expansion curve
follows the lawPF 1  02 ^ c. Compression is to 40 Ib., according to the law PV 1 03 =c.
Disregard rounded corners. The boiler pressure is 130 Ib., the steam leaving the
boiler is dry, the steam at the throttle being 95 per cent dry and at 120 Ib. pressure.
The boiler evaporates 26,500 Ib. of steam per hour ; 2000 Ib. of steam are supplied to
the jackets at 120 Ib. pressure. The engine runs jetcondensing, the inlet water
weighing 530,000 Ib. per hour at 43.85 F., the outlet weighing 554,000 Ib. at 90 P.
The coal burned is 2700 Ib. per hour, its average heat value being 14,000 B. t. u.
Compute as follows :
40. The mean effective pressure and indicated horse power. (NOTE. The work
quantities under the curves must be computed with much accuracy.)
(Ans., 68.57 Ib.; 1196.8 hp.)
41. The cylinder dimensions of the engine. (Ans., 30.24 by 48 in.)
42. The heat supplied at the throttle per pound of cylinder and jacket steam, and
the B. t. u. consumed per Ihp. per minute ; the engine being charged with heat above
the temperatures of the respective discharges.
43. The dry steam consumption per Jhp.hr., thermal efficiency, and work per
pound of dry steam.
44. The Carnot efficiency, the Clausius efficiency, and the efficiency ratio, taking
the limiting conditions as at the throttle and the condenser outlet,
45. The cylinder feed steam consumption computed as in Art. 500 ; the consump
tion thus computed but assuming x = 0.80 at release, z= 1.00 at compression. Com
pare with Problem 43.
46. The percentage of steam lost by leakage (all leakage occurring between the
boiler and the engine); the transmissive efficiency ; the unaccountedfor losses.
47. The duty, the efficiency of the plant, and the boiler efficiency.
48. The heat transfers and the loss of heat by radiation, as in Art. 504, assuming
x 1.00 at compression. Compare the latter with the unaccountedfor heat obtained
In Problem 46.
49. The value of the mechanical equivalent of heat which might be computed
from the experiment, (Jns., 720.)
362 APPLIED THERMODYNAMICS
50. Explain the meaning of the figure 2068.84 in Art. 503.
51. Revise Fig. 233, showing the arrangement of machinery and piping if a sur
face condenser is used.
52. A locomotive weighing 2uO,000 Ib. carries, normally, 60 per cent of its weight
on its drivers. The cylinders are 19"X26", the wheels 66" in diameter. What is
the maximum boiler pressure that can be profitably utilized ? If the engine has a
traction increaser that may put 12,000 Ib. additional weight on the drivers, what
maximum boiler pressure may then be utilized ?
53. Represent Fig. 217 on the PV diagram.
54. Find the steam consumption in Ib. per Ihp.hr. of an ideal engine working in
the Clausius cycle between absolute pressures of 150 Ib. and 2 Ib., the steam contain
ing 2 per cent of moisture at the throttle. What is the thermal efficiency ?
55. What horse power will be given by the engine in Problem 10 if the ratio of
expansion is made (a) 5, (b) 3 ?
56. If an engine use dry steam at 150 Ib. absolute pressure, what change in
efficiency occurs when the back pressure is reduced from 2 to  Ib. absolute, if the
ratio of expansion is 30 ? If the ratio of expansion is 100 ?
CHAPTER XIV
THE STEAM TURBINE
512 The Turbine Principle. Figure 235 shows the method of using steam in
a typical impulse turbine. The expanding nozzles discharge a jet of steam at high
velocity and low pressure against
the blades or buckets, the im
pulse of the steam causing ro
tation. We have here, not
expansion of high pressure steam
against a piston, as in the ordi
nary engine, but utilization of
the kinetic energy of a rapidly
flowing stream to produce move
ment. One of the assumptions
of Art. 11 can now no longer
hold. All of the expansion oc
curs in the nozzle ; the expansion
j i A 4.u / , j Fro SSS Arts. 512, 524, 536. De Laval Turbine
produces velocity, the velocity does me ' el ^ Nozzles>
work. The lower the pressure
at which the steam leaves the nozzle, the greater is the velocity attained. It will
presently be shown that to fully utilize the energy of velocity, the buckets must
themselves move at a speed proportionate to that of the steam. This involves ex
tremely high rotative speeds.
The steps in the design of an impulse turbine are (a) determination
of the velocity produced by expansion, (6) computation of the nozzle
dimensions necessary to give the desired expansion, and (c) the propor
tioning of the buckets.
513. Expansive Path. There is a gradual fall of pressure while the
steam passes through the nozzle. With a given initial pressure, the pres
sure and temperature at any stated point along the nozzle should never
change. There is, therefore, no tendency toward a transfer of heat be
tween steam and walls. Further, the extreme rapidity of the movement
gives no time for such transfer ; so that the process in the nozzle is truly
adiabatic, although friction renders it nonisentropic. The first problem
of turbine design is then to determine the changes of velocity, volume,
temperature or dryness, and pressure, during such adiabatic expansion,
for a vapor initially wet, dry, or superheated ; the method may be accu
363
364 APPLIED THERMODYNAMICS
rate, approximate (exponential), or graphical. The results obtained are
to include the effect of nozzle friction.
514. The Turbine Cycle. Taking expansion in the turbine as adiabatic
and as carried down to the condenser pressure, the cycle is that of Clansius,
and is theoretically more efficient than that of any ordinary steam engine
working through the same range. The turbine is free from losses due to
interchange of heat tcitJi the icalls. The practical losses are four:
(a) Friction in the nozzles, causing a fall of temperature without the
performance of work ;
(&) Incomplete utilization of the kinetic energy by reason of the
assumed blade angles and residual velocity of the emerging jet (Art. 528);
(c) Friction along the buckets, increasing as some power of the stream
speed ;
(rZ) Mechanical friction of journals and gearing, and friction between
steam and rotor as a whole.
515. Heat Loss and Velocity. In Fig. 236, let a fluid flow adiabatically
from the vessel a through the frictionless orifice b. Let the internal en
ergy of the substance be e in a and E in 6; the
velocities v and V\ the pressures p and P\ and
the specific volumes w and W. If the velocities
could be ignored, as in previous computations,
the volume of each pound of fluid in a would
decrease by w in passing out at the constant
pressure^; and the volume of each pound of
FIG aril Art 515. Flow fl u jfl i n i W0 uld increase by W at the constant
ioug n ee pressure P. The net external work done would
be PW2M, the net loss of internal energy e E, and these two quan
tities would be equal. With appreciable velocity effects, we must also
consider the kinetic energies in a and b ; these are
and Zf;
2ff 2&
and we now lave
H=T+I+W+V,
(T+I)+W+V=0,
2g " ' ' 2g'
or = DW
HEAT DROP AND VELOCITY
365
Let X, Z7, H~, JR, and x, u, Ji, r, be the dry ness, increase of vol
ume during vaporization, heat of liquid, and internal latent heat, at
P W'diidpw respectively ; let * be the specific volume of water ; then
for expansion of a vapor from pw to P JF within the saturated region,

in which q, Q represent total heats of wet vapor above 32 degrees.
If expansion proceeds from the superheated to the saturated ret/ion,
y1
in which n = u 4 s is the volume of saturated steam at the pressure p,
w is the volume of superheated steam, and
p(w n)
is the internal energy measured above saturation.* This also re
duces to q Q f s(p P), where q is the total heat in the super
heated steam, and the same form of
expression will be found to apply to
expansion wholly in the superheated
region. The gain in kinetic energy
of a jet due to adiabatic expansion to
a lower pressure is thus equivalent to
the decrease in the total heat of the
steam plus the work which would be
required to force the liquid back F^ 337  Art '
against the same pressure head. In
Fig. 237, let al, AB, CD, represent the three paths. Then the
losses of heat are represented by the areas dale, deABc, deCDfc*
* For any gas treated as perfect, the gain of internal energy from t to T is
tf J. ~~ VJ Q JL ~~ tj ~" ~~~ ~~ ^ * ~ Uj ^ ,
!/ y ~~ * y "~* ^
or in this case, since internal energy is gained at constant pressure,
 Adiabatic Heat
366 APPLIED THERMODYNAMICS
The term s(pP) being ordinarily negligible, these areas also rep
resent the kinetic energy acquired, which may be written
V 2 r v 2
In the turbine nozzle, the initial velocity may also, without serious
error, be regarded as negligible; whence
=70 or 7= V50103.2(g Q) =223.84Vg :l Q feet per second.
20
516. Computation of Heat Drop. The value of q Q may be determined
for an adiabatic path between stated limits from the entropy diagram,
Fig. 175, or from the Mollier diagram, Fig. 177. Thus, from the last
named, steam at 100 Ib. absolute pressure and at 500 F. contains 1273
B. t. u. per pound; steam 85 per cent dry at 3 Ib. absolute pressure
contains 973 B. t. u. Steam at 150 Ib. absolute pressure and 600 F. con
tains 1317 B. t. u. If it expand adiabatically to 2.5 Ib. absolute pressure,
its condition becomes 88 per cent dry, its heat contents 1000 B. t. u., and
the velocity produced is
223.84 V317 = 3980 ft. per second.
517. Vacuum and Superheat. The entropy diagram indicates the nota
ble gain due to high vacua and superheat. Comparing dry steam expanded
from 150 Ib. to 4 Ib. absolute pressure with the same steam superheated
to 600 and expanded to 2.5 Ibs. absolute pressure, we find q Q in the
former case to be 248 B. t. u., and in the latter, 317 B. t. u. The corre
sponding values of V are 3330 and 3980 ft. per second. The turbine is
peculiarly adapted to realize the advantages of wide ratios of expansion.
These do not lead to an abnormally large cylinder, as in ordinary engines;
the "toe" of the Clausius diagram, Fig. 184, is gained by allowing the
steam to leave the nozzle at the condenser pressure. Superheat, also, is
not utilized merely in overcoming cylinder condensation 5 it increases the
available " fall " of heat, practically without diminution.
518. Effect of Friction. If the steam emerging from the nozzle were brought
back to rest in a closed chamber, the Mnetic energy would be reconverted into
heat, as in a wiredrawing process, and the expanded steam would become super
heated. Watkinson has, in fact, suggested this (1) as a method of supei heating
steam, the water being mechanically removed at the end of expansion, before re
conversion to heat began. In the nozzle, in piactice, the friction of the steam
against the walls does partially convert the velocity energy back to heat, and the
heat drop and velocity are both less than in the ideal case.
The efficiencies of nozzles vary according to the design from 0.90 to 0.97. The
corresponding variation in ratio of actual to ideal velocity is 0.95 to 0.99.
EFFECT OF NOZZLE FRICTION
367
In Fig. 238, for adiabatic expansion from j>, v, q, to P, V, Q, the
velocity imparted is
223 84 V?^ P
During expansion from p, v, g, to P^ Vi, Qi,
the velocity imparted is
223.84 V^ft
Since Fi exceeds F, the steam is more nearly
dry at Fi; i.e. Q l exceeds Q. The loss of
energy due to the path pvq P^ViQi as
compared with puy PVQ, is
FIG. 238. Art. 518 Abiabatic
Expansion with and without
Friction,
in which X 2 is the difference of the squares of the velocities at Q and ft.
This gives X 2 = 50103.2 (Q l  Q). In Fig. 239, let NA be the adiabatic
path, NX the modified path due to fric
tion. NZ represents a curve of constant
total heat ; along this, no work would be
done, but the heat would steadily lose its
availability. As NX recedes from NA
toward NZ, the work done during expan
sion decreases. Along NA, all of the heat
lost (area FHNA) is transformed into
work: along NZ, no heat is lost and no
. . , ,, r>nrT*m j
work 1S done ^ the areas BFHNQ and
BFZD being equal. Along NX, the heat
transformed into work is BFHNC  BFXE = FHNA CAXE, less
than that during adiabatic expansion by the amount of work converted
back to heat. Considering expansion from _ZVto Z 9
IE !o
FIG. 239 Art. 518. Expansive
Path as Modified by Friction.
F= 223.84 Vq=~& = 0,
since q = ft. Nozzle friction decreases the heat drop, the final velocity
attained, and the external work done.
519. Allowance for Friction Loss. For the present, we will assume
nozzle friction to reduce the heat drop by 10 per cent. In Fig. 240, which
is an enlarged view of a portion of Fig. 177, let AB represent adiabatic
(isentropic) expansion from the condition A to the state B. Lay off
'
368
APPLIED THERMODYNAMICS
and draw the line of constant heat CD.
H
Then D is the equivalent final
state at the same pressure
as that existing at B, and
AC represents the heat
drop corrected for friction.
Similarly by laying off
FIG. 240. Arts 519, 524, 3;i3, rJl\ 3;U The Steam Path
of the Tuibine
and drawing GE to inter
sect the 35lb. pressure
line, we find the point E
on the path AD of the
steam through the nozzle.
We may use the new heat
drop thus obtained in de
termining "T; or generally,
N if m is the friction loss,
and
If m = 0.10, F =
= 223.84
212.42 vq 
520. Analytical Relations. The influence of friction in determining the final
condition of the steam may be examined analytically. For example, let the initial
condition be wet or dry ; then friction will not ordinarily cause superheating, so
that the steam will remain saturated throughout expansion. Without friction, the
final dryness X Q would he given by the equation (Art. 392),
Friction causes a return to the steam of the quantity of heat m(q Q). This in
creases the filial dryness by  W^S/, making it
'
>o
If the initial condition is superheated to t g , and the final condition saturated,
adiabatic expansion would give
f, _ # n /
and friction would make the final condition
T flog, 3,+ ' + * log, f j j + m(q 
NOZZLE PROPORTIONS
369
If the steam is superheated throughout expansion, we have for the final tem
perature T st without friction,
log. 3, + \
=
in which the value of k Q must be obtained by successive approximations.
521. Rate of Flow. For a flow of G pounds per second at the velocity F, when
the specific volume is W, the necessary crosssectional area of nozzle is F = .
The values of W and V may be
read or inferred from the heat
chart or the formulas just given.
In Fig. 241 (2), let ab represent
frictionless adiabatic expansion
on the TN plane, a'b' the same
process on the PV plane. By
finding q a and values of Q at
various points along ab, we may
obtain a series of successive
values of V. The correspond
ing values of W being read from
a chart or computed, we plot the
curve MN, representing the re
lation of specific volume and
velocity throughout the expan FIG 241 Art. 521. Graphical Determination of
sion. Draw yy' parallel to W, Nozzle Area,
making Oy = G, to some con
venient scale. Draw any line OD from to MN, intersecting yy f at k. From
similar triangles, yk : yO : : On : nD, or yk =  F.
To find the prewnre at any specified point on the nozzle, lay off yk = F> draw
OkD, Dn, and project z to the PT plane. The minimum value of F is reached
when OD is tangent to j\TN. It becomes infinite when V = 0. The conclusion
that the crobssectionul area of the nozzle reaches a minimum at a certain stage in the
expansion will be presently verified.
522. Maximum Flow (2a). For a perfect gas,
yl'~ yr
k
If the initial velocity be negligible, we have, as the equation of flow (Art. 515),
9W PW
yl yl yl
and since
(pwPW);
Edt
370 APPLIED THERMODYNAMICS
Then.
From Art. 521,
Taking the value of V at
we obtain
G=
This reaches a maximum, for air, when P p 0.5274 (3). The velocity is then
equal to that of sound. For dry steam, on the assumption that y = 1.135, and
that the above relations apply, the ratio for maximum flow is 0.577.
Using the value just given for the ratio P p, with y = 1.402, the equation
for G simplifies to
the equation of flow of a permanent gas, which has been closely confirmed by
experiment. With steam, the ratio of the specific heats is more variable, and the
ratio of pressures has not been as well confirmed experimentally. Close approxi
mations have been made. Claike (4), for example, shows maximum flow with
saturated steam to occur at an average ratio of 0.56. The pressure of maximum
flow determines the minimum or throat diameter of the nozzle, which is independ
ent of the discharge pressure. The emerging velocity may be greater than that
in the throat if the steam is allowed to further expand after passing the throat.
The nozzle should in all cases continue beyond the throat, either straight or ex
panding, if the kinetic energy is all to be utilized in the direction of flow.
In all cases, the steam velocity theoretically attained at the throat of the nozzle
will be 1450 ft per second.
523. Experiments. Many experiments have been made on the flow of fluids
through *nozzles and orifices. Those of Jones and Rathbone (5), Rosenhain (6),
Gutermuth (7), Napier (8), Rateau (9), Hall (10), Wilson (11), Kunhardt (12),
Buchner (13), Kneass (14), Lewicki (15), Durley (16), and chiefly, perhaps, those
of Stodola (17), should be studied. There is room for further advance in our
knowledge of the friction losses in nozzles of various proportions. There are sev
eral methods of experimentation : the steam, after passing the orifice, may be con
densed and weighed; the pressure at various points in the nozzle may be measured
by side orifices or by a searching tube ; or the reaction or the impulse of the steam
at its escape may be measured. The velocity cannot be measured directly.
TYPES OF TURBINE
371
A greater rate of flow is obtainable through an orifice in a thin plate (Fig.
242) than through an expanding nozzle (Fig. 243). For pressures under 80 lb.,
with discharge into the atmosphere, the plain oiifice is more efficient
in producing velocity. For wider pressure ranges, a divergent
nozzle is necessary to avoid deferred expansion occurring after
emergence. Expansion should not, however, be carried to a pres
sure lower than that of dischaige. The rate of flow, but not the
emeiging velocity, depends upon the shape of the inlet; a slightly
rounded edge (Fig. 243) gives the greatest rate ; a greater amount ^ IG 342. Art.
of rounding may be less desirable. The experimentally observed 523. Diverg
critical pressure ratio ( , Art. 522 J ranges with various fluids mg n Ce *
from, 0.50 to 0.85. Maximum flow occurs at the lower ratios with rather sharp
corners at the entrance, and at the higher ratios when a long divergence occurs
beyond the throat, as in Fig. 243. The "most efficient"
nozzle will have different proportions for different pressure
ranges. The pressure is, in general, greater at all points
along the nozzle than theory would indicate, on account of
243. Arts. 523, friction ; the excess is at first slight, but increases more and
525 Expanding more rapidly during the passage. Most experiments have
necessarily been made on very small orifices, discharging to
the atmosphere. The fiiction losses in larger orifices are probably less. The
experimental method should include at least two of the measurements above
mentioned, these checking each other. The theory of the action in the nozzle
has been presented by Heck (18). Zeuner (19) has discussed the flow of gases to
and from the atmosphere (20), both under adiabatic and actual conditions, and
the efflux of gases in general through orifices and long pipes.
524. Types of Turbine. The single stage impulse turbine of Fig.
235 is that of De Laval. Its action is illustrated in Fig. 244. The
pressure falls in the nozzle, and remains
PRESSURES
constant in the buckets. The Curtis and
......... Rateau turbines
use 'a series of
wheels, with ex
panding nozzles
between the va
FIG. 244.
Art. 524. De Laval
Turbine.
FIG. 245.
rious series (Figs.
245, 246). The steam is only partially ex
Art. 524. Curtis panded in each nozzle, until it reaches the
last one. Such turbines are of the multi
stage impulse type. During passage through the blades, the ve
locity decreases, while the pressure remains unchanged. In the
372
APPLIED THERMODYNAMICS
pressure turbine of Parsons, there are no expanding nozzles ; the
steam passes successively through the stationary guide vanes OS g,
_ FBESSUSES^ and movable wheel buckets, TFJ w. Fig. 247.
^ A gradual fall of pressure occurs, the buck
"T*SI!! OF ets being at all times full of steam. In
impulse turbines, the buckets need not be
full of steam, and the pressure drop occurs
FIG. 246 Art. 524. Rateau i n the nozzle only.
Turbme " A lower rotative speed results from the
**
use of several pressure stages with expanding nozzles
total heat drop of 317 B. t. u., in Art.
516, be divided into three stages by three
sets of nozzles. The exit velocity from G
each nozzle, corrected for friction, is
^ = 2180 ft. per sec
Let the
Arts. 524, 533. Parsons
Turbine.
then 212.
ond, instead of 3980 ft. per second; lay
ing off in Fig. 240 the three equal heat
drops, we find that the nozzles expand between 150 and 50, 50 and
13, and 13 and 2.5 Ib. respectively. The rotative speeds of the wheels
(proportional to the 'emerging velocities), Art. 52S ; are thus reduced.
525. Nozzle Proportions ; Volumes. The specific volume W of the.
steam at any point along the path AD, Fig. 240, having been obtained
from inspection of the entropy chart, or from the equation of condition,
and the velocity V at the same point having been computed from the
WQ
heat drop, the crosssectional area of the nozzle, in square feet, is F= 
(Art. 521). Finding values of jPfor various points along the expansive
path, we may plot the nozzle as in Fig. 243, making the horizontal inter
vals, abj be, cd, etc., such that the angle between the diverging sides is
about 10, following standard practice.* It has been shown that I 1 reaches
a minimum value when tlie pressure is about 0.57 of the initial pres
sure, and then increases as the pressure falls further. If the lowest
pressure exceeds 0.57 of the initial pressure, the nozzle converges toward
the outlet. Otherwise, the nozzle converges and afterwards expands, as
in Fig. 243. Let, in such ease, o be the minimum diameter, the outlet
diameter, L the length between these diameters; then for an angle of
10 between the sides, ~ = L tan 5, or L = 5.715(0 o).
2i 2
* A variable taper may be used to give constant acceleration of the steam
VELOCITY DIAGRAMS
373
526. Work Done. The work done in the ideal cycle per pound
of steam is 778(2 Q) footpounds. Since 1 horse power = 1,980,000
footpounds per hour, the steam consumption per hp.hr. is theoreti
cally 1,980,000 r 778(2  <?) = 2545 s (y  <?). If H is the effi
ciency ratio of the turbine, from steam to buckets, and e the
efficiency from steam to shaft, then the actual steam consumption
per indicated horse power is 2545 5 E(q Q), and per brake horse
power is 2545 f e(q Q*) pounds. The modifying influences of nozzle
and bucket friction in determining ]3 are still to be considered.
527. Relative Velocities. In Fig. 248, let a jet of steam strike
the bucket A at the velocity t;, the bucket itself moving at the speed
u. The velocity of the steam rela
tive to the bucket is then repre
sented in magnitude and direction
by V. The angles a and e made
with the plane of rotation of the
bucket wheel are called the absolute
entering and relative entering angles
respectively. Analytically, sin e = v
rr rp, . J , FIG. 248. Art. 527. Velocity Diagram,
sin a 5 v. 1 he stream traverses
the surface of the bucket, leaving it with the relative velocity a/,
which for convenience is drawn as x from the point 0. Without
bucket friction, x = V. The
angle / is the relative angle of
exit. Laying off w, from 2, we
find Y as the absolute exit ve
locity, with g as the absolute
angle of exit. Then, if x = V,
To include the effect of nozzle
and bucket friction, we proceed
as in Fig. 249, decreasing v to
VI m of its original value
(Art. 519), and making x less than F'by from 5 to 20 per cent, as
in ordinary practice. As before, sin e = v sin a*~V\ but for a bucket
friction of 10 per cent, sin^ = 0.9 F"sin/f Y.
FIG. 240, Arts. 527, 532, 534. Velocity
Corrected for Friction.
374
APPLIED THERMODYNAMICS
FIG. 250 Arts 528,529.
Rotative and Thrust
Components.
528. Bucket Angles and Work Done. In. Fig. 250, the absolute
velocities v and Y may be resolved into components ab and db in the
direction of rotation, and ac and de at right
angles to this direction. The former compo
nents are those which move the wheel ; the lat
ter produce an end thrust on the shaft. Now
ab 4 M (Id being negative) is the change in
velocity of the fluid in the direction of rotation ;
it is the acceleration; the force exerted per
pound is then
(ab + M)*ff= s (ab 4 Brf) 4 32.2
= (y cos a 4 Fcos y) f 32. 2.
This force is exerted through the distance u
feet per second ; the work done per pound of steam is then.
u(v cos a 4 T"cos5r)7 32.2 footpounds. This, from Art. 526, equals
778 U (2 <?) whence
J= (z; cos a 4 rcos^)* 25051.6(2 <?)
The efficiency is thus directly related to the bucket angles.
To avoid splashing, the entrance angle of the bucket is usually
made equal to the relative entering angle of the jet, as in Fig. 251.
(These formulas hold only when the sides of the
buckets are enclosed to prevent the lateral
spreading of the stream.) In actual turbines,
Id (Fig. 250) is often not negative, on account
of the extreme reversal of direction that would
be necessary. With positive values of Id, the
maximum work is obtained as its value ap
proaches zero, and ultimately it is uv cos #~32.2.
o
Since the kinetic energy of the jet is , the
efficiency
2  cos a.
FIG 251. Art 528
Velocities and Bucket
Angles.
5? from steam to buckets then becomes
In designing, we may either select an exit bucket angle
which shall make Id equal to zero (the relative exit velocity being
tangential to the surface of the bucket), or we may choose such an
angle that the end thrust components de and ca^ Fig. 250, shall bal
VELOCITY EFFICIENCY
375
ance. In marine service, some end thrust is advantageous ; in
stationary work, an effort is made to eliminate it. This would be
accomplished by making the entrance and exit bucket angles equal,
for a zero retardation by friction. With friction considered, the
angle of exit 1C, in Fig. 251, must be greater than the entering an
gle e. In any case, where end thrust is to be eliminated, the rota
tive component of the absolute exit velocity must be so adjusted as
to have a detrimental effect on the economy.
529. Effect of Stream Direction on Efficiency. Let the stream strike
the bucket in the direction of rotation, so that the angle a = 0, Fig. 250,
the relative exit velocity being perpendicular
to the plane of the wheel. The work done is
_v
kinetic energy is The
efficiency, 2 u
u
becoines a maximum at
0.50 when u =  With a ciipshaped vane, as
_j
in the Pelton wheel, Pig. 252, complete reversal
of the jet occurs 5 the absolute exit velocity,
ignoring friction, is v2u. The change in FIG. 252 Arts. 529, 536. Pel
velocity is v + v 2 u = 2(v ?*), and the work ton Bucket<
is 2u(v u) f g, whence the efficiency, 7" ? becomes a maximum
at 100 per cent when u = ^ Complete reversal in turbine buckets is im
practicable. ^
530. SingleStage Impulse Turbine. The absolute velocity of steam enter
ing the buckets is computed from the heat drop and nozzle friction losses. In a
u turbine of this type, the speed of the
v ! buckets can scarcely be made equal
to half that of the steam; a more
usual proportion is 0.3. The velocity
u thus seldom exceeds 1400 ft. per
second. Fixing the bucket speed and
the absolute entering angje of the
steam (usually 20) we determine
graphically the entering angle of the
bucket. The bucket may now be de
signed with equal angles, which would
eliminate end thrust if there were no
FIG. 253. Art. 530. Bucket Outline. friction, or, allowance being made for
376
APPLIED THERMODYNAMICS
friction, either end thrust or the rotative component of the absolute exit velocity
may be eliminated. The normals to the tangents at the edges of the buckets being
I drawn, as ec, Fig. 253,
the radius r is made
equal to about 0.965 ec.
The thickness t may
be made equal to 0.2
times the width kl.
The bucket as thus
drawn is to a scale as
yet undetermined;
the widths kl vary in
practice from 0.2 to
1.0 inch. (For a study
of steam trajectories
and the relation there
of to bucket design,
see Roe, Steam Tur
bines, 1911.)
It should be noted
that the back, rather
than the front, of the
bucket is made tan
gent to the relative
velocity V. The work
per pound of steam
being computed from
the velocity diagram,
and the steam con
sumption estimated
for the assumed out
put, we are now in a
position to design the
nozzle.
531. Multistage
Impulse Turbine. If
the number of pres
sure stages is few, as
in the Curtis type, the
heat drop may be di
vided equally between
the stages. In the
Bateau type, with a
large number of
FIG. 254. Art. 531. Curtis Turbine. (General Electric Company ) Stages, a proportion
ately greater heat drop
occurs in the lowpressure stages. The corresponding intermediate pressures are
determined from the heat diagram, and the various stages are then designed as
DESIGN OF MULTISTAGE TURBINE
377
separate singlestage impulse turbines, all having the same rotative speed. The
entrance angles of the fixed intermediate blades in the Curtis turbine are equal to
those of the absolute exit velocities of the steam. Their exit angles may be
adjusted as desired; they may be equal to the entrance angles if the latter are not
too acute. The greater the number of pressure stages, the lower is the economical
limit of circumferential speed; and if the number of revolutions is fixed, the smaller
will be the wheel. Figure 254 shows a form of Curtis turbine, with five pressure
stages, each containing two rows of moving buckets. The electric generator is at
the top.
532. Problem. Preliminary Calculations for a Multistage Impulse Tnrline.
To design a 1000 (brake) hp. impulse turbine with three pleasure stages, having
two moving wheels in each pressure stage. Initial pressuie, 130 Ib. absolute;
temperature, 600 F. ; final pressure, 2 Ib. absolute; entering stream angles, 20;
peripheral velocity, 500 ft. per second ; 1200 revolutions per minute.
By reproducing as in Fig. 240 a portion of the Molher heat chart, we obtain
the expansive pat,h AB, and the heat drop is 1316.6  987.5 = 329.1 B. t. u. Divid
ing this into three equal parts, the heat drop per stage becomes 329.1 3 = 109.7
B. t. u. This is without correction for friction, and we may expect a somewhat
unequal division to appear as friction is considered. To include friction in deter
mining the change of condition during flow through the nozzle, we lay off, in Fig.
240, AH = 109.7, HG = , and project GE, finding/* = 50, t = 380, at the out
lets of the first set of nozzles. The velocity attained (with 10 per cent loss of
available heat by friction) is v = 212.42 V109.7 = 2225 ft. per second
t u *n f
FIG. 255. Art. 532. Multistage Velocity Diagram.
We now lay off the velocity diagram, Fig. 249, making a =20, ^ = 500,
v=2225. The exit velocity x may be variously drawn; we will assume it so that
378 APPLIED THERMODYNAMICS
the relative angles e and/ are equal, and, allowing 10 per cent for bucket friction,
will make x 0.9 F. For the second wheel, the angle a' is again 20, while v', on
account of friction along the stationar} T or guide blades, is 0.9 Y. After locating
F', if the angles e 1 and/ 7 were made equal, there would in some cases be a back
ward impulse upon the wheel, tending to stop it, at the emergence of the jet along
T. On the other hand, if the angle/' weie made too acute, the stream would be
unable to get away from the moving buckets. With the particular angles and
velocities chosen, some backward impulse is inevitable. AVe will limit it by mak
ing/' = 30. The rotative components of the absolute velocities may be computed
as follows, the values being checked as noted from the complete graphical solution
of Fig. 255 :
ab = v cos 20 = 2225 x 0.93969 = 2090.81. (2080)
cd = cz  rfz = 0.9 Fcos/ u 0.9 Fcose  u = 0.9(2090.81  500) 500 = 931.73.
(925)
ef= eg cos20 = 0.9 c#*cos20 = 0.9 x 1158 x 0.93969 = 979. (975)
U = km  Im = 500  x 1 cos 30 = 500  0.9 V cos 30
= 500  (0.9 x 596.2 f x 0.80603)= 36,
The work per pound of steam is then ("* + "* +/"*') = 30fiG x 50 = 61500
v O < w / O**'i
footpounds, in the first stage. This is equivalent to 61,500 778 = 79.2 B. t. u.
The heat drop assumed foi this stage was 109.7 B. t. u. The heat not converted
Into work exists as lesidual velocity or has been expended in overcoming nozzle
and bucket friction and thus indirectly in superheating the steam. It amounts
to 109 7  79 2 = 30.5 B, t. u.
Returning to the construction of Fig. 240, we lay off in Fig. 256 aw, = 79.2
B. t. u. and project no to r?, finding the condition of the steam after passing the
first stage buckets. Bucket friction has moved the state point from m to o, at
which latter point Q = 12:37.2, p = 50, t = 414. This is the condition of the steam
which is to enter the second set of nozzles. These nozzles are to expand the steam
down to that pressure at which the ideal (adiabatic) heat drop from the initial
condition is 2 x 109.7 = 219.4 B. t. u. Lay off ae = 219.4, and find the line eg of
12 Ib. absolute pressure. Drawing the adiabatic op to intersect eg, we find the
heat drop for the second stage, without friction, to be 1237.2 1120 = 117.2 B. t. u.,
giving a velocity of 21 2.42 Vl 17.2 = 2299.66 ft. per second.
* To find cgr, we Lave
cb = Fcos e = 2090.81  500 = 1590.81, bj = v sin a = 2225 x S4202 = 760.99,
F= ^cb 2 + ty* = Vi5iio.81* + 700.W* = 1705, T: = 9 F= 0.9 x 1765 = 1688.5,
ch = 3 sin/ = 1588 5 sin e = 1688.5^ = 1588.5 700  99 = 685,
_ _ F 1765
eg = V'ch 2 + hf = Veg^+MTfl 2 = 1158.
t To find F', we have
gf= v' sin 20 = Tsin 20 = 0.9 x 1158 x 0.34202 = 355,
ft/= tf u = 979  500 = 470, F/ = rftf* + gf* =^ / 479 2 + 35? = 596.2.
STEAM PATH, MULTISTAGE TURBINE
379
The complete velocity diagram must now be drawn for the second stage, fol
lowing the method of Fig. 255. This gives for the rotative components, ab  2160.97,
cd = 994.87, ef= 1032.59, LI = 8.06. (There is no backward impulse from kl in
this case.) The work per pound of steam is
500(2160.97+994.87+1032.59+8.06) =
or 83.76 B. t, u. Of the available heat drop, 117.2 B. t. u., 33.44 have been ex
pended in friction, etc. Laying off, in Fig. 256, pq = 33.44, and projecting qr to
meet pr, we have r as the state point
for steam entering the third set of nozzles.
Here p = 12, *i=223, <?'i=115344. In
expanding to the final condenser pressure,
the ideal path is rs, terminating at 2 Ib.
absolute, and giving an uncorrected heat
drop of Q r & = 1153.441039 = 114.44
B. t. u. The velocity attained is
212.42 VlU.44 =2271.83 feet per second.
A third velocity diagram shows the
work per pound of steam for this
stage to be 63,823 footpounds, or 82.04
B.t.u. We are not at present con
cerned with determining the condition
of the steam at its exit from the third
stage,
The whole work obtained from a
pound of steam passing through the three
stages is then 79 .2 +83. 76 +82.04= 245.0
B. t. u. (20a). The horse power required
is 1000 at the brake or say 10000.8 =
FIG. 256. Art. 532. Steam Path, Multi
stage Turbine.
1250 hp. at the buckets. This is equivalent to 1250 X
1980000
778
' 3,181,250 B. t. u.
per hour. The pounds of steam necessary per hour are 3,181,2504245.0=12,974.
This is equivalent to 12.97 Ib. per brake hp,hr., a result sufficiently well confirmed
by the test results given in Chapter XV.
GW
Proceeding now to the nozzle design, we adopt the formula F= from Art.
521. It will be sufficiently accurate to compute crosssectional areas at throats
and outlets only. The path of the steam, in Fig. 256, is as follows: through the
first set of nozzles, along am; through the corresponding buckets, along mo; thence
alternately through nozzles and buckets along ou, ur, n>, vt. The points u, v 3 etc.,
are found as in Fig. 240. It is not necessary to plot accurately the whole of the
paths am, ou, rv] but the condition of the steam must be determined, for each
nozzle, at that point at which the pressure is 0.57 the initial pressure (Art. 522).
The three initial pressures are 150, 50, and 12; the corresponding throat pressures
are 85.5, 28.5. and 6.84. Drawing these lines of pressure, we lay off, for example,
, project xy to wy, and thus determine the state y at the throats of the
380
APPLIED THERMODYNAMICS
first set ot nozzles. The corresponding states are similarly determined for the
other nozzles. We thus find,
at y, p = 85.5, t = 474, at m, p = 50, t = 380,
q = 1260.5 ; q = 1217.87 ;
at A, p = 28.5, t = 313, at u, p = 12, x = 0.989,
q = 1192 ; q = 1131.72 ;
at B, p = 6.84, ar = 0.9835, at u, ^ = 2, a; = 0,932,
= 1118; 5=1050.44.
We now tabulate the corresponding velocities and specific volumes, as below.
The former are obtained by taking V = 223.84 V^  q 2 ; the latter are computed from
the Tumlirz formula, W = 0.5963  0.256.
Thus, at the throat of the first nozzle,
V = 223.84 V1316.8  12(50.5 = 1683 ; while W = 0.5963 4GO + 474 _ 0.256 = 6.26.
80. 5
In the wet region, the Tumlirz formula is used to obtain the volume of dry
steam at the stated pressure and the tabular corresponding temperature ; this is
applied to the wet vapor : W w = 0.017 + x( W  0.017) . The tabulation f ollows.
At y, V = 1683, W = 6.26 ; at m, V = 2225, W = 9.724 :
at A V= 1507, W = 15.92 ; at ti, F= 2299, TT= 32.24;
at B, V = 1330, TF = 53.92 ; at v, 7 = 2271, W = 162.62.
The value of G 9 the weight of steam flowing per second, is 12,974 3600 = 3.604 Ib.
For reasonable proportions, we will assume the number of nozzles to be 16 in the
first stage, 42 in the second, and 180 in the third. The values of G per nozzle for
the successive stages are then 3.604 16 = 0.22525, 3.604  42 = 0.08581 and
3.604 ^ 180 = 0.02002. We find values of F as follows :
Aty,
at m,
0.22525 x 6.26
1683
0.22525 x 9.724
2225
0.08581 x 15.92
= 0.000839; at u,
= 0.000989; at 3,
= 0.000903; at u,
0.08581 x 32.24
2299
0.02002 x 53.92
1330
0.02002x162.62
= 0.001205 ;
= 0.000809;
= 0.00144.
' 1507 ' ' 2271
Completing the computation as to the last set of nozzles only, the throat
area is 0.000809 sq. ft, that at the outlet being 0.00144 sq. ft. These corre
spond to diameters of 0.385 and
0.515 in. The taper may be uniform
from throat to outlet, the sides mak
ing an angle of 10. This requires
a length from throat to outlet of
(0.515  0.385)  2 tan 5 = 0.742 in.
The length from inlet to throat may
be one fourth this, or 0.186 in., the
FIG. 267. Axt.532.mrd Stage Nozzle. ^f * i*^ l^* ^ oT^'
The nozzle is shown in Fig. 257.
The diameter of the bucket wheels at midheight is obtained from the rotative
speed and peripheral velocity. If d be the diameter,
3.1416 d x 1200 = 60 x 500, or d = 7.98 feet.
PRESSURE TURBINE 381
The forms of bucket are derived from the velocity diagrams. For the first
stage, we proceed as in Art. 530, using the relative angles e and /given in Fig. 255
for determining the angles of the backs of the moving blades, and the absolute
angles for determining those of the stationary blades.
533. Utilization of Pressure Energy. Besides the energy of impulse
against the wheel, unaccompanied by changes in pressure, the steam may
expand while traversing the buckets, producing work by reaction. This
involves incomplete expansion in the nozzle, and makes the velocities of
the discharged jets much less than in a pure impulse turbine. Lower
rotative speeds are therefore practicable. Loss of efficiency is avoided by
carrying the ultimate expansion down to the condenser pressure. In the
pure pressure turbine of Parsons, there are no expanding nozzles ; all of
the expansion occurs in the buckets (Art. 524). (See Fig. 247.) Here
the whole useful effort is produced by the reaction of the expanding steam
as it emerges from the working blades to the guide blades. No velocity is
given up during the passage of the steam ; the velocity is, in fact, increasing,
hence the name reaction turbine. The impulse turbine, on the contrary,
performs work solely because of the force with which the swiftly moving
jet strikes the vane. It is sometimes called the velocity turbine. Turbines
are further classified as horizontal or vertical, according to the position of
the shaft, and as radial flow or axial flow, according to the location of the
successive rows of buckets. Most pressure turbines are of the axial flow
type.
534. Design of Pressure Turbine. The number of stages is now large. The
heat drop in any stage is so small that the entering velocity is no longer negligible.
The velocity of the steam will increase continually throughout the machine, being
augmented by expansion more rapidly than it is decreased by friction. If the
effective velocity at entrance to a row of moving blades is Fi, increasing to F a by
reason of expansion occurring in the blades, the energy of reaction, available for
7 2 2_7j2
performing work, is  . The effective velocity entering the stationary blades
being Fa, and increasing to V by expansion therein, energy is produced equal to
7 4 z_y 3 2
  , which is given up to the following set of moving blades, in the shape of an
impulse. Each moving blade thus receives an impulse at its entrance end and a
reaction at its outlet end. By making the forms and angles of fixed and moving
blades the same, the work done by impulse equals the work done by reaction, or
In Fig. 259, lay off the horizontal distance F0 } representing the aggregate axial
length of four drums composing a pressure turbine. The peripheral speeds of
drums vary from 100 to 350 ft. per sec., increasing as the pressure decreases and
382
APPLIED THERMODYNAMICS
as the size of the machine increases, and being generally less in marine than in
stationary service The successive drum diameters and peripheral speeds frequently
have the ratio A/2 : 1 (21) Assume, in this case, that the peripheral speed of the
first drum is 130 it per sec., and that
A of the last drum 350 ft per sec. The
* usual plan is to increase the successive
i drum speeds at constant ratio. This
makes the speeds of the blades on the
intermediate drums 181 and 251 ft per
sec , respectively.
The steam velocity will be usually
between If and 3 times the blade ve
locity. it will increase more rapidly as
Art. 534, Piob. 17. Design of the f ower pressures are reached The
Pressure Turbine. yalue of thifl ratlo should vary between
about the same limits for each drum.
The curve EA is sketched to represent steam velocities assumed: the ordinate
FE may be 130X2 = 260 ft. per second, and the ordinate OA say 973 ft. per sec.
The shape of this cuive is approximately hyperbolic.
It is now desirable to lay off on the axis FO distances representing approximately
the lengths of the various drums. An empirical formula which facilitates this is
C
Fro. 259.
where %=number of rows of blades when the blade speed is u ft. per sec.,
C = a constant, =1,500,000 for marine turbines, =2,600,000 for turbogenerators.
When (as in our case) u is different for different drums, we have
ni being the number of stages on a drum of blade velocity wi> developing the s pro
portion of the total power. The power developed by the successive drums increases
toward the exhaust end : let the division in this case be }, , 1, f , of the total respect
ively. Then for (7 = 2,600,000,
2,600,000 1
~ X 6"~ 2b}
2,600,000 1 
~ X '
2,600,000
2,600,000 3
350~ X 8
The total number of stages is then approximately 60. The distances FC, CD, DB,
BO, are then laid off, equal respectively to !, i, $ and & of FO, At any point
like G, then, the steam velocity is ZG and the blade velocity is that for the drum
in question: for G, for example, it is 181 ft. per sec.
Knowing the steam velocity and peripheral velocity for any state like <?, we
construct a velocity diagram as in Fig. 249, choosing appropriate angles of entrance
and exit. In ordinary practice, the expansion in the buckets is sufficient, not
PRESSURE TURBINE
383
withstanding friction, to make the relative exit and absolute entrance angles and
velocities about equal. (This equalizes the amounts of work done by impact
and by reaction.) In such case, we have the simple graphical construction of
Fig. 260.
Since abbc, db=*be, and ad=ec, we ob
tain
. u(ah+he) ad(hc + hd)
W0rk '
Drop the perpendicular bh, and with h as
a center describe the arc aj. Draw dg per
pendicular to ac. Then
dg 2 = adXdc = ad(dh+hc), and
footpounds, or
B. t. u.
FIG. 260 Art. 634, Prob. 18. Velocity
Diagram, Pressure Turbine.
This result represents the heat converted
into work at a stage located vertically in
line with the point G, Fig. 259, Let this heat be laid off to some convenient
scale, as GH. Similar determinations for other states give the heat drop curve
IJKELMNOP. The average ordinate of this curve is the average heat drop or
work done per stage. If we divide the total heat drop obtained by the average
drop per stage, we have the number of stages, the nearest whole number being taken.*
Suppose the machine to be required to drive a 2000 kw. generator (2400 kw. overload
capacity) at 175 to. initial absolute pressure and 50 of superheat, the condenser pressure
being 1 Ib absolute, the r. p. m. 3600, the generator efficiency 0.94 and the losses as follows:
steam friction, 25; leakage, 06, windage and bearings, 0.16; residual velocity in
exhaust, 0.03. The theoretical heat drop is 1227890=337 B. t. u. The drop
corrected for steam friction is 337X0.75 =253 B. t. u. The average ordinate of the
heat drop curve in Fig. 259 being 4.16 B. t. u., the corrected number of stages is
253
=61 (nearest whole number) instead of 60. The curve of heat drops may now
4.16
be corrected for the necessary revised numbers of stages in the various drums: thus,
253
the whole heat drop being 253 B. t. u., that in the first drum must be =42 2
6
B.\ u. The average heat drop per stage for the first drum being (average ordinate
42.2
of U) 1.56 B, t. u., the number of stages on that drum is ~ = 27 (instead of 26).
r  1.56
For the other drums, proceeding in the same way, the numbers of stages work out
as before, 16, 10 and 8.
The aggregate of losses exclusive of steam friction is 0.25. The heat available
for producing power is then 253X0.75 = 190 B. t. u. per Ib. of steam. With the
given generator efficiency, the weight of steam required per kw.hr. is
2545 X 1.34
190X0.94 5
= 19.0.
* Dividing the total heat drop at a state in a vertical line through C by the average
drop per stage from F to C, we have the number of stages on the first drum.
384
APPLIED THERMODYNAMICS
At normal rate, the weight of steam used at the overload condition is
19.0X2400
3600
12.67 Ib. per sec.
535. Specimen Case. To determine the general characteristics of a pressure
turbine operating between pressures of 100 and 3 5 Ib., with an initial superheat
of 300 F., the heat drop being reduced 25 per cent by friction. There are to he
3 drums, and the heat drop is to be equally divided between the drums. The per
ipheral speeds of the successive drums are 160, 240, 320 ft. per second. The rela
tive entrance and absolute exit velocities and angles are equal; the absolute entrance
angle is 20. The turbine makes 3000 r. p. m. and develops 2500 kw. with losses
between buckets and generator output of 65 per cent.
^ri_
i A ^
3s& Y.,0
I x
1 ^0 \J$
1 V if*
1290
i \
~J~ ~*2Q .r$
I \
>i *
"I"
c
1270
T 2%
33**"
X
j[
ft \
1
XJ
T
< t
< J
%
1246
"*" i
rX
^
jj
y
u ' (*
9
I
\t
o '
*j
*0
^s
^
122Q l
i
r~ ~3
h
^
^
^
\J
^
1210
3l
'> '
s.
s^
i
s
T
zd
V
<*
T
\
flu ^
V
so
S
V
V
9
dp
Sy 
J
X
~Zfj
1g
1
T
*D
^2
4
F iJ
Tfr
T
. 260 a. Art. 535. Expansion Path, Pressure Turbine.
PRESSURE TURBINE
385
In Fig. 260 a, the expansive path is plotted on a portion of the total heat
entropy diagram. The total heat drop is shown to be 1342 1130 = 212 B. t. u.,
and the heat drop per drum is 212  3 = 70 B. t. u. In Fig. 260 b, lay off to any
scale the equal distances ab, be, cd, and the vertical distances ae, bg, ci, rep
resenting the drum speeds. Lay ofE also ak, bm, co, equal respectively to
1 x (ae, bg, ci), and al, bn, cp, equal respectively
ale
FIG. 260 b. Art. 535 Elements of Pressure Turbine.
of entrance absolute velocities is now assumed, so as to lie wholly within the area
llsntpuvowmx. Figure 260 c shows the essential parts of the velocity diagram
for the stages on the first drum. Here ab represents aq in Fig. 260 b> ad represents
ae, the angle bad is 20, and (^} 2 = f^ZV =3.12 B. t. u. is the heat drop
\lo8.o/ \158.o/
for the first stage in the turbine. Making ac represent by and drawing dc, ch, af,
= 3.70 B. t. u. as the heat drop for the last stage on
we find ( rz TVT
\lo8.3/ Vlob.o/
the first drum. For intermediate stages between these two, we find,
IlHTIAt AnSOTUTE
VELOCITY
OEDINATB PROM
d
HEAT DROP,
B. T.U
ab = 350
de  279.7
3.12
356J
282.8
3.20
362J
285.9
3.26
368f
289.0
3.34
375
202.1
3.40
381J
295.3
3.48
387J
298.4
3.56
393 i
301.5
3.63
ac = 400
df= 304.7
3.70
386
APPLIED THERMODYNAMICS
In Pig. 260 5, we now divide the distance ab into 8 equal parts and lay off to
any convenient vertical scale the heat drops just found, obtaining the heat drop
curve zA. The average ordinate of this curve is 3.41 and the number of stages on
the first drum is 70 j 3.41 = 21 (nearest whole number). The number of stages
FIQ. 260 c. Art. 535, Velocity Diagram, Pressure Turbine.
on the other drums is found in the same way, the peripheral velocity ad, Fig.
260 c, being different for the different drums. The diameter d of the first drum is
given by the expression
3000^ = 60X160 or d =
The weight of steam flowing per second is
2500 x 1.34 x 2545
0.65x212x3600
_, 71
1/ * 11D '
PRESSURE TURBINE 387
In the first stage of the first drum, the condition of the steam at entrance to
the guide blades is (Fig 260 a) # = 1342, p = 100; at exit from the moving blades,
it is H = 1338 59, p = 98. From the total heatpressure diagram, or by computa
tion, the corresponding specific volumes are 6.5 and 6.6. The volumes of steam
flowing are then 6.5 X 17.1 = 111 and 6.6 X 17.1 = 113 cu ft per second. The absolute
steam velocities are (Fig 260 6) 350 and 356i ft. per second. The axial components
of these velocities (entrance angle 20) are 034202X350 = 120, and 0.34202 X356
= 122 The drum periphery is 1 02 X3 1416 = 3 2 f t. If the blade thicknesses occupy
J this periphery and the width for steam passage between the buckets is constant,
the width for passage of steam is f X3 2 =2 133 ft., and the necessary height of fixed
buckets is = 434 ft. or 5 2 in. at the beginning of the stage'and
2. loo X 1^0 2133X122
= 0.434 ft. or 5 2 in. at the end. The fixed blade angles are determined by the
velocities be and ab, Fig. 260, those of the moving blades by bd and be. There is
no serious error involved in taking the velocitv and specific volume as constant
throughout a blade. The height of the movmg buckets should of course not be less
than that of the guide blades; this may be accomplished by increasing the thick
ness of the former The blade heights should be at least 3 per cent of the drum
diameter, if excessive leakage over tips is to be avoided. The clearance over tips
varies from 0.008 to 0.01 inch per foot of drum diameter. Blade widths vary from
 to 1J in , with centertocenter spacing from If to 4 ins.
The method of laying out the blades is suggested in Fig 260 d. Let ab be the
absolute steam velocity at entrance to a row of moving blades, cb the blade velocity.
Then the relative velocity ac determines the enter
ing angles at c and e The movmg blade is made
with a long straight tapering tail, in which expan
sion occurs after the steam passes the point r. Let
hjj parallel with the center line of the expanding
portion of the blade (fa), represent the velocity
attained at the outlet of this blade, and let jk again
represent the blade velocity. Then hk represents
the absolute velocity of exit and determines the
entering angles of the following fixed blades, on and
ml being parallel with hk. Finally, since the steam
must emerge from the fixed blades with a velocity
parallel with ab, we draw pq parallel with ab,
determining the direction of the expanding posi FJQ. 260 d. Art 535 Blading
tion (beyond s) of the fixed blade. The angles abc of Pressure Turbine,
and kjk are made equal, and range between 20
and 30.
It should be noted that the velocities indicated by the curve qr, Fig. 260 6, are
those of the steam at exit from the fixed blades and entrance to the moving blades.
The diagram of Fig. 260 gives the absolute velocity of the steam entering the next set
of fixed blades.
COMMERCIAL FORMS OF TURBINE.
536. De Laval; Stumpf. Figure 235 illustrates the principle of the De Laval
machine, the working parts of which are shown in Fig. 261. Entering through
divergent nozzles, the steam strikes the buckets around the periphery of the wheel
b. The shaft c transmits power through the helical pinions a, a, which drive the
gears e, e> e t e, on the working shafts /, /. The wheel is housed with the iron cas
388
APPLIED THERMODYNAMICS
ing g. This is a horizontal singlestage impulse turbine with a single wheel. Its
rotative speed is consequently high; in small units, it reaches 30,000 r. p. m. It is
b.iilt principally in small sizes, from 5 to 300 h.p. The nozzles make angles
of 20 with the plane of the wheel; the buckets are symmetrical, and their angles
DE LAVAL TURBINE
389
range from 32 to 36, increasing with the size of the unit. For these proportions,
the most efficient values of u would be about 950 and 2100 for absolute steam veloci
ties of 2000 and 4400 feet per second, respectively; in practice, these speeds are
not attained, u ranging from 500 to 1400 feet per second, according to the size.
The high rotative speeds require the use of gearing for most applications. The
helical gears used are quiet, and being cut right and lefthand respectively they
practically eliminate end thrust on the shaft. The speed is usually reduced in the
proportion of 1 to 10. The high rotative speeds also prevent satisfactory balanc
ing, and the shaft is, therefore, made flexible ; for a 5hp. turbine, it is only J
inch in diameter. The bearings h, /are also arranged so as to permit of Rome
movement. The pressure of steam in the wheel case is that of the atmosphere or
condenser, all expansion occurring in the nozzle. A centrifugal governor controls
the speed by throttling the steam supply and by opening communication between
the wheel case and atmosphere when necessary.
The nozzles of the De Laval turbine are located as in Fig. 235. Those of the
Stumpf, another turbine of this class, are tangential, while the buckets are of the
Pelton form (Fig. 252), and are milled in the periphery of the wheel. A very
large wheel is employed, the rotative speeds being thus reduced. In a late form
of the Stumpf machine, a second stage is added. The reversals of direction are so
extreme that the fluid friction must be excessive.
537. Curtis Turbine. This is a multistage impulse turbine, the principle of
operation having been shown in Fig. 245. In most cases, it is vertical ; for marine
applications, it is necessarily made
horizontal. Figure 262 illustiates
the stationary and moving blades
and nozzles. Steam enters through
the nozzle A, strikes a row of mov
ing vanes at a, passes from them
through stationary vanes B to
another row of moving vanes at e,
then passes through a second set
of expanding nozzles at li to the
next pressure stage. This particu
lar machine has four pressure
stages with two sets of moving
buckets in each stage. The direc
tion of flow is axial. The number
of pressure stages may range from
two to seven. From two to four
velocity stages (rows of moving
buckets) are used in each pressure
stage. In the twostage machine,
the second stage is disconnected
when the turbine runs noncon
densing, the exhaust from the first
stage being discharged to the at
mosphere. Governing is effected
FIG. 262. Art. 637. Curtis Turbine.
390
APPLIED THERMODYNAMICS
by automatically varying the number of nozzles in use for admitting steam to the
first stage. A step bearing carries the whole weight of the machine, and must be
supplied with lubricant under heavy piessure ; an hydiauhc accumulator system is
commonly employed.
538. Rateau Turbine. This is a hoiizontal, axial flow, multistage impulse
turbine. The number of pressure stages is very laige from twentyfive upward.
There is one velocity stage in each pressure stage. Very low speeds are, theiefore,
possible. Figure 203 shows the general airangement ; the tranveise partitions e, e
form cells, in which i evolve the wheels/, /, the nozzles are merely slots in the
partitions. The blades aie pressed out of sheet steel and riveted to the wheel.
The wheels themselves are of thin pressed steel.
FIG. 2G3 Art. 538. Rateau Turbine.
539. WestinghouseParsons Turbine. This is of the axial 'flow pressure type,
and horizontal. The steam expands through a large number of successive fixed
and moving blades. In Fig. 204, the steam enters at A and passes along the vari
ous blades toward the left; the movable Buckets are mounted on the three drums,
and the fixed buckets project inward from the casings. The diameters of the
drums increase by steps ; the iuci easing volume of the steam within any section is
accommodated by varying the bucket heights. The balance pistons P, P, P are
used to counteract end thrust. The speed is fairly high, and special provision
must be made for it in the design of the bearings. Governing is effected by inter
mittently opening the valve T r ; this valve is wide open whenever open at all.
The length of this machine is sometimes too great for convenience. To over
come this, the " doubleflow " turbine receives steam near its center, through
expanding nozzles which supply a simple Pelton impulse wheel. This utilizes
a large proportion of the energy, and the steam then flows in both directions
axially, through a series of fixed and moving expanding buckets. Besides reduc
ing the length, this arrangement practically eliminates end thrust and the neces^
sity for balance pistons.
APPLICATIONS OF TURBINES
391
392 APPLIED THERMODYNAMICS
540. Applications of Turbines. Turbolocomotives have been experimented
with in Germany ; the direct connection of the steam turbine to highpressnre
rotary air compressors has been accomplished. In stationary work, the diiect
driving of genei ators by turbines is common, and the high rotative speeds of the
latter have cheapened the former. At high speeds, difficulties may be experi
enced with commutation; so that the turbine is most successful with aJteinating
current machines. When driving pumps, turbines permit of exceptionally high
lifts with good efficiencies for the centrifugal type, and low first costs. For low
pressure, highspeed blowers, the turbine is an ideal motor. (See Art. 239.) The
outlook for a gas turbine is not promising, any gas cycle involving combustion at
constant pressure being both practically and thermodynamically inefficient.
The objections to the turbine in marine application have arisen from the high
speed and the difficulty of reversing. A separate reversing wheel may be em
ployed, and graduation of speed is generally attained by installing tuibines 111
pairs. A small reciprocating engine is sometimes employ ed for maneuvering at
or near docks. Since turbines are not well adapted to low rotative speeds, they
are not recommended for vessels rated under 15 or 16 knots. The advantages ot
turbooperation, in decreased vibration, greater simplicity, smaller and more deeply
immersed propellers, lower center of gravity of engineroom machinery, decreased
size, lower first cost, and greater unit capacity without excessive size, have led to
extended marine application. The most conspicuous examples are in the Cunard
liners Lusitania and Mauretama. The former has two highpi essure and two low
pressure main turbines, and two astern turbines, all of the Parsons type (22).
The drum diameters are respectively 96, 140, and 104 in. An output of 70,000 hp.
is attained at full speed.
541. The Exhauststeam Turbine. From the heat chart, Fig. 177, it is
obvious that sfceam expanding adiabatically f rom 150 Ib. absolute pressure and
600 F. to 1.0 Ib. absolute pressure transforms into work 365 B. t. u. It has been
shown that in the ordinary reciprocating engine such complete expansion is unde
sirable, on account of condensation losses. The final pressure is rarely below 7 Ib.
absolute, at which the heat converted into work in the above illustration is only
252 B. t. u. The turbine is particularly fitted to utilize the remaining 113 B. t. u.
of available heat. The use of lowpressure turbines to receive the exhaust steam
from reciprocating engines, has, therefore, been suggested. Some progrebs has
been made in applying this principle in plants where the engine load is intermit
tent and condensation of the exhaust would scarcely pay. With steel mill en
gines, steam hammers, and similar equipment, the introduction, of a lowpressure
turbine is decidedly profitable. The variations in supply of steam to the tuibine
are offset by the use of a regenerator or accumulator, a castiron, watersprayed
chamber having a large storage capacity, constituting a " fly wheel for heat," and
by admitting live steam to the turbine through a Deducing valve. When a sur
plus of steam i caches the accumulator, the pressure rises; as soon as this falls,
some of the watei is evaporated. The maximum pressure is kept low to avoid
back pressure at the engines. A steam consumption by the turbine as low as
35 Ib. per brake hp.hr. has been claimed, with 15 Ib. initial absolute pressure and
a final vacuum of 26 in. Other good results have been shown in various trials
(23). (See Art. 552.) Wait (24) has described a plant at South Chicago, 111., in
EXHAUST STEAM TURBINES 393
which a 42 by 60 inch double cylinder, reversible rollingmil I engine exhausts to an
accumulator at a pressure 2 or 3 Ib. above that of the atmosphere. This delivers
steam at about atmospheric pressure to a 500 kw. Rateau turbine operated with
a 28m vacuum. The steam consumption of the turbine was about 35 Ib. per
electrical hp hr , delivered at the switchboard.
The S S. Turbinia, in 1897, was fitted with lowpressure turbines receiving the
exhaust from reciprocating engines and operating between 9 Ib. and 1 Ib. absolute.
One third of the total power of the vessel was developed by the turbines, although
the initial pressure was 160 Ib.
542. Commercial Considerations. The best turbines, in spite of their thermo
dynaimcally superior cycle, have not yet equalled in thermal efficiency the best
reciprocating engines, both operating at full load. (This refers to work at the cylinder.
The heat consumption referred to work at the shaft has probably been brought as
low, with the turbine, as with any form of reciprocating engine ) The combination
of reciprocating engine and turbine (Art. 552) has probably given the lowest con
sumption ever reported for a vapor engine. The average turbine is more economical
than the average engine; and since the mechanical and fluid friction losses are
disproportionately large, it seems reasonable to expect improved efficiencies as
experimental knowledge accumulates.
The turbine is cheaper than the engine; it weighs less, has no fly wheel, requires
less space and very much less foundation. It can be built in larger units than a
reciprocating cylinder. Power house buildings are cheapened by its use; the
cost of attendance and of sundry operating supplies is reduced. It probably depre
ciates less rapiflly than the engine. The wide range of expansion makes a high
vacuum desirable; this leads to excessive cost of condensing apparatus. Similarly,
superheat is so thoroughly beneficial in reducing steam friction losses that a con
siderable investment in superheaters is necessary* The choice as between the
turbine and the engine must be determined with reference to all of the conditions,
technical and commercial, including that of load factor. Turbine economy cannot
be measured by the indicator, but must be determined at the brake or switchboard,
and should be expressed on the heat unit basis (B t u. consumed per unit of output
per minute).
For results of trials of steam turbines, see Chapter XV.
(1) Tram. Inst. Engrs and Shipbuilders in Scotland, XLVI, V. (2) Berry,
The TemperatureEntropy Diagram, 1905. (2 a) For the general theory of fluid
flow, see Cardullo, Practical Thermodynamics, 1911, Arts, 5560; Goodenough,
Principles of Thermodynamics, 1911, Arts. 148150, 153; for empirical formulas,
see Goodenough, op cit. } Art. 154. (3) To show this, put the expression in .
v
the brace equal to m, and make p 0; then (   , which may be solved
for any given value of y. (4) Thesis, Polytechnic Institute of Brooklyn, 1905.
(5) Thomas, Steam Turbines, 1906, 89. (6) Proc. Inst. Civ. Eng,, CXL, 199.
(7) Zetts. Ver. Deutsch. Ing , Jan. 16, 1904. (8) Rankine, The Steam Engine, 1897,
344. (9) Experimental Researches on the Flow of Steam, Brydon tr.; Thomas, op. tit.,
106. (10) Thomas, op. tit., 123. (11) Engineering, XIII (1872). (12) Trans.
A.S.M. E., XI, 187. (13) MUM. uber Forschungsarb., XVIII, 47. (14) Practice
and Theory of the Injector, 1894, (15) Peabody, Thermodynamics, 1907, 443.
394 APPLIED THERMODYNAMICS
(16) Trans. A. S. M. E., XXVII, 081. (17) Stodola, Steam Turbines. (18) The
Steam Engine, 1905, I, 170. (19) Technical Thermodynamics, Klem tr., 1907: I,
225; II, 153. (20) Trans. A. S. M. E , XXVII, 081. (20 a) For a method for
equalizing the three quantities of work, see Caidullo's paper, " Energy and Pressure
Drop in Compound Steam Turbines," Jour. A. S M. E., XXXIII, 2. (21) See
H. F. Schmidt, in The Engineer (Chicago), Dec. 16, 1907; Trans. Inst Engrs. and
Shipbuilders in Scotland, XLXIX. (22) Power, November, 1907, 770. (23) Trans.
A. S. M. E., XXV, 817; Ibid, XXXII, 3, 315. (24) Proc. A. I. E. E., 1907.
SYNOPSIS OF CHAPTER XIV
The turbine utilizes the velocity energy of a jet or stream of steam.
Expansion in a nozzle is adiabatic, but not isentropic , the losses in a turbine are due
to residual velocity, friction of steam through nozzles and buckets and mechanical
friction.
JS + PW+ =e+pw + ^,oT^ = qQ, approximately ;
2(7 ly <*g
whence V = 223 .84 \ 'q  Q.
The complete expansion secured in the turbine warrants the use of exceptionally high
vacuum.
Nozzle friction decreases the heat converted into work and the velocity attained;
F= 212.42 V^Q.
The heat expended in overcoming friction reappears in drying or superheating the
steam.
F # , which reaches a minimum at a definite value of  Tor steam, this value
V P
is about 0.57. If the discharge pressure is less than 57 p, the nozzle converges to
a "throat" and afterward diverges.
The multistage impulse turbine uses lower rotative speeds than the single stage.
The diverging sides of the nozzle form an angle of 10 ; the converging portion may be
one fourth as long.
Steam consumption per Ihp.hr. = 2545 > JE(q  ).
The rotative components of the absolute velocities determine the work ; the relative
velocities determine the (moving) bucket angles. Bucket friction may decrease
relative velocities by 10 per cent during passage. Work = (0 cos a Ycosg*) .
ff
Efficiency = E = Work 778(7 $) . Bucket angles may be adjusted to equalize
end thrust, to secure maximum work, or may be made equal
For a rightangled stream change, maximum efficiency is 0.50 ; with complete reversal,
it is 1.00. TVith practicable buckets, it is always less than 1.0.
The backs of moving buckets are made tangent to the relative stream velocities.
The angles of fixed blades are determined by the absolute velocities.
In the pure pressure turbine, expansion occurs in the "buckets. No nozzles are used.
Turbines may be horizontal or vertical, radial or axial flow, impulse or pressure type.
In designing a pressure turbine,  = 0.33 to 0.67. The heat drop at any stage may
equal f O 2 5 Fig. 200, The number of stages is the quotient of the whole heatj
PROBLEMS 395
drop, corrected for friction; by the mean value of this quantity. Friction through
buckets may be from 20 to 30 per cent. The accumulated heat diop to any stage
is ascertained and the condition of the steam found as in Pig. 240 Typical
design, Arts. 534, 535.
Commercial forms include the De Laval, singlestage impulse :
Stumpf , single or twostage impulse, with Pelton buckets.
Curtis, multistage impulse, usually vertical, axial flow.
Bateau, multistage impulse, axial flow, horizontal, many stages.
WestinghouseParsons, pressure type, axial flow, horizontal ; sometimes of the
" double flow " form.
Marine applications involve some difficulty, but have been satisfactory at high speeds.
The turbine may utilize economically the heat rejected by a reciprocating engine. A
regenerator is sometimes employed.
The best recorded thermal economy has been attained by the reciprocating engine ;
but commercially the turbine has many points of superiority.
PROBLEMS
1. Show on the 7W diagram the ideal cycle for a turbine operating between pressure
limits of 140 Ib. and 2 lb., with an initial temperature of 600 F. and adiabatic
(isen tropic) expansion. What is the efficiency of this cycle ?
(Ana., efficiency is 0.24 )
2. In Problem 1, what is the loss of heat contents and the velocity ideally
attained ?
3. In Problem 1, how will the efficiency and velocity be affected if the initial
pressure is 150 lb.? If the initial temperature is 600 F.? If the final pressure is 1 lb.?
4. Solve Problems 1, 2, and 3, making allowance for friction as in Art. 519.
5. Compute analytically, in Problem 3, first case, the condition of the steam after
expansion, as in Art 520, assuming the heat drop to have been decreased 10 per cent
by friction. (Ans , dry ness =0.877.)
6 An ideal reciprocating engine receives steam at 150 lb. pressure and 550 F.,
and expands it adiabatically to 7 lb. pressure. By what percentage would the
efficiency be increased if the steam were afterward expanded adiabatically in a turbine
to 1.5 lb. pressure. (Ans. 9 47 per cent.)
7. Steam at 100 lb. pressure, 92 per cent dry, expands to 16 lb. pressure. The
heat drop is reduced 10 per cent by friction. Compute the final condition and the
velocity attained. (Ans^ dryness= 0.846 ; velocity = 2375 ft. per sec.)
8. In Problem 5, find the throat and outlet diameters of a nozzle to discharge
1000 lb. of steam per hour, and sketch the nozzle.
(Ans. t throat diameter =0.416 in.)
p
9. Check the value = 0.5274 for maximum flow in Art. 522.
P
396 APPLIED THERMODYNAMICS
10. Check the equation of flow of a permanent gas, in Art. 522.
11. If the efficiency in Problem 5, from steam to shaft, is 0.60, find the steam
consumption per brake hp hr, and the thermal efficiency.
12. In Problem 5, let the peripheral speed be it =480, the angle a =20, and find
the work done per pound of steam in a singlestage impulse turbine (a) with end
thrust eliminated, (&) with equal relative angles. Allow a 10 per cent reduction of
relative velocity for bucket friction.
13 In Problem 12, Case (&), what is the efficiency from steam to work at the
buckets ? (Item J7, Art. 526.) Find the ideal steam consumption per Ihp.hr.
14. Sketch the bucket in Problem 12, Case (6), as in Art. 530.
15. Compute the wheel diameters and design the firststage nozzles and buckets
for a twostage impulse turbine, with two moving wheels in each stage, as m Art. 532,
operating under the conditions of Problem 5, the capacity to be 1500 kw., the enter
ing stream angles 20, the peripheral speed 600 ft. per second, the speed 1500 r. p. m.,
the heat drop reduced 0.10 by nozzle friction. Arrange the bucket angles to give the
highest practicable efficiency,* the stream velocities to be reduced 10 per cent by
bucket f notion. State the heat unit consumption per kw.minute.
16. In Problem 5, plot by stages of about 10 B*t.u. the N'T expansion path in a
pressure turbine in which the heat drop is decreased 0,25 by bucket friction.
17. In Problem 16, the drums have peripheral speeds of 150, 250, 350. Construct
a reasonable curve of steam velocities, as in Fig. 259, the velocity of the steam enter
ing the fiist stage being 400 ft. per second, and the outputs of the three drums
as t, J, }.
18. In Problem 17, let the absolute entrance angles be 20 7 and let the velocity
diagram be as in Pig. 260. Find the work done in each of six stages along each drum.
Find the average heat drop per stage, and the number of stages in each drum, the
total heat drop per drum having been obtained from Problem 16.
19. The speed of the turbine in Problem IS is 400 r.p.m. Find the diameter of
each drum.
20. In Problems 1619, the blades are spaced 2" centers. The turbine develops
1500 kw. Find the heights of the moving blades for one expansive state, assuming
losses between buckets and generator of 45 per cent. Design the moving bucket.
21. Sketch the arrangement of a turbine in which the steam first strikes a Pelton
impulse wheel and then divides ; one portion traveling through a threedrum pressure
rotor axially, the other through a twopressure stage velocity rotor with three rows of
moving buckets in each pressure stage, also axially, the shaft of the velocity turbine
being vertical.
22. Compare as to effect on thermal efficiency the methods of governing the
De Laval, Curtis, and WestinghouseParsons turbines*
23. Detemtine whether the result given in Art. 541, reported for the S.S.
is credible.
* The angle / must not be less than 24 in any case.
CHAPTER XV
RESULTS OF TRIALS OF STEAM ENGINES AND STEAM TURBINES
543. Sources. The most reliable original sources of information as to con
temporaneous steam economy are the Transactions or Proceedings of the various
national mechanical engineering societies (1). The reports of the Committee of
the Institution of Mechanical Engineers on Marine Engine Trials aie of special
interest (2). The Alsatian experiments on superheating have already been le
f erred to (Art. 443). The works of Barrus (3) and of Thomas (4) present a maso
of results obtained on reciprocating engines and turbines respectively. The
investigations of Isherwood are still studied (5). The Code of the American Society
of Mechanical Engineers (Trans. A. S. M. E. t XXIV) should be examined.
543 a. Steam Engine Evolution. The Cornish simple pumping engines (9)
which developed from those of the original Watt type had by 1840 shown dry steam
rates between 16 and 24 Ib. per Ihp.hr. They ran condensing, with about 30 Ib.
initial pressure, and ratios of expansion between 3 and 1, and were unjacketed.
Excessive wiredrawing and the singleacting balanced exhaust (which produced
almost the temperature conditions of a compound engine) led to a virtual absence
of cylinder condensation.
The advantage of a large ratio of expansion was understood, and was supposed
to be without definite limit until Isherwood (1860) demonstrated that expansion
might be too long continued, and that increased condensation might arise from
excessive ratios. Early compound engines, without any increase in expansion
over the ratios common in simple engines, failed to produce any improvement,
steam rates attained being around 19 IK As higher boiler pressures (150 Ib )
became possible, the ratio of expansion of 14, then adopted for compounds, promptly
reduced steam rates to 15 Ib. These have been gradually brought dovn to 12 Ib.
in good practice. The 5400 hp. Westinghouse compound of the New York Edison
Co., with a 5.8 : 1 cylinder ratio, 185 Ib. steam pressure and 29 expansion^, reached
the rate of 11.93 Ib.
Triple engines, using still higher ratios of expansion, soon attained steam rates
around 12 \ Ib. The best record for a triple with saturated steam seems to be 11 05
Ib., reached by the Hackensacfc, N. J., pumping engine, with 188 Ib. throttle pressure
and 33 expansions.
Quadruple engines, and engines with superheat, have shown still better results:
see Arts. 549c, 549d, 550.
544. Limits and Measures of Efficiency. Art. 496 gives expressions
for the Clausius (EJ and relative (E R ) efficiencies corresponding with
397
398 APPLIED THERMODYNAMICS
given steam rates and pressure and temperature " conditions. The
efficiency of the turbine cannot exceed E r That of the reciprocating
engine has for a still lower limit the Rankine efficiency, which is with
saturated steam,
where pi upper pressure, Ib. per sq. in., absolute;
P2=tenninal pressure, Ib. per sq. in., absolute (end of expansion) ;
p 3 Blower pressure, Ib. per sq. in., absolute (atmosphere or
condenser) ;
Xi =initial dryness (beginning of expansion);
0:2= terminal dryness (end of expansion);
vi =specinc volume at pressure pi m ,
7; 2 =specific volume at pressure p%;
hi =heat of liquid at pressure pi;
7z, 2 =heat of liquid at pressure p 2 ;
7z, 3 =heat of liquid at pressure pz]
TI ^internal heat of vaporization at pressure p\]
r 2 =mtemal heat of vaporization at pressure p%]
LI =latent heat of vaporization at pressure pi.
With the regenerative cycle (Art. 550) the Carnot efficiency is the
limit. With superheated steam, the Rankine cycle efficiency is
144
where ,H"=total heat in the superheated steam, B. t. u.;
fi"2=total heat above 32 at the end of adiabatic expansion;
^2=specific volume of the actual steam at the end of adiabatic
expansion;
2>2= pressure of steam at the end of adiabatic expansion, Ib, per
sq. in. ;
Pa=lowest pressure, Ib. per sq. in.;
h 3 heat of liquid at the pressure p 3 .
The efficiency ratio E R is almost always between 0.4 and 0,8;
in important practice, between 0.5 and 0.7. Attention is called
* The backpressure p 9 of best efficiency is not necessarily the lowest attainable.
RESULTS OF TRIALS
399
to the table, Art. 551. Average values of the efficiency ratio seem
to be: Condensing. Noncondensing.
Simple 0.4 0.6
Compound 0.5 . 65
Triple 0.6 0.8 (Art. 5490).
With saturated steam, it is from 0.15 to 2 higher in noncondensing
than in condensing engines, and increases by 0.05 to 0.1 as the number
of expansive stages increases from 1 to 2 or from 2 to 3. With high
superheat, E R seems to be between 0.6 and 0.7 for both condensing and
noncondensing engines having either one or two expansive stages.
The figures given for saturated steam are increased 0.03 to 0.05 by
jacketing. The steam rate (Ib. dry steam per hp.hr.) is scarcely a
precise measure of performance, and is of very little significance
when superheat is used. Results should preferably be expressed in
terms of the thermal efficiency or B. t. u. consumed per Ihp.min.
(See Art. 551.)
545. Variables Affecting Performance. Some of these can be weighed from
thermodynamic considerations alone: but in all cases it is well to confirm computed
anticipations from tests The essentially thermodynamic factors are:
(a) Initial pressure (Art. 549 e)\ (6) Dryness or superheat (Arts. 549/, 549 ff);
(c) Backpressure (Art. 549 /z); (d) Ratio of expansion (Art 549 ).
The factors influencing relative efficiency, to be considered primarily from experi
mental evidence, are
(e) Wiredrawing (Art. 549.7); (/) Cylinder condensation including :
Leakage (Art. 549 fc);
(h) Compression 1
(i) Clearance / (Art '
(1) Jacketing (Art. 549m);
(2) Superheating (Art. 5490);
(3) Multiple expansion and reheating
(Arts. 549m, 549 n);
(4) Speed, Size (Art. 549 o);
(5) Ratio of expansion (Art. 549 i) ;
0") valve action (Arts. 546548 &,
549 o, 551).
SUMMARY OF TESTS
546. Saturated Steam: Simple Noncondensing Engines, without Jackets.
Steam Kate,
S& ^
Initial
Lb
Gage
Ratio of
oa
Type of Valve.
Pres
R p m
Size, Hp
Expan
ss
sure,
Lb
sion.
Avge.
Timits
III
as
Single, automatic, high
compression
70100
100300
20100
34
324
3038
0.141
0.55
Double automatic
7580
50150
4
30
134
63
Fourvalve, nonreleas
ing
100
below 225
above 50
34J
29
2632
0.15
58
Fourvalve, releasing.
100
below 100
above 75
34i
26
2428
15
0.65
400
APPLIED THERMODYNAMICS
547. Saturated Steam, Simple Condensing Engines, without Jackets: Improved
Valve Gear.
Valves
Initial
Gage
Pressure,
Lb.
R p m
Size, Hp
Ratio of
Expan
sion
Steam Rate, Lb
(Approximate)
Average
Range
*1
E R
Nonreleasing
Releasing. .
90110
90100
below 225
below 100
over 60
over 100
35
3J5
24
21*
2226
19J23i
268
266
35
40
548#. Saturated Steam, Compound Noncondensing Engines, without Jackets.
Valve.
Initial
Gage
Pressure,
Lb
R p m
Size, Hp
Stpam
Rate, Lb.
(Approximate)
*/
E R
Single, automatic
110165
120
130
250300
265
100
50250
165
350450
23 6
23 2
21 9
20 9
167
162
166
0.166
63
67
70
72
Double, automatic. . , .
Fourvalve, releasing , ,
Willans
5485. Saturated Steam, Compound Condensing, without Jackets, Normal
Cylinder Ratio,
Initial
(Approximate)
Valve.
Gage
Pressure,
Lb.
R p. m
Size, Hp
Steam
Rate, Lb
, t
ER
Single, automatic
110130
200300
100500
19 1
275
43
Double, automatic. . . .
120
160170
100300
16 3
275
50
Fourvalve, releasing. . . .
100150
underlOO
abovelOO
14 6
278
56
(15).
. Saturated Steam, Triple Expansion, without Jackets (12), (13), (14),
Back Pressure.
Initial
Gage
Pressure,
Lb
Steam Rate, Lb.
(Approximate)
E;
E*
Noncondensing (8)
1!
Average.
12i
^
Range.
11J45
0.169
0.295
80
61
Condensing
12^200
RESULTS OF TEIALS
401
5495. Jacketed Engines, High Grade, Saturated Steam, Compounds Usually
with Reheaters.
Steam Us
ite, Lb.
Type.
Jacketed
Same Type of
Engine,
Unjacketed
Small, noncondensing simple, 5 exp., 75 Ib. gage
pressure . ...
25
2632J
Simple condensing, 120150 Ib. pressure. . .
Woolf compound, condensing, 16 exp., 12 r p. m.,
120 Ib. pressure
1720
13.6
1926
rinmpoiinfi nonnondensing
19
20 923 6
Compound condensing, ordinary cylinder ratio *
(Saving due to jackets, 1J to +10 9 per cent:
per cent of total steam consumed in jackets,
about 5.0.)
Compound condensing, high cylinder ratio, 150
175lb. pressure, about 30 expansions, 8 to
14 per cent of total steam used in jackets and
reheaters
13.5
11.9
14.619.1
Triple condensing, 85 to 175 Ib. pressure, 25 to 33
expansions
11 0511 75
11 7515
* One engine gave, with jackets, 13.85; without jackets, 14 99.
5490. Superheated Steam, Reciprocating Engines.
Type.
Steam
Rate, Lb
B t u.
per
Hp mm
Approxi
mate
Clausms
Efficiency
Relative
Efficiency.
Simple noncondensing, no jackets, slight
superheat
23
Simple noncondensing, no jackets, 620 F .
Simple condensing, 800 hp., 4 exp., 65 Ib.
pressure, 450 F
15.3
16
319
317
0.182
259
66
52
Simple condensing, 620 F
11 6
234
27
67
doTTipoupd noncondensing, locomotive. . . .
17 6
Compound condensing, 500 F
12 9
253
291
57
Compound condensing, 620 F
10.6
224
0.3
'0 63
Compound condensing, 45 hp., 600lb. pres
sure, 800 F. (19)
10 8
246
375
46
Triple condensing, 500
10 9
221
0.299
0.64
Triple condensing, 620
9 6
200
309
69
402 APPLIED THERMODYNAMICS
549J. Comparative Tests, Saturated and Superheated Steam.
Type.
Steam Rate, Lb
B. t. u. per Hp mm.
Saturated
Super
heated
Saturated
Super
heated
Compound condensing, 150lb pressure,
9 56
8 99
213246
247
225
199223
205
192
Compound condensing, 140lb. pressure,
superheated 400 (18)
Compound condensing, 130lb. pressure,
fliirbprhpntpd 307
13 84
11 98
(126 r. p m. f 250 hp , 32 exp )(11)
5490 Initial Pressure. Increased pressures have been so associated with
development in other respects that it is difficult to show by experimental evidence
just what gain in economy has been due to increased pressure alone. Art. 546
gives usual steam rates from 2428 Ib. for simple noncondensing engines of the
best type, in this country, with initial pressures around 100 Ib, In Germany, where
pressures range from 150 to 180 Ib., the corresponding rates are between 19 and 23
Ib. per Ihp hr.
549/1 Initial Dryness. The efficiency of the Clausius or Rankine cycle is greater
as the initial dryness approaches 1.0 (Art 417). No considerable amount of moisture
is ever brought to the engine in practice, and tests fail to show any influence on
dry steam consumption resulting from variations in the small proportion of entrained
water.
. Superheat. There is no thermodynamic gain when superheating is less
than 100, because the steam is then brought to the dry condition by the time
cutoff is reached Tables 549 c and 549 d show that heat rates for compound
engines with low superheat are around 250 B. t. u , and for triples about 220 B. t. u.,
while with high superheat the compound or the triple may reach about 200 B. t. u.
With high superheat, exceeding 200 F,, some gain due to temperature is realized
in addition to the elimination of cylinder condensation. To properly weigh the
effect of high superheat, all steam rates given for saturated steam should be
reduced to the heat unit equivalent. This is done in the table shown at the top of
page 403
Adequate superheating thus causes a large gain in simple engines, either condens
ing or noncondensing In either case, the simple engine using superheated steam
is as economical as the ordinary compound engine using saturated steam, so that
superheat may be regarded as a substitute for compounding. The best compounds
and triples with superheat are (though in a less degree) superior to the same types
of engine using saturated steam.
RESULTS OF TRIALS
403
Type.
Steam
Rate, Lb
Initial
Absolute
Preraure,
Lb.
Feed
Tempera
ture,
B t u
per
Ihp mm.
B. t u
per
Ihp mm ,
Super
heated
Per Cent
Gain by
Super
heating.
Simple noncondensing,
best
26
100
200
434
319
26
Simple condensing, best
Compound noncondens
inff ....
21J
22
110
120
150
200
383
380
234
332 (loco
39
13
Compound condensing,
ordinary
Compound condensing,
high cylinder ratio (see
Art. 5496)
15
1213
150
175
150
150
268
213247
motive)
224253
192223
517
022
Triple condensing, aver
age
12
175
150
224
205221
19
5497i Back Pressure. This is best investigated by considering the difference
in performance of condensing and noncondensing engines. Arts. 546549 c give:
Steam Rate, Lt
s per Ihp.hr.
Per Cent
Type.
Noncondens
ing
Condensing.
Saving Due to
Condensing.
f Simple nonreleasing.
Saturated Steam, 1 Simple, releasing. . .
not jacketed 1 Compound (average)
L Triple
29
26
22 2
18 5
24
21.5
16 7
12 5
17
17
17
32
_, , ( Simple
25
18 5
26
Saturated steam, 1 ^ , , ,
. . , , \ Compound (usual
jacketed. \ ^ \
19
(average)
13 5
29
Superheated / Simple, average
19.15
13 8
28
steam. 1 Compound, average . . .
17.6
11.4
35
The arithmetical averages give about the results to be expected:
(1) Condensing saves 24 per cent in simple engines, 27 per cent in compounds,
32 per cent in triples;
(2) Condensing is relatively more profitable when jackets or superheat are used.
549t. Ratio of Expansion. This has been discussed in Art. 436. Since engines
are usually governed (i. e., adapted to the external load) by varying the ratio of
expansion, a study of the variation in efficiency with output is virtually a study of
the effect of a changing ratio of expansion. (The question of mechanical efficiency
(Arts. 554^558) somewhat complicates the matter.) Figure 266 gives the results
404
APPLIED THERMODYNAMICS
of such an investigation. The shape of the economy curve is of great importance.
A flat curve means fairly good economy over a wide range of probable loads. The
43
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50 00 70 80 90 100 110 120 130
LOAD PER CENT OF RATING
FIG. 266a. Art. 649i. Efficiency at Various Ratios of Expansion.
flatness varies with different types of engine. A few typical curves are given in
Fig, 266a. Curve I is from a singleacting Westinghouse compound engine, run
ning noncondensing Curve II is
from the same engine, condensing
(Trans A S. M . E , XIII, 537).
Curve III is for the 5400hp.
Westinghouse compound condensing
engine mentioned in Art. 543a.
Curve IV is for a small fourvalve
simple noncondensing engine : curve
V for a singlevalve highspeed simple
noncondensing engine
If we regard the usual ratio of ex
pansion in a compound as 16, in a
simple engine, 4, and in a triple or
highratio compound as 30, with cor
responding steam rates of_15, 26, and
12J (condensing engines), we obtain
the curves of Fig. 266 6, showing a
steady gain of efficiency as the total
ratio of expansion increases, provid
ing two stages of expansion are used
when the ratio exceeds some value
8 10 a . a . Jt s between : * T* W K W" that
RATIO OF EXPANSION no considerable further gain can be
FIG. 2666. Art. 549i. Efficiency and Ratio of expected by increasing either the
Expansion. ratio of expansion or the number of
RESULTS OF TRIALS 405
549;'. Wiredrawing. None of the tests above quoted applies to throttling
engines. Cutoff regulation is now almost universal. Moderate throttling may be
desirable at high ratios of expansion (Art. 426) A large part of the 8 per cent differ
ence in steam consumption between the singlevalve and doublevalve engines
of Art. 546 (15 per cent in Art. 548 6) is due to the partial throttling action of the
single valve at cutoff. The difference between the performances of fourvalve
engines with and without releasing gear is very largely due to the comparative
absence of wiredrawing in the former. This difference is 10 per cent in Art. 546
Leakage. The average steam rate ascertained on engines which had
run from 1 to 5 years without refitting of valves or pistons (7) was 39.3 Ib. This
was for simple singlevalve noncondensing machines, for which the figure given in
Art 546 is 32 J. Some of the difference was due to the fact that the engines tested
ran at light loads ( to f normal: see Art. 549 i and Fig. 266) but a part must*also
have been due to leakage resulting from wear In 65 tests reported by Barrus,
the average steam rate of engines known to have leaking valves or pistons was
4 8 per cent higher than that of those which were known to be tight. Leakage
is less in compound than in simple engines. (See Art. 452.)
549 1. Compression, Clearance. The theory of compression has been discussed
(Art. 451). Highspeed engines have more compression than those running at low
speed. The compression in compound engines is less than that in simple engines.
There is an amount of compression (usually small) at which for a given engine and
given conditions the efficiency will be a maximum. No general results can be given,
The maximum desirable compression occurs at a moderate cutoff: at other points
of cutoff, compression should be less Within any range that could reasonably
be prescribed, the amount of compression does not seriously influence efficiency
Clearance is a necessary evil, and the waste which it causes is only partially
offset by compression. Designers aim to make the amount of clearance (which
depends upon the type and location of the valves) as small as possible. The pro
portion of clearance in steam engines of various types is given in Art. 450. The
differences between the steam rates of single valve and Corliss valve engines, shown
in Arts. 546 to 548 &, already mentioned as partly due to wiredrawing, are also
in part attributable to differences in clearance.
549m. Jackets. The saving due to jackets may range from nothing (or a slight
loss) up to 20 per cent or more. Art. 549 6 shows minimum savings of 6 to 9 per cent
and maximum of 19 to 23 per cent, for one, two or three expansive; stages. Yet
there are undoubtedly cases where jackets have not paid, and they are not usually
applied (excepting on pumping engines) in American stationary practice today.
The best records made by compounds and triples have been in jacketed engines.
This is with saturated steam. With superheat, jackets are not warranted. The
proportion of steam used in jackets (of course charged to the engine) ranges usually
between 0.03 and 0.08, increasing with the number of expansive stages. Jacketing
pays best at slow speeds and hiejh ratios of expansion.
Reheaters for compound engines can scarcely be discussed separately from
jackets. It is difficult to get an adequate amount of transmitting surface without
making the receiver very large. The objection to the reheater is the same as that
to the jacket increased attention is necessary in operation and maintenance.
There is an irreversible drop of temperature inherent in the operation of the reheater.
406 APPLIED THERMODYNAMICS
549 n. Multiple Expansion. The tables already given furnish the following:
UNJACKETED ENGINES
Steam Rate, Lb. per Ihp hr 
Condensing Noncondensing.
No. of expansion stages .1 2 3 123
Type
Singlevalve .... 19 1 32J 23 6 . .
Doublevalve . . 16 3 . . 30 23 2
Fourvalve, nonreleasing. 24 . . 29 .
Fourvalve, releasing . 21* 14 6 12 5 26 21 9 18 5
Superheat, good valve 11 616 10 612 9 9 610 9 15 323 17 6
The noncondensing engine \vith a cheap type of valve is 23 to 27 per cent more
economical in the compound form than when simple. (The noncondensing compound
is on other grounds than economy an unsatisfactory type of engine, see American
Machinist, Nov. 19, 1891 ) In fourvalve releasing engines, noncondensing, the
compound saves 16 per cent over the simple and the triple saves 16 per cent over
the compound. The same engines, condensing, give a saving of 32 per cent by com
pounding and an additional saving of 14 per cent by triple expansion With super
heat, noncondensing, the compound is from 15 per cent worse to 23 per cent better
than the simple engine Condensing, the compound saves 15 per cent over the simple
and the triple saves 13 per cent over the compound.
High Ratio Compounds have been discussed in Art. 473 The tests in Art. 548 b
include only compound engines of normal cylinder ratio. The following results
have been attained with wider ratios :
Lbs per Ihp hr.
150lb pressure, 26 exp , ratio 7:1 12 45 (jacketed)
150lb pressure, 120 r. p. m , 33 exp 12 1 (head jacketed)
130lb. pressure. 126 r. p. m., 32 exp 11 98 (jacketed)
These figures are practically equal to those reached by triple engines. They
are due to (a) high ratios of expansion, (6) jacketing, and (c) the high cylinder ratio
5499. Speed and Size: Efficiency in Practice. None of the tests shows a steam
rate below 16.3 Ib. at speeds above 140 r. p. m. Low rotary speed is essential to the
highest thermal efficiency. Between very wide limits say 100 or 200 to 2500 hp.
the size of an engine only slightly influences its steam rate. Very small units
are wasteful (some directacting steam pumps have been shown to use as much as
300 Ib. of steam per Ihp hr)(6) and very large engines are usually built with such
refinement of design as to yield maximum efficiencies.
All figures given are from published tests. It is generally the case that poor
performances are not published. The tabulated steam rates will not be reached
in ordinary operation: first, because the load cannot be kept at the point of
maximum efficiency (Art. 549 1} nor can it be kept steady and second, because
under other than test conditions engines will leak Probably few bidders would
guarantee results, even at steady load, within 10 per cent of those quoted. In
estimating the probable steam rate of an engine in operation, this 10 per cent should
first be added, correction should then be made for actual load conditions, based
on such a curve as that of Fig. 266, and an additional allowance of 5 per cent or
upward should then be made Tor leakage.
RESULTS OF TRIALS
407
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INDICATED HORSE POWLR
FIG. 266. Arts. 549i, 556. Test of Rice and Sargeut Engine (10).
550. Quadruple Engines: Regenerative Cycle,
ances on record with saturated steam
have been made in quadrupleexpansion
engines. The Nordberg pumping engine
at Wildwood (16), although ot only
6.000,000 gal. capacity (712 horse power)
and jacketed on barrels of cylinders
only, gave a heat consumption of 186
B. t. u. with 200 Ib. initial pressure and
only a fair vacuum. The high efficiency
\\ as obtained by drawing off live steam
from each of the receivers and trans
tcrrmg its hightemperature heat direct
of the boiler feed water by means of
coil heaters. Heat was thus absorbed
more nearly at the high temperature
Some of the best perform
\.
. 267. Ait. 550. Nordberg Engine Diagrams
2B8_78
^THROTTLE
limit, and a closer approach made to the Carnot cycle than in the ordinary en
gine. Thus, in Fig. 267, BCDS represents the Clausius cycle. The heat areas
lil HE) gKJh, NMLg represent the withdrawal of steam from the
various receivers, these amounts of heat being applied to heating
the water along Bd, de, ef. The heat imparted from without is tben
only cfCDE. The work area DHIJKLMRS hag been lost, but
the much greater heat area ABfc has been saved, so that the effi
ciency is increased. The cycle is regenerative 5 if the number of
steps were infinite, the expansive path would be DF, parallel to
BO, and the cycle would be equally efficient with that of Carnot.
The actual efficiency was 68 per cent of that of the Carnot cycle.
The steam rate was not low, being increased by the system of
drawing off steam for the heaters from 11.4 to 12.26; but the leal
efficiency was, at the time, unsurpassed. A later test of a Nord
berg engine of similar construction, used to drive au air com
pressor, is reported by Hood (17). Here the combined diagrams
were as in Fig. 268. Steam was received at 257 Ib. pressure, the
vacuum being rather poor.
At normal capacity, 1000
hp. ; the mechanical effi
ciency was 90.35 per cent,
and the heat consumption
^ t , 169 29 B. t. u
13.85 RECEIVER
M.24CONDENSER
FIG. ?6$. Art. 550 Hood Compressor Diagrams.
408 APPLIED THERMODYNAMICS
551. Summary of Best Results, Reciprocating Stationary Engines.
Lbs Steam n Q Cy B. t.u. per
perlhp.hr. ^ 4Q6) Ihp.mm.
Saturated steam, simple, noncondensing,
single valve, without j ackets. . . . 32 0510 55 548
Saturated steam, simple, noncondensing,
double valve, without jackets . . 30 0.63 502
Saturated steam, simple, noncondensing,
four valve, releasing, without jackets . . 26 . 65 434
Saturated steam, simple, noncondensing,
with jackets .............. 25 68 418
Saturated steam, compound noncondensing,
without jackets ...... . .22 0.630.72 353
Saturated steam, simple condensing, four
valve releasing, without jackets ...... 21 040 383
Saturated steam, compound noncondensing,
with jackets, four valve ....... 19 0.710.82 305
Saturated steam, compound condens
ing, normal ratio, single valve, no
jackets .................... 19 0.43 359
Saturated steam, simple condensing, with
jackets .................. 18J 0.45 330
Superheated, compound noncondensing
(locomotive) ........ ....... 17J 580 72 332
Superheated (620 F.) steam, simple, non
condensing ....... ... 15 0,66084 319
Saturated steam, compound condensing, four
valve, no jackets .............. 14J 56 274
Saturated steam, compound condensing,
normal ratio, four valve, with jackets. . 13J . 500 . 60 255
Saturated steam, triple condensing, no
jackets ...................... 12J 0.61 234
Saturated steam, high ratio compound con
densing, jacketed .................. 12 0.63 226
Superheated (620 F.) steam, simple, con
densing ........ . ................. HJ 0. 67 234
Saturated steam, triple condensing, with
jackets .......................... llf 0.66 205
Superheated (620 F.) steam, compound con
densing ....................... 10J 0.63 224
Superheated (620 F.) steam, triple con
densing ....................... 9} 0.69 200
Saturated steam, quadruple, condensing ..... * 169
* Efficiency is 77 per cent, that of the Carnot cycle between the same extreme
temperature limits.
TURBINES 409
552. Turbines. With pressures of from 78.8 to 140 lb.,* and vacuum from
24.3 to 26 4 in , steam rates per brake horse power of 18.0 to 23 2 have been obtained
with saturated steam on De Laval turbines. Dean and Main (20) found correspond
ing ratea of 15.17 to 16 54 with saturated steam at 200 lb. pressure, and 13.94 to 15.62
with this steam superheated 91.
Parsons turbines, with saturated steam, have given rates per brake horse power
from 14 1 to 18 2, with superheated steam, from 12 6 to 14 9. This was at 120
lb pressure. A 7500kw. unit tested by Sparrow (21) with 177.5 lb. initial pressure,
95.74 of superheat, and 27 in. of vacuum, gave 15 15 lb of steam per kw.hr. Bell
reports for the Lusitama (22) a coal consumption of 1.43 lb. per horse power hour
delivered at the shaft. Denton quotes (23) 10.28 lb. per brake horse power on a
4000 hp. unit, with 190 of superheat (214 B t. u. per minute); and 13.08 on a 1500
hp. unit using saturated steam. A 400kw unit gave 11 2 lb. with 180 of super
heat. A 1250kw. turbine gave 13.5 lb. with saturated steam, 12.8 with 100 of
superheat, 13.25 with 77 of superheat (24). (All per brake hp.hr.)
A Rateau machine, with slight superheat, gave rates from 15.2 to 19.0 lb.
per brake horse power. Curtis turbines have shown 14.8 to 18.5 lb. per kw.hr.,
as the superheat decreased from 230 to zero, and of 17.8 to 22.3 lb. as the back
pressure increased from 08 to 28 lb. absolute. Kruesi has claimed (25) for a
5000kw Curtis unit, with 125 of superheat, a steamrate of 14 lb. per kw.hr.;
and for a 2000kw. unit, under similar conditions, 16.4 lb.
A 2600kw. BrownBo veri turboalternator at Frankfort consumed 11.1 lb. of
steam per electrical horsepowerhour with steam at 173 lb. gauge pressure, super
heated 196 and at 27.75 ins. vacuum. The 7500kw. ALLis crosscompound engines
of the Interborough Rapid Transit Co., New York, with 190 lb. gauge pressure and
25 ins vacuum (saturated steam) used 17.82 lb. steam per kw.hr. When exhaust
turbines were attached (Art. 541) the steam rate for the whole engine became between
13 and 14 lb. per kw.hr., or (at 28 ins. vacuum) the B. t. u. consumed per kw>
min., ranged from 245 to 264; say, approximately from 156 to 168 B. t. u. per
Ihp.min , which was better than any result ever reached by a reciprocating engine
or a turbine alone Heat unit consumptions below 280 B. t. u. per kw.min. (190
per Ihp.min.) have been obtained in many turbine tests.
553. Locomotive Tests. The surprisingly low steam rate of 16.60 lb. has
been obtained at 200 lb. pressure, with superheat up to 192. This is equivalent
bo a rate of 17.8 lb. with saturated steam. The tests at the Louisiana Purchase
Exposition (20) showed an average steam, rate of 20.23 lb. for all classes of engines
tested, or of 21.97 for simple engines and 18.55 for compounds, "with steam pres
sures ranging from 200 to 225 lb. These results compare most favorably with any
obtained from highspeed noncondensing stationary engines. The mechanical
3/ficiency of the locomotive, in spite of its large number of journals, is high ; in
bhe tests referred to, under good conditions, it averaged 88.3 per cent for consoli
iation engines and 89.1 per cent for the Atlantic type. The reason for these high
efficiencies arises from the heavy overload carried in the cylinder in ordinary ser
vice. The maximum equivalent evaporation per square foot of heating surface
varied from 8 55 to 16.34 lb. at full load, against a usual rate not exceeding 4.0 lb.
n stationary boilers ; the boiler efficiency consequently was low, the equivalent
evaporation per pound of dry coal (14,000 B. t. u.) falling from 11.73 as a maxi
num to 6.73 as a minimum, between the extreme ranges of load. Notwithstand
* Pressures in this chapter, unless otherwise stated, are gauge pressures
410
APPLIED THERMODYNAMICS
ing this, a coal consumption of 2.27 Ib. per Ihp.hr. has been reached. These trials
were, of course, laboratory tests; road tests, reported by Hitchcock (27), show less
favorable results ; but the locomotive is nevertheless a highly economical engine,
considering the conditions under which it runs*
554. Engine Friction. Excepting in the case of turbines, the figures given
refer usually to indicated horse power, or horse power developed by the steam in
the cylinder. The effective horse power, eseited by the shaft, or brake horse
power, is always less than this, by an amount depending upon the friction of the
engine. The ratio of the latter to the former gives the mechanical efficiency, which
may range from 85 to 0.90 in good piactice with rotative engines of moderate
size, and up to 0.965 in excep tional cases. (See Art. 497.) The brake horse power is
usually determined by measuring the pull exerted on a friction brake applied to the
belt wheel. When an engine drives a generator, the power indicated in the cylinder
may be compared with that developed by the generator,
and an overall efficiency of mechanism thus obtained. The
difficulties involved have led to the general custom, in
turbine practice, of reporting steam rates per kwhr.
Thurston has employed the method of driving the engine
as a machine from some external motor, and measuring
the power required by a transmission dynamometer.
In directdriven pumps, air compressors and re
frigerating machines, the combined mechanical efficiency
is found by comparing the indicator diagrams of the
steam and pump cylinders. These efficiencies are
high, on account of the decrease in number of bearings,
crank pins, and crosshead pins.
Art 555. Engine
Friction.
r700
_100.
555. Variation in Friction. Theoretically, at^ 10  269 
least, the friction includes two parts: the initial
friction, that of the stuffing boxes, which remains practically constant ; and the
Ijad friction, of guides, pins, and bearings, which varies with the initial pressure
and expansive ratio. By plotting
concurrent values of the brake horse
power and friction horse power, we
thus obtain such a diagram as that
of Tig. 269, in which the height ab
represents the constant initial fric
tion, and the variable ordinate xy
the load friction, incieasing in arith
metical proportion with the load.
It has been found, however, that in
practice the total friction is more
affected by accidental variations in
lubrication, etc., thau by changes in
load, and that it may be regarded as
practically constant,_for a given en
gine, at all loads.
^^
20
40
BRAKE HORSE POWCT
FIG. 270. Art. 555. Willans Line for Varying
Initial Pressure.
MECHANICAL EFFICIENCY
411
The total steam consumption of an engine at any load may then be regarded
as made up of two parts : a constant amount, necessary to overcome friction ; and
a variable amount, necessary to
do external work, and varying
with the amount of that work.
Willans found that this latter
part varied in exact arithmeti
cal proportion with the load,
1.2200
/
A
1800
1600
/
*s
/
A
y
1200
^
y
/
8004
/
7
100.
10 20 30 40 50 00 70 8
90 100 110 1*20
ELECTRICAL HORSE POWtR
with the
when the output of the engine
was varied by changing the initial
pressure; a condition repre
sented by the Willans line of
Fig. 270 (28). The steam rate
was thus the same for all loads,
excepting as modified by fric
tion. (Theoretically, this
should not hold, since lowering
of the initial pressure lowers
the efficiency.) When the load
is changed by varying the ratio
of expansion, the corrected steam rate tends to decrease with increasing ratios,
and to increase on account of increased condensation; there is, however, some
gain up to a certain limit, and the Willans line must, therefore, be concave up
ward. Figure 271 shows the practically straight line obtained from a series of
tests of a Parsons turbine. If the line for an ordinary engine were perfectly
straight, with varying ratios of expansion, the indication would be that the gain
by more complete expansion was exactly offset by the increase in cylinder con
densation. A jacketed engine, in which the influence of condensation is largely
eliminated, should show a maximum curvature of the Willans line.
FIG. 271.
Art. 556, Prob. 10. Willans Line for a
Parsons Turbine.
559. Variation in Mechanical Efficiency. With a constant friction loss, the
mechanical efficiency must increase as the load increases, hence the desirability
of running engines at full capacity. This is strikingly illustrated in the locomotive
(Art. 554). Engines operating at serious variations in load, as in street railway
power plants, may be quite wasteful on account of the low mean mechanical
efficiency.
The curve in Fig. 266 gives data for the " Total " curve of Fig. 271a, which is
plotted on the assumption that the horse power consumed in overcoming friction
is 100, and the corresponding total weight of steam 1000 Ib. per hour. Thus, at
700 Ihp., the steam rate from Fig. 266 is 12.1 Ib., and the steam consumed per hour
is 8470 Ib. The corresponding ordinate of the second curve in Fig. 271a is then
(8470  1000) + (700  100) =7470 *600 = 12.45,
where the abscissa is 600.
412
APPLIED THERMODYNAMICS
13,000
12,000
3 11,000
= 10,000
0,000
8,000
7,000
6,000
5,000
100 200 300 400 500 600 700 800 900
BRAKE HORSE POWER
>FiG. 271a. Art. 556. Effect of Mechanical Efficiency.
557. Limit of Expansion. Aside from cylinder condensation, engine friction
imposes a limit to the desirable range of expansion Thus, in Fig 272, the line
ab may be drawn such that the constant
pressure Oa represents that necessary to
overcome the friction of the engine. If
expansion is carried below ab, say to c, the
force exerted by the steam along be will be
less than the resisting force of the engine,
and will be without value. The maximum
desirable expansion, irrespective of cylinder
condensation, is reached at 6.
FIG. 272 Art 557. Engine Friction
and Limit of Expansion.
558. Distribution of Friction. Experi
menting m the manner described in Art,
555, Thurston ascertained the distribution
of the friction load by successively removing
various parts of the engine mechanism.
Extended tests of this nature, made by
Carpenter and Preston (29) on a horizontal engine indicate that from 35 to 47 per
cent of the whole friction load is imposed by the shaft bearings, from 22 to 49 per
cent by the piston, piston rod, pins, and slides (the greater part of this arising from
the piston and rod), and the remaining load by the valve mechanism.
(1) Trans. A. 8 M. E , Proc. Inst, Jf. E , Zeits. Ter Deutsch. Ing., etc. (See
The Engineering Diciest, November, 1908, p. 542.) (2) Proc. Inst. Mech. Eng., from 1889.
(3) Engine Tests, by Geo. H. Barrus. (4) Steam Turbines, 1900, 208207. (5) Be
searches in Experimental Steam Engineering. (6) Peabody, Tliermoaynamics, 1907,
244 , White, Jour. Am. Sue. Ifav. Engrs., X. (7) Trans, A. S. M. E. t XXX, 6, 811.
PROBLEMS 413
(8) Ewing, The Steam Engine, 1006, 177. (9) Denton, The Stevens Institute Indi
cator, January, 1905. (10) Trans. A. JS. M. E., XXIV, 1274. (11) Denton, op. cit.
(12) Ewing, op cit., 180. (13) Trans. A S. M. E., XXI, 1018. (14) Ibid., XXI, 327.
(15) J&M , XXI, 793. (10) J6 M f,XXI,181. (17) Hid., XXVIII, 2, 221. (18) Ibid.,
XXV, 2G4. (19) Ibid , XXVIII, 2, 226. (20) Thomas, Steam Turbines, 1906, 212.
(21) Power, November, 1907, p. 772. (22) Proc. List. Nav. Archls., Apnl 9, 1908.
(23) Op cit. (24) Trans, A. S. M E , XXV, 745 et seq. (25) Power, December,
1907. (20) Locomotive Tests and Exhibits, published by the Pennsylvania Railroad.
(27) Ttans. A S. M. E., XXVI, 054. (28) Mm. Proc. Inst. G. E., CXIV, 1893.
(29) Peabody, op. cit., p. 29G.
SYNOPSIS OF CHAPTER XV
Sources of information : development of steam engine economy.
Limit of efficiency (Rankme cycle) , with the regenerative engine, the Carnot cycle;
with the turbine, the Clausius cycle. Efficiency vs. steam zate.
Variables affecting performance :
Efficiency vanes directly with initial pressure ;
is independent of initial dryness ;
is increased by high superheat (superheat is a substitute for compounding)^
varies inversely as the back pressure, and is greater in condensing than in
noncondensing engines ;
reaches a maximum at a moderate ratio of expansion and decreases for
ratios above or below this ;
varies directly with the number and independence of valves ;
may be seriously reduced by leakage or high compression ;
is usually somewhat increased by jacketing;
increases with the number of expansive stages, though more and more
slowly ;
is low in very small engines or at very high rotative speeds ;
in ordinary practice is below published records.
Typical figures for reciprocating engines and turbines, with saturated and super
heated steam, simple vs. compound, condensing vs. noncondensing, with and
without jackets, triple and quadruple regenerative.
PROBLEMS
(See footnote, Art. 552.)
1. Find the heat unit consumption of an engine using 30 Ib. of dry steam per
Ihp.hr. at 100 Ib. gauge pressure, discharging this steam at atmospheric pressure.
How much of the heat (ignoring radiation losses) in each pound of steam is rejected ?
What is the quality of the steam at release ?
(Ans., a, 504.4 B. t. u. per minute ; 6, 1088.8 B. t.u. ; c, 93.6 per cent.)
2. What is the mechanical efficiency of an engine developing 300 Ihp., if 30 hp.
is employed in overcoming friction ? (Ans., 90 per cent.)
3. Why is it unprofitable to run multiple expansion engines noncondensing ?
414 APPLIED THERMODYNAMICS
4. Find the heat unit consumptions corresponding to the figures in Art. 552 for
De Laval turbines, assuming the vacuum to have been 27 in. *
(Aiis., a, 295 , 6, 286 B t. u. per minute.)
5. Find the heat unit consumption for the 7500kw. unit in Art. 552.
(AM., 296.3 B.t.u.)
6. Estimate the probable limit of boiler efficiency of the plant on the S.S.
Lusttama, if the coal contained 14,200 B. t. u. per Ib.
{Ana., if engine thermal efficiency were 0.20, mechanical efficiency 0.90, the
boiler efficiency must have been at least 0,69 )
7. Determine from data given in Art. 553 whether a coal consumption of 2.27
Ib. per. Ihp.hr. is credible for a locomotive.
8. Using values given for the coal consumption and mechanical efficiency, with
how little coal (14.000 B. t. u. per pound), might a locomotive travel 100 miles at a
speed of 50 miles per hour, while exerting a pull at the drawbar of 22,0001b. ? Make
comparisons with Problem 8, Chapter n, and determine the possible efficiency from
coal to drawbar.
9. An engine consumes 220 B t. u. per Ihp.min., 360 B. t. u. per kw.min. of
generator output. The generator efficiency is 0.93. What is the mechanical
efficiency of the directconnected unit ? (Ans., 88 per cent.)
10. From Fig. 271, plot a curve showing the variation in steam consumption per
kw.hr. as the load changes.
11. An engine works between 150 and 2 Ib. absolute pressure, the mechanical
efficiency being 0.75. What is the desirable ratio of (hyperbolic) expansion, friction
losses alone being considered, and clearance being ignored ?_ (Ans., 12.25.)
12. If the mechanical efficiency of a rotative engine is 0.85, what should be its
mechanical efficiency when directly driving an air compressor, based on the minimum
values of Art. 558 ? (Ans^ 0.94.)
13. In the jacket of an engine there are condensed 310 Ib. of steam per hour,
the steam being initially 4 per cent wet. The jacket supply is at 150 Ib. absolute
pressure, and the jacket walls radiate to the atmosphere 52 B t. u. per minute. How
much heat, per hour, is supplied by the jackets to the steam in the cylinder ? '
14. A plant consumes 1.2 Ib. of coal (14,000 B. t. u. per Ib.) per brake hp.hr.
What is the thermal efficiency ?
* Vacua are measured downward from atmospheric pressure. One atmosphere
14.690 Ib per square inch= 29.921 inches of (mercury) vacuum. If p = absolute
pressure, pounds per square inch, 0= vacuum hi inches of mercury,
as ~ >
CHAPTER XVI
THE STEAM POWER PLASTT
560. Fuels. The complex details of steam plant management arise
largely from differences in the physical and chemical constitution of
fuels. "Hard" coal, * for example, is compact and hard, while soft coal is
friable ; the latter readily breaks up into small particles, while the f orfner
maintains its initial form unless subjected to great intensity of draft.
Hard coal, therefore, requires more draft, and even then burns much less
rapidly than soft coal ; and its low rate of combustion leads to important
modifications in boiler design and operation in cases where it is to be used.
Soft coal contains large quantities of volatile hydrocarbons ; these distill
from the coal at low temperature, but will not remain ignited unless the
temperature is kept high and an ample quantity of air is supplied. The
smaller sizes of anthracite coal are now the cheapest of fuels, in propor
tion to their heating value, along the northern Atlantic seaboard ; but the
supply is limited and the cost increasing. In large city plants, where
fixed charges are high, soft coal is often commercially cheaper on account
of its higher normal rate of combustion, and the consequently reduced
amount of boiler surface necessary. The sacrifice of fuel economy in
order to secure commercial economy with! low load factors is strikingly
exemplified in the "double grate" boilers of the Philadelphia Rapid
Transit Company and the Interborough Rapid Transit Company of New
York (1).
561. Heat Value. The heat value or heat of combustion of a fuel is determined
by completely burning it in a calorimeter, and noting the rise in temperature of the
calorimeter water. The result stated is the number of heat units evolved per pound
with products of combustion cooled down to 32 F. Fuel oil has a heat value
upward of 18,000 B. t. u. per pound, its price is too high, in most sections of the
country, for it to compete with coal. Wood is in some sections available at low
cost; its heat value ranges from 6500 to 8500 B. t. u. The heat values of com
mercial coals range from 8800 to 15,000 B. t. u. Specially designed furnaces are
usually necessary to burn wood economically.
* A coal may be called famZ, or anthracite, when from 89 to 100 per cent of its
combustitle is fixed (nonvolatile, uncombined) carbon. If this percentage is between
83 and 89, the coal is semi^bituminou^ ; if less than S3, it is bituminous, or soft.
415
416
APPLIED THERMODYNAMICS
TABLE COMBUSTION DATA FOB VARIOUS FUELS
Symbol
Equivalent Reaction f
B t u
per Lb
Hydrogen . ...
H
H 2 +0 = H 2 O
62, lOOt
Carbon
C
C+OCO
4,450
Carbon .
C
C+0 2 = C0 3
14,500
Carbon monoxide.
CO
CO+0C0 2
4,385
Acetylene
C 2 H 2
C 2 H 2 +0 6 =2C0 2 +H 2
21,4001
Methane
CH 4
CH 4 +04 = C0 2 2H 2
23.842J
Ethylene
C 2 H 4
C 2 H4fO 6 =2C0 2 +2H 2 O
21,250t
Sulphur . ...
S
SK) 2 =S0 2
4,100
Gasolene* . .
CeHu
C 6 Hu+0 19 =6C0 2 +7H 2
1 9,000 1
* Gasolene IB a variable mixture of hydrocarbons, CeHu being a probable approximate formula
t The number of atoms m the molecule is disregarded
j These figures represent the 4< high values " When hydrogen, or a fuel containing hydrogen,
is burned, the maximum heat is evolved if the products of combustion are cooled below the tem
perature at which they condense, so that the latent heat of vaporization is emitted The *' low
neat value " would be (970 4 XHJ) B t u less than the high value when w is the weight in pounds
of steam formed during the combustion, if the final temperatures of the products of combustion
were the same in both " high " and " low " determinations When the products of combustion
are permanent gases there is no distinction of heat values
Computed Heat Values. When a fuel contains hydrogen and carbon
only, its heat value may be computed from those of the constituents. If oxygen
also is present, the heat of combustion is that of the substances uncombined with
oxygen. Thus in the case of cellulose, C 6 Hi O & , the hydrogen is all combined with
oxygen and unavailable as a fuel. The carbon constitutes the yVu = 0.444 part
of the substance, by weight, and the computed heat value of a pound of cellulose
is therefore 0.444X14,500 = 6430 B. t. u.
The heat of combustion of a compound may, however, differ from that of the
combustibles which it contains, because a compound must be decomposed before
it can be burned, and this decomposition may be either exothermic (heat emitting)
or endothenric (heat absorbing). In the case of acetylene, C 2 H 2 , for example, if
the heat evolved in decomposition is 3200 B. t. u., the " high " heat of combustion
is computed as follows:
C =fX14,500 =13,400
E=AX62,100  4,790
Heat of decomposition = 3,200
Heat value
=21,390
With an endothennic compound the heat of combustion will of course be less
than that calculated from the combustibles present
Suppose 0.4 cu. ft. of gas to be burned in a calorimeter, raising the temperature
of 10 Ib. of water 25 F. The heat absorbed by the water is 10X25 =250 B. t. u ,
and the heat value of the gas is 250r0.4=#25 B. t. u. per cu. ft. If the tempera
ture of the gas at the beginning of the operation were 40 F., and its pressure 30.5
ins. of mercury, then from the relation
PV^pv 30.57 29.920
T t' 40+460 32+460'
EFFICIENCY OF COMBUSTION
417
we find that a cubic foot of gas under the assigned conditions would become 1 001
cu. ft. of gas under standard conditions (32 F. and 29 92 in. barometer) The
heat value per cubic foot under standard conditions would then be 625 T 1 001 = 624 4
B. t. u.
These are the " high " heat values. Suppose, during the combustion, & Ib.
of water to be condensed from the gas, at 100 F. Taking the latent heat at 970.4
and the heat evolved m cooling from 212 to 100 at 112 B. t. u., the heat con
tributed during condensation and cooling would be 05(970 4f112) =54 12 B t. u.,
and the " low " heat value of the gas under the actual conditions of the experiment
would be 62554.12 =570.88 B t. u
The tabulated " heat value " of a fuel is usually the amount of heat liberated
by 1 Ib. thereof when it and the air for combustion are supplied at 32 F and atmos
pheric pressure, and when the products of combustion are completely coolerl down
to these standard conditions. In most applications, the constituents are supplied
at a temperature above 32 F., and the products of combustion are not cooled down
to 32 F. Two corrections are then necessary: an addition, to cover the heat
absorbed in raising the supplied fuel and air from 32 to their actual temperatures,
and a deduction, equivalent to the amount of heat which would be liberated by the
products of combustion in cooling from their actual condition to 32.
562. Boiler Room Engineering. While the limit of progress in steam engine
economy has apparently been almost realized, large opportunities for improvement
are offered in boiler operation. This is usually committed to cheap labor, with
insufficient supervision. Proper boiler operation can often cheapen power to a
greater extent than can the substitution of a good engine for a poor one. New
designs and new test records aie not necessary. Efficiencies already reported
equal any that can be expected; but the attainment of these efficiencies in ordi
nary operation is essential to the continuance in use of steam as a power produc
ing medium.
563. Combustion. One pound of pure carbon burned in air uses 2.67
Ib. of oxygen, forming a gas consisting of 3.67 Ib. of carbon dioxide and
8.90 Ib. of nitrogen.
If insufficient air
is supplied, the
amount of carbon
dioxide decreases,
some carbon mon
oxide being
formed. If the air
supply is 50 per
cent, deficient, no
carbon dioxide can
(theoretically at
least) be formed.
With, air in excess,
additional free
, . AJR SUPPLY. PERCENTAGE OP AM'T THEOR._NECeSSARY FOR COMBUSTION
oxygen and mtro FJQ 373 Arts. 563, 564. Air Supply and Combustion.
418 APPLIED THERMODYNAMICS
gen will be found in the products of combustion. Figure 273 illus
trates the percentage composition by volume of the gases formed by
combustion of pure carbon in varying amounts of air. The propor
tion of carbon dioxide reaches a maximum when the air supply is just
right.
564. Temperature Rise. In burning to carbon dioxide, each pound of
carbon evolves 14, 500 B. t u. In burning the carbon monoxide, only 4450
B. t. u. are evolved per pound. Let W be the weight of gas formed per
pound of carbon, ./Tits mean* specific heat, Tt the elevation of tempera
ture produced ; then for combustion to carbon dioxide, T t = and
4450 .
for combustion to carbon monoxide, T t = . The rise of tempera
ture is much less in the latter case. As air is supplied in excess, W
increases while the other quantities on the righthand sides of these equa
tions remain constant, so that the temperature rise similarly decreases.
The temperature elevations are plotted in Fig. 273. The maximum rise
of temperature occurs when the air supply is just the theoretical amount.
565. Practical Modifications. These curves ti;uly represent "the
phenomena of combustion only when the reactions are perfect. In
practice, the fuel and air are somewhat imperfectly mixed, so that 1 we
commonly find free oxygen and carbon monoxide along with carbon
dioxide. The presence of even a very small amount of carbon monoxide
appreciably reduces the evolution of heat. The best results are obtained
by supplying some excess of air; instead of the theoretical 11.57 lb.,
about 16 lb. may be supplied, in good practice. In poorly operated
plants, the air supply may easily run up to 50 or even 100 lb., the
percentage of carbon dioxide, of course, steadily decreasing. Gases
* JSTis quite variable for wide temperature ranges. (See Art. 63.) In general, it
may be written as a& or as adb&e 2 where a, & and c are constants and t the
temperature range from some experimentally set state. For accurate work, then
= \Kdt= \adt r*6fttt
Jti Jti Jti
the last term disappearing when 2T may be written as a function; of the first power
only of the temperature.
STEAM BOILERS 419
containing 10 per cent of dioxide by volume are usually considered
to represent fair operation.
566. Distribution of Heat. Of the heat supplied to the boiler by the fuel, a
part is employed in making steam, a small amount of fuel is lost through the grate
bars, some heat is transferred to the external atmosphere, and some is carried away
by the heated gases leaving the boiler. This last is the important item of loss.
Its amount depends upon the weight of gases, their specific heat and temperature.
The last factor we aim to fix in the design of the boiler to suit the specific rate of
combustion; the specific heat we cannot control; but the weight of gats is determined
solely by the supply of air, and is subject to operating control.
Efficient operation involves the minimum possible air supply in
excess of the theoretical requirement; it is evidenced by the percentage
of carbon dioxide in the discharged gases. If the air supply be too
much decreased, however, combustion may be incomplete, forming
carbon monoxide, and another serious loss will be experienced, due
to the potential h j?at carried off by the gas.
567. Air Supply and Draft. The draft necessary is determined by the
physical nature of the fuel; the air supply, by its chemical composition. The
two are not equivalent; soft coal, for example, requires little draft, but ample air
supply. The two should be subject to separate regulation. Low grade anthracite
requires ample draft, but the air supply should be closely economized. If forced
draft, by steam jet, blower, or exhauster, is employed, the necessary head or
pressure should be provided without the delivery of an excessive quantity or
volume of air.
Drafts required vary from about 0.1 in. of water for freeburning soft coal to
1.0 in. or more for fine anthracite. A chimney is seldom designed for less than
0.5 in,, nor forced blast apparatus for less than 0.8 in.
568. Types of Boiler. Boilers are broadly grouped as firetube or
watertube, internally or externally fired. A type of externally fired water
tube boiler has been shown in Fig. 233. In this, the Babcock and Wilcox
design, the path of the gases is as described in Art. 508. The feed water
enters the drum 6 at 29, flows downward through the back water legs
at a, and then upward to the right along the tubes, the high tem
perature zone at 1 compelling the water above it in tubes to rise. Figure
274 shows the horizontal tubular boiler, probably most generally used in this
country. The fire grate is at S. The gases pass over the bridge wall 0, under
the shell of the boiler, up the back end F", and to the right through tubes run
ning from end to end of the cylindrical shell. The tubes terminate at C, and
420
APPLIED THERMODYNAMICS
the gases pass up and away Feed water enters the front head, is carried in the
pipe about two thirds of the distance to the back end, and then falls, a compensating
I
upward current being generated over the grate. This is an eternally fired firetube
boiler. Figure 275 shows the wellknown locomotive boiler, which is internally fired.
The coldest part of this boiler is at the end farthest from the grate, on the exposed
sides. The feed is consequently admitted here. Figure 276 shows a boiler com
monly used in marine service. The grate is placed in an internal furnace ; the
gases pass upward in the back end, and return through the tubes. The feed pipe
is located as in horizontal tubular boilers.
STEAM BOILERS
421
569. Discussion.
The internally fired boiler requires no brick furnace
setting, and is compact.
The watertube boiler is
rather safer than the fire
tube, and requires less
space. It can be more
readily used with high
steam pressures. The im
portant points to observe
in boiler types are the
paths of the gases and of
the water. The gases
should, for economy, im
pinge upon and thoroughly
circulate about all parts
of the heating surface;
the circulation of the
water for safety and large
capacity should be posi
tive and rapid, and the
cold feed water should be
introduced at such a point
as to assist this circula
tion.
There is no such thing
as a "most economical
type" of boiler. Any
type may be economical
if the proportions are
right. The grade of fuel
used and the draft attain
able determine the neces
sary area of grate for a
given fuel consumption.
The heating surface must
be sufficient to absorb the
heat liberated by the fuel.
The higher the rate of
combustion (pounds of fuel
burned per square foot of
grate per hour), the greater
the relative amount of
heating surface necessary.
422
APPLIED THERMODYNAMICS
LONGJTUDINAL SECTION
FIG. 276, Ait. 5fJ8 Marine Boiler. (The Bigelow Company.)
Rates of combustion, range from 12 Ib, with, low grade hard coal and
natural draft up to 30 or 40 Ib. with soft coal ; * the corresponding ratios
of heating surface to grate surface may vary from 30 up to CO or 70.
The best economy has usually been associated with high ratios. The
rate of evaporation is the number of pounds of water evaporated per
square foot of heating surface per hour; it ranges from 3.0 upward, de
pending upon the activity of circulation of water and gases, f An effective
heating surface usually leads to a low fluegas temperature and relatively
small loss to the stack. Small tubes increase the efficiency of the heat
ing surface but may be objectionable with certain fuels. Tubes seldom
exceed 20 ft. in length. In watertube boilers, the arrangement of tubes
is important. If the bank of tubes is comparatively wide and shallow,
the gases may pass off without giving up the proper proportion of their
heat. If the bank is made too high and narrow, the grate area may be
* Much, higher rates are attained in locomotive practice ; and in torpedo boats,
with intense draft, as much as 200 Ib. of coal may be burned per square foot of grate
per hour.
f Former ideas regarding economical rates of evaporation and boiler capacity are
being seriously modified. Bone has found in " surface combustion " with gas fired
boilers an efficiency of 0.94 to be possible with an evaporation rate of 21,6 Ib.
Power, Nov. 21, 1911, Jan. 16, 1912.)
STEAM BOILER ECONOMY 423
too much restricted. The gases must not be allowed to reach the flue
too quickly.
570. Boiler Capacity. A boiler evaporating 3450 Ib. of water per
hour from and at 212 F. performs 970.4X778X3450 =2,600,000,000
footpounds of work, or 1300 horse power. No engine can develop
this amount of power from 3450 Ib. of steam per hour; the power
developed by the engine is very much less than that by the boiler which
supplies it. Hence the custom or rating boilers arbitrarily. By defini
tion of the American Society of Mechanical Engineers, a boiler horse
power means the evaporation of 31J Ib. of water per hour from and at
212 F. This rating was based on the assumption (true in 1S7G, when
the original definition was established) that an ordinary good engine
required about this amount of steam per horse power hour. This
evaporation involves the liberation of aboujfc 33,000 B. t. u. per hour.
Under forced conditions, however, a boiler may often transmit as many
as 25 B. t. u. per' square foot of surface per hour per degree of tem
perature difference on the two sides of its surface.
571. Limit of Efficiency. The gases cannot leave the boiler at a
lower temperature than that of the steam iu the boiler. Let t be the
initial temperature of the fuel and air, x the temperature of the steam,
and T the temperature attained by combustion ; then if W be the
weight of gas and K its specific heat, assumed constant, the total
heat generated is WK(T ), the maximum that can be utilized is
WK(T a;), and the limit ol efficiency is
Tx
Tt
In practice, we have as usual limiting values T= 4850, #= 350, =60;
whence the efficiency is 0.94 a value never reached in practice.
572. Boiler Trials. A standard code for conducting boiler trials has
been published by the American Society of Mechanical Engineers (2).
A boiler, like any mechanical device, should be judged by the ratio of the
work which it does to the energy it uses. This involves measuring the
fuel supplied, determining its heating value, measuring the water evaporated,
and the pressure and superheat, or wetness, of the steam. The result
is usually expressed in pounds of dry steam evaporated per pound of coal
from and at 212 F., briefly called the equivalent evaporation.
Let the factor of evaporation be F. If W pounds of water are fed to
the boiler per pound of coal burned, the equivalent evaporation is FW.
424 APPLIED THERMODYNAMICS
If C be the heating value per pound of fuel, the efficiency is 970 FW + C.
Many excessively high values for efficiency have been reported in con
sequence of not correcting for wetness of the steam; the proportion
of wetness may range up to 4 per cent in overloaded boilers. The
highest wellconfirmed figures give boiler efficiencies of about S3 per
cent. The average efficiency, considering all plants, probably ranges
from 0.40 to 0.60,
573. Checks on Operation. A careful boiler trial is rather expensive, ""and
must often interefere with the operation of the plant. The best indication of cur
rent efficiency obtainable is that afforded by analysis of the flue gases It has
been shown that maximum efficiency is attained when the percentage of carbon
dioxide reaches a maximum Automatic instruments are in use for continuously
determining and recording the proportion of this constituent present in flue gases.
575. Chimney Draft. In most cases, the high temperature of the flue gases
leaving the boiler is utilized to produce a natural upward draft for the maintenance
of combustion. At equal temperatures, the chimney gas would be heavier than the
external air in the ratio (n+l] I sn, where n is the number of pounds of air supplied
per pound of fuel. If T, t denote the respective absolute temperatures of air and
T /n I 1\
gas, then, the density of the outside air being 1, that of the chimney gas is (  )
At 60 F., the volume of a pound of air is 13 cu. ft. The weight of gas in a chimney
of crosssectional area A and height H is then
The " pressure head," or draft, due to the difference in weight inside and outside
is, per unit area,
This is in pounds per square foot, if appropriate units are used ; drafts are, how
ever, usually stated in " inches of water," one of which is equal to 5.2 Ib. per square
foot. The force of draft therefore depends directly on the height of the chimney;
and since n f 1 is substantially equal to n, maximum draft is obtained when T t
is a minimum, or (since T is fixed) when t is a maximum; in the actual case,
however, the quantity of gas passing would be seriously reduced if the value of t
were too high, and best results (3), so far as draft is concerned, are obtained when
*.r::2512.
To determine the area of chimney: the velocity of the gases is, in feet per
second,
v = V2~h = 8.03 V^ = 8.03 V 7
h being the head corresponding to the net pressure p and density d of the gases in
the chimney. Also
4 T ( n+1 \
13\ n )'
CHIMNEY DESIGN 425
Then if C Ib. of coal are to be burned per hour, the weight of gases per second is
,, ,
3600 ' their V lume 1S
and the area of the chimney, in square feet, is
A slight increase may be made to allow for decrease of velocity at the sides. The
results of this computation will be in line with those of ordinary " chimney tables,"
if side friction be ignored and the air supply be taken at about 75 Ib. jper pound
of fuel.
576. Mechanical Draft. In lieu of a chimney, steamjet blowers or fans may
be employed. These usually cost less initially, and more in maintenance. The
ordinary steamjet blower is wasteful, but the draft is independent of weather con
ditions, and may be greatly augmented in case of overload. The velocity of the
air moved by a/<w is ,_ _
v = v2 gh,
where 7i is the head due to the velocity, equal to the pressure divided by the
density. Then
If a be the area over which the discharge pressure p is maintained, the work
necessary is w = pav =
We may note, then, that the velocity of the air and the amount delivered
vary as the peripheral speed of the wheel, its pressure as the square, and the
power consumed as the cube, of that speed. Low peripheral speeds are
therefore economical in power. They are usually fixed by the pressure
required, the fan width being then made suitable to deliver the required
volume.
577. Forms of Fan Draft. The air may be blown into a closed fire room or
ash pit or the flue gases may be sucked out by an induced draft fan. In the last
case, the high temperature of tho gases reduces the capacity of the fan by about
one half; i.e., only one half the weight of gas will be discharged that would be
delivered at 60 F. Since the density is inversely proportional to the absolute
temperature, the required pressure can then be maintained only at a considerable
increase in peripheral speed; which is not, however, accompanied by a concordant
increase in power consumption Induced draft requires the handling of a greater
weight, as well as of a greater volume of gas, than forced draft; the necessary pres
sure is somewhat greater, on account of the fnctional resistance of the flues and
passages; high temperatures lead to mechanical difficulties with the fans. The
difficulty of regulating forced draft has nevertheless led to a considerable applica
tion of the induced system.
578. Furnaces for Soft Coal; Stokers. Mechanical stokers are often used when
soft coal is employed as fuel Besides saving some labor, in large plants at least,
they give more perfect combustion of hydrocarbons, with reduced smoke produc
426
APPLIED THERMODYNAMICS
tion. Figure 277 shows, incidentally, a modern form of the old " Dutch oven "
principle for soft coal firing. The flames are kept hot, because they do not strika
the relatively cold boiler surface until combustion is complete. Fuel is fed alter
nately to the two sides of the grate, so that the smoking gases from one side meet
the hot flame from the other at the hot baffling " wing walls " a, &. The principle
FIG. 277. Arts. 578, 579. Sectional Elevation of Foster Superheater combined with Boiler
and Kent Wing Wall Furnace. (Power Specialty Company )
FIG. 278. Arts. 578,579. Babcoek and Wilcox Boiler with Chain Grate Stoker and,
Superheater.
SUPERHEATERS
427
involved in the attempt to abate smoke is that of all mechanical stokers, which
may be grouped into three general types. In the chain grate, coal is carried forward
continuously on a moving chain, the ashes being dropped at the back end. The
gases from the fresh fuel pass over the hotter coke fire on the back portion of the
grate. (See Fig. 278.) The second type comprises the vndined grate stokers.
The high combustion chamber above the lower end of the grate is a decided advan
tage with many types of boilers. The smoke is distilled off at the " coking plate."
The underfeed stoker feeds the coal by means of a worm to the under side of the fire,
and the smoke passes through the incandescent fuel. All stokers have the advantage
of making firing continuous, avoiding the chilling effect of an open fire door, Airing
soft coal furnaces not associated with stokers, one of the best known is the Hawley
down draft. In this, there are two grates, coal being fired on the upper, through
which the draft is downward. Partially consumed particles of coal (coke) fall
through the bars to the lower gate, where they maintain a steady high temperature
zone through which the smoking gases from the upper grate must pass on their
way to the flue,
579. Superheaters ; Types. Superheating was proposjd at an early date, and
given a decided impetus by Hirn. After 1870, as higher steam pressures were
introduced, superheating was partially abandoned. Lately, it has been reintro
FlG. 279. Art. B79. Cole Superheater. (American Locomotive Company )
duced, and the use of superheat is now standard practice in France and Germany,
while being quite widely approved in this country. Superheaters may be sepa
rately fired, steam from a boiler being passed through an entirely separate machine,
or, as is more common, steam may be carried away from the water to some space
428
APPLIED THERMODYNAMICS
provided for it within the boiler setting or flue, and there heated by. the partially
spent gases. When it is merely desired to dry the steam, the "superheater" may
be located in the flue, using waste heat only. When any considerable increase
of temperature is desired, the superheater should be placed in a zone of the
furnace where the temperature is not less than 1000 F. With a difference m
mean temperature between gases and steam of 400 F , from 4 to 5 B t. u may be
transmitted per degree of mean temperature difference per square foot of surface
per hour (4) . According to Bell, if 8 ~ amount of superheat, deer. F , T = temperature
of flue gases reaching superheater, ^tem
perature of saturated steam, x sq. ft. of
superheater surface per boiler horsepower;
108f
FIG. 280.
The location of the Babcock and Wilcox
superheater is shown in Fig. 277; a similar
arrangement, in which the chain grate
stoker is incidentally represented, is shown
in Fig 278. In locomotive service (in which
superheat has produced unexpectedly
large savings) Field tubes may be em
ployed, as in Fig 279, the steam emerging
Art. 579. Superheater Element. frQm ^ bmlfff ftt ^ an d passing through
(Power Specialty Company.) thc header b to the small tubes c, c, c, in
the fire tubes d,d,d(5).
A typical superheater tube or " element " is shown in Fig 280. This is made
double, the steam passing through the annular space. Increased heating surface
is afforded by the cast iron rings a, a. In some singletube elements, the heating
surface is augmented by internal longitudinal ribs. The tubes should be located
so that the wettest steam will meet the hottest gases.
580. Feedwater Heaters. In Fig. 233, the condensed water is returned directly
from the hot weU 24, by way of the feed pump IV, to the boiler. This water is
seldom higher in temperature than 125 F. A considerable saving may be effected
by using exhaust steam to further heat the water before it is delivered to the boiler.
The device for accomplislung this is called the feedwater heater. With a condens
ing engine, as shown, the water supply may be drawn from the hot well and the
necessary exhaust steam supplied by the auxiliary exhausts 27 and 31; I Ib. of
steam at atmospheric pressure should heat about 9.7 Ib. of water through 100.
Accurately, W(xL+ hqJ=w(Qq), in which W is the weight of steam condensed,
x is its dryness, L its latent heat, h its heat of liquid, and w is the weight of feed
water, the initial and final heat contents of which are respectively q and (?. The
heat contents of the steam after condensation are q Q . Then
With noncondensing engines, the exhaust steam from the engines themselves is
used to heat the cold incoming water.
FEED WATER HEATERS
429
581. Types. Feedwater heaters may be either
" open," the steam and water mixing, or
" closed," the heat being transmitted through
the surface of straight or curved tubes, through
which the water circulates. Figure 281 shows a
closed heater; steam enters at A and emerges
at JS, wator enters at C, passes through the
tubes and out at D. The openings E, E are
for drawing off condensed steam. An open
heater is shown in Fig. 282. Water enters
through the automatically controlled valve a,
steam enters at 6. The water drips over the
trays, becoming finely divided and effectively
heated by the steam. At c there is provided
storage space for the mixture, and at d is a bed
of coke or other absorbent material, through
which the water filters upward, passing out at e.
The open heater usually makes the water rather
hotter, and lends itself more readily to the re
claiming of hot drips from the steam pipes,
returns from heating systems, etc., than a heater
of the closed typo. Live steam is sometimes
used for feed water heating ; the greater effective
ness of the boilerheating surface claimed to arise
from introducing the water at high temperature
has been disputed (6) ; but the high temperatures
possible with live steam are of decided value in
removing dissolved solids, and the waste of steam
may be only slight. Closed heaters are, of course,
used for this service, as also with the isodiabatic
multiple expansion cycle described in Art. 550,
Removal of some of the suspended and dissolved
FIG. 281. Art. 581. Wheeler
Feed Water Heater.
FIG. 282. Art. 581. Open Feed
Heater.
(Harrison Safety Boiler "Works.)
solids is also possible in ordinary openexhaust steam
heaters. Various forms of feed water filters are used,
with or without heaters.
582. The Economizer. This is a feedwater
heater in which the heating medium is the waste gas
discharged from the boiler furnace. It may increase
the feed temperature to 300 F. or more, whereas no
ordinary exhaust steam heater can produce a tem
perature higher than 212 F. The gam by heating
feed water is about 1 B. t. u. per pound of water for
each degree heated, or since average steam contains
1000 B. t. u. net, it is about 1 per cent for each 10
that the temperature is raised; precisely, the gain is
(# /0rQ, in which Q is the total heat of the steam
gained from the temperature of feed to the state at
evaporation and h and H the total heats in the water
before and after heating. If T, t be the temperatures
430
APPLIED THERMODYNAMICS
of flue gases and steam, respectively, W the weight, and K the mean specific heat
of the gases (say about 0.24), then the maximum saving that can be effected by a
peifect economizer is WK(T t).
Good operation decreases W and T
and thus makes the possible sav
ing small. A typical economizer
installation is shown in Fig. 283;
arrangement is always made for
bypassing the gases, as shown, to
permit of inspecting and cleaning.
The device consists of vertical cast
iron tubes with connecting headers
at the ends, the tubes being some
times staggered so that the gases
will impinge against them. The
external surface of the tubes is
kept clean by scrapers, operated
from a small steam engine. The
tubes obstruct the draft, and some
form of mechanical draft is em
ployed in conjunction with econo
mizers. From 3J to 5 sq. ft. of
economizer surface are ordinarily
used per boiler horse power. The
rate of heat transmission (B. t. u.
per square foot per degree of mean
temperature difference per hour) is
usually around 2.0.
583. Miscellaneous Devices.
A steam separator is usually placed
on the steam pipe near the engine.
This catches and more or less
thoroughly removes any condensed
steam, which might otherwise cause
damage to the cylinder. Steam
meters are being introduced for
approximately indicating the
amount of steam flowing through
a pipe. Some of them record their
indications on a chart. Feedwater
measuring tanks are sometimes in
stalled, where periodical boiler
trials are a part of the regular
routine. The steam loop is a de
vice for returning condensed steam
direct to the boiler. The drips are
piped up to a convenient height,
and the down pipe then forms a radiating coil, in which a considerable amount of
condensation occurs. The weight of this column of water in the down pipe offsets
CONDENSERS 431
a corresponding difference in pressure, and permits the return of drips to the
boiler even when their pressure is less than the boiler pressure. The ordinary
steam trap merely removes condensed water without permitting the discharge of
un condensed steam. Oil separators are sometimes used on exhaust pipes to keep
back any traces of cylinder oil.
534. Condensers. The theoretical gain by running condensing is shown by
the Carnot formula (2 1 t) + T. The gain m practice may be indicated on the
PV diagram, as in Fig. 284 The shaded area represents
work gamed due to condensation; it may amount to 10
or 12 Ib. of mean effective pressure, which means about
a 25 per cent gain, in the case of an ordinary simple
engine.* This gain is principally the result of the intro
duction of cooling water, which usually costs merely the
power to pump it; in most cases, some additional powor
is needed to drive an air pump as well.
In the surface condenser the steam and the water do
not come into contact, so that impure water may be used, jp I(J> 2 84. Art. 584. Sav
as at sea, even when the condensed steam is returned to ing Due to Condensation,
the boilers, f Such a condenser needs both air and
circulating pumps. The former ordinarily carries away air, vapor and condensed
steam. In some cases, the discharge of condensed steam is separately cared for
and the dry vacuum pump (which should always be piped to the condenser at a point
as far as possible from the steam inlet thereto) handles only air and vapor.
The amount of condensing surface should be computed from Orrok's formula
(Jour. A. S. M. E., XXXII, 11):
\vhere Z7 = B. t. u. transmitted per sq. ft. of surface per hour per degree of mean
temperature difference between steam and water;
C = a cleanliness coefficient (tubes), between 1.0 and 0.5;
r ratio of partial pressure of steam to the total absolute pressure in the
condenser, depending on the amount of air present, and varying
from 1.0 to 0;
m = Si coefficient depending upon the material of the tubes; 1.0 for copper,
0.63 for Shelby steel, 0.98 for admiralty, etc., ranging down to 0.17 for
a tube vulcanized on both sides, all of these figures being for new
metal. Corrosion or pitting may reduce the value of m 50 per cent;
V velocity of water in tubes, ft. per sec., usually between 3 and 12;
Tin mean temperature difference between steam and water, deg. F. For
T m = 18.3 (corresponding with 28" vacuum, 70 temperature of inlet
water , 90 temperature of outlet water), this becomes 435Cr 8 mVy.
The former approximate expression of Whitham (T^was
* In the case of the turbine, good vacuum is so important a matter that extreme
refinement of condenser design has now "become essential.
f There is always an element of danger involved in returning condensed steam
from reciprocating engines to the "boilers, on account of the cylinder oil which it
contains.
432
APPLIED THERMODYNAMICS
where S was the condenser surface in sq. ft , W the weight of steam condensed,
Ib. per hour, L the latent heat at the temperature T of the steam, and t the mean
temperature of circulating water between inlet and outlet. With the same nota
tion, Orrok's formula gives
~_WL WL
U(Tty
nearly.
With C=0.8, r = OS, m = 0.50, TV, = 18.2, T = 16, V becomes approximately ISO,
as in the Whitham formula
Let u, U be the initial and final temperatures of the water; then the weight w of
water required per hour is WL7(U u). The weight of water is often permitted
to be about 40 times the weight of steam, a considerable excess being desirable.
The outlet temperature of the water in ordinary surface condensers will be from
5 to 15 below that of the steam. The direction of flow of the water should
be opposite to that of the steam.
The jet condenser is shown in Fig. 285. The steam and water mix in a chamber
above the air pump cylindei, and this cylinder is utilized to draw in the water, if
the lift is not excessive. Here U = T; the supply of water necessary
is less than in surface condensers. With ample water supply, the
surface condenser gives the better vacuum. The boilers may be
fed from the hot well, as in Fig 233 (which shows a jet condenser
installation), only when the condensing water is pure.
The siphon condenser is shown in Fig 286 Condensation occurs
in the nozzle, a, and the fall of water through b produces the
vacuum. To preserve this, the lower end of the discharge pipe must
be sealed as shown. The vacuum would draw water up the pipe 6
and permit it to flow over into the engine, if it were not that the
length cd is made 34 ft. or more, thus giving a height to which
the atmospheric pressure cannot force the water. Excellent results
have been obtained with these con
densers without vacuum pumps.
In some cases, however, a " dry"
vacuum pump is used to remove
air and vapor from above the
nozzle. The device is then called
a barometric condenser. The vacuum
will lift the inlet water about 20
ft., so that, unless the suction head
is greater than this, no water sup FIG. 285. Art. 584. Horizontal Independent Jet
ply pump is required after the Condenser,
condenser is started.
Either the jet or the siphon (or barometric) condenser requires a larger air pump
than a surface condenser. Experience has shown that there will be present 1 cu.
ft. of free air (Art 187) per 10 to 50 cu. ft. of water entering the air pump of a surface
condenser or per 30 to 150 cu. ft. of water entering a jet or barometric condenser.
The surface condenser air pump handles the condensed steam only; the other
condensers add the circulating water (which mav be 20 to 40 times this) to the steam.
The volume of air at the low absolute pressure prevailing in the condenser is large,
and the necessity for reducing the partial pressure due to air has led to the employ
ment of pumps still larger than the influence of air volume, alone would warrant.
CONDENSERS
433
(For a discussion of air pump design and the importance of clearance in connection
with high vacuums, see Caidullo, Practical Thermodynamics , 1911, p. 210.)
585. Evaporative Condensers; Cooling Towers. Steam has occasionally been
condensed by allowing it to pass through coils over which fine streams, of v ater
trickled. The evaporation of the
water (which may be hastened by a
fan) cools the coils and condenses the
steam, which is drawn off by an air
pump. With ordinary condensers
and a limited water supply cooling
towers are sometimes used. These
may be identical in construction
with the evaporative condensers,
excepting that warm water enters
the coils instead of steam, to be
cooled and used over again, or
they may consist of open wood
mats over which the water falls
as in the open type of feedwater
heater. Evaporation of a portion
of the water in question (which
need not bo a, large proportion of
the whole) and warming of the
air then cools the remainder of
the water, the cooling being facili
tated by placing the mats in a
cylindrical tower through which FlG m Art. C8A.Bulkley Iniectoi Condenser.
there is a rapid upward current of
air, naturally or artificially produced (8). The cooling pond (8a) is equivalent to a
tower.
586. Boiler Feed Pump. This maybe either steanidriven or powerdriven
(as may also be the condenser pumps). Steamdriven pumps should be of the
duplex type, with plungers packed from the outside, and with individually acces
sible valves. If they are to pump hot water, special materials must be used for
exposed parts. The power pump has usually three singleacting water cylinders.
There is much discussion at the present time as to the comparative economy of
steam and powerdriven auxiliaries. The steam engine portion of an ordinary
small pump is extremely inefficient, while powerdriven pumps can be operated, at
little loss, from the main engines. The general use of exhaust steam from aux
iliaries for feedwater heating ceases to be an argument in their favor when econo
mizers are used ; and in large plants the difference in cost of attendance in favor
of motordriven, auxiliaries is a serious item.
587. .The Injector. The pump is the standard device for feeding stationary
boilers; the injector, invented by Giffard about 1858, is used chiefly as an auxil
iary, although still in general application as the prime feeder on locomotives. It
434
APPLIED THERMODYNAMICS
consists essentially of a steam nozzle, a combining chamber, and a delivery tube.
In Fig 287, steam enters at A and expands through B, the amount of expansion
being regulated by the valve C. The water enters at D, and condenses the
steam in Ef. We have here a rapid adiabatic expansion, as in the turbine; the
ve'ocity of the water is augmented by the impact of the steam, and is in turn con
veited into pressure at F. In starting the injector, the water is allowed to flow
away through G ; as soon as the velocity is sufficient, this overflow closes. An in
jector of this form will lift the water from a reasonably low suction level ; when
the water flows to the device by gravity, the valve C may be omitted.
FIG. 287 Art 587. Injector.
A selfstarting injector is one in which the adjustment of the overflow at G is
automatic. The ejector is a similar device by which the lifting of water from
a we,!! or pit against a moderate delivery head (or none) is accomplished. The
siphon condenser (Art. 584) involves an application of the injector principle. The
double injector is a series of successive injectors, one discharging into another.
588. Theory. Tet x, L, h be the state of the steam, .fftheheat in the
water, and v its velocity; Q the heat in the discharged water at its veloc
ity V. The heat in one pound of steam is xL + h; the heat in one pound
of water supplied is H f and its kinetic energy v 2 5 2 g j the heat in one
pound of discharge is Q, and its kinetic energy F 2 s 2 0. Let each pound
of steam draw in y pounds of water ; then
THE INJECTOR 435
v 2 V 2
The values of and may ordinarily be neglected, and
~~ QH '
In another form, y(Q J/)= xL + h Q, or the heat gained by the water
equals that lost by the steam. This, while not rigidly correct, on account
of the changes in kinetic energy, is still so nearly true that the thermal
efficiency of the injector may be regarded as 100 per cent ; from this stand
point, it is merely a livesteam feedwater heater.
589. Application. The formula given shows at once the relation between
steam state, water temperature, and quantity of water per pound of steam. As
the water becomes initially hotter, less steam is required ; but injectors do not
handle hot water well. Exhaust steam may be used in an injector : the pressure
of discharge is determined by the velocity induced, and not at all by the initial
pressure of the steam ; a large steam nozzle is required, and the exhaust injector
will not ordinarily lift its own water supply.
590. Efficiency. Let S be the head against which discharge is made ;
then the work done per pound of steam is (! + ?/)$ footpounds ; the
efficiency is /S(l + #)* (xL + h Q), ordinarily less than one per cent.
This is of small consequence, as practically all of the heat not changed to
work is returned to the boiler. Let W be the velocity of the steam issuing from
the nozzle; then, since the momentum of a system of elastic bodies remains con
stant during impact, W + yv = (1 + y) V. The value of W may be expressed in
terms of the heat quantities by combining this equation with that in Art. 588. The
other velocities are so related to each other as to give orifices of reasonable size.
The practical proportioning of injectors has been treated by Kneass (9).
(1) Finlay, Proc. A. I. E. E., 1907. (2) Trans A. S. M. E., XXI, 34. (3) Ran
kme, The Steam Engine, 1897, 289. (4) Longridge, Proc. Inst. M. E., 1896, 175.
(5) Trans. A. S. M. E., XXVIII, 10, 1606. (6) Bilbrough, Power, May 12, 1908,
p. 729. (7) Trans. A. S. M. E., IX, 431. (S) Bibbins, Trans. A. S. M. E. t XXXI,
11; Spangler, Apphed Thermodynamics, 1910, p. 152. (8a) Cardullo, Practical
Thermodynamics, 1911, p. 264. (9) Practice and Theory of the Injector.
SYNOPSIS OF CHAPTER XVI
Hard coal requires high, draft ; soft coal, a high rate of air supply.
In spite of its higher cost, commercial factors sometimes make soft coal the cheaper
fuel.
Heating values: fuel oil, 18,000; wood, 65008500; coals, 880015,000; B.t.u. per
Ib. Method of computing heat value.
The proportion of carbon dioxide in the flue gases reaches a maximum when the air
supply is just right ; this is also the condition of maximum temperature and
theoretical efficiency.
436 APPLIED THERMODYNAMICS
Advance in steam power economy is a matter of regulation of air supply j economy
may be indicated by automatic records of carbon dioxide.
Types of boiler : watertube, horizontal tubular, locomotive, marine ; conditions of
efficiency.
Attention should be given to the circulation of the gases and water.
A boilT fcp.34J Ib, of water per hour from and at 212 P., approximately 33,000
B. t. u. per hour.
Limit of efficiency = % . say ^94. . never reached in practice.
T t
Boiler efficiency = 5 usually 0.40 to 0.00 , may be 0.83.
Furnace efficiency = "**** . Heating surface efficiency = ^at in steam .
heat in fuel neat in gases
en*.** *** = jr[i(ll) j is
Fan draft : w= ^/2gh, p = , W=  a<bS 3 slow speeds advantageous.
2 Q 2 g
In induced draft, the fan is between the furnace and the chimney ; in forced draft, it
delivers air to the ash pit.
Mechanical stokers (inclined grate, chain grate, underfeed), used with soft coal, aim
to give space for the hydrocarbonaceous flame without permitting it to impinge on
cold surfaces.
Superheaters may be located in the flue, or, if much superheating is required, may be
separately fired. About 5 B. t. u. may be the transmission rate.
Feedwater heaters may be open or closed: w = TC^ "~ g) ; for open heaters, q = Q.
Q~q
The economizer uses the waste heat of the flue gases : saving per pound of fuel
= WK(T t). From 3 to 5 sq.. ft. of surface per boiler hp.
Condensers may be surface, jet, evaporative, or siphon, w = WL+('O' u),
S = W. Lr U(Tt); U = 630 r \ . The siphon condenser may operate with
* OT*
out a vacuum pump.
The use of steamdriven, auxiliaries affords exhaust steam for feedwater heating.
The injector converts heat energy into velocity: y= ^ \ efficiency =
PROBU3MS
1. One pound of pure carbon is burned in 16 Ib. of air. Assuming reactions to
be perfect, find the percentage composition of the flue gases and the rise in tempera
ture, the specific heats being, C0 2 , 0,215 ; N, 0.245 ; 0, 0.217.
2. A boiler evaporates 3000 Ib. of water per hour from a feedwater temperature
of 200 3T. to dry steam at 160 Ib. pressure. What is its horse power?
3. In Problem 1, what proportion of the whole heat in the fuel is carried away
PROBLEMS 437
m the flue gases, if their temperature is 600 F., assuming the specific heats of the
gases to be constant ? The initial temperature of the fuel and air supplied is F.
4. The boiler in Problem 2 burns 350 Ib. of coal (14,000 B. t.u. per pound) per
hour. What is its efficiency ?
5. In Problem 1, if the gas temperature is 600 F., the air temperature 60 F.,
compare the densities of the gases and the external air. What must be the height of
a chimney to produce, under these conditions, a draft of 1 in. of water ? Find the
diameter of the (round) chimney to burn 5000 Ib. of coal per hour. (Assume a 75
Ib. air supply in finding the diameter.)
6. Two fans are offered for providing draft in a power plant, 15,000 cu. ft. of
air being required at 1J oz. pressure per minute. The first fan has a wheel 30 in. in
diameter, exerts 1 oz. pressure at 740 r. p. m., delivers 405 cu. ft. per minute, and
consumes 0.10 hp., both per inch of wheel width and at the given speed. The second
fan has a 54inch wheel, runs at 410 r. p. m. when exerting 1 oz. pressure, and
delivers 726 cu. ft. per minute with 0.29 hp., both per inch of wheel width and at the
given speed. Compare the widths, speeds, peripheral speeds, and power consump
tions of the two fans under the required conditions.
7. Dry steam at 350 F. its superheated to 450 F. at 135 Ib. pressure. The flue
gases cool from 900 F. to 700 F. Find the amount of superheating surface to pro
vide for 3000 Ib. of steam per hour, and the weight of gas passing the superheater.
If 180 Ib. of coal are burned per hour, what is the air supply per pound of coal ?
8. Water is to be raised from 60 F. to 200 F. in a feedwater heater, the weight
of water being 10,000 Ib. per hour. Heat is supplied by steam at atmospheric pres
sure, 0.95 dry. Find the weight of steam condensed (a) in an' open heater, (Z>) in a
closed heater. Find the surface necessary m the latter (Art. 584).
9. In Problem 3, what would be the percentage of saving due to an economizer
which reduced the gas temperature to 400 F. ?
10. An engine discharges 10,000 Ib. per hour of steam at 2 Ib. absolute pressure,
0.95 dry. Water is available at 00 F. Find the amount of water supplied for a jet
condenser. Find the amount "of surface, and the water supply, for a surface con
denser in which the outlet temperature of the water is 85 F. If the surface con
denser is operated with a cooling tower, what weight of water will theoretically be
evaporated in the tower, assuming the entire cooling to be due to such evaporation.
(N. B. A large part of tho cooling is in practice effected by the air.)
11. Find the dimensions of the cylinders of a triplex singleacting feed pump to
deliver 100,000 Ib. of water per hour at 60 F. at a piston speed of 100 ft. per minute
and 30 r. p. m.
12. Dry steam at 100 It), pressure supplies an injector which receives 3000 Ib. of
water per hour, the inlet temperature of the water being 60 F. Find the weight of
steam used, if tho discharge temperature is 165 F.
13. In Problem 12, the boiler presBure is 100 Ib. What is the efficiency of the
injector, considered as a pump ?
14. In Problem 12, the velocity of the entering water is 12 ft. per second, that of
the discharge is 114 ft. per second. Find the velocity of the steam leaving the
discharge nozzle.
15. What is the relation of altitude to chimney draft ? (See Problem 12, Chapter
438 APPLIED THERMODYNAMICS
16. Circulating water pumped from a surface condenser to a cooling tower loses
4J per cent of its weight by evaporation and is cooled to 88 I\ If the loss is made
up by city water at 55, fed continuously, what is the temperature of the water
entering the condenser ?
17. Steam at 100 Ib. absolute pressure and 500 F. is used in an open feedwater
heater to warm water from 60 to 210. How much water will be heated by 1 Ib.
of steam ?
18. Steam at 150 Ib. absolute pressure, 2 per cent wet, passes through a super
heater which raises its temperature to 500 F. How much heat was added to each
Ib. of steam ?
19. 20,000 Ib, of steam at 150 Ib. absolute pressure, 2 per cent wet, are super
heated 200 in a separatelyfired superheater of 0.70 efficiency. What weight of coal,
containing 14,000 B, t. u, per Ib,, will be required ?
CHAPTER XVII
FIG. 288., Art 501, Still.
DISTILLATION FUSION LIQUEFACTION OF GASES
VACUUM DISTILLATION
591. The Still. Figure 288 represents an ordinary still, as used for
purifying liquids or for the recovery of solids in solution by concentration.
Externally applied heat evaporates the liquid in A, which is condensed at
g B. All of the heat ab
 sorbed in A is given up at
B to the cooling water;
the only wastes, in theory,
arise from radiation. Con
ceive the valve c to be
closed, and the space from
the liquid level d to this
valve to be filled with satu
rated vapor, no air being
present in any part of the
apparatus. Then when the
value c is opened, a vacuum will gradually be formed throughout the
system, and evaporation will proceed at lower and lower temperatures.
Since the total heat of saturated vapor decreases with decrease of
pressure, evaporation will thus be facilitated. In practice, however, the
apparatus cannot be kept free from air ; and, notwithstanding the opera
tion of the condenser, the vacuum would soon be lost, the pressure increas
ing above that of the atmosphere. This condition is avoided by the use
of a vacuum pump, which may be applied at e, removing air only; or, in
usual practice, at/, removing the condensed liquid as well. Evaporation
now proceeds continuously at low pressure and temperature. The possi
bility of utilizing lowtemperature heat now leads to marked economy.
Solutions are usually assumed to contain about 5 per cent of their volume of
free air. The condenser, if of the jet type, should be designed to handle about 150
times the water volume of actual air; if of the surface type (which must be used
when the distilled product is to be recovered), about 100 times the water volume.
592. Application. Vacuum distillation is employed on an important scale in
sugar refineries, soda process paperpulp mills, glue works, glucose factories, for
the preparation of pure water, and in the manufacture of gelatine, malt extract,
439
APPLIED THERMODYNAMICS
MULTIPLEEFFECT EVAPORATION
441
wood extracts, caustic soda, alum, tannin, garbage products, glycerine, sugar of
milk, pepsin, and licorice. In most cases, the multipleeffect apparatus is employed
(Art. 594).
593. Newhall Evaporator. This is shown in Fig. 289. Steam is used
to supply heat ; it enters at A, and passes through the chambers A 1 , A 2 9
to the tubes B, B. After passing through the tubes, it collects in the
chambers C 2 , C l , from which it is drawn off by the trap D. The liquid
to be distilled surrounds the tubes. The vapor forms in E, passes around
the baffle plate F and out at G. The concentrated liquid is drawn off from
the bottom of the machine.
594. Multipleeffect Evaporation. Conceive a second apparatus
to be set alongside that just described ; but instead of supplying
FIG. 290. Art. 595. Triple Effect Evaporator.
steam at A, let the vapor emerging from 6 of the first stage be
piped to A in the second, and let the liquid drawn off from the hot
442
APPLIED THERMODYNAMICS
torn of the first be led into the second ; then further evaporation may
proceed without the expenditure of additional heat, the liquid being
partially evaporated and the vapor partially condensed by the inter
change of heat in the second stage, the pressure in the shell {outside
the tubes) being less than that in the first stage. The construction will
be more clearly understood by reference to Fig. 290 (la).
595. Yaryan Apparatus. Here the heat is applied outside the tubes,
the liquid to be distilled being inside. The liquid is forced by a pump
through a small orifice
at the end of the tube,
breaking into a fine
spray during its pas
sage. The fine sub
division and rapid
movement of the
liquid facilitate
the transfer of heat.
The baffle plates E,
E, Tig. 291, serve to
separate the liquid and no. 291, Art. 595. Yaryan Evaporator,
its vapor, the former
settling in the chamber b, the latter passing out at c. Figure 290 shows a
"tripleeffect" or threestage evaporator; steam (preferably exhaust
steam) enters the shell of the first stage. The liquor to be evaporated
enters the tubes of this stage, becomes partly vaporized, and the separated
vapor and liquid pass off as just described. From the outlet c, Fig. 291,
the vapors pass through an ordinary separator, which removes any ad
ditional entrained liquid, discharging it back to &, and then proceed to
the shell of the second stage. Meanwhile the liquid from the chamber b
of the first stage has been pumped, through a hydrostatic tube which
permits of a difference in pressure in two successive sets of tubes, into the
tubes of the second stage. As many as six successive stages may be
used; * the vapors from the last being drawn off by a condenser and
vacuum pump. The liquid from the chamber b of the last stage is at
maximum density.
596. Condition of Operation. The vapor condensed in the various
shells is ordinarily water, which in concentrating operations may be
* The number of effects that can be used is limited by the difference in tempera
ture of steam supphed and final condensate discharged.
MULTIPLEEFFECT EVAPORATION 443
drawn off and wasted, or, if the temperature is sufficiently high,
employed in the power plant. The condenser is in communication
with the last tubes, and, through them, with all of the shells and tubes
excepting the first shell; but between the various stages we have the
heads of liquid in the chambers b, which permit of carrying different
pressures in the different stages. A gradually decreasing pressure
and temperature are employed, from first to last stage; it is this which
permits of the further boiling of a liquid already partly evaporated in a
former effect. The pressure in the tubes of any stage is always less
than that in the surrounding shell; the pressure in the shell of any
stage is equal to that in the tubes of the previous stage.
597. Theory. Let TFbe the weight of dry steam supplied; the
heat which it gives up is WL. Let w be the weight of liquid enter
ing the first stage, H its heat, and h and I the heat of the liquid and
latent heat corresponding to the pressure in the firststage tubes. If
x pounds of this liquid are evaporated in the first stage, the heat
supplied is xl + w(li .fl), theoretically equal to WL m > whence
x= \WLwQi ny\ s 1.
Then x pounds of vapor enter the shell of the second stage, giving
up the heat xl. The weight of liquid entering the tubes of the
second stage is w x. Let the latent heat and heat of liquid at
the pressure in the tubes of this stage be m and i: then the heat ab
sorbed, if y pounds be evaporated, is ym + (w oi)(i A), the last
term being negative, since i is less than h. Tlien
y = [xl (w #)0* /O] *" m "
Consider now a third stage. The heat supplied may be taken at ym ;
the heat utilized at
zM+ (w x y)(J
(z being the weight of liquid evaporated, AT its latent heat, and I the
corresponding heat of the liquid),
whence z = \_yrn (w x y)(I 01 "*" ^
The analysis may be extended to any number of stages.
598. Rate of Evaporation. Ordinarily, the evaporated liquid is an aqueous
solution; the total evaporation per pound of steam supplied increases with the
number of stages, being practically limited by the additional constructive expense
444 APPLIED THERMODYNAMICS
and radiation loss. For a tripleeffect evaporator, the total evaporation per W
pounds of steam supplied is a? + y + a. Let W = 1, and let the steam be siipplied
at atmospheric pressure, the vacuum at the condenser being 0.1 Ib. absolute, and
the successive shell pressures 14 7, 8.1, 1.5. The pressures in the tubes are then
8.1, 1.5, and 1 : whence L= 970.4, /= 987.9, h = 151.3, m = 1027.8, *=81.9, 7=6.98,
M = 1048.1. Let H be 100, the liquid being supplied at 132 F. A definite re
lation must exist between w and W, in order that the supply of vapor to the last
effect, y t may be sufficient to produce evaporation, yet not so great as to burden the
apparatus; this is to be detei mined by the degree of conceiitiation desired in any
particular case, whence x + y + z = (/) w, in which (/) represents the proportion of
liquid to be evaporated. Let (/) = 1.0, as is practically the case in the distillation
of water; then w = x + y+z. We now have, x =0.982  0.0521 ?, y=O.S8 + 0.0211 w,
a = 0.726 + 0.094 w, x + y + z = w = 2.588 4 0.063 w, whence w = 2.76. This is
equivalent to about 27.6 Ib. of water evaporated per pound of coal burned under
the best conditions. By increasing the number of effects, evaporation rates up to
37 Ib. have been attained in the tripleeffect machine. A sextupleeffect apparatus
has given an evaporation of 45 Ib. of water per Ib. of combustible in the coal.
599. Efficiency. The heat expended in evaporation is in this case xl+ym+zM
=3080 B. t. u. The heat supplied by the steam was WL = 970.4 B. t. u. The
efficiency is, therefore, apparently 3 18, a result exceeding unity. A large amount
of additional heat has, however, been furnished by the substance itself, which is
delivered, not as a vapor, but as a liquid, at the condenser.
600. Water Supply. The condenser being supplied per pound of steam
supplied to the first stage with v pounds of water, its heat increasing from
n to N, the heat interchange is zM=v(tf~n), whence, v=zM+ (#71), the
liquid being discharged at the boiling point corresponding to the pressure
in the condenser. In this case, for JTn = 25, v = 40.2 Ib., or the water
supply is 40.2 * 2,76 = 14.5 Ib. per pound of liquid evaporated. Some ex
cess is allowed in practice : the greater the number of effects, the less, gen
erally speaking, is the quantity of water required.
601. The Goss Evaporator. This is shown in Pig. 292. Steam enters
the first stage F from the boiler G, say at 194 Ib. pressure and 379 F.
The liquid to be evaporated (water) here enters the last stage A, say at
62 F. 5 the boiling of the liquid in each successive stage from F to A
produces steam which passes to the interior tube of the next succeeding
stage, along with the water resulting: from condensation in the interior tube
of the previous stage. The condensed steam from the first stage, is, how
ever, returned to the boiler, which thus operates like a househeating boiler,
with closed circulation. Let 1 Ib. of liquid be evaporated in F\ its pressure
and temperature are so adjusted that, in this case, the whole temperature
range between that of the steam (379 F.) and that of the liquid finally dis
charged from A (213 F.) is equally divided between the stages. The
THE GOSS EVAPORATOR
445
446 APPLIED THERMODYNAMICS
amount of vapor produced in any stage may then be computed from the
heat supplied for the assigned temperature and corresponding pressure.
Finally, in A, no evaporation occurs, the incoming liquid being merely
heated; and it is found that the weights of discharged liquid and incoming
liquid are equal, amounting each to 4.011 lb. The steam supplied by the
boiler may be computed ; in F } we condense steam at 379 !F., at which its
latent heat per pound is 845.8. It is assumed that 3 per cent of the heat
supplied in each effect is lost by evaporation; the available heat in each pound
of steam supplied is then 0.97 x 845.8 = 820.426. This heat is expended in
evaporating 1 Ib. of water at 312.6 to dry steam at 345.8% requiring
1187.44  282.26 = 905.18 B. t. u., for which = 11 lb  of steam ai ' e
8 JO. 43
required. The whole evaporation for the sixeffect apparatus is =
3.646 lb. per pound of steam. For the second effect, E, the heat supplied
is LW 8 = 870.66, gross, or 0.97 x 870.66 = 844.54, net. The heat utilized
is 1.873(282.22 248.7) +(0.873 x 895.18) =844.54. In D, the heat supplied
is 0.97 [(0.873 x 3126 ) + 1(316,98  282 22)] = 790.8 ; that utilized is
2.633(248.7 215.3) + (076x918.42) = 790.8. The heat interchange is
perfect ; it should be noted that the liquid to be evaporated and the heat
ing medium are moving in opposite direction) This involves the use of a
greater amount of heating surface, but leads, 'o higher efficiency, than the
customary arrangement. An estimated ecoi omy of 60 lb. of water per
pound of coal is possible with seven stages (1).
The Petleton evaporator, instead of reducing the pressure over the liquid to
permit of easier vaporization, mechanically compressed the vapor previously removed
and thus enabled it to further vaporize the remaining liquid. Steam was used to
start the apparatus. The vapor generated was compressed by a separate pump to
a higher pressure and temperature and was then passed back through a coil in
contact with the residual liquid. Here it gave up its heat and was condensed and
trapped off. Enough additional vapor was thus produced to maintain operation
without the further supply of steam. With an efficient pump, the fuel consump
tion may be less than half that ordinarily reached in triple effect machines.
FUSION
602. Change of Volume during Change of State. The foimuia,
T dP
was derived in Art. 368. The specific volume of a vapor below the criti
cal temperature exceeds that of the liquid from which it is produced;
FUSION 447
dT
consequently V v has in all cases a positive value, and hence must
UiJL
be positive; i.e. increase of pressure causes an increase in temperature.
It is universally true that the boiling points of substances are increased by
increase of pressure, and vice versa, at points below the critical tempers
ture. If for any vapor we know a series of corresponding values of V> L,
T, and v, we may at once find the rate of variation of temperature with
pressure.
603. Fusion. The same expression holds for the change of state de
scribed as fusion ; the Carnot cycle, Pigs. 162, 1C3, may represent melting
along ab, adiabatic expansion of the liquid along be, solidification along
crt, and adiabatic compression of the solid to its melting point along da.
In this case, V does not always exceed v ; it does for the majority of sub
stances, like wax, spermaceti, sulphur, stearine, and paraffin, which con
tract in freezing ; and for these, we may expect to find the melting point
raised by the application of pressure. This has, in fact, been found to be
the case in the experiments of Bunsen and Hopkins (2). On the other
hand, those few substances, like ice, cast iron, and bismuth, which expand
in freezing, should have their melting points lowered by pressure \ a result
experimentally obtained, for ice, by Kelvin (3) and Moussou (4). The
melting point of ice is lowered about 0.0135 F, for each atmosphere of
pressure. The expansion of ice in freezing is of practical consequence. A
familiar illustration is afforded by the bursting of water pipes in winter.
604. Comments. As the result of a number of experiments with nonmetallic
substances, Person (5) found the following empirical formula to hold :
in which L is the latent heat of fusion, C, c are the specific heats in the
liquid and solid states respectively, and T the Fahrenheit temperature of fusion.
Another general formula is given for metals. A body may be reduced from the
solid to the liquid state by solution. This operation is equivalent to that of fusion,
but may occur over a wider range of temperatures, and is accompanied by the ab
sorption of a different quantity of heat. The applications of the fundamental
formulas of thermodynamics to the phenomena of solution have been shown by
Kirchofi (6). The temperature of fusion is that highest temperature at which the
substance can exist in the solid state, under normal pressure. The latent heat of
fusion of ice has a phenomenally high value.
448
APPLIED THERMODYNAMICS
LIQUEFACTION OF GASES
605. Graphical Representation. In Fig. 293, let a represent the
state of a superheated vapor. It may be reduced to saturation, and
liquefied, either at constant pressure, along acd>
the temperature being reduced, or at constant
temperature along ale, the pressure being in
creased. After reaching the state of satura
tion, any diminution of volume at constant
temperature, or any de
crease in temperature at
FIG 293 Art. 606 Lique constant volume, must
faction of Superheated , . . , , .
Vapor. produce partial lique
faction. Constant tem
perature liquefaction is not applicable to gases
having low critical temperatures. Thus, in
Fig, 294, ab is the liquid line and cd the FIG 294. Art. 605 Liquefac
,,. ,. , T  i , i , tion and Critical Temperature,
saturation curve of carbon dioxide, the two
meeting at the critical temperature of 88 F. From the state e
this substance can be liqueiied only by a reduction in temperature.
With "permanent" gases, having critical temperatures as low as
200 C., an extreme reduction of temperature must be effected
before pressure can cause liquefaction.
606. Early Experiments. Monge and Clouet, prior to 1SOO, had liquefied sul
phur dioxide, and Northmore, in 1805, produced liquid chlorine and possibly also
sulphurous acid, in the same manner as was adopted by Faraday, about 1823, in
liquefying chlorine, hydrogen sulphide, carbon dioxide, nitrous oxide, cyanogen,
ammonia, and hydrochloric acid gas. The apparatus consisted simply of a closed
tube, one end of which was heated, while the other was plunged in a freezing mix
ture. Pressures as high as 50 atmospheres were reached. Colladon supplemented
this apparatus with an expansion cock, the sudden fall of pressure through the
cock cooling the gas ; and in Cailletet's hands this apparatus led to useful results,
Thilorier, utilizing the cooling produced by the evaporation of liquid carbon diox
ide, first produced that substance in the solid form. Natterer compressed oxygen
to 4000 atmospheres, making its density greater than that of the liquid, but with
out liquefying it. Faraday obtained minimum temperatures of 166 F. by the
use of solid carbon dioxide and ether in vacuo.
607. Liquefaction by Cooling. Andrews, in 1849, recognizing the
limiting critical temperature, proposed to liquefy the more permanent
LIQUEFACTION OF GASES
449
Art 607 Lique
faction by Pressure and
Cooling.
gases by combining pressure and cooling. Figure 295 stows the
principle involved. Let the gas be com ^
pressed isothermally from P to <2, expanded
through an orifice along ai, recompressed to
c, again expanded to d, etc. A single cycle
might suffice with carbon dioxide, while
many successive compressions and expansions
would be needed with a more permanent gas. FIG.
The process continues, in all cases, until the
temperature falls below the critical point;
and at x the substance begins to liquefy. The action depends upon
the cooling resulting from unrestricted expansion. With an abso
lutely perfect gas, no cooling would occur ; the lines ab, cd, etc.,
would be horizontal, and this method of liquefaction could not be
applied. The " perfect gas," in point of fact, could not be liquefied.
All common gases have been liquefied,
608. Modern Apparatus. Cailletet and Pictet, independently, in 1877,
succeeded in liquefying oxygen. The Pictet apparatus is shown in
Fig. 296. The jacket a was filled with liquid sulphur dioxide, from which
the vapor was drawn off by a pump, and delivered to the condenser 5.
The compressor c redelivered this
substance in the liquid condition
to the jacket, cooling in d a quan
tity of carbon dioxide which was
itself compressed in e and used as
a cooling jacket for the oxygen
gas in /. The oxygen was formed
in the bomb g, and expanded
through the cock A, producing a
_ fall of temperature which, sup
G. 296. Art. 608, Prob. 7. Cascade System, plemented by the cooling effect
of the carbon dioxide, produced
liquid oxygen. The series of cooling agents used suggested the name
cascade system.
609. Dewar's Experiments. Dewar liquefied air in 1884 and nitrogen about
1892. In 1895 he solidified air by free expansion, producing a jellylike substance.
In 1896 he obtained liquid hydrogen, by the use of which air and oxygen were
solidified, forming white masses. A temperature of  396.4 P. was obtained.
Dewar's final apparatus was that of Pictet, but compressors were used to deliver
450
APPLIED THERMODYNAMICS
the gases to the liquefying chamber, and ethylene was employed in place of car
bon dioxide. >
610. Regenerative Process ; Liquid
Air. The fall of temperature ac
companying a reduction in pressure
has been utilized by Linde (7) and
others in the manufacture of liquid
air. In the first form of apparatus,
shown in Fig. 297, air was com
pressed to about 2000 Ib. pressure in
a threestage machine A, and after
cooling in B was delivered to the
inner tube of a double coil <7, through
which it passed to the expansion
valve D. Here a considerable fall
of temperature took place. The
cooled and expanded air then passed
, , , , L . . ,, FIG 297. Art. 610. Liquefaction of Air.
back through the outer tube or the
coil, cooling the air descending the inner tube, and was discharged
at F. The effect was cumulative, and after a time liquid air was
deposited in JS. In the present type of machine, the compressor
takes its supply from F, a decided improvement. The regenerative
principle has been adopted in the recent forms of apparatus of
Hampson, Solvay, Dewar, and Tripler.
The latent heat of evaporation of air at atmospheric pressure is about 140
B. t. u. (8). In its commercial form, it contains small particles of solid carbon
dioxide; when these are removed by filtration, the liquid becomes clear. The
boiling point of nitrogen is somewhat higher than that of oxygen; fairly pure
liquid oxygen may, therefore, be obtained by allowing liquid air to partially
evaporate (9). The cost of production of liquid air has been carefully estimated
in one instance to approach 22 cents per pint (10).
(1) Trans. A. S. M. E., XXV, 03. The steam table used was Peahody's, 1890 ed.
The temperatures noted on 3Tig. 292 are approximate : those in the text are correct.
(1 a) See the paper by Newhall, before the Louisiana Sugar Planters' Association,
June 13, 1907. (2) Rep B. A., 1854, II, 56. (3) Ph\l. Mag., 1850: III, xxxvii, 123.
(4) Deschanel, Natural Philosophy (Everett tr ), 1893, II, 331. (5) Ann. de Chem.
et de Phys., November, 1849. (6) Pogg Ann., 1858. (7) Zeuner, Technical Thermo
dynamics (Klein), II, 303313; Trans. A. S. M. E , XXI, 156. (8) Jacobus and
Dickerson: Trans. A. S. M. E., XXI, 166. (9) See the very complete paper by
Bice, Trans. A. S. M. E., XXI, 156, (10) Tests of a Liquid Air Plant, Hudson and
Garland; University of Illinois Bulletin, V. 16.
SYNOPSIS 451
SYNOPSIS OF CHAPTER XVII
Distillation
The still is a device for purifying liquids or recovering solids by partial evaporation.
By evaporation in vacuo, the heat consumed may be reduced in many important
applications : waste heat may be employed.
Steam may supply the heat ; in the Newhall apparatus, the steam circulates through
tubes.
In the Yaryan apparatus, the steam surrounds the tubes.
The vapors rising from the solution may supply the heat required in a second " effect,"
provided that the solution there is under a less pressure than in the first stage.
As many as six stages are used, the pressure on the solution decreasing step by step.
Jfeporattmjur effect: x = ^ W (hH) . y = xl( w  x^iK) .
I m
_ym(wxy) (i
M
In a typical case, the tripleeffect machine gives an evaporation of 2.76 Ib. per pound of
steam.
~fUT
Water required at the condenser per pound of liquid evaporated =
(.ZV n)
In the Goss evaporator, the steam and the solution move in opposite directions ; this
increases the necessary amount of surface, but also the efficiency. Petleton
evaporator.
Fusion
The formula V v = ( ^ applies to fusion. The melting points of substances
may be either raised or lowered by the application of pressure, according as the
specific volume in the liquid state is greater or less than that in the solid state.
The melting point of ice is lowered about 0.0135 IT. per atmosphere of pressure imposed.
Zr=((7c)(:r+ 256) for nonmetallic substances.
Liquefaction of Gases
A vapor below the critical temperature may be liquefied either at constant pressure or
at constant temperature.
No substance can be liquefied unless below the critical temperature.
A few common substances have been liquefied by the use of pressure and freezing
mixtures.
A further lowering of temperature is produced "by free expansion.
Liquefaction may be accomplished with actual gases by successive compressions and
free expansions.
The Pictet apparatus (cascade system) employed the latent heat of vaporization of
successive fluids to cool more volatile fluids.
The regenerative system provides for the free expansion of a highly compressed gas
previously reduced to atmospheric temperature. This is used in manufacturing
liquid air.
452 APPLIED THERMODYNAMICS
PROBLEMS
1. Water entering a still at 40 F. is evaporated, (a) at atmospheric pressure,
(6) at 2 Ib. absolute pressure. What is the saving m heat in the latter case ? What
more important saving is possible ?
2. Water entering a doubleeffect evaporator at 80 F. is completely distilled, the
steam supplied being dry and at atmospheric pressure, the pressure in the secondstage
shell being 8 Ib. and that in the secondstage tubes 1 Ib. Cooling water is available at
60 F. The temperature of the circulating water at the condenser outlet is 80.
Find the steam consumption per pound of water e\aporated and the cooling water
consumption, if the vacuum pump discharge is at 85 F.
3. In Fig. 292, take temperatures as given ; assume one pound of water to be com
pletely evaporated in F, and complete condensation to occur in the inner tube of each
effect, and compute, allowing 3 per cent for radiation, as in Art. 601 :
(a) The weight of steam condensed in F.
(6) The weight of steam evaporated in E, and of water delivered to K.
(c) The weight of boiler steam used per pound of water evaporated in the whole
apparatus. Use the steam tables on pp 247, 248.
4 The weight of one cubic foot of H 2 at 32 F. and atmospheric pressure being
57.5 Ib. as ice and 62.42 Ib. as water, and the latent heat of fusion of ice being 142
B. t. u., find how much the melting point of ice will be lowered if the pressure is
doubled (Art. 003).
5. The specific heat of ice being 504, find its latent heat of fusion at 32 F. froir
Art. 604.
6. How much liquid air at atmospheric pressure would be evaporated in freezing
1 Ib. of water initially at 00 F. ?
7. In a Pictet apparatus, Fig. 296, 1 Ib. of air is liquefied at atmospheric pressure,
free expansion having previously reduced its temperature to the point of liquefaction.
The condensation is produced by carbon dioxide, which, evaporates m the jacket with
out change of temperature, at such a pressure that its latent heat of vaporization is
200 B. t. u. How many pounds of carbon dioxide are evaporated ? This dioxide is
subsequently liquefied, at a higher pressure and while dry (latent heat = 120), and
cooled through 100 F. Its specific heat as a liquid may be taken as 0.4, The lique
faction and cooling of the carbon dioxide are produced by the evaporation of sulphur
dioxide (latent heat 220 B.t. u.). What weight of sulphur dioxide will be evap
orated per pound of air liquefied ? Why would the operation described be imprac
ticable ?
8. From Art. 245, find the fall of temperature at expansion in a Lmde air machine
in which the air is compressed to 2000 Ib. absolute and cooled to 60 F., and then ex
panded to atmospheric pressure. How many complete circuits must the air make in
order that the temperature may fall from 60 F. to 305 F., if the same fall of tem
perature is attained at each circuit ?
9. Plot on the entropy diagram the path of ice heated at constant pressure from
400 F. to 32 P F. , assuming the specific heat to be constant, and then melted at
atmospheric pressure. How will the diagram be changed if melting occurs at a pres
sure of 1000 atmospheres ?
PROBLEMS 453
Plot a curve embracing states of the completely melted ice for a wide range of
pressures. Construct lines analogous to the constant dryness lines of the steam
entropy diagram and explain their significance.
10. At what temperature will the latent heat of fusion of ice be ? What would
be the corresponding pressure ?
CHAPTER XVIII
MECHANICAL REFRIGERATION
611* History. Refrigeration by "freezing mixtures" has been practiced for
centuries. Patents covering mechanical refrigeration date back at least to 1835 (1),
In the first machines, ether was the working substance, and the cost of operation
was high. Pictet introduced the use of sulphur dioxide and carbon dioxide. The
transportation of refrigerated meats began about 1873 and developed rapidly after
1880, most of the earlier machines using air as a working fluid. The possibility
of safely shipping refrigerated fresh fruits, milk, butter, etc., has revolutionized
the distribution, of these food products ; and, to a large extent, refrigerating pro
cesses have eliminated the use of ice in breweries, packing houses, fish and meat
markets, hotels, etc. The two important applications of artificial refrigeration at
present are for the production of artificial ice and for cold storage.
612. Carnot Cycle Reversed. In Fig. 298, let the cycle be
worked in a counterclockwise direction. Heat is absorbed along
dc and emitted along ba; the latter quantity of heat exceeds the
former by the work expended, abed. The object of refrigeration
is to cool some body. This cooling may be produced by a flow of
FIG, 298. Art. 612. Reversed Carnot Cycle.
heat from the body to the working fluid along d c. Cyclic action is
possible only under the condition that the working fluid afterward
transfer the heat to some second body along la. The body to be
454
REGENERATIVE REFRIGERATION 455
cooled is called the vaporizer ; the second body, which in turn re
ceives heat from the working fluid, is the cooler. TLe heat taken
from the vaporizer is ndcN\ that discharged to the cooler is nabN.
The function of the machine is to cause heat to pass from the vapor
izer to a substance warmer than itself; i.e. the cooler. This is
accomplished without contravention of the second law of thermo
dynamics, by reason of the expenditure of mechanical work. The
refrigerating machine is thus a heat pump.
The Carnot cycle, with a gas as the working fluid, would lead to an exces
sively bulky machine (Art. 249). Early forms of apparatus therefore embodied
the regenerative principle (Art. 257). This
is illustrated in Fig. 299.
Without the regenerator, air would
be compressed adiabatically from 1 to
2, cooled at constant pressure along
2 3, expanded adiabatically along 3 4,
and allowed to take up heat from the
body to be refrigerated along 4 1. In
practice, this heat is partly taken from
the body, and partly from other sur , f%nn . _ rt _
,. Jy , . r J n , , . FIG. 299. Art. 612. Regenerative
rounding objects after the working Refrigeration.
air has left this body, say at 5. The
absorption of heat along 51 then effects no good purpose. If, however,
this part of the heat be absorbed from the compressed air at 3, that
body of air may be cooled, in consequence, along 3 6, so that adiabatic ex
pansion will reduce the temperature to that at 7, lower than that at 4.
This is accomplished by causing the air leaving the cooler to come into
transmissive contact with that leaving the vaporizer. The effect of the
regenerator is cumulative, increasing the fall of temperature at each step ;
but since the expansion cylinder must be kept constantly colder as expan
sion proceeds, a limit soon arises in practice.
In Kirk's machine (1863), a compressing cylinder was used for the operation c5,
Fig. 298, and two expansive cylinders for the operation ad, one receiving the air
from each end of the compressor cylinder. The pressure throughout the cycle was
kept considerably above that of the atmosphere, and temperatures of 39 F, were
obtained. The regenerator consisted of layers of wire gauze located in the pis
tons (2). The air machines of Hargreaves and Inglis (1878), Tuttle and Lugo,
Lugo and MoPherson, Hick Hargreaves, Stevenson, Haslam, Lightfoot, Hall, and
Cole and Allen, have been' described by WallisTayler (3). The BellColeman ma
chine may be regarded as the forerunner of all of these, although many variations
in construction and method of working have been introduced.
456
APPLIED THERMODYNAMICS
613. BellColeman Machine. This is the Joule air engine of Art. 101,
reversed. It operates in the net cycle given by an air compressor and an
air engine, as in Art, 213. In Pigs. 300 and 301, is the room to be
cooled, A a cooler, M a compressor, and JV an expansive cylinder (air
engine). In the position shown, with the pistons moving toward the left,
air flows from (7 to M at the temperature T ( . On the return stroke, the
valve a closes, the air is compressed along c6, Fig. 301, and the valve a
Fia. 300. Art 613. BellColeman
Machine.
FIG 301. Arts. 013, C14, 016, 622, 623.
Eeyeised Joule Cycle.
opens, permitting of discharge into A along be, at the temperature T b .
The operation is now repeated, the drawing in of air from to If being
represented by the line /c. Meanwhile an equal weight of air has been
passing from A to JTat the temperature T a , less than T b on account of the
action of the cooler, along ea, expanding to the pressure in O along ad,
reaching the temperature T# lower than that in <7; and p t assing into at
constant pressure along df. The work expended in the compressor cylinder
is/c&e; that done by the expansion cylinder is/ead; the difference, abed,
represents work required from witJiout to permit of the cyclic operation.
If the lines ad, be, are isodiabaties,
Suitable means are provided for cooling the air in the compressor cylinder, so as to
avoid the losses due to a rise of temperature (Art. 195)., and also for drying the
air entering the expansion cylinder.
The expansion cylinder is necessary for the operation. Free expansion of the air
through a valve from p e to p/ would be unaccompanied Jby any drop in temperature.
614. Analysis of Action. Let air at 147 Ib. pressure and 60 P.,
at 0, Pig. 301, expand adiabatieally behind a piston along ad, until
its pressure is 14.7 Ib. Its temperature at d is
T d = T a
/ P \ ~~u~
519.6 H (10)
02875
269 absolute or  191 P.
BELLCOLEMAN MACHINE
457
Let this cold air absorb heat along dc at constant pressure, until its
temperature rises to F. Then let it be compressed adiabatically
until its pressure is again 147 lb. ; along cb. Since
T T /519 G\
7j? =V' !T 6 =459.6( ^) =890 absolute, or 430 F.
J. d * c \ jO J /
The air now rejects heat at constant pressure along ba to cold water,
or some other suitable agent, and the action recommences. In
practice, the paths ad and be are very nearly adiabatic, but if n<y,
the changes of temperature are less than those just computed.
615. Entropy Diagram. Let aenfbc, Pig. 302, represent the pv and nt
diagrams of a BellColeman machine working in two compressive stages.
Choosing the point c on the entropy plane arbitrarily as to entropy, but in
its proper vertical location, we plot the line of constant pressure ca up to
the line of temperature at a. Then ae is drawn as an adiabatic, intersected
FIG 302 Art. 615.  Twostage Joule Cycle.
by the constant pressure curve ne, with nf, cb, and bf as the remaining
paths. The area aenfbc measures the expenditure of work to effect the
process. Along ca, theoretically, heat is taken from the cold chamber to
the extent cgha. The work expended in singlestage compression would
have been camb. We have then the following ratios of heat extracted to
work expended:
singlestage compression, ^^L  twostage compression, ^ a u
& ** r camb aenfbc
616. Work of Compression. In Fig. 301, for M pounds of air
circulated per minute, the heat withdrawn from the cold chamber
along dc is Q = Mk(T c T d ). The work expended in compression is
Mn
458 APPLIED THERMODYNAMICS
fp \ft~i T
If compression is adiabatic, n = y, fpM n ^irn PJ 7
v*c/ c
and T7 C = JfftZ^  lU m(Zi  20. Similarly,
y
for the engine (clearance being ignored in both cases), W E =
 2^), The net work expended is then
We might also write, heat delivered to the cooler =q =MJc(T b T a ),
Practical imperfections will increase the power consumption 30 to
50 per cent above this.
617. Cooling Water. The heat carried away at the cooler must be
equal to the heat extracted along dc plus the heat equivalent of the net
work expended; it is
Mk(T c  T d + T b  T c  T a + T d ) = Mk(T> T a ),
as the path indicates. Let the rise in temperature of the cooling water
be Tt: then the weight of water required is Mk(T b T a ) + (Ti).
618. Size of Cylinders. At ^revolutions per M pounds of air
circulated, the displacement per stroke of the doubleacting com
pressor piston must be, ignoring clearance, D = MV C *%N=>
The same air mus