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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 gas-engine 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 



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. 

NEW YORK, 1913. 


" 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 non-condensing 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. 


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 text-books 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 so-called 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 


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 multiple-effect 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 re-presented. 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. 

NEW YORK, August, 1910. 














The cold air engine: cycle, temperature fall, preheaters, design of 
engine: the compressor: cycle, form of compression curve, 
jackets, multi-stage 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- 


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. 


Formation at constant pressure: saturated steam: mixtures: 
superheated steam: paths of vapors: vapors in general: steam 
cycles: steam tables. 




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 

"Economy, condensing and non-condensing, of various commercial 
forms with saturated and superheated steam: mechanical effi- 


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, multiple-effect evaporation. 


Change of volume during change of state, pressure-temperature 
relation, latent* hoat of fusion of ice. 


Preswure and cooling, critical temperature, cascade system, regen- 
erative apparatus. 


Air machines: reversed cycle, Bell-Coleman machine, deiw air 
apparatus, coefficient of performance, Kelvin warming machine: 
vapor-compression machines: the cycle, choice of fluid, ton- 
nage rating, ice-melting effect, design of compressor: the absorp- 
tion system, heat balance: methods and fiolde of application: ice- 
making; commercial efficiencies. 


F = Fahrenheit; 
C = Centigrade; 
R = R6aumur; 
- Radiation (Art. 25); 
= gas constant for air = 53,36 fUb. 

= ratio of expansion; 
P,p = pressure: usually Ib. per sq, in. 

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; 


-7= -specific heat; 





-=-= 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- 

I - specific heat at constant volume; 


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, =horse-power; 
d= density; 

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. 



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 stored-up 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 



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. 


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 far-reaching 
applications in physios and chemistry. In its mechanical aspects, it deals 


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 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 


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 non-cardinal 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* 


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 


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 


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 
so-called "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 well-defined 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 non-existent, 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 270-290 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 


Solids, in Science, Nov. 5, 1909, p. 654. (4) Quarterly Mathematical Journal, 
April, 1855. (5) Darling: Heat for Engineers, 208. 


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. 
Thermodynamics : the mechanical theory of heat ; in its engineering applications, the 

science of heat-motor efficiency. 

Heat intensity, temperature : definition of, measurement of ; pyrometers. 
Thermometric scales: Fahrenheit, Centigrade, Reaumur; fixed points and their 


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; 



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.) 



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 

Thus, 5 Ib. of water at 200 mixed with 1 Ib. at 104 gives 6 Ib. at 
184; the hotter water having lost 80 " pound-degrees," 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). 



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 gram-calorie 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 

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, 


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 

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. [62-6r3 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 + zZv-m =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 left-hand 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, 


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 foot-pounds = 
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 kilogram-meters 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= foot-pounds 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 


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 foot-pounds 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 foot-pounds 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 


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 foot-pounds 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 = KV-J = (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. 


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, 1878-79. (10) Phil. Trans. Boy. 
Soc , 1893. (11) Phil. Trans , 1897. (12) History of European Thought, II, 137. 
(13) K. Wormell: Thermodynamics, 1886. 


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. 


1. How many Centigrade heat units are equivalent to one calorie? (Ana., 

2. Find the number of gram-calories 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.) 


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 kilogram-meters 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 !T-P(14-0.000002 P) I" 0.031 (~\ 3 -0.0052~| . 

L \ / I 

!., PF=0.5962 T-P(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? 



38. Boyle's (or Marietta's) Law. The simplest thermodynamie 
relations are those exemplified by the so-called permanent gases. 
Boyle (Oxford, 1662) and Mariotte (1676-1679) 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- 

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 well-known 
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. 






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 Gay-Lussac 
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 

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 Gay-Lussac or of Charles (1). Davy had announced that the 
coefficient of expansion of air was independent of the pressure. Gay-Lus- 
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 



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. 




FIG. 4. Arts. 42, 48. Coefficient of Pressure. 


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 Gay-Lussac, 
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 de-note 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, 


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 


experiments of Regnault and Rudberg, 

T= 1.365, 


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 Gay-Lussac (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- 

FIG. 7. Art. 48. 
Preston Air 


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 - 491-4 = 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 (000-800 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. 
Air Ther- 

Any deductions which might be made from these sym- 
-- riii A n-p /wiirqA he rigorously true only 


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 foot-pounds. 
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 , 

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. 



times given, PV= rT -+ M, in which Mia the molecular weight and r 
a constant having the same value for all sensibly perfect gases. 











Carbon dioxide 





C0 2 




C 2 H fl O 

Carbon monoxide 





(C 2 H 6 ) 2 

Sulphur dioxide . . . 
Chlo'rofQTrn. . T . . 




S0 2 
CHC1 3 





Olefiant gas 



CH 2 



SteRTY) . . ,.,.,.. 



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(d-d') 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 


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. 

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, 

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 

PV-WRT t- 
PV-W&l, V ~WRTP 


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). 

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. 


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 = j-N = ~JV =* [ N ' m w ^ c ^ ^ ^ tne totia ^ number of molecules 


in the vessel. Let q be the area of the limiting surface. Then the force per unit 

of aurfaw-p-F+g-^ + a-^.whencepB-^-g-w*, 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 

(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, 
104-107. (8) Op. eft., 593-595. (9) Nichols and Franklin, The Elements of Physics, 
I, 199-200. (10) Ibid., 199 ; Wormell, Thermodynamics, 167-161. (11) Over de 
Continuiteit van den Gas en Vloeistoestand, Leinden, 1873, 76 ; tr. by Roth, Leipsic, 

Boyle's law, pw = PF: deviations. 
Dalton's law, Avogadro's principle. 
Law of Gay-Lussac 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. 


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 : 


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 1-gallon 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 oxy-hydrogen 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 


cent of the weight of air has been removed ? (14.697 Ib. per sq. in. =29. 92 ins. 

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 2-mile 
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 2-mile 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.) 



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; 


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 = dH-s- 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). 


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 


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 12.387-7-491.4 = 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 53.36-7-778 = 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. 


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, 



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 l-R l-k-R i-v- -* .y. 

A-Z-.B. i-k t. t -y k _ E y 

fFor air, this gives ' 237 ^ ^ = 1.402.) 

9^7^ ^"J-t5v / 


~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 

Regnault's result was 2375. 

= R -- = (53.15 - 772) x (1.4 - 0.4) = 0.239. 
y 1 



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 
foot-pounds. 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 

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- 

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. 


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. 

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 


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=(k-R)(T-t}. 

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(T-t) 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* 


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 



quantities. R may by either 63.36 or -ZZT-- The student should discern whether 


heat units or foot-pounds are intended. (4) Zeuner, Technical Thermodynamics, 
Klein tr., I, 121. (5) Ibid., loc. tit. (6) Ewing: The Steam Engine, 1906. (7) 
Wormell, Thermodynamics. 


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. 


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, 

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.) 


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.) 




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. 12-14, 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^ 




twice; and the corresponding lines AB, AC, and AD are always 
equal in length. 


FIG. 11 



x^ o 


Art. 82. FIG 
ictive Dia- 

V t 

...A. D . 



! 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 




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. 16-18. 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. 12-14, 16-18). The path of a 
substance is sometimes called its process curve, and its thermal line, a 

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 




(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 




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 


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 

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- 


FIG. 23. Art. 89. 
External Work in 
Closed Cycle. 


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 foot-pounds. 
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 

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 + J-h W, F+I= 0, whence H= W, the expenditure of heat 
being equivalent to its sole effect. 

If the work is measured in foot-pounds, 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 

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.), 

(7-53.36X491.4 = 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. 


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 

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, 


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 


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, 



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;. 


When F= infinity, P = 0, and the work is indeterminate by this expression; but 

we may write W = -^~ (l - } = -PL. [~l _ f-HV" 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, 

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 non-conducting 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 


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). 


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) 


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) 


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, 


* See, for example, Appendix A to Vol. HI of Nichols and Franklin's Elements of 


V being in feet per second and w in Ibs. per cubic foot. When a volume of gas 
of cross-section =n and length I is subjected to the specific pressure increment dp, 
producing the extension (negative compression) dl, 



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, 

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, 

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 = 


For t = 530, p = 100, P = 400, this becomes 


F = 

32.2 x 53.36 x 530 x 800 x 144 




= 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 




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- ' 


FIG. 29. Art. 108. Isodiabatics. 


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 




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). 



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 


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 H-W*=(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. 


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 ; G-ay- 
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 

" * rt-rr 

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 


= 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, 



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 



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. 



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. 



In Fig. 32, let a&, ctf, e/be curves thus drawn. Their general equation is pv n = C, 


= 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 

FIG. 33. 

Art 116. Negative 

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. 


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 = (V-v)-v', and tan 7>0JB = UL - = 7X- NR = (p-P) -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. 



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- 


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, 



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 


FIG. 39. Art 123 Represen- 
tation of Heat Absorbed. 



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 


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 


I + cs 

where H is the heat evolved in combustion. During any part of the subsequent 

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 - = --, 

i-Mo * T 

= = f J-Y* 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 non-gaseous 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 



W= 0.24 x 4000 x 778 { 1 - (j^) "} = 1,000 ft-lb. 

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, 

= -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. 


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. 


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 

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 heat-insulated 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- 


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). 

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 ^ 


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 


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 = 1593-Ji^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 


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 self-sustaining ; 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, 71-77, 437. 
(4) Zeuner, Technical Thermodynamics, Klein tr., I, 156. (5) Ripper, Steam Engine 
Theory and Practice, 1895, 17. 


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 

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. 


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 . 







1 30103 




2 30103 




1 30103 



Definitions; log x or com log s-n, 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: 

log (a-j-6) =log a log b. zfr^tyx. 

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) =0,74-0.1114 = 0.6286. 

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= -0.6286-1.3714 
-log (0.2352 =0.02' 37 ). 



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, 


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 



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? 


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 water-jacketed 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 

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)i-35=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- 



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 

absorbed; to 


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. 



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 non-conduct- 
ing, while the head is 
a perfect conductor. 

The piston itself is 
FIG. 43. Arts. 132, 138. Operation of the Carnot Cycle. -, , , 

a non-conductor 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 non-conductor. 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 , 


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. 133-136, 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, 


lor the adiabatie da, 

T I V \i- 
~ = ( .iJ? ) 
t \\ &/ 

T / T 7 " \ i~v 
and - = ( ~5 J for the adiabatie Ic \ 


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 


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. 


' a 

136. Second Derivation. The external work done under the two adiabaties 
, da is 

y-i y-i 

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 


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 

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 self-acting 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- 


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 


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 


Off /"i 

decreased to ^ ft 4fi =0.43, or about one-half. (A boiler is of course 
not a heat engine.) 

(1) Alexander, Treatise on Thermodynamics, 1893, 38-40. (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, 


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 - = J-5; ^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. 


1. Show how to express the efficiency of any heat-engine 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 non-reversible 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 non-condensing 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,) 


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 horse-power (33,000 foot-pounds) 
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 kilowatt-hour. 
(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 double-acting, and makes 100 r.p.m., what is its 
horse power? 



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 

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 



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. 


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 (T-T 3 ) = 3(T-2 r7 1 ) and (T-T 3 ) 
= 2(!T-2 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 Tuermodynamu-s. 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 waste-heap of the Universe." Griffiths, 


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 


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 

^' "^ ^ ~~\ f ?h' 
~ h + (111 ~~ A + rlh 

Tor e = z(T f)> we ^a-J write e zdt or s = -f, giving 


* = 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)N-abfe' 

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 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. 


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. 


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 


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 foot-pounds. 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. 


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 


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.), 


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 pathpvi-8 =c for air (2=0.1689). 

(Ans., 0.084.) 


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. 

Find the amount of heat converted to work in the cycle, if the working substance is 
air. (4ns., 4175 B.t.uJ 



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.* 

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 


FIG. 4:7* Arts. 158, 163, ]71. En- 
tropy Diagram. 


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.. 


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 





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. 



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- 

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 


dH_ I riff I rlH__~ 
^r~ \* "F + P "F-' 
C/^i *Jo 

while for the cycle ADCDA, 


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 f-m- 



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 = dff-i- 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= PV-r- 72, while t = pv -*- 72, we have 

n = k log e (P V+- pv) . 

The expression Jclog e (T-r-) is the one most 
commonly used for calculating values of the hori- 
zontal coordinate for polytropic paths. The 
expression dn = dH-t- 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 JV-axis 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. 


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 

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 


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, 


= 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. 

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 


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, 


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. 


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, theta-phi diagram. 
The TN diagram is now commonly called the temperature-entropy 
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. 


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 


is Jc log fl -^ The difference of entropy between 

* i 

1 and 2, irrespective of the path, is 



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,, 


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 text-books, 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. 45i-458. 



FIG. 56. 

Art. 175. Irreversible 


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 : 


(a) In a reversible operation, the sum of the entropies of the participating substances 
is unchanged. During a reversible change, the temperatures of the heat-absorbing 
and heat-emitting 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, 


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 


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 heat-insulated 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 Temperature-Entropy 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. 


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 

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- 

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. 


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 ; non-isentropic adiabatics. 
In reversible operations, the aggregate entropy of the participating substances is 

During irreversible operations, the aggregate entropy increases, and J <0. 

The loss of work due to increase of entropy is nT\ du>d. 



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 020-0.0004 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.) 


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 


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. 



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. Two-stage 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 paddle-wheel 
blowers, commonly employed in ventilating plants, move large yolumes of air at 
very slight pressures, usually a fraction of a pound, and the thermodynamio 




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."' 


3 \g \^ 



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. 180-183, 189, 222, 223, 
226, Prob. 6. Air Engine Cycles. 


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 
back-pressure line CD. Then, q = P~pv+ F, and the work would be 
pv log e (F-4- v). For complete adiabatic expansion, the work would be 


-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 



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 10-hp. 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 72-hp. 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. 


188. Volume of Cylinder. If n be the number of single strokes per 
minute of a double-acting 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 foot-pounds 
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 

through by the piston per minute = Wv=nV = WR- f whence 


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 single-acting 10-hp. 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. 



= 61.31. 

Work per stroke =jABi + iBCm - EDmk -jFEk 

D ,r T. x . P*V*-PoVG r> (V V\ PrVr-P*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 foot-pounds. 

The actual engine will then give 0.9 x 5803.2 D = 5222.88 D foot-pounds per stroke 
or 5222.88 D x 100 foot-pounds per minute, which is to be made equal to 10 hp., or 

b 114.7 


FIG. 59. Art. 190. Design of Air Engine. 

to 10 x 33,000 foot-pounds. 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 


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 1.0-r- 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. 


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 right-hand 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- 



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 

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 



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 cast-iron 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. 



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. 


Art. 200. Cooling by Jet 

Art. 20Q. Card from Colladon 

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 half-inch 
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 

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. 



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 




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 heat-conducting power is 


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 


>v-PF , PV 
-. nAIEi = . 


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 



p, expressed in foot-pounds. For isothermal compression; as along 
AD, IACL=pv log e (F-s-fl), 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(^ 


y-1 0-1 

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=H-r-(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. Multi-stage Compression. The effective method of securing a 
low value of n is by multi-stage 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 multi-stage 
compression, the air follows the path 
AB up to a moderate pressure, as at 
), and is then discharged and cooled 

Art. 205. Multi-stage Com- 

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. 



a second cylinder, is adiabatically compressed along HF, ejected and 
cooled along ]?G/-, and finally compressed in still another cylinder 

along G-H. 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. Two-stage Com- 
pressor Indicator Diagram. 

about 1 .25. Figure 70 shows the diagram from a two-stage 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 two-stage 

Biedler compressor with spray in- 

jection, AB being an isothermal 

and A an adiabatic. 

JFio 71. Arts. 205, 214. Two-stage 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. Multi-stage 
compression does not pay unless the intercooling is thoroughly effective. It seldom 
pays when the pressure attained is low. Incidental advantages in multi-stage 
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 cast-iron cylindrical shell, with cast- 



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 cast-iron inter- 

FIG. 72. Art. 207. Allis-Chalmers 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. 


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.- Ingersoll-Seigeant Vertical 
present after the last stage of com- Intercooler, 


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 

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 foot-pounds 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 text-books, 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. 


211. Range of Stages in Multi-stage 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 

Differentiating with respect to P, we obtain 




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 four-stage 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 two-stage 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 three-stage 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. 



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 


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 well-designed 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 over-all 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 



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 


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 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, 



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 

FIG. 77. Ait. 220. Combined Entropy Diagrams. 

221. Multi-stage 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 



FIG. 78. Art. 221. Three-stage Com- 
pression and Expansion. 

FIG. 79. Art. 221. Entropy Diagram, 
Multi-stage Compression. 



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 three-stage, 
with perfect intercooling and aftercooling 
and preheating and with no drop of pres- 
sure in transmission. BbA and AliB 
would be the diagrams with single-stage 
acliabatic compression and expansion. 


Fm NO Art 22] Three-stage 
Compression and Expansion. 

FIG 81. Art 221. Thiee-stage Compression and Expansion. 


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 (Vc-V 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 




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 

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 Multi-stage Compression. Since the 
effect of multi-stage 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 single-stage 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 



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 

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 non-isothermal 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 BC-t-BE, 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. 



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 jPF-s- T ' =f>v -s- t, the volume drawn 
into the compressor per minute is V=pvT-t- 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 double-acting, 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 

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, 



2464+- [1-0.02 

+ 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 double-acting 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. Two-stage 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 first-stage 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 low-pressure 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 
double-acting single-stage 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, 




V a = ( ^+= \ 1.04 D = 0.2158 D. 
4 0.04 D = 0.1829 D, 
1.04 D = 0.9872 ZX 

But j4-B = 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 double-acting, 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 




= 144*[(114.7x0.1758 )+ 

-(13.7 x O.S473)- 

^ J 


= 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 


' = 715. 


234. Design of a Two- 
stage Machine. With condi- 
tions as in the preceding, con- 
sider a two-stage 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 Two-stage 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). 


sure is V114.7 x 13.7 = 39.64, whence the first-stage discharge pressure and the 
second-stage 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 = 1-04 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 />, 


/ 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 oot-pounda or 10,404,475 foot-pounds per minute, equivalent to 
315.3 horse power. 


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 


volume at this point will then be V z = i-fV 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 
(Fs-Fj?) =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. 


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 


- (39.14X0 3329; ,^4-7X0 OW-gQ 14XO.OB11)] 

=816 5 horse power. 

The total horse power for the two-stage 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 two-stage compression, the saving 

i-r-t e ftQO 

in power is ^ "^ = 12 of the power expended in single-stage compression; 

that the low-pressure cylinder of the two-stage machine is somewhat smaller than 
the cylinder of the single-stage compressor; and that, in the two-stage 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 single-stage 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) . 


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. Steam-driven 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 cut-off on the steam cylinder. Power- 
driven compressors may be operated by electric motor, belt, water wheel, or in 




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 single-stage 
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 single-stage 
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 high-grade service, in large units, the standard form is the cross-compound 
two-stage machine, the low-pressure 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 6-stage 
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. 



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 


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 

The Leavitt hydraulic piston compressor at the Calumet and Hecla copper 
mines, Michigan, has double-acting 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 

FIG. 90. 

Art. 241. Taylor Hydraulic 



her. The shaft is lined with a riveted or a cast-iron 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 

242. Details of Construction. The standard form of cylinder for large machines 
is a two-piece 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 



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 
semi-rocking 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 high-pressure cylin- 
ders of multi-stage com- 
pressors, as they become 
fully open too late in the 
stroke to give a suffi- 
ciently free discharge. 
In Fig. 93 Corliss valves 


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. (AUis-Ohalmers Oo.) 


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. 


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 6-inch pipe,/= 0.00435; for 8-inch 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 


a smaller scale, by Stookalper. For cast-iron 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 well-designed 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 

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 
three-stage 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 


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, 303-313: Trans. A. S. M. E. t XXI, 156. 


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 - ; O-PF) -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 Uip-hr. 

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= 

Multi-stage 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. 


Work, neglecting clearance (single cylinder), = T ^ r =-]; 

The area under the compression curve is called the ioork of compression. 
Minimum work, in two-stage 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 multi-stage 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: multi-stage 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, multi-stage 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 


A typical problem shows a saving of 12 per cent by two-stage 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 cross-compound two-stage 

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. 


Cylinder barrels and jackets are separate castings. Access to water space must be 

Poppet, mechanical inlet, Corliss, and mechanical discharge valves are used. 

Compressed Air Transmission 
Loss in pressure = 0.00000936 n-l-rd 

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 > 

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 


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 double-acting 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 10-lb. 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 four-stage 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 four-stage compressor and two-stage air 


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 smgle-stage 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 
double-acting 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 12-m. 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, 



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 hot-air 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 




was estimated that a steam engine of equal power would have required 
only a single cylinder, 4 ft in diameter and of 10-ft. 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 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. . 


Another expression for the work, since 



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 


~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 


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 


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 


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 


~P fT\ vT 
be obtained; hence ^< ( ] . The efficiency of the Carnot cycle may of course 

be written as 1 





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 

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 


- 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 



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" T-T \ /" 77" _L 7" \ 

\ n <u ~r -"ic -fljc -H-aj) (/"ai r -Miejj 

which may be expanded as in Arts. 251, 252. 


FIG. 99. Arts. 233, 256. Reit- 
linger Cycle. 

FIG. 100. Arts 253, 250, 257, 258 f 
259. Reitlmger Cycle, Entropy 

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. 



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 


q -_^ _ 

T-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 

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 


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. 
95-100, 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 

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. Hot-air Cycles. 


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 



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 
non-conducting 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. 


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 loose-fitting 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. 



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 loose-fitting plunger; in double-acting 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 
Laubereau-Schwartzkopfl: 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. 



FIG. 106. Arts. 204, 205, 267. Stirling Cycle. 

264. Pressure-Volume Diagram. The cycle of operations described in 
Art. 261 is that of Fig. 106, ABCD* Considering the cushion air, the 



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 non-conformity 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, 

and the efficiency is 


T ' 

that of the Carnot cycle. Losses through the regenerator and by imper- 

fection of cycle reduce this in prac- 


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. 



267. Importance of the Regenerator. Without the regenerator, the non- 
reversible Stirling cycle would have an efficiency of 

(P.- P,)^ log. ? 


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 50-hp. 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 Hot-air Engine. In 1833, Ericsson constructed an unsuccess- 
ful hot-air engine in London. About 1855, he built the steamer Ericsson, of 2200 
tons, driven by four immense hot-air 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 non-conductor. 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 



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 

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. 


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 


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- 


sure (EBCF-r- 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 foot-pounds =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 Hot-air 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 
hot-air 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 heat-transmitting 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 


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, 314-323. (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, 1883-1884; Min. Proc. Inst. 
C. E., 1845,1854. 


The hot-air 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. 


; the Carnot cycle leads to either excessive pressures or an enormous 



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 

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. 


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. 


(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. 


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 





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 four-cycle 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 twenty-six 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) non-compressive 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 400-hp. four-cylinder engine was in use on gas made from anthracite coal. At 
the present time, double-acting 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 25-inch pipe line, at 500 Ib. 
initial pressure and 250 Ib. terminal pressure, carrying 40,000,000,000 cu. ft. of 



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 


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 by-product 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). 



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 brick-lined cylindrical shell, set over a water-sealed 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 red-hot 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. 


At E are openings for the bars used to agitate the fire. At F are peep- 

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 
" under-feed," " inverted combustion/ 7 or " down-draft " types : causing the trans- 
formation of the tar to fixed gases. On account of the difficulty of tar removal, 
anthracite coal or coke or semi-bituminous, non-caking 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. By-product 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. 




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 



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 


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 


follows : to heat the gas 1 F. there are required 


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. 

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 


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- 



FIG. 116. Art. 287. Single-acting 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.} 


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. 


287. Four-cycle Engine. A gas engine of one of the most commonly used 
types is shown in Fig. 116. This represents a single-acting 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 double-acting, 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.m-TheOttoCrcle. 

being at c, Fig. 118, the gas is ignited and practically instantaneous 
combustion ogqurs at constant volume, giving the line <?(7, An out 



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 double-acting engine of this type, 
there would be two working strokes in every four. 

FIG. 110. Arts. 1289-201, -TO, 3#J. Two-cycle Gas Engine. 
(From "The Gas Engine," t>y Cecil P Poole, with the permission of the Hill Publishing Company ) 

289. Two-stroke 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 single-acting 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 


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 left-hand 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 two-stroke 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. Two-stroke Fig. 121 COrre- 
y cle - spends with the 

negative loop deal of Fig. 118. Aside from FIG. 13L Art. 291. Two-stroke 
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. 



diagrams for the two-cycle and four-cycle 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 = PtVjfVJ-v and P f V f = P d F 6 "F^, giving 

a- P* W V^ - P Vf VJ-* 


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 



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 
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 












JO .20 .30 .40 .50 .60 .70 .80 .90 LOO 

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 


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 



FIG. 1226 Art. 2955. 

FIG. 122e. Art 


FIG. 122d. Art. 295b. 
Let P c = 14.7, P d = 147, P tf = 294. 


n-y 0.561 

n - 1 0.963 

log 0.10 
= 0.582. 






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 

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: 



Atkinson, 1 

T-t_ 3440 -520 
T ~ 3440 

.. n . 620,. 

= 0.85. 

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 

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 


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 two-cycle 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 two-cycle 



FIG. 127. Arts, 2<>9, P>02. Bray ton 

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 

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. 


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) 

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. 

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. 


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 equal-leugth-stroke 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) 


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 = Tc-Tn 

r r r n / 7 T " 

1 E - 1 c J-C 

306. Oil Engines : The Diesel Cycle. Oil engines may operate in either 
the two-stroke or the four-stroke 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 four-cycle 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 pre-ignition 
in the supply tank, the high-pressure air used to inject the oil must 
be cooled. The cylinder is water-jacketed. Figure 131 shows a three- 
cylinder engine of this type; Fig. 132, its actual indicator diagram, 

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. 


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.) 


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 



But 2>= r- rancl Z^; whence 


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- 


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 



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 


Inspection of the diagram shows that the efficiency decreases as the load 

(For a description of the Junkers engine, see the papers by Junge, in 
Power, Oct. 22, 29, Nov. 5, 1912 ) 

P T 


FIG. 132a. Art. 307&. Constant-pressure 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. 122-128, 130. The cycles are, 

FIG. 133. Art. ^07. Diesel Engine 



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 


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 
four-cycle 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. 


charge may be safely introduced at that point, by outside compression 
on the two-cycle 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 twenty-four hours, and 
has developed a thermal efficiency (to water) under test of about 
22 per cent. (See American Machinist, Jan. 5, 1911.) 


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 

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 high-speed engines; these are positive in their 


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, 


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 
two-cycle engines, this varies from 18 to 21 Ib., and for four-cycle 
engines, from 12 to 14 Ib., both absolute. 

The pre-compression 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 pre-compression 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 non-compressive 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 





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 = - 


- = 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- 

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 


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- 



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 



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. 



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 pre-ignition, 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 






FIG. 140. ArL. C.I. 7i-.c of Ii^...L_. 
(From Poole'a " The Gas Engine," by permission of the Hill Publishing Company.) 












320- ^K-^ 10,500 


10 11 12 13 14 15 16 17 18 

Ignition Advance, Per Cent 
FIG. 140a. Art. 321. Mixture Strength and Ignition Point. 


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 left-hand 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. Self-ignition 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 self-induced 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 self-ignition. 

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). 




141. Art. 323. After 

'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; after-burning 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 


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 

valve-actuating 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 



FIG. 144. Ait SCO Thi-ottlod 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 pre-ignition, 
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. 




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 


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. 



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 two-cycle 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, 


FIG 149. Art. 329. Actual and Ideal Gas Engine Diagrams. 


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 





(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), 




a al S-J 

The external work done during the expansion is 

J/ 6 
Idt = I (CL -J- oi) dt'=' ct \t% ti) ^2 ~ ^i)^ or 
x 2 

n-1 "' 

where n has the value given in Equation (3). 

We may find a simple expression for n by combining these equations : 


The efficiency of the Otto cycle debf, Fig. 122, may now be written 


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 


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. 

a-c 0.1620 -0.23327 c 0.23327 ,.. . _ , A _ 

= - 01620 - = -- 4il ; a~oi62 L44; lo &*-3-7- 

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 15030-13.32-0.0327= 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+(3.27x-0.131) = 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 


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, 7360-12.3 -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 

. . 


The value of n is thus about the same for this curve as for that formerly 
considered, and (approximately), in Fig. 122, 


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 


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) 

= 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. 


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 


then write m. e. p. = W - ( V 4 - 7 e ); but from Art. 295, W = Q[I - (fr) * ] 5 



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 four-cycle^ 
two-cylinder, double-acting 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 (0-619 - 
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. 332-335. Design of Gas 

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 : 


= 613, 


> 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. 


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 two-cylinder, four-cycle, double-acting engine, all of the strokes are work- 
ing strokes ; the foot-pounds 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 0.2045y-H(12 X 1.2045)1 
144XU.4S^ 04Q2 J 

=7630 D fooir-pounds. 

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 : 


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 


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 
four-cylinder^ four-cycle, single-acting 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- 

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, 


The quotient 1234 -f- 123'4' = the diagram factor Then write 
1( ? T3} * diagram factor x Jf= -JSL^ 


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/ = 1-35 Ib. II 2 per Ib. C 6 H 14 
11 x W = 11-82 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 
foot-bounds. This is equivalent to '^ * = 1540 B. t u. per minute or to 


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 


of these rules is, brake hp. == -, where n is the number of four-cycle cylinders of 


diameter d inches, running at a piston speed of 1000 ft. per minute, 


336. Otto Cycle Oil Engines. This class includes, among many others, the 
Mietz and Weiss, two-cycle, and the Daimler, Priebtman, and Hornsby-Akroyd, 
four-cycle. In the last-named, 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 



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 hot-tube ignition without a timing valve, the hot tube serving as a vaporizer. 
Extraordinarily high speeds are attained. 

337. Modern Gas Engines : the Otto. The present-day small Otto engine is ordi- 
narily single-cylinder and single-acting, 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 Crossley-Otto, a leading British type, hot-tube ignition is used, and 
the large units have two horizontal opposed single-acting cylinders. In the 
Andrews form, tandem cylinders are used, the two pistons being connected by 
external side rods. 



338. The Westinghouse Engine. This has recently been developed in very 
large units. Figure 132 shows the \\oiking side of a two-cylinder, tandem, 
double acting engine, representing the mlt valves on top of the cylinders. 

FIG. 152. Arts. 338, 330. Westinghouse Gas Engine. Two-cylinder Tandem, Four-cycle. 

Smaller engines are often built vertical, with one, two, or three single-acting 
cylinders. All of these engines are four-cycle, with electric ignition, governing 
by varying the quantity and proportions of the admitted mixture. Sections of 
the cylinder of the Riverside horizontal, tandem, double-acting engine are shown in 
Fig. 15;?. It has an extremely massive frame. The Allis-Chalmers engine is built 
in laige units along similar general lines. Thirty-six 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 direct-connected to electric generators. 

339. Two-cycle Engines. In these* the explosions are twice as frequent as 
with the four-cycle engine, and cooling is consequently more difficult. With an 
equal number of cylinders, single- or double-acting, the two-cycle engine of course 
gives better regulation. The first important two-cycle 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 single-acting 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 double-acting 
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 air-cooled cylinder is often 
used, with gasoline fuel. Occasionally, tv\o cyhndeis are employed. The engine 




is four-cycle 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 air-cooled 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, four-c\cle, single-acting, 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 hot-tube 
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. Two-cycle engines 
have been introduced. The fuel in this country is usually gasoline. For launch 
engines, the two-cycle 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 
over-all 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 


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 well-adjusted 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 (3-1) 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, fom-cycle 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 kilowatt-hour 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, 


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: 





7V/ n-cyvlc 

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 single-acting engine are of course relative!} 71 greater 
than those for an ordinary double-acting 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 by-product 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. 



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 



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 

= 12.64. 

Figure 155, from Reeve (42), is from a similar four-cycle 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 Hornsby-Akroyd 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 


FIG. 156. Art. 347. Diagrams for Hornsby-Akroyd Engine. 


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 cut-off 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 "Jiitor-miss" 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 pre-ignition 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 hit-or-miss system may 
be regarded as entirely inapplicable to large engines. The other 
practicable methods sacrifice the efficiency. Further than this, the 


governing influence is exerted during; the suction stroke, one full revolu- 
tion (in four-cycle 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 four-cycle 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 double-beat 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 four-cycle 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 self-starting. 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 gear-faced 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 multiple-cylinder 
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 


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 direct-driven 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 water-spray injection during expansion has been 
abandoned, and air cooling is practicable only in small sizes (say, for diameters 
less than 5-inch). The cylinder, piston, piston rod, and valves must usually be 
thoroughly water-jacketed * 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 non-condensing 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 stand-by 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 single-acting four-cycle 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 twin-tandem double-acting cylinders. 


stored gas has been used for driving river steamers in France, England, and 

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, 
119-121. (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. 342-343. (41) Trans. A. S. M. E., XIX, 491. 
(42) Ibid., XXIV, 171. (43) Lucke, Gas Engine Design, 1905. (44) Op. tit., 


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 

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 ics and two lines of constant volume; the 

engine may operate in either thefour-s'rftkc eyrie or the two-stroke cycle. 
In the two-stroke 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 J-n) Tb T n 
T T 1 

or, with complete expansion, " "" <l 


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 rri g rn 

JL e JLo J. e 

intermediate between that of the ordinary Otto and the Atkinson. 


The Diesel cycle: isothermal combustion; efficiency = 1 -- L \ a/ - =!; increases 
as ratio of expansion decreases. . yRT a log e 

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 pre-ignition ; the 
range of compression, \\hich determines the efficiency, depends upon this as 
well as upon the pre-coinpiesaion 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 

Analysis with specific heats variable. 


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. 


Current Forms 

Otto cycle oil engines include the Mietz and TVeiss, two-cycle, and the Daimler, Priest- 
man, and Hornsby-Akroyd, four-cycle. 

Modem forms of the Otto got* engine include the Otto, Foos, Crossley-Otto, and 

The TTestinghouse, Riverside, and Allis-Chalmers engines are built in the largest sizes. 

Two-cycle 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 
four-stroke 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 hit-or-miss 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 ; 
stand-by 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 self-starting ; 
cylinder must be cooled ; low ratio of expansion ; non-reversible ; no overload 
capacity ; no available by-product heat for process work in manufacturing plants, 


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, 


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 four-cycle 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 four-cycle, two-cylinder, double-acting, 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. 



modified as described in Arts 309-328, 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 two-cycle. 

18. Find the size of cylinders of a four-cylinder, four-cycle, single-acting 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 two-cycle 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. 


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 pre-compression 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 , cut-off occurs when the volume is u ~~ * + F, Fig. 134.) 


26. Under the conditions of Art. 835, develop a relation between piston displace- 
ment in cubic inches per minute, and Ihp., lor four cylinder four-cycle 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 six-cylinder four-cycle engine, single-acting, 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 2-cycle 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. 


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 

* 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. 



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. 


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 ; 


(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. -f-J 

* 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. 


(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 

(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 Dwelshauevers-Dery (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 pressure-tempera- 
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.) 


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 


are compared. Evaporation under these conditions is described as 

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. Pressure-temperature 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- 

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. 


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 


where p is in pounds per square inch, c = 0.256, 5 = 0.5962 and T is in degrees 


More recent experiments by Knoblauch, Linde, and Klebe (1905) (22) give 

the formula 

j-0 5962 r-p(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 

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. 




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 

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 



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 feed-water 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. 



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 

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 i-s-2 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. 


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. 



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 (T-L)(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. 


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. 




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 pressure-tempera- 
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. 


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 

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 88-5 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. 



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- 


S 85 ' 



I 7fi 




FIG. 168. Art 380. Critical Temperature. 

FIG. 109. Art. 381. Evaporation at 
Constant Volume. 


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. 


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, 






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 




29 849 




29 803 




29 740 




29 654 




29 533 




29 365 




29 136 




28 829 




28 420 




27 885 




27 190 




26 300 




25 169 








21 991 




19 822 








13 961 




10 093 




5 471 







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 


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 14.697-0.1217 = 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 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 wick-covered ("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 wet-bulb 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 

question has therefore a relative humidity of o~74i~^'^" ^ e va ^ ue f -^ * or ^^ 

air is not 53.2, but 

a subordinate relation being 


- 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 


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 super-saturated 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. 

The internal energy of the mixture is 

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, ^H-p2(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(k-l) log* j 

as before; ^e-r 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 


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 

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 . 

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, 

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, 


<=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 


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 f|f) +144X82.3(508-4.845) [ =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 

j (0.1 

2 0.1689X2.3 log + o 0686X2.3 log 


and the check value for heat absorbed is (460 +314) X (1.49-1. 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, 

pv-py pv ~py+w 

W ~ n-l > n ~ W 

the equivalent value of n is 

1441 (200 5X4.845)-(195.3X5.08)} + (8.45X778) 

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 

.1^220X2 9-0085X80^ . 


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) 

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 

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 


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 


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 

7-814.0+2(34 59+72.64+0.043X119) -1255(1258) B.t.u. 

The entropy computed as before mixing Is 

after mixing, it is 

0.4627+ (HJ2-L1492) +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: 


Then 2'=% where 0/=26.79, the volume of saturated steam at 212. At 


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 heat-rejection is thus the feature 
which makes the use of a fluid mixture of practical interest. This is the more 
important, since from a power-producing 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 

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. 



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 : 



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 

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 



Amp(n+l) (l+^p hyT-n- 

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 


being the water line and cd the saturation curve. Its total heat is then 
H c -i-k(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 non-conducting nozzle, with reduction 


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 

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 


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. 


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 



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 short-dotted lines are plotted from Zeuner's equation, and the 
long-dotted 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 absoj-bed 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. 



396. External Work. The work during adiabatic expansion from 
PVto pV) assuming pv n = PF", is represented by the formula 


71-1 ' 

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 




/ I 




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 right-hand 
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 


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. 



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. Temperature-entropy 

for Steam. 



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 j-s- 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 





















i fc a " g * i i * l ". 




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 



80 100 & UO MO 180 SOO *2D iltt 2<KI 2W 300 SSO MO SCO 3*0 400 4204401004 

FIG. 185, Art, 399, PruWw*, Total Heavpressure Diagram, 


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 so-called total heat-pressure 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 pressure-temperature 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. 


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 foot-pounds, 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-* dT-P ^dx 

= J778 [for + C (l _ *)] _ p|ZJ dT + ( 77SL -P^\ dx. 
Moreover, E = (/) T, x, whence 



(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. 


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) 



+-*-. ( 

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 


* 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. 


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 pressure-temperature 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. 314-316, have the same foundation 

The present writer (in Vapor* for Heat Engines, D. Van No-strand Co., 1911) 
has computed the entropies and prepared temperature-entropy 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 

= S.40<9 


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. 


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 / 


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 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 pressure-temperature formula, permit of the 
computation of the latent heat from equation (D), Art, 400. An empirical formula 


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). 



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. 


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 

diagram as 


The external 

work done per pound of steam 

T t 
is L - ; or if the steam at I 


is wet, it is xL 


If the 



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 


FIG. 180. Arts. 407, 408, 410 T 412, 413. 
The Steam Power Plant. 



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 

409. Efficiency. 

In Fig. 181, cycle dele, the efficiency is 
_ ft, Ji a + L b xjj f 

But x c = 

% if the specific heat of the 


liquid be unity. Then letting 7, L refer to the state J, and t, I to 
the state <?, the efficiency is 

T-t+ L 

which is determined s0ZeZ# by tJie temperature limits Tand t. For 
steam initially wet, the efficiency is 


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 -h-x t Lsi 

a result identical with the numerator of the first expression in Art. 

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 


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 

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, ^-s-r*, Fig. 181, with given cyclic limits, the greater 
is the efficiency. 

412. NoB-expansive 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/ 



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 


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 


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 

py-pv^ ^ 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 



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 



FIG. 183. Arts. 417, 441, 442. Seventeen Steam Cycles. 



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. 



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. 


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 non-expansive 
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. 


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 
TW-T* 353.1 -,120.15 

= 0.88, 


!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. 


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. ~> foot-pounds (Art. 407); 
along eb is 144 x 140 x (Fi 0.017) = 64,300 foot-pounds ; 


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, *,= 


= [0.5072-2.3 (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- 

The whole work of the cycle is 64anft ~ : ^ 8 ' 5 " 374 + 100.76 = . 


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- 


The work along ek Is 14i x 140 X 0.90(T" 6 - 0.017)= 57J70 foot-powids. 
The work along kJ is A -f Xtf* A x jrj-r 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 foot-pounds; 
along eb? as in IFI, iw $4,300 foot-poundt ; 

along td is^(F fc - T*) = 144 x 2 x (3.21& - 0.017)= 9$2 foot-pound*. 
The wA^ tcorjfe efttie cycle is 

- 338.5 - 922 = 63,039.5 foot-pounds = 81.0$ B. 



VI. Non-expansive 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 foot-pounds* 
The work along Kd is 

J>(r*- 0.017)= 144 x 2 x 0.90 x (3.219 - 0.17)= 829.8 foot-pounds. 
The whole work of the cycle is 

57,531.5 - 829.8 = 56,701.7 foot-pounds = 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 foot-pounds. 
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- 

The whole work of the cycle is 63,961.5 -1- 236,800 - 49,900 = 250J61.5 foot- 

(Otherwise 1433 log, - 0.695 (2^- /)= 312 B. t. u. = 42, 000 foot-pounds 

(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 foot-pounds. 
The work along bu is 144 x 16[(140 x 3.219) - (31.84 x 13)] = 85,800 foot- 


The work along qd, as in III, is 37 40 foot-pounds. 
The whole work of the cycle is 

63,961.5 + 85,800 - 3740 = 146,021,5 foot-pounds = 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- 
plete-expansion Fambour cycle and the lower as a non-expansive 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 


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 


(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 foot-pounds. 
The work done along bx is 

P b (V, - T 5 ) = 144 x 140(5.96 - 3.219)= 55,000 foot-pounds. 
The w?0r& cfone atony a; is 

x 5J>51.1 x 13 = 81 j 00 foot-pounds. 

(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 foot-pounds. 
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 non-ezpansively. The (7r(w fta/ alwrbed 



The j<?rA: a/on^ <?, eb 9 bx, as in X, is 118,961.3 foot-pounds. 

The worfc along pd is 2 x 144 X (5.96 - 0.017)= 1716 foot-pounds. 

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 


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. 


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 font-pounds f 
The work done ahmy ly is 144 x 140 X (6.81 - 3 219) = 72,200 foot-pounds. 

(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 foot-pounds. 
The 7r&0/e ttorl' of the cycle is 

63,961.5 + 72,200 + 81,100 - 3740 = 213,521.5 foot-pounds = 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 foot-pounds. 

The zconfc atofl' *rf is 2 x 144 x (6.81 - 0.017) = 1952 foot-pounds. 

The tota/ wor/fc' is 136,161-5 - 1952 = 134,209.5 foot-pounds = 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. 

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 


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, 



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; non-expansive, 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 non-expansive cycle showing the maximum waste. 

Comparing the superheated cycles, we have the following 
efficiencies : 













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 non-expansive 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 finally-dry-steam 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. 



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 


= 226.95(1.5747 - 0.1749)= 317.5 : quotient = ^|i^ = 2.29, 

L 278.8-5-130.3 = 2.0 

II. 259.44-5-128.2 = 2. 

III. 187.29-13 = 14.4. 

IV. 169.1 + 13 =13.0. 
V. 81.05-* 8.219 = 25.1. 

VL 73.0-5-2.9=25.1. 

VII. E18-f- 173.5 =1.84. 

VIII. 188.2-13 = 14.5. 

IX, 400.8^-166.5 = 2.445. 


X. 253.5 -i- 13 = 19.45. 

XL 150.6 -f- 5.96:= 25.3. 

XII. 458.5 -r- 173.5 = 2.65. 

XIII. 275-13 = 21.1. 

XIV. 172,7 -6.81 =25.4. 
XV. 386,2-186.1 = 2.075. 

XVI. 275.3-13 = 21.1. 

XVII. 182,2-5-7.18 = 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 non-expansive 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 pressure-drop 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 : 







r* = 13 - 3.219 = 4.04 

P ff - P 9 - 26.3 

Superheat I 


r, = 18 -* 5.06 =2.185 

PL P q = 49.1 

Superheat II 


V, = IS ^- 0.81 = 1.91 

Pjf P 9 = 68.3 

Superheat III 


F^=13-7.18 =1.815 

P T - P f = 63.3 


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 non-condensing 
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, 


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 pressure-temperature 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 



adding the values of k(T-t) 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. 


hirmiiBAT, *F 








Absolute Pre&Rnre- 
Lbfl. per Square Inch 


' * = 141.7 
V = 357.8 
' # = 1122.6 









n = 2.0060 







' t = 166 1 







F= 186.1 








1 IT =1133.2 







n = 1.9486 







(t = 280.1 

















n = 1.7402 







' t = 367.8 








# = 1208 4 







n = 1.6294 







' t = 393.1 








' # = 1213.8 








n = 1.6031 







' 1 = 398 5 








1 #=1217.3 








n = 1.5978 







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.) 




(Condensed from Steam Tables and Dirrr/ntmn, by Marks and Duvis, with the permit 
sion of the publishers, Messrs Longmans, Green, & Co.) 










n t 





















































61 80 













1 ];](>. 5 











018 2 







































1140 5 






































































































































































































































































































1 6741 



















, 1.2720 




























649. 2 
















































(Condensed from Steam Tables and Diagrams, by Marks and Davis, with the permis- 
sion of the publishers, Me&srs. Longmans, Green, & Co.) 



r i h L n 


n u 

e i n s 




231 4 ! 922.0 




1 2435 i 1 6505 




2526 921.7 


843.1 0.4147 

1 2402 

1 0549 




253.0 920.8 









5*55.1 911>.0 





1 6519 






























1 6475 
























262 1 




















2(54 3 






















1178 2 














































302.0 6.29 










6.20 j 272.6 
















1 6296 









































80 i 312.0 












286. 3 






1 6151 






















































































i 1191.0 






























































































































(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), 149-2(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, 1419-1432. (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, 207-208 (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 Katie-Industrie, 1895, 66-85. (48) Op. cit., II, App. L. (49) JfcirAjn** a 
froid, Paris, 1878. (60) Elleau and Ennis, Jour. Frank. Inst., Mar., Apr., 1K98 ; 
Dietrici, Zeits. Kalte-Ind., 1904. (51) Op. cit., II, App. XL VI. (52) Zrit*.fur die 
gesamte Kalte-Industrie, 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, Physitetbache-chemische 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, 404-407. (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)3-620. (09) WM. AnnaUen, (4), 
XVIII, 1905, 73&~756; (4), XXIII, 1907, 809-845. (70) Marks and Davis, op. cit., 5. 
(71) Trans. A. S. M. E., May, 1908. 

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 

Superheated vapor is *n imperfect gas, produced by adding heat to a dry saturated vapor. 


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. 

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 


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^h-pressure 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, + %,(:r-o. r = ^ rL r- -Hi-J3i=-*C^-afi) + ^(2l-r.). 

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 : 


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 pressure-temperature relation and the characteristic equation are given, -we 
may compute L for various temperatures, and the specific heat of the vapor. 



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 = 26.4T-1S4P 23 . Pressure-tem- 
perature relation. L = 176- 0.27(2-32). 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 


Sankine cycle (incomplete expansion) determination of efficiency, with steam 
initially wet or dry. 

tfbn-opanszoe cycle: efficiency = (fr-lX 3 **- - 017 ). 

1483 log. 0.695(r-0 
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,. 


The superheated tables give /*, F, H, f, for various superheats at various pressures ; all 
values depending on H^t, w, and kp. 


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.) 


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 ? 


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 


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 heat-pressure 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 81-37, 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. 


7 T 48. A 10-gal. 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 

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 

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 10-w[(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 heat-pressure 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 


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 

59. Represent on the entropy diagram the temperature of inversion of a dry 

60. Sketch the Molher Diagram (Art, 399) from T=0 to JBT=r400, n = to 7i = 0.5. 



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 to-and-fro 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 left-hand 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 





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. 


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 pressure-drop 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 non-resistant 
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.; 



at p-100, 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. 


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 cut-off, &, 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. 


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 cut-off 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 
non-condensing ttnjacketed cylinders) by the equation 

* constant, n*0.&c- 0.465, 

where x is the proportion, of dryness at cut-off: the value of n being 
independent of the initial pressure or ratio of expansion. The initial 


wetness is thus the important factor in determining the rate of reevapora- 
tion during expansion. With steam very dry at cut-off (due to jacket- 
ing or superheat) heat may be lost throughout expansion. In ordinary 
cases, the condensation which may occur after cut-off, 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 cut-off. 

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 L-Li+h-h^H-Hi, 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 cut-off to decrease from 20.2 to 5.0 as the speed increased from 
122 to 401 r. p. m., the steam consumption per Ihp-hr. 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 cut-off 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- 


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, 
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 cut-off, 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 cut-off 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: 




I-TEA.W CuNMrMrrmv. 



Poi"KI*s PEC laV-IIB 





Zo/r ! ///i/1 

Loring and Emery 
Willans (9) 





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 : 



CtTT*>rr T Fm* CENT. 


Pl-nCETTAl/E >F ( 












16 : 








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 



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. 


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 non-jacketed engines is 


in which M is the proportion of steam condensed at cut-off, 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 

























































































































(T in the formula is equal to the difference in constants corresponding with the 
highest and lowest absolute pressures in the cylinder.) 


absolute pressure per square inch at cut-off, 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 steam-jackets, (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 
low-pressure 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 high-pressure 
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. Hot-air 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 



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 


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 cut-off, 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 well-known 
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 cut-off, following such a path as debxbkl, 
Fig. 183. With a greater amount of superheat, the steam may remain 



FIG. 190. Art. 441. Effect 
of Jackets at Various Ex- 
pansion Ratios. 



dry or even superheated at cut-off, giving the paths debzijf, deblzA. 
The minimum amount of superheat ordinarily necessary to give dry- 
ness at cut-off 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 cut-off 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 






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^.1-5-36 = 0.73, while that of the non-super- 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 


FIG. 193. Art. 443, Prob. 7. Steam 
Economy in Relation to Superheat. 


(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 
(cut-off) 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. 


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 


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 double-acting engine 2 LN feet, 
and the work per minute is 2 mALN foot-pounds, 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 cut-off 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 cut-off in both cylinders, without jackets or 
separate expansion valves . . . 0.67 to 0.77; 

Fast-running 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 cut-off control. Jackets raise the value by 0.05 to 0.15. Extremely early 
cut-off 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 four-valve 
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. 


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 ~4i-i* X 0.15 

More exactly, calling the clearance volume 0.04, the length of the diagram is 1.0, 
the volume at cut-off is 0.29, and the maximum volume attained is 1.04. The 
mean effective pressure is 

f- 6Xl -^-16 (1.04-0.08) 

- (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 


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 non-condensing 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, 



clearance volume included, IB , The proportion of clearance (always ex- 


pressed in terms of the piston displacement) is . The clearance in actual engines 


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 smooth-running 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. 


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 cut-off 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 cut-off. 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 cut-off. 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. 


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. ' 



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. 


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 


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 
cut-off 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 right-hand 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, 



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 non-expansive 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 Ihp-hr., respectively. 


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 non-conducting 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 cut-off) is approximately, 



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- 



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 multiple-expansion engine may have any number of cylinders ; a triple expan- 
sion engine may, for example, have five cylinders, a single high-pressure cylinder 
discharging its steam to two succeeding cylinders, and these to two more. In a 
multiple-expansion engine, the first is called high-pressure cylinder and the last 
the low-pressure 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 cross-compound. This last is the type most 
commonly used in high-grade 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 two-expansion 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, 



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, multiple-expansion 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 high-pressure cylinder may be carried to as high a 
pressure as is desnable without beginning it so early as to greatly i educe the woik 

(3) The low-pressure 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 - 

low-pressure 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 

double-acting 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 high-pressure 

cylinder, the fall of pressure along CD being due to the increase in volume of 

the steam, now passing into the low-pressure cylinder and forcing its piston out- 
ward. EFGH shows the action in the low-pres- 
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. 



the stroke of the pistons and not actual steam volumes. The low-pressure dia- 
gram has been reversed for convenience Some expansion in the low-pressure 
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 

464# . Receiver Engine. In this more modern form the steam passes 
from the high-pressure cylinder to a closed chamber called the receiver, 
and thence to the low-pressure 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 
cross-compound. In all cases the receiver engine has the characteristic 
advantage over the Woolf type that the low-pressure cylinder need not 
receive steam during the whole of the working stroke, but may have a 
definite point of cut-off, 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 cut-off on the low-pressure cylinder 
(Art. 467). 

Receiver volumes vary from to 1 times the high-pressure cylinder 

4645. Reheating. A considerable gain in economy is attained by 
drying or superheating the steam during its passage through the 







FIGS. 215 and 216. Art. 4646. Effect of Reheaters and Jackets (25). 

receiver, by means of pipe coils supplied with high-pressure 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 



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 high-pressure 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 high-pressure 
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 low-pressure diagrams from a pTrm.11 compound*engine. These are again 


FIG. 210. Art. 466. Compound Engine 

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 high-pressure 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. 



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 low-pressure 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 low-pressure 
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 Cut-off. 

exhaust from the high-pressure 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 high-pressure exhaust valve closes (at E) and while the high-pressure 
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 cut-off 
in the low-pressure 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 cut-off in the low-pressure cylinder raises the lower limit of tempera- 



FIG. 206. Art. 468. Effect of Changing Low- 
pressure Cut-off 

ture in the high-pressure 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 
cut-off on the high-pressure cylinder greatly increases the output of 
the low-pressure cylinder while (the receiver pressure being raised) 
scarcely affecting its own output. 

In Fig. 206, is shown the result of varying low-pressure cut-off in 
a tandem receiver engine with drop, the low-pressure clearance being 

exaggerated for clearness. 
The high-pressure diagram 
is fabcde, the low-pressure is 
ghjkl, p f =p d = p a and p e =p. 
Low-pressure cut-off occurs at 
h (point e in the high-pressure 
diagram). If this event occur 
earlier, the corresponding 
point on the high-pressure 
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 low-pressure 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 low-pressure cut-off now occurs at p m = 
p n . The same results will be found with cross-compound engines. 

The total output of the engine is very little affected by changes 
in low-pressure cut-off: but (contrary to the result in simple cylinders) 
the output of the low-pressure cylinder varies directly as its ratio 
of expansion. With delayed cut-off, the low-pressure cylinder performs 
a decreased proportion of the total work. 

When the load changes in a compound engine which has a fixed 
point of low-pressure cut-off , equality of work distribution becomes 
impossible. The output of the engine should be varied by varying 
the point of high-pressure cut-off. Equal distribution of the work 
should then be accomplished by variation of low-pressure cut-off. 
The two points of cut-off 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 low-pressure cut-off at 
which, with the given receiver volume and cylinder ratio, drop will be 
eliminated. (Arts. 475-478), 



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., high-pressure cylinder; 
effective pressure, Ib. per sq. in., low-pressure cylinder; 

=# A = ratio of expansion, high-pressure cylinder; 

=Ri = ratio of expansion, low-pressure 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) 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. 


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 high-pressure clearance 
^hd-r-dc, Fig 213), the apparent ratio of 
expansion in the high-pressure cylinder is 


FIG. 213. Arts. 471-473. Design Similarly, the apparent ratio of expansion 

Diagram i Compound Engine. j^ the low-pressure cylinder is 

Ri' = 

where dj=--T ls the low-pressure clearance. Engines are usually designed by 

specifying the whole apparent ratio of expansion, (dD-\-gf)-s-ab. 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- 


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, 


E-P* DO* f\r\ \Pmh-"- ft -\-pml A. i) . 

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 

, 2fLNA l 

in which ^ describes what is called the "high-pressure mean-effective 
pressure referred to the low-pressure 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 low-pressure 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 no-clearance 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 


474. Cylinder Ratio. Ratio of Expansion. Non-condensing 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 low-pressure 
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 high-pressure 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 high-pressure 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 (two-stage) 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 


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 Low-pressure Cut-off. Tandem Compound. In Fig. 
205, let ABCD be a portion of the indicator diagram of the high-pressure 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 high-pressure 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 low-pressure 
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 cut-off in the low-pressure cylinder. The 
diagram EFSHI maj 7 be completed, the curve FS being hyperbolic. 

476. Analytical Method. Let the volume of high-pressure cylinder be taken 
as unity, that of the receiver as R, that of the low-pressure cylinder as L. Let x 
be the fraction of its stroke completed by the low-pressure piston at cut-off, and let 
p be the pressure at release from the high-pressure cylinder, equal to the receiver 
pressure at the moment of admission to the low-pressure cylinder. The volume 
of steam at this moment is 1+-K; at low-pressure cut-off, it is 1 -\-R-i-xL x If 
expansion follows the law pv = PV t and P be the pressure in the low-pressure cylinder 
at cut-off, 

-x), or P^ 

The remaining quantity of steam in the high-pressure 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 



Combining the two values of P, we find 


477. Cross-compound: Cranks at Right Angles. In Fig. 208, let dbC be a 

portion of the high-pressure 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 high-pressure cylinder. This 
continues until admission to the low-pressure 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 high-pressure 



cylinder. At the point C of the cycle, the high-pressure crank is at i, the low-pres- 
sure "crank 90 ahead or behind it. When the high-pressure crank has moved from 
^ to m, the volume of steam in that cylinder is represented by the distance hn, the 
low-pressure crank is at o and the volume of steam in the low-pressure 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 high-pressure exhaust line and v is a point 
on the low-pressure 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 high-pressure exhaust line 

FIG. 208, Arts. 477-479. Elimination of Drop, Cross-compound 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 high-pressure cylinder and 
receiver will make I the final state. At I the high-pressure 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 high-pressure cycle at which cut-off occurs in the low-pressurq 
cylinder. By drawing a horizontal line through H to intersect vD, we find the point 
of cut-off J on the low-pressure diagram. If we regard the initial state as that when 
admission occurs to the low-pressure cylinder, then at low-pressure cut-off the 

high-pressure cylinder will have completed the -^- proportion of a full stroke. 


Modifications of this construction permit of determining the point of cut-off for no 
drop in triple or quadruple engines with any phase relation of the cranks. 

478. Cross-compound Engine: Analytical Method. In this case, the fraction 
of the stroke completed at low-pressure cut-off is different for the two cylinders. 
Let X be the proportion of the high-pressure stroke occurring between admission and 
cut-off in the low-pressure cylinder. Proceeding as before, the volume of the steam 
at low-pressure admission is 0.5 +R, and that at low-pressure cut-off is 0.5 X+R 
-}-xL. The volume of steam in the high-pressure cylinder and the receiver at the 
end of the high-pressure exhaust stroke is R; the volume just after low-pressure 
cut-off 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 



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 

-y=0.5 + ^-: 


In Pig. 209, we have the crank 
circles corresponding to the 
discussed movements. If Ow 
and Ox are at right angles, 
then for a high-pressure pis- 
ton displacement Oy, we have 
the corresponding low-pres- 
sure displacement kz. If these 
displacements be taken as at 
low-pressure cut-off, 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\ = ( 4-J 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 high-pressure 
cylinder, that the fluctuations of pressure along the release lines are scarcely notice- 
able. The fall of pressure during admission to the low-pressure 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 



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 steam-ether 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 
ra-Biuaiy Vapor Eu- Ihl) _ br> . ft ^ favorab le result than that obtainable from 
glne> good .steam engines of that time. Several attempts have 



been made to revive the binary vapor engine on a small scale, the most important 
recent experiments are those of Josse (27), on a 200-hp. engine using steam 
at 160 Ib. pressure and 200 of superheat, including four cylinders. The first three 
cylinders constitute an ordinary triple-condensing steam engine, a vacuum of 20 
to 25 in of mercury being maintained in the low-pressure 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 high-grade 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. 


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 to-and-fro 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, 



cessiye gas engine diagrams. "Outside-spiing" indicators are a recent type, in 
which the spring is kept away from the hot steam. The Ripper mean-pressure 
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. 



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 cut-off scarcely the usual condition of things. 

487. Deductions. By taking a "full-load" 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 cut-off 
is shown in c ; late cut-off, 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 


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 cut-off; 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 


' Barrel or tank 

Jet - 


External Barrus 

Barrus Continuous 

() 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 



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. Surface-condensing 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 

ing Calorimeter. 



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 

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 heat-augmentation 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 . 

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 


accuracy and of giving accurate results in a test of shorter duration. Where 
the engine is designed to operate non-condensing, 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(H-h )(T-ty 

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 



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 72-in. 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. 


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 = 

From Art. 491> A>+^o = &+& (Tfy or in this case, 866.5 x + 322.47 
= 1155.84 + 0.48* (302.1-242.3) 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. 


(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, 



(&) Ratio of (<*)-*-&) = 0.1621 -4- 0.265 = 0.61. 

500. Steam Consumption from Diagram. The inaccuracy of such estimates 
will be shown. In the high-pressure 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. 

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 high-pressure diagrairs 
is 10.6 Ib., or 26 per cent less than that which the test shows. The same process 
may be applied to the low-pressure 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. 


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 

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 

M/1+ \ (JL-JL\ 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 X-x = 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 


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 foot-pounds of work done in the 
cylinder per 100 pounds of coal burned (sometimes and preferably the number 
of foot-pounds 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 2068.84-J-1338,62 = 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 

Dwelshauvers-Dery (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 feed-water temperature, 



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 -(St-S^ 

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 

x = , 


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 non-condensing 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 



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 fly-wheel. 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). 


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 hot-water steam engines occasionally employed for locomotion (45). 

507. Classification of Engines. Commercially important types may be con- 
densing or non-condensing. They are classified as right-hand or left-hand, 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 multiple-expan- 

FIG. 23$. Art- 607- An^le-Compouud Engine. (American Ball Engine Company.) 



sion, with all the successive stages and cylinder arrangements made possible in 
the latter case. They may be single-acting or double-acting ; the latter is the far 
more usual arrangement. They may be rotative or non-rotative. The direct-acting 
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. Cross-compound vertical engines are often direct-connected 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 



the interesting angle-compound, in which a horizontal high-pressure 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, direct-con- 
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 
semi-tangye. 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 semi-tangye. 

A slow-speed 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 
slide-valve 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 medium-speed engines has, however, been in- 
troduced, in which the independent valve action of 
the Coiliss type has been retained, and the promptness 
of cut-ofE only attainable by a releasing gear has been 
approximated. In some cheap high-speed engines 
governing is effected simply but uneconomically by 
throttling the steam supply. Such engines may have 
shallow continuous frames or the sub-base, as in Fig. 
220, which represents the large class of automatic 
high-speed engines in which regulation is effected by 
automatically varying the point of cut-off. Figure 230 
shows three sets of indicator diagrams from a com- 
pound engine of this type, running non-condensing 
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 slow-speed 



releasing gear, in Fig. 218. All of the so-called " 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 


3och Cylinder Heo 
Back Cyl Head 

Steam pipe 

feom Flanq* 
^Throttle Valve 
Planished 5te) Laqqinq 
Heat insulating Filling 

iss STeomVolve Chamber 1 

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 ) 



which the four events of the stroke may be independently adjusted, and this type 
is often tised at moderately high speeds. Sharpness of cut-off 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 high-grade 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- 
tional-flow piston-exhaust 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 non-condensing a special piston valve, working ID 



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 concave-ended 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. 



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 non-condensing, 
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 



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- 


preaching a heavy grade, may utilize a higher boiler pressure or a later cut-off 
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 high-pressure elements, with a very large low-pressure cylinder 
placed under the boiler between the wheels. In the Cole compound, two outside 
low-pressure cylinders receive steam from trwo high-pressure 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: two-cylinder 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. G-reater 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 power-producing 
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 Temperature-Entropy 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, 1850-1851. (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- 


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. 103-107; Carpen- 
ter, op. Git , pp. 41-55, 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. 

Practical Modifications of the Rankine Cycle 

With valves moving instantaneously at the ends of the stroke, the engine would 
operate in the non-expansive cycle. The introduction of cut-off 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 cut-off 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= -^i-vl . 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 cut-off by 10 per cent. The actual expan- 

sion curve averages PV=pv. M.E.P.=-Pj> with the RanMne 


form of cycle. H . P . , 2 X dlagm itrtorXiiwUjr Diagnim faotor =0 .5 to 


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 non-condensing 

(4) Clearance varies from 2 to 15 per cent. "Real" and "apparent" ratios of 

(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 

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 low-pressure cylinder received steam through- 
out the stroke as discharged by the high-pressure 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 low-pressure 
cylinder works expansively. Early cut-oS in the low-pressure cylinder increases 
its proportion of the load, and is practically without effect on the total work of 
the engine. 


The approximate point of low-pressure cut-off to eliminate drop may "be graphically 

or analytically determined for tandem and cross-compound 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 non-condensing, 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 high-pressure 
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 cut-off 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 high-pressure 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 

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 = i-Z_i __ 

superheating : XQ = -"--) . limits of capac j ty . 

JBarrus : XQ = - - =-^ - - ; 


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 


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 cut-off 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 


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 : non-condensing or condensing, light-hand or left-hand, simple 
or multiple expansion ; single-acting or double-acting ; rotative or non-rotative , 
duplex or single ; horizontal, vertical, or inclined , locomotive, stationary (pump- 
ing, mill, power plant), or marine , "belted, direct-connected, or rope-driven ; air 
compressors ; girder, tangye, or semi-tangye frames ; slow, medium, or high speed ; 
throttling, automatic, four-valve, 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; 

two-cylinder and four-cylinder compounds , the balanced compound \ high econ- 
omy of locomotive engines. 


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- 

6. Find the percentage of initial condensation at J cut-off in a non-condensing 
engine using dry steam, running at 100 r. p. m. with a pressure at cut-ofE 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 feed-water 
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 double-acting 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.) 


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 

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 non-expansive 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 non-expansive 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 cut-off. 

(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., non-condensing, at J cut-off, the initial pressure being 100 Ib. 
and the piston speed 300 ft. per minute. The engine is double-acting. 

22. Sketch a possible cylinder arrangement for a quadruple-expansion 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 


engine and in the high-pressure 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 cross-compound double-acting 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 cut-off 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 foot-pounds. What is the 
mean effective pressure if the engine is 30"X48" ? What is the horse power of this 
engine if it runs double-acting 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 OP-axis. 

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. 


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 double-acting 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 jet-condensing, 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 unaccounted-for 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 unaccounted-for heat obtained 
In Problem 46. 

49. The value of the mechanical equivalent of heat which might be computed 
from the experiment, (Jns., 720.) 


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 ? 



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 non-isentropic. 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- 



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 


2g " '- ' 2g' 
or = DW 



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, 


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 


The term s(p-P) 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 

=7-0 or 7= V50103.2(g -Q) =223.84Vg :l Q feet per second. 

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. 



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 

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 




and draw the line of constant heat CD. 


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 35-lb. 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, 


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 



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 - 



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 cross-sectional 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 crobs-sectionul 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, 

y-l'~ y-r 


If the initial velocity be negligible, we have, as the equation of flow (Art. 515), 

9W PW 

y-l y-l y-l 

and since 





From Art. 521, 

Taking the value of V at 

we obtain 


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. 



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 


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 

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 



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. 2-46 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 

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 

heat drop, the cross-sectional 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 



526. Work Done. The work done in the ideal cycle per pound 
of steam is 778(2 Q) foot-pounds. Since 1 horse power = 1,980,000 
foot-pounds 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. 



FIG. 250 Arts 528,529. 
Rotative and Thrust 

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 foot-pounds. 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. 


Since the kinetic energy of the jet is , the 

2 - cos a. 

FIG 251. Art 528 
Velocities and Bucket 

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- 



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 

kinetic energy is The 

efficiency, 2 u 


becoines a maximum at 

0.50 when u = - With a ciip-shaped vane, as 

in the Pelton wheel, Pig. 252, complete reversal 
of the jet occurs 5 the absolute exit velocity, 
ignoring friction, is v-2u. 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. Single-Stage 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 



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 

531. Multi-stage 
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 low-pressure stages. The corresponding intermediate pressures are 
determined from the heat diagram, and the various stages are then designed as 



separate single-stage 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 Multi-stage 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. Multi-stage 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 


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. 

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 

foot-pounds, 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. 



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 -& = 1153.44-1039 = 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 foot-pounds, 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 


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 


' 3,181,250 B. t. u. 

per hour. The pounds of steam necessary per hour are 3,181,2504-245.0=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. 

Proceeding now to the nozzle design, we adopt the formula F= from Art. 

521. It will be sufficiently accurate to compute cross-sectional 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 



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 : 

at m, 

0.22525 x 6.26 

0.22525 x 9.724 

0.08581 x 15.92 

= 0.000839; at u, 
= 0.000989; at 3, 
= 0.000903; at u, 

0.08581 x 32.24 

0.02002 x 53.92 


= 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 mid-height 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. 


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 

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 



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 


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 turbo-generators. 

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 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- 



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- 

. 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 

foot-pounds, 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 


=61 (nearest whole number) instead of 60. The curve of heat drops may now 

be corrected for the necessary revised numbers of stages in the various drums: thus, 


the whole heat drop being 253 B. t. u., that in the first drum must be =42 2 


B.\ u. The average heat drop per stage for the first drum being (average ordinate 


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. 



At normal rate, the weight of steam used at the overload condition is 


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. 


i A ^ 

3s& Y.,0 

I x 

1 ^0 \J$ 

1 V- if* 


i \ 

~J~ ~*2Q .r$ 

I \ 

>i * 




T 2% 




ft \ 




< t 

< J 



"*" i 





u ' (* 




o ' 





122Q l 


r~ ~3 









'> ' 











flu ^ 








Sy - 










F iJ 



. 260 a. Art. -535. Expansion Path, Pressure Turbine. 



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 


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 ( r-z TVT 

\lo8.3/ Vlob.o/ 

the first drum. For intermediate stages between these two, we find, 





B. T.U 

ab = 350 

de - 279.7 




















393 i 



ac = 400 

df= 304.7 




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 


_, 71 
1/ * 11D ' 


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 heat-pressure 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 center-to-center 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. 


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- 



ing g. This is a horizontal single-stage 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 



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 left-hand 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 5-hp. 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 multi-stage 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 two-stage machine, 
the second stage is disconnected 
when the turbine runs non-con- 
densing, the exhaust from the first 
stage being discharged to the at- 
mosphere. Governing is effected 

FIG. 262. Art. 637. Curtis Turbine. 



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, multi-stage impulse 
turbine. The number of pressure stages is very laige from twenty-five 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. Westinghouse-Parsons 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 " double-flow " 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. 




540. Applications of Turbines. Turbo-locomotives have been experimented 
with in Germany ; the direct connection of the steam turbine to high-pressnre 
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, high-speed 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 
turbo-operation, in decreased vibration, greater simplicity, smaller and more deeply 
immersed propellers, lower center of gravity of engine-room 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 high-pi 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 Exhaust-steam 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 low-pressure 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 low-pressure 
turbine is decidedly profitable. The variations in supply of steam to the tuibine 
are offset by the use of a regenerator or accumulator, a cast-iron, water-sprayed 
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 


which a 42 by 60 inch double cylinder, reversible rolling-mil 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 28-m 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 low-pressure 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 Temperature-Entropy Diagram, 1905. (2 a) For the general theory of fluid 
flow, see Cardullo, Practical Thermodynamics, 1911, Arts, 55-60; Goodenough, 
Principles of Thermodynamics, 1911, Arts. 148-150, 153; for empirical formulas, 

see Goodenough, op cit. } Art. 154. (3) To show this, put the expression in . 


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. 


(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. 


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 


JS + PW+ =e+pw + ^-,oT^ = q-Q, approximately ; 
2(7 ly <*g 

whence V = 223 .84 \ 'q - Q. 

The complete expansion secured in the turbine warrants the use of exceptionally high 

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 


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 multi-stage 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*) -. 


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 right-angled 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 


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, single-stage impulse : 

Stumpf , single- or two-stage impulse, with Pelton buckets. 
Curtis, multi-stage impulse, usually vertical, axial flow. 
Bateau, multi-stage impulse, axial flow, horizontal, many stages. 
Westinghouse-Parsons, 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. 


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.) 


9. Check the value = 0.5274 for maximum flow in Art. 522. 



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 single-stage 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 first-stage nozzles and buckets 
for a two-stage 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 16-19, 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 three-drum pressure 
rotor axially, the other through a two-pressure 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 Westinghouse-Parsons 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. 



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 single-acting 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 



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 


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 

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. 



to the table, Art. 551. Average values of the efficiency ratio seem 

to be: Condensing. Non-condensing. 

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 non-condensing 
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 
non-condensing 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) Wire-drawing (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. 546-548 &, 
549 o, 551). 

546. Saturated Steam: Simple Non-condensing Engines, without Jackets. 

Steam Kate, 

S& ^ 




Ratio of 


Type of Valve. 


R p m 

Size, Hp 









Single, automatic, high 










Double automatic 







Four-valve, non-releas- 



below 225 

above 50 






Four-valve, releasing. 


below 100 

above 75 








547. Saturated Steam, Simple Condensing Engines, without Jackets: Improved 
Valve Gear. 



R p m 

Size, Hp 

Ratio of 


Steam Rate, Lb 





E R 

Releasing. . 


below 225 
below 100 

over 60 
over 100 







548#. Saturated Steam, Compound Non-condensing Engines, without Jackets. 



R p m 

Size, Hp 

Rate, Lb. 



E R 

Single, automatic 




23 6 
23 2 
21 9 
20 9 



Double, automatic. . , . 
Four-valve, releasing , , 

5485. Saturated Steam, Compound Condensing, without Jackets, Normal 
Cylinder Ratio, 





R p. m 

Size, Hp 

Rate, Lb 

, t 


Single, automatic 




19 1 



Double, automatic. . . . 




16 3 



Four-valve, releasing. . . . 




14 6 




. Saturated Steam, Triple Expansion, without Jackets (12), (13), (14), 

Back Pressure. 


Steam Rate, Lb. 




Non-condensing (8) 












5495. Jacketed Engines, High Grade, Saturated Steam, Compounds Usually 
with Reheaters. 

Steam Us 

ite, Lb. 



Same Type of 

Small, non-condensing simple, 5 exp., 75 Ib. gage 
pressure . ... 



Simple condensing, 120-150 Ib. pressure. . . 
Woolf compound, condensing, 16 exp., 12 r p. m., 
120 Ib. pressure 



rinmpoiinfi non-nondensing 


20 9-23 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- 
175-lb. pressure, about 30 expansions, 8 to 
14 per cent of total steam used in jackets and 



Triple condensing, 85 to 175 Ib. pressure, 25 to 33 

11 05-11 75 

11 75-15 

* One engine gave, with jackets, 13.85; without jackets, 14 99. 

5490. Superheated Steam, Reciprocating Engines. 


Rate, Lb 

B t u. 
Hp -mm 



Simple non-condensing, no jackets, slight 


Simple non-condensing, no jackets, 620 F . 
Simple condensing, 800 hp., 4 exp., 65 Ib. 
pressure, 450 F 





Simple condensing, 620 F 

11 6 




doTTipoupd non-condensing, locomotive. . . . 

17 6 

Compound condensing, 500 F 

12 9 




Compound condensing, 620 F 




'0 63 

Compound condensing, 45 hp., 600-lb. pres- 
sure, 800 F. (19) 

10 8 




Triple condensing, 500 

10 9 




Triple condensing, 620 

9 6 





549J. Comparative Tests, Saturated and Superheated Steam. 


Steam Rate, Lb 

B. t. u. per Hp -mm. 





Compound condensing, 150-lb pressure, 

9 56 

8 99 



Compound condensing, 140-lb. pressure, 
superheated 400 (18) 
Compound condensing, 130-lb. 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 24-28 Ib. for simple non-condensing 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 

. 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 
cut-off 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 non-condensing 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. 




Rate, Lb 



B t u 
Ihp -mm. 

B. t u 

Ihp -mm , 

Per Cent 
Gain by 

Simple non-condensing, 







Simple condensing, best 
Compound non-condens- 
inff .... 





332 (loco- 


Compound condensing, 
Compound condensing, 
high cylinder ratio (see 
Art. 5496) 








Triple condensing, aver- 







5497i Back Pressure. This is best investigated by considering the difference 
in performance of condensing and non-condensing engines. Arts. 546-549 c give: 

Steam Rate, Lt 

s per Ihp.-hr. 

Per Cent 




Saving Due to 

f Simple non-releasing. 
Saturated Steam, 1 Simple, releasing. . . 
not jacketed 1 Compound (average) 
L Triple 

22 2 
18 5 

16 7 
12 5 


_, , ( Simple 


18 5 


Saturated steam, 1 ^ , , , 
. . , , \ Compound (usual 
jacketed. \ ^ \ 


13 5 


Superheated / Simple, average 


13 8 


steam. 1 Compound, average . . . 




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 



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 









f ai 


I M 














- . 



Z. 28 
-< 25 




& 24 


S" 00 








u - 
















50 00 70 80 90 100 110 120 130 


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 single-acting Westinghouse compound engine, run- 
ning non-condensing Curve II is 
from the same engine, condensing 
(Trans A S. M . E , XIII, 537). 
Curve III is for the 5400-hp. 
Westinghouse compound condensing 
engine mentioned in Art. 543a. 
Curve IV is for a small four-valve 
simple non-condensing engine : curve 
V for a single-valve high-speed simple 
non-condensing 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 
high-ratio 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 


549;'. Wire-drawing. None of the tests above quoted applies to throttling 
engines. Cut-off 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 single-valve and double-valve engines 
of Art. 546 (15 per cent in Art. 548 6) is due to the partial throttling action of the 
single valve at cut-off. The difference between the performances of four-valve 
engines with and without releasing gear is very largely due to the comparative 
absence of wire-drawing 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 single-valve non-condensing 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). High-speed 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 cut-off: at other points 
of cut-off, 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 wire-drawing, 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 to-day. 
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. 


549 n. Multiple Expansion. The tables already given furnish the following: 


-Steam Rate, Lb. per Ihp -hr - 

Condensing Non-condensing. 

No. of expansion stages .1 2 3 123 


Single-valve .... 19 1 32J 23 6 . . 

Double-valve . . 16 3 . . 30 23 2 

Four-valve, non-releasing. 24 . . 29 . 

Four-valve, releasing . 21* 14 6 12 5 26 21 9 18 5 

Superheat, good valve 11 6-16 10 6-12 9 9 6-10 9 15 3-23 17 6 

The non-condensing engine \vith a cheap type of valve is 23 to 27 per cent more 
economical in the compound form than when simple. (The non-condensing compound 
is on other grounds than economy an unsatisfactory type of engine, see American 
Machinist, Nov. 19, 1891 ) In four-valve releasing engines, non-condensing, 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, non-condensing, 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. 

150-lb pressure, 26 exp , ratio 7:1 12 45 (jacketed) 

150-lb pressure, 120 r. p. m , 33 exp 12 1 (head jacketed) 

130-lb. 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 direct-acting 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. 











' . 

- . a 

















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 quadruple-expansion 
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 high-temperature 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 


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 



FIG. ?6$. Art. 550- Hood Compressor Diagrams. 


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, non-condensing, 

single valve, without j ackets. . . . 32 051-0 55 548 
Saturated steam, simple, non-condensing, 

double valve, without jackets . . 30 0.63 502 

Saturated steam, simple, non-condensing, 

four valve, releasing, without jackets . . 26 . 65 434 

Saturated steam, simple, non-condensing, 

with jackets .............. 25 68 418 

Saturated steam, compound non-condensing, 

without jackets ...... . .22 0.63-0.72 353 

Saturated steam, simple condensing, four 

valve releasing, without jackets ...... 21 0-40 383 

Saturated steam, compound non-condensing, 

with jackets, four valve ....... 19 0.71-0.82 305 

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 non-condensing 

(locomotive) ........ ....... 17J 58-0 72 332 

Superheated (620 F.) steam, simple, non- 

condensing ....... ... 15 0,66-084 319 

Saturated steam, compound condensing, four 

valve, no jackets .............. 14J 56 274 

Saturated steam, compound condensing, 

normal ratio, four valve, with jackets. . 13J- . 50-0 . 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. 


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 7500-kw. 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 400-kw unit gave 11 2 lb. with 180 of super- 
heat. A 1250-kw. 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 
5000-kw Curtis unit, with 125 of superheat, a steam-rate of 14 lb. per kw.-hr.; 
and for a 2000-kw. unit, under similar conditions, 16.4 lb. 

A 2600-kw. Brown-Bo veri turbo-alternator at Frankfort consumed 11.1 lb. of 
steam per electrical horse-power-hour with steam at 173 lb. gauge pressure, super- 
heated 196 and at 27.75 ins. vacuum. The 7500-kw. ALLis cross-compound 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 high-speed non-condensing 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 



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 over-all efficiency of mechanism thus obtained. The 
difficulties involved have led to the general custom, in 
turbine practice, of reporting steam rates per kw-hr. 
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 direct-driven 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 



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. 





FIG. 270. Art. 555. Willans Line for Varying 
Initial Pressure. 



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, 


















10 20 30 40 50 00 70 8 

90 100 110 1*20 


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 

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. 





3 11,000 

= 10,000 






100 200 300 400 500 600 700 800 900 


>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, 208-207. (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. 


(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. 


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 

non-condensing 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. non-condensing, with and 
without jackets, triple and quadruple regenerative. 

(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 non-condensing ? 


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 7500-kw. 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 direct-connected 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 ~- > 



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 (non-volatile, uncombined) carbon. If this percentage is between 
83 and 89, the coal is semi^bituminou^ ; if less than S3, it is bituminous, or soft. 






Equivalent Reaction f 

B t u 
per Lb 

Hydrogen . ... 


H 2 +0 = H 2 O 

62, lOOt 





Carbon . 


C+0 2 = C0 3 


Carbon monoxide. 


CO+0-C0 2 



C 2 H 2 

C 2 H 2 +0 6 =2C0 2 +H 2 



CH 4 

CH 4 +04 = C0 2 -|-2H 2 



C 2 H 4 

C 2 H4-f-O 6 =2C0 2 +2H 2 O 


Sulphur . ... 


S-K) 2 =S0 2 


Gasolene* . . 


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 =f|X14,500 =13,400 

E=AX62,100 - 4,790 

Heat of decomposition = 3,200 

Heat value 


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 250-r0.4=#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' 



-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 4-f-112) =54 12 B t. u., 
and the " low " heat value of the gas under the actual conditions of the experiment 
would be 625-54.12 =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 


oxygen and mtro- FJQ 373 Arts. 563, 564. Air Supply and Combustion. 


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 

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 right-hand 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. 


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 free-burning 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 fire-tube or 
water-tube, 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 



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 


upward current being generated over the grate. This is an eternally fired fire-tube 
boiler. Figure 275 shows the well-known 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. 



569. Discussion. 

The internally fired boiler requires no brick furnace 

setting, and is compact. 
The water-tube 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- 

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. 




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 flue-gas 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 water-tube 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.) 


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 
foot-pounds 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 



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. 


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 well-confirmed 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 -s-n, 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 cross-sectional 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 

To determine the area of chimney: the velocity of the gases is, in feet per 

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 )' 


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, steam-jet blowers or fans may 
be employed. These usually cost less initially, and more in maintenance. The 
ordinary steam-jet 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 

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- 



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, 




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 



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; 


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 single-tube 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. Feed-water 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 feed-water 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+ h-qJ=w(Q-q), 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 non-condensing engines, the exhaust steam from the engines themselves is 
used to heat the cold incoming water. 



581. Types. Feed-water 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 boiler-heating 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 

(Harrison Safety Boiler "Works.) 

solids is also possible in ordinary open-exhaust steam 
heaters. Various forms of feed- water filters are used, 
with or without heaters. 

582. The Economizer. This is a feed-water 
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 
(# /0-rQ, 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 



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 
by-passing 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. Feed-water 
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 


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 



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 



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 WL-7-(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. 



(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 feed-water 
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 


586. Boiler Feed Pump. This maybe either steani-driven or power-driven 
(as may also be the condenser pumps). Steam-driven 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 single-acting water cylinders. 
There is much discussion at the present time as to the comparative economy of 
steam- and power-driven auxiliaries. The steam engine portion of an ordinary 
small pump is extremely inefficient, while power-driven pumps can be operated, at 
little loss, from the main engines. The general use of exhaust steam from aux- 
iliaries for feed-water 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 motor-driven, 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 



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 self-starting 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 


v 2 V 2 

The values of and may ordinarily be neglected, and 

~~ Q-H ' 

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 live-steam feed-water 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 (! + ?/)$ foot-pounds ; 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. 


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 

Heating values: fuel oil, 18,000; wood, 6500-8500; coals, 8800-15,000; 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. 


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 : water-tube, horizontal tubular, locomotive, marine ; conditions of 


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. 

Feed-water heaters may be open or closed: w = TC^ "~ g) ; for open heaters, 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. L-r- U(T-t); U = 630 r \ . The siphon condenser may operate with- 

* OT* 

out a vacuum pump. 
The use of steam-driven, auxiliaries affords exhaust steam for feed-water heating. 

The injector converts heat energy into velocity: y= ^ \ efficiency = 


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 feed-water 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 


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 54-inch 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 feed-water 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 single-acting 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 


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 feed-water 
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 separately-fired superheater of 0.70 efficiency. What weight of coal, 
containing 14,000 B, t. u, per Ib,, will be required ? 


FIG. 288., Art 501, Still. 


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 low-temperature 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 paper-pulp mills, glue works, glucose factories, for 
the preparation of pure water, and in the manufacture of gelatine, malt extract, 





wood extracts, caustic soda, alum, tannin, garbage products, glycerine, sugar of 
milk, pepsin, and licorice. In most cases, the multiple-effect 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. Multiple-effect 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- 



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 
"triple-effect" or three-stage evaporator; steam (preferably exhaust 
steam) enters the shell of the fir