PRINCIPLES
OF
THERMODYNAMICS
BY
G. A. GOODENOUGH, M.E.
PROFESSOR OF THERMODYNAMICS IN THE UNIVERSITY
OF ILLINOIS
SECOND EDITION, REVISED
COPYRIGHT, 1911
HY
HENRY HOLT AND COMPANY
PREFACE
THIS book is intended primarily for students of engineering.
Its purpose is to provide a course in the principles of thermo
dynamics that may serve as an adequate foundation for the
advanced study of heat engines. As indicated by the title,
emphasis is placed on the principles rather than on the appli
cations of thermodynamics. In the chapters on. the technical
applications the underlying theory of various heat engines is
quite fully developed. The discussion, however, is restricted
to ideal cases, and questions that involve the design, operation,
or performance of heat engines are reserved for a second
volume.
The arrangement of the subject matter and the method of
presentation are the result of some twelve years' experience in
teaching thermodynamics. Briefly, the arrangement is as fol
lows : In the first six chapters, the fundamental laws are
developed and the general equations of thermodynamics are
derived. The laws of gases and gaseous mixtures are dis
cussed in Chapters VII and VIII, and this discussion is fol
lowed immediately by the technical applications in which
gaseous media play a part. A discussion of the properties
of saturated and superheated vapors is likewise followed by the
technical applications that involve vapor media.
Some of the features of the book to which attention may
be directed are the following :
1. The method of presenting the fundamental laws. In
this treatment I have followed very closely the development
in. Bryan's thermodynamics. The second law is made identical
with the law of degradation of energy, the connection between
i vpGiToT'C'iVnliftr ann Inau rr mrjiln nili'f'.sr ic nniiTforl rm+: anrl Inr
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derived. By this method of presentation, a definite concep
tion of the meaning and scope of the second law is obtained,
and the difficulties that usually surround the definition of
entropy are removed.
2. The discussion of saturated and superheated vapors.
The experiments in the Munich laboratory and the researches
of Professor Marks and Dr. Davis have furnished now and
accurate data on the thermal properties of saturated and super
heated steam. In Chapters X and XI a concise but fairly
complete account of these important researches is given. Kno
blauch's experiments on specific volumes have been correlated
with the experiments on specific heat by means of the Cluusius
relation (^A =  AT(~\ and equations for the specific heat,
\opj T \oj_ j p
entropy, energy, and heat content of superheated steam arc
thereby deduced. These results have not hitherto been pub
lished.
3. The discussion of the flow of fluids and of throttling
processes. The applications of the throttling process are so
important from all points of view that a separate chapter is
devoted to them.
4. The treatment of gaseous mixtures, Chapter VIII. An
attempt is made to present in concise form the principles and
methods required in the accurate analysis of the internal com
bustion engine.
5. The note on the interpretation of differential expressions,
Art. 23. This important topic should be discussed fully in
calculus, but experience shows that students rarely have a
grasp of it. In thermodynamics the exact differential has
extensive applications; hence it seems desirable to include
a rather complete explanation of exact and inexact differentials
and their connection with thermodynamic magnitudes. A
f this article should ">aWB the student
PREFACE V
The text is illustrated by numerous solved problems, and
exercises are given at the ends of the chapters and elsewhere.
Many of the exercises require only routine numerical solutions,
but others involve the development of principles.
References are given to the treatment of various topics in
standard works and to original articles. It is not expected
that undergraduate students will make extensive use of these
references, but it is hoped that instructors and advanced
students will find them helpful.
In writing this book I have consulted many of the standard
works on thermodynamics, and have made free use of whatever
material suited my purpose. I desire to acknowledge my
special indebtedness to the works of Bryan, Preston, Griffiths,
Zeuner, Chwolson, Weyrauch, and Lorenz, and to the papers
of Dr. H. N. Davis. To Mr. John A. Dent I am indebted
for assistance in the construction of the tables and in the
revision of the proof sheets. Mr. A. L. Schaller also gave
valuable assistance in getting the book through the press.
G. A. GOODENOUGH.
URBANA, ILL., July, 1911.
CONTENTS
CHAPTER I
ENERGY
AB T. PAGE
1. Scope of Thermodynamics 1
2. Energy 1
3. Mechanical Energy 2
4. Heat Energy 3
5. Other Forms of Energy . . . 5
6. Transformations of Energy 5
7. Conservation of Energy 6
8. Degradation of Energy 7
9. Units of Energy 8
10. Units of Heat 9
11. Relations between Energy Units 10
CHAPTER II
CHANGE OF STATE. THERMAL CAPACITIES
12. State of a System 15
13. Characteristic Equation 16
14. Equation of a Perfect Gas 17
15. Absolute Temperature 18
16. Other Characteristic Equations ....... 20
17. Characteristic Surfaces 20
18. Thermal Lines 21
19. Heat absorbed during a Change of State 22
20. Thermal Capacity: Specific Heat 24
21. Latent Heat 26
22. Relations between Thermal Capacities 27
23. Interpretation of Differential Expressions 28
CHAPTER III
THE FIRST LAW OF THERMODYNAMICS
ART. .57
27. External Work .
28. Integration of the Energy Equation ' '
29. Energy Equation applied to a Cycle Process '>'
30. Adiabatic Processes '
31. Isodynamic Changes ' "
32. Graphical Representations
CHAPTER IV
THE SECOND LAW OF THERMODYNAMICS
33. Introductory Statement ' i; 
31 Availability of Energy ' i( j
35. Reversibility '*'
36. General Statement of the Second Law '^
37. Cavnot's Cycle ; " )(
38. Carnot's Principle '>
39. Efficiency of the Carnot Cycle ">'
40. Available Energy and Waste <r > (
41. Entropy ; " lf
42. Second Definition of Entropy <'<
43. The Inequality of Clausius <>
44. Summary l!
45. Boltzmann's Interpretation of the Second Law .... (5:
CHAPTER V
TEMPERATURE ENTROPY REPRESENTATION
46. Entropy as a Coordinate
47. Isothermals and Adiabatics
48. The Curve of Heating and Cooling
49. Cycle Processes
50. The Rectangular Cycle
51. Internal Frictional Processes
52. Cycles with Irreversible Adiabatics
53. Heat Content
CHAPTER VI
GENERAL EQUATIONS OF THERMODYNAMICS
54. Fundamental Differentials '
55. The Thermodynamic Relations
56. General Differential Equations.
57. Additional Thermodynamic Formulas ....
CONTENTS ix
CHAPTER VII
PROPERTIES OF GASES
ART. PARE
59. The Permanent Gases 89
60. Experimental Laws 89
61. Comparison of Temperature Scales !)1
62. Numerical Value of B 92
63. Forms of the Characteristic Equation 93
64. General Equations for Gases 94
65. Specific Heat of Gases 90
66. Intrinsic Energy 97
67. Heat Content . 99
G'8. Entropy of Permanent Gases . ' 100
69. Constant Volume and Constant Pressure Changes . . . .101
70. Isothermal Change of State 102
71. Adiabatic Change of State 102
72. Polytropic Change of State 104
73. Specific Heat in Polytropic Changes 106
74. Determination of n . . . 108
CHAPTER VIII
GASEOUS COMPOUNDS AND MIXTURES. COMBUSTION
75. Preliminary Statement Ill
76. Atomic and Molecular Weights Ill
77. Relations between Gas Constants 112
78. Mixtures of Gases. Dalton's Law . ' 114
79. Volume Relations 116
80. Combustion :' Fuels ..... , . 117
81. Air required for Combustion. Products of Combustion . .119
82. Specific Heat of Gaseous Products 123
83. Specific Heat of a Gaseous Mixture 125
84. Adiabatic Changes with Varying Specific Heats .... 126
85. Temperature of Combustion ........ 127
CHAPTER IX
TECHNICAL APPLICATIONS. GASEOUS MEDIA
86. Cycle Processes 133
87. The Carnot Cycle 134
88. Conditions of Maximum Efficiency . 135
89. Isoadiabatic Cycles 136
00. r.la.ssifina.f.inn nf Air "Rnmiifis . . . . . . . .137
ART.
93. Analysis of Cycles
94. Heating by Internal Combustion ....
95. The Otto Cycle
96. The Joule, or Brayton, Cycle
97. The Diesel Cycle
98. Comparison of Cycles
99. Closer Analysis of the Otto Cycle ....
100. Air Refrigeration
101. Air Compression
102. Water Jacketing
103. Compound Compression
104. Compressedair Engines
105. rSDiagram of Combined Compressor and Engine
CHAPTER X
SATURATKD VAPORS
106. The Process of Vaporization
107. Functional Relations. Characteristic Surfaces ....
108. Relation between Pressure and Temperature ....
109. Expression for ^
at
110. Energy Equation applied to Vaporization
111. Heat Content of a Saturated Vapor
112. Thermal Properties of Water Vapor
113. Heat of the Liquid
114. Latent Heat of Vaporization
115. Total Heat. Heat Content
116. Specific Volume of Steam
117. Entropy of Liquid and of Vapor
118. Steam Tables IHi
119. Properties of Saturated Ammonia 1H<
120. Other Saturated Vapors IS
121. Liquid and Saturation Curves IS
122. Specific Heat of a Saturated Vapor . . . . , .IS
123. General Equation for Vapor Mixtures IS
124. Variation of x during Adiabatic Changes . . . . .IS
125. Special Curves on the jTSplane IS
126. Special Changes of State IS
127. Approximate Equation for the Adiabatic of a Vapor Mixture . 1H
CHAPTER XI
SUPERHEATED VAPOKS
CONTENTS xi
AHT. VMF.
130. Equations of van der Waals and Clausius 200
131. Experiments of Knoblauch, Linde, and Klebe .... 201
132. Equations for Superheated Steam 203
133. Specific Heat of Superheated Steam 204
134. Mean Specific Heat 210
135. Heat Content. Total Heat 210
13G. Intrinsic Energy 214
137. Entropy 215
138. Special Changes of State 21(5
139. Approximate Equations for Adiabatic Changes .... 220
140. Tables and Diagrams for Superheated Steam . . . .221
141. Superheated Ammonia and Sulphur Dioxide . . . .225
CHAPTER XII
MIXTURES OF GASES AND VAPORS
1 12. Moisture in the Atmosphere 228
143. Constants for Moist Air 230
1 14. Mixture of Wet Steam and Air 232
145. Isothermal Change of State 232
140. Adiabatic Change of State 233
147. Mixture of Air with Highpressure Steam 236
CHAPTER XIII
THE FLOW or FLUIDS
148. Preliminary Statement 243
149. Assumptions 214
150. Fundamental Equations 244
151. Special Forms of the Fundamental Equation .... 247
152. Graphical Representation 247
153. Flow through Orifices. Saint Venant's Hypothesis . . .252
154. Formulas for Discharge 255
155. Acoustic Velocity 257
156. The de Laval Nozzle 258
157. Friction in Nozzles 262
158. Design of Nozzles 264
CHAPTER XIV
THROTTLING PROCESSES
CONTENTS
162. The Expansion Valve . ~7'J
163. Throttling Curves i27!i
164. The Davis Formula for Heat Content 274
165. The JouleThomson Effect . . !275
166. Characteristic Equation of Permanent Gases .... 1277
167. Linde's Process for the Liquefaction of Gases .... USD
CHAPTER XV
TECHNICAL APPLICATIONS, VAVOII MKDIA
The Steam Engine
168. The Carnot Cycle for Saturated Vapors . . . . . l2K:{
169. The Rankine Cycle i>S I
170. The Rankine Cycle with Superheated Steam .... 12 SI 5
171. Incomplete Expansion 288
172. Effect of Changing the Limiting Pressures 12H!)
173. Imperfections of the Actual Cycle !>!)()
174. Efficiency Standards ;2<)1
The Steam Turbine
175. Comparison of the Steam Turbine and Reciprocating Engine . I2!ll
176. Classification of Steam Turbines 'j!>r>
177. Compounding o<)i;
178. Work of a Jet [ ~n } $
179. Singlestage Velocity Turbine ;}00
180. Multiplestage Velocity Turbine ;}()o
181. Turbine with both Pressure and Velocity Stages .... Ml
182. Pressure Turbine ; ) () ~
183. Influence of High Vacuum my
Refrigeration with Vapor Media
184. Compression Refrigerating Machines . . . . ;50 ( s
185. Vapors used iu Refrigeration ' ' ;no
186. Analysis of a Vapor Machine . . . . ' * ' an
SYMBOLS
NOTE. The following list gives the symbols used in this book. In
a magnitude is dependent upon the "weight of the substance, the si
letter denotes the magnitude referred to unit weight, the capital letter
same magnitude referred to M units of weight. Thus q denotes the ]
absorbed by one pound of a substance, Q = Mq, the heat absorbed b;
pounds.
J, Joule's equivalent.
A, reciprocal of Joule's equivalent.
M, weight of system under consideration.
t, temperature on the F. or the C. scale.
T, absolute temperature.
p, pressure.
v, V, volume.
y, specific weight ; also heat capacity.
u, U, intrinsic energy of a system,
z, 7, heat content at constant pressure.
s, S, entropy.
W, external work.
q, Q, heat absorbed by a system from external sources.
h, fl, heat generated within a system by irreversible transformatioj
work into heat.
c, specific heat.
c v , specific heat at constant volume.
c p , specific heat at constant pressure.
k, ratio c p /c v .
B, constant in the gas equation pv = BT.
R, universal gas constant.
n, exponent in equation for polytropic change, p V n = C.
m, molecular weight,
oi, 03..., atomic weights.
H m , heating value of a fuel mixture.
x, quality of a vapor mixture (p. 165).
q', heat of the liquid.
q", total heat of saturated vapor.
iv SYMBOLS
v', v", specific volume of liquid and of vapor, respectively,
w', u", internal energy of liquid and of vapor, respectively.
s', s", entropy of liquid and of vapor, respectively.
i', i", beat content of liquid and of vapor, respectively.
c', c", specific heat of liquid and of vapor, respectively.
<, humidity.
w, velocity of flow.
w e , acoustic velocity.
J 1 , area of crosssection of channel.
z, work of overcoming friction in the flow of fluids.
p m , critical pressure (flow of fluids).
/JL, JouleThomson coefficient.
77, efficiency of a heat engine.
N, steam consumption per h.p.hour.
PRINCIPLES OF THERMODYNAMICS
CHAPTER I
ENERGY
1. Scope of Thermodynamics. In the most general sense,
thermodynamics is the science that deals with energy. Since
all natural phenomena, all physical processes, involve manifes
tations of energy, it follows that thermodynamics is one of
the most fundamental and farreaching of sciences. Thermo
dynamics lies at the foundation of a large region of physics
and also of a large region of chemistry ; and it stands in a
more or less intimate relation with other sciences.
In a more restricted sense, thermodynamics is that branch of
physics which deals specially with a form of energy called heat.
It deals with transformations of heat energy into other forms
of energy, develops the laws that govern such transformations,
and investigates the properties of the media by which the
transformations are effected. In technical thermodynamics the
general principles thus developed are applied to the problems
presented by the various heat motors.
In this volume the general principles of thermodynamics are
developed so far as is essential to give a firm foundation for the
technical applications in engineering practice. The scope of
the book does not permit a discussion of the methods of inves
tigation that are employed so fruitfully in physics and chem
istry.
2 ENERGY I' :UAI>  l
ing to rest, that is, in changing its state as regards velocity;
a body in an elevated position can do work in changing ii
position; a heated metal rod is capable of doing mechanical
work when it contracts in cooling. In each case sonic change,
in the state of the body results in the doing of work ; hence, in
each case the body in question possesses energy.
Energy, like motion, is purely relative. It is impossible to
give' a numerical value to the energy of a system without
referring it to some standard system, whose energy we ^niay
arbitrarily assume to be zero. For example, the energy of the
waterman elevated reservoir is considered with re Terence to
the energy of an equal quantity at some chosen lower level.
The kinetic energy of a body moving with a definite velocity
is compared with that of a body at rest on the earth's .surface,,
and having, therefore, zero velocity relative to the earth. The
energy of a pound of steam is referred to that of a pound of
water at the temperature of melting ice.
3. Mechanical Energy is that possessed by a body or system
due to the motion or position of the body or system relative to
some standard of reference. Mechanical kinetic energy is thai
due to the motion of a body and is measured by the product
\ mv\ where m denotes the mass of the body and v its velocity
relative to the reference system. It should be observed that
2 mv z is a scalar, not a vector, quantity and it must be considered
positive in sign. Hence, if a system consists of a number of
masses m 15 ra 2 , , m n moving with velocities v v v z , , v, n
respectively, the total kinetic energy of the system is the sum
 (ra^ 2 + w 2 v 2 2 4 ... + m n y, 2 ) =  Swy 2 ,
independently of the directions of the several velocities.
The mechanical potential energy of a system is that due to
AET. 4] HEAT ENERGY 3
4. Heat Energy. Heat is tlie name given to an active agent
postulated to account for changes in temperature. It is ob
served that when two bodies are placed in communication, the
temperature of the warmer falls, that of the colder rises, and the
change continues until the two bodies attain the same tempera
ture. To account for this phenomenon we say that heat flows
from the hotter to the colder body. The fall of temperature of
one body is due to the loss of heat, while the rise in tempera
ture of the other is due to the heat received by it. It is to be
noted that the change of temperature is the thing observed and
that the idea of heat is introduced to account for the change,
just as in dynamics the idea of force is introduced to account
for the observed motion of bodies. Whatever may be the
nature of heat, it is evidently something measurable, something
possessing the characteristics of quantity.
In the old caloric theory, heat was assumed to be an impon
derable, allpervading fluid which could pass from one body to
another and thus cause changes of temperature. The experi
ments of Rumford (1798), Davy (1812), and Joule (1840)
shattered the caloric theory and established the modern me
chanical theory, of which the following is a brief outline.
Heat may be generated by the expenditure of mechanical
work. Familiar examples are shown in the heating of journals
due to friction, the heating of air by compression, the develop
ment of heat by impact, etc. Conversely, work may be ob
tained by the expenditure of heat, as exemplified in the steam
engine and other heat motors. Joule's experiments established
the fact that a definite relation exists between the heat gener
ated and the work expended ; thus to produce a unit of heat a
definite amount of work is required, no matter in what particu
lar way the work is done. Heat and mechanical energy are
therefore equivalent in a certain sense. Either may be produced
at the expense of the other, and the ratio between the quantity of
one produced and the quantity of the other expended is always
the same. The conclusion is evident that heat is not a sub
4 ENERGY t ( ' JIAI>  J
Heat energy, like mechanical energy, may bo either of tlm
kinetic or the potential form. Denoting the mass of si mole
cule by m and the velocity by v, the kinetic energy of the mole
cule is I mv z . In a given system the different molecules are
moving" with different velocities and in different directions ;
nevertheless, the summation
2  mv z
extended to all the molecules of the system gives the thermal
kinetic energy of the system. If we denote by c 2 the mean
square of the velocities of the molecules, we have
where M denotes the mass of the system. Considerations de
rived from the kinetic theory of gases show that the lempera
ture of the system is a function of 6 >2 ; hence, since the kinetic,
energy is directly proportional to 6' 2 , it follows that the tempera
ture of a system is a measure of its thermal kinetie energy.
Whenever the temperature of a body rises, wo infer that, the
kinetic energy has increased, and that the mean velocity of the
molecules is greater than before.
Potential thermal energy is due to the relative position of
the molecules of the system. The addition of heat to a body
usually results in the expansion of the body. The molecules
are moving with higher speeds than before the addition of heat,
and on the whole they are farther apart. To separate them
against their mutual attractions requires the expenditure of
work; conversely, in coming back to the original configura
tion the molecules will do work. Hence, the work expended
in separating the molecules is stored in the system as potential
energy.
As long as the body remains in the same state of aggregation,
the potential energy it is capable of storing is small. ^ lint if a
body changes its state of aggregation, it may, during the pro
cess, store a large amount of potential energy. Consider, for
example, the melting of ice. To nimn ^
QUJL WVjU U.JL V; WJU U1XU DVJJ.1U. JH_/O C1/JLJLIL V^XlCtil tf 14.J.tI J. U \J\J UUclU UJL UliC 11 U L111I.
water. The heat is therefore stored as potential energy. In the
same manner when water is transformed into steam, work is
done in forcing apart the molecules against their cohesive forces,
and this work is stored as potential energy.
5. Other Forms of Energy. In addition to heat and mechani
cal energy, there are other forms of energy that require consid
eration. The energy stored in fuel or in explosives may be
considered potential chemical energy. Electrical energy is
exemplified in the electric current and in the electrostatic charge
in a condenser. Other forms of energy are due to wave motions
either in ordinary fluid media or in the ether. Sound, for
example, is a wave motion usually in air. Light and radiant
heat are wave motions in the ether.
The vibratory forms of energy are neither kinetic nor potential,
but rather periodic alternations between the two. To illustrate
this statement, let us consider the motion of a pendulum bob.
In its lowest position the bob has zero potential energy and
maximum kinetic energy ; as it rises its velocity decreases ;
therefore, its kinetic energy also decreases, while its potential
energy simultaneously increases and reaches a maximum at the
end of the swing when the kinetic energy is zero. This same
alternation from kinetic to potential and back occurs in vibrating
strings, water waves, and, in fact, in all wave motions.
6. Transformations of Energy. Attention has been called
to the generation of heat energy by the expenditure of mechani
cal work. This is only one of a great number of energy changes
that are continually occurring. We see everywhere in every
day life one kind of energy disappearing and another form
simultaneously appearing. In a power station, for example,
the potential energy stored in the coal is liberated and is used up
in adding heat energy to the water in the boiler. Part of this
heat energy disappears in the engine and its equivalent appears
as mechanical work. Finally, this work is expended in driving
a generator, and in place of it appears electric energy in the
form of the current in the circuit. We say in such cases that
\Jil\J J.Uiil1.
ing are a few familiar examples of energy transformations ;
many others will occur to the reader.
Mechanical to heat : Compression of gases ; friction; im
pact.
Heat to mechanical : Steam engine ; expansion and contrac
tion of bodies.
Mechanical to electrical : Dynamo ; electric machine.
Electrical to mechanical : Electric motor.
Heat to electrical : Thermopile.
Electrical to heat : Heating of conductors by current.
Chemical to electrical : Primary or secondary battery.
Electrical to chemical : Electrolysis.
Chemical to thermal : Combustion of fuel.
7. Conservation of Energy. Experience points to a general.
principle underlying all transformations of energy.
The total energy of an isolated system remains constant and
cannot be increased or diminished ly any phi/xi.aal prwxm'x
whatever.
In other words, energy, like matter, can be neither created
nor destroyed ; whenever it apparently disappears it has been
transformed into energy of another kind.
This principle of the conservation of energy was lirst defi
nitely stated by Dr. J. R. Meyer in 1842, and 'it soon received
confirmation from the experiments of Joule on the mechanical
equivalent of heat. The conservation law cannot be proved
by mathematical methods. Like other general principles in
physics, it is founded upon experience and experiment. So
far, it has never been contradicted by experiment, and it may
be regarded as established as an exact law of nature.
A perpetual motion of the first class is one that would sup
posedly give out energy continually without any corresponding
expenditure of energy. That is, it would create enerU from
nothing. ^ A perpetualmotion engine would, therefore, <n've out
an unlimited amount of work without fuel or other external
supply of energy. Evidently such a machine would violate the
conserve law ; and the statement that perpetual motion of
ART. SJ DEGRADATION OF ENERGY 7
the first class is impossible is equivalent to the statement of the
conservation principle at the beginning of this article.
8. Degradation of Energy. While one form of energy can
be transformed into any other form, all transformations are not
effected with equal ease. It is only too easy to transform
mechanical work into heat ; in fact, it is one duty of the
engineer to prevent this transformation as far as possible.
Furthermore, of a given amount of work all of it can be trans
formed into heat. The reverse transformation, on the other
hand, is not easy of accomplishment. Heat is not transformed
into work without effort, and of a given quantity of heat only a
part can be thus transformed, the remainder being inevitably
thrown away. All other forms of energy can, like mechanical
energy, be completely converted into heat. Electrical energy,
for example, in the form of a current, can be thus completely
transformed. Comparing mechanical and electrical energy, we
see that they stand on the same footing as regards transforma
tion. In a perfect apparatus mechanical work can be com
pletely converted into electrical energy, and, conversely, electric
energy can be completely converted into mechanical work.
We are thus led to a classification of energy on the basis of
the possibility of complete conversion. Energy that is capable
of complete conversion, like mechanical and electrical energy,
we may call highgrade energy; while heat, which is not capable
of complete conversion, we may call lowgrade energy.
There seems to be in nature a universal tendency for energy
to degenerate into a form less available for transformation.
Heat will flow from a body of higher temperature to one of
lower temperature with the result that a smaller fraction of it
is available for transformation into work. Highgrade energy
tends to degenerate into lowgrade heat energy. Thus work is
degraded into heat through friction, and electrical energy is
rendered unavailable when transformed into heat in the con
ducting system. Even when one form of highgrade energy is
substances, the difference being due to tJio Jioat developed dur
ing the reaction. As Griffiths aptly says: "Each time we.
alter our investment in energy, we have thus to pay a commis
sion, and the tribute thus exacted can never bo wholly recovered
by us and must be regarded, not as destroyed, but us tinown on
the wasteheap of the Universe."
The terms degradation of energy, dissipation of energy, and
thermodynamic degeneration are applied by different/ writers to
this phenomenon that we have just described. We may for
mally state the principle of degradation of energy as follows :
Every natural process is accompanied ly a certain rfi't/i'ddiitinn.
of energy or tliermodynamic degeneration.
The principle of the degradation of energy denies the. possi
bility of perpetual motion of the second class, which may be de
scribed as follows : A mechanism with friction is inclosed in a
case through which no energy passes. Let the mechanism be
started in motion. Because of friction, work is converted into
heat, which remains in the system, since no energy passes
through the case. Suppose now that the heat thus produced
can be transformed completely into work ; then the work may
be used again to overcome friction and the heat thus produeed
can be again transformed into work. "We then have a perpetual
motion in a mechanism with friction without the addition of
energy from an external source. Such a mechanism does not
violate the conservation law, since no energy is created. It,
however, is just as much of an absurdity as the perpetual mot ion
of the firstclass because it violates the principle of degradat ion.
We shall discuss the degradation principle more at length in
a subsequent chapter.
9. Units of Energy. According to the conservation law,
the quantity of energy remains unchanged through all trans
formations. Hence, a single unit is sufficient for the measure
ment of energy whatever its form may be. This unit is f urnished
by the erg, the absolute unit of work in the C. G. S. system, or
by the joule, which is 10* ergs. It would save much confusion
ART. 10] UNITS OF HEAT 9
arid annoyance if a single unit, as the joule, were used for all
forms of energy. Unfortunately, however, the joule is ordina
rily used in connection with electrical energy only, and other
units are used for other forms of energy. The following are
the units generally employed.
For mechanical energy:
1. The footpound (or in the metric system, the kilogram
meter). This is the unit ordinarily employed by
engineers.
2. The horsepowerhour, which is equal to 1,980,000 foot
pounds. This unit is most convenient for ex
pressing large quantities of work. It should be
noted that although the word " hour " is included in
the name, the time element is in reality lacking,
and the horsepowerhour is a unit of work, not a
unit of power.
For heat energy :
1. The British thermal unit (B. t . u.).
2. The calorie.
The accurate definition of these thermal units and the means
employed in establishing them demand special consideration.
10. Units of Heat. Obviously heat may be measured by
observing the effects produced by it upon substances. Two of
the most marked effects are : (1) rise of temperature ; (2)
change of state of aggregation, as in the melting of ice or
vaporization of water. Hence, we have two possible means of
establishing a unit of heat :
1. The heat required to raise a given mass of a selected
substance, as water, through a chosen range of temperature
may be taken as the unit.
2. The quantity of heat required to change the state of
aggregation of some substance, as, for example, to melt a given
weight of ice, may be taken as the unit.
20 G on the same scale. This thermal unit is called tho ffram
calorie, or the small calorie. If the weight of water in taken an
1 kilogram, the resulting unit is the kilogramcalorie or largo
calorie. This is the unit employed by engineers. ^
The British thermal unit is defined as t/w heat rjur.'d to
raise the temperature of 1 pound of water from l>3 to C>4' J /*'.
The method of establishing thermal units by tho rise <>i tem
perature of water is open to one serious objection, namely :
The energy required to raise the temperature of water one
degree is quite different at different temperatures. Thu^ the,
number of ioules required to raise a given mass of water from
0tol0. or from !>'.)" to
100 C. is considerably
larger than the immbc.r
of joules required to
raise tho same mass from
40 to 41 C. The curve.,
Fig. 1, shows graphically
the energy required per
degree riso of tempera
ture from to 100" (I
It follows that we may
have a number of different thermal units depending upon tho
temperature adopted in the definition. By many physicists
the 15calorie is used. This is the heat required to raise the
temperature of a gram of water from 14 C. to If)! C. In
recent years there has been a tendency to unite on the so
called mean calorie, which may be denned as the ^  ff part, of the
heat required to raise a gram of water from (J. to 100" (1.
The 17calorie, as denned by Griffiths, is practically equal
to the mean calorie. Corresponding to the mean caloric, is the
mean B.t. u., which is T 7 of the heat required to raise the
temperature of one pound of water from 32 to 212 F. This
is equal to the B. t. u. at 63.
1.008
L006
1.004
1.002
1.000
0.988
0.996
1
\
A
\
/
\
7*
/
\
/
,
/
O u
V
4
O u
u u
X
8
u u
11
\
X
FIG. 1,
11. Relations between Energy Units. The relation lwtwo.au.
the joule, the absolute unit of energy, and any of the grswita
ART. 11] RELATIONS BETWEEN ENERGY UNITS H
hour, is readily derived when the value of the constant g is
given. By international agreement g is taken as
980.665 = 32.174^
sec
The second value is obtained by means of the conversion factor
1cm. = 0.3937 in.
Bearing in mind the definition of the erg, we have
1 kilogrammeter = 98066500 ergs
= 9.80665 joules.
Now making use of the relation 1 kg. = 2.204622 Ib. and the
preceding relation between the units of length, we readily find
the relation
1 footpound = 1.3558 joules,
or 1 joule =0.73756 footpound.
The numerical relation between the thermal unit and the
joule, that is, the number of joules in one gramcalorie, is called
the mechanical equivalent and is denoted by J. The determi
nation of this constant has engaged the efforts of physicists
since 1843.*
In this work two experimental methods have been chiefly
employed : (1) The direct method, in which mechanical energy
is transformed directly into heat. (2) The indirect method, in
which heat is produced by the expenditure of energy in some
form other than mechanical. Usually electrical energy is thus
transformed.
The earliest experiments were those of Joule (1843), using
the direct method. Work was expended in stirring water by
means of a revolving paddle. From the rise of temperature
of the known weight of water, the heat energy developed could
be expressed in thermal units; and a comparison of this quan
tity with the measured quantity of work supplied gave imme
diately the desired value of J.
Professor Rowland (18781879) used the same method, but
by driving the paddle wheel with a petroleum engine he was
to the water, and the influence of various corrections was cor
respondingly decreased. Rowland's results are justly tfivcm
great weight in deducing the finally accepted value of '/.
Another result of the highest value in that [omul by Rey
nolds and Moorby (1897). The work, of a 100 horsepower
engine was absorbed by a hydraulic brake. Water entered
the brake at or near C. and was run through it at a rale that,
caused it to emerge at a temperature of about 1.00" ( . In this
way the mechanical equivalent of the heat required to raise (lie.
temperature of one pound of water from O u to 100" (!. was
determined.
Of the experiments by the indirect method those of ( irifliths
(1893), Schuster and Gannon (1H!)4), and dallendar and
Barnes (1899) deserve mention. In each set of experiments
the heat developed by an electric current was measured and
compared with the electrical energy expended.
From a careful comparison of the results of the most trust
worthy experiments, Griffiths has decided that the, most prob
able value of Jia 4.184. That is, taking the 17 r ^ramealorie,
1 gramcalorie = 4.184 joules.
By the use of the necessary reduction factors, we, obtain (he
following relations :
1 kg. calorie = 426.65 kilogrammeters.
1 B. t. u. = 777.64 footpounds.
For ordinary calculations, the values 427 and 77S, respectively,
are sufficiently accurate.
In writing some of the general equations of thennodynamies
it is frequently convenient to use the reciprocal of J. ' This is
denoted by the symbol A ; that is, A = X We may reard ,1
as the heat equivalent of work; thus
1 ft.lb. = A B. t. u.
When the horsepowerhour is taken as the unit of work, we
have
As 1980000
ART. 11] RELATIONS BETWEEN ENERGY UNITS 13
Hence, 1 h.p.hr. = 2546.2 B. t..u.,
a relation that is frequently useful.
EXERCISES
1. If the thermal unit is taken as the heat required to raise the tempera
ture of 1 pound of water from 17 to 18 C., what is the value of / in foot
pounds? ''.',' '.<."' J ''
2. In the combustion of a pound of coal 13,200 B. t. u. are liberated. If
Y per cent of this heat is transformed into work in an engine, what is the
coal consumption per horsepowerhour?
3. A gas engine is supplied with 11,200 B.t. u. per horsepowerhour.
Find the percentage of the heat supplied that is usefully employed.   : ?
4. In a steam engine 193 B. t. u. of the heat brought into the cylinder
by each pound of steam is transformed into work. Find the steam con
sumption per horsepowerhour. /.'. /
5. The metric horsepower is denned as 75 kilogrammeters of work per
second. Find the equivalent in kilogramcalories of a metric horsepower
hour.
6. Find the numerical relations between the following energy unite :
() Joule and B.t. u.
(1>) Joule and metric h.p.hr.
(c) B. t. u. and kg.meter
(rf) h.p.minute and B.t.u.
7. A unit of power is the watt, which is defined as 1 joule per second.
1 kilowatt (lew.) is 1000 watts. Find the number of B. t. u. in a kw.hr. ;
the number of footpounds in a watthour.
8. A Diesel oil engine may under advantageous conditions transform as
high as 38 per cent of the heat supplied into work. If the combustion of a
pound of oil develops 18,000 B.t.u., what weight of oil is required per h.p.hr.?
REFERENCES
THE MECHANICAL THEORY OF HEAT
Rumford: Phil. Trans., 1798, 1799.
Davy : Complete works 2, 11.
Black : Lectures on the Elements of Chemistry 1, 33.
Verdet: Lectures before the Chemical Society of Paris, 1862. (See Ront
geu's Thermodynamics, 3, 29.)
CONSERVATION AND DEGRADATION ov KNKIUJY
Helmholtz : Uber die Erhalfcung der Kraft. Hc.vlin, IS 17.
Thomson (Lord Kelvin) : Ediub. Trans. 20, 2(il, iiK!) (1*51); Phil. M;itf. (I)
4 (1852).
Griffiths: Thermal Measurement of Energy, Lucluro I.
Preston : Theory of Heat, 80, 030.
Planck: Treatise on Thermodynamics (Ogg), !<).
UNITS OF ENERGY. Tin; MKCIIAWOAI, KQUIVAI.KNT
Rowland: Proc. Amer. Acad. 15, 75. 1HSI).
Reynolds and Moorby : Phil. Trans. 190 A, Ml . 1 HfW.
Schuster and Gannon: Phil. Trans. 186 A, 11 r>. 1K!)5.
Barnes: Phil. Trans. 199 A, 140. 1002. Proo. Royal Son. 82 A, :!!><>.
1910.
Griffiths : Thermal Measurement of Energy.
Chwolson : Lehrbuch der Physik 3, 414.
Wiukelmann : Handbuch der Pliysik 2, i5!{7.
Marks and Davis : Steam Tables and Diagrams, !)!2.
CHAPTER II
CHANGE OF STATE. THERMAL CAPACITIES
12. State of a System. A therm odynamic system, may be
defined as a body or system of bodies capable of receiving and
giving out heat or other forms of energy. In general, we shall
assume such a system at rest so that it has no appreciable ki
netic energy due to velocity. As examples of thermodynamic
systems, we may mention the media used in heat motors : wa
ter vapor, air, ammonia, etc.
We are frequently concerned with changes of state of systems,
for it is by such changes that a system can receive or give out
energy. We assume ordinarily that the system is a homogeneous
substance of uniform density and temperature throughout ;
also that it is subjected to a uniform pressure. Such being the
case, the state of the substance is determined by the mass, tem
perature, density, and external pressure. If we direct our
attention to some fixed quantity of the substance, say a unit
mass, we may substitute for the density its reciprocal, the vol
ume of the unit mass ; then the three determining quantities
are the temperature, volume, and pressure. These physical
quantities which serve to describe the state of a substance are
called the coordinates of the substance.
In all cases, it is assumed that the pressure is uniform over
the surface of the substance in question and is normal to the
surface at every point ; in other words, hydrostatic pressure.
We may consider this pressure in either of two aspects : it
may be viewed as the pressure on the substance exerted by some
external agent, or as the pressure exerted ly the substance on
whatever bounds it. For the purpose of the engineer, the lat
ter view is the most convenient, and we shall always consider the
pressure exerted by instead of on the substance. The pressure
is always stated as a specific pressure, that is, pressure per unit
15
pound per square foot.
The volume of a unit weight of the substance is the spi'nifit:
volume. Ordinarily volumes will bo expressed in cubic, feet,
and specific volumes in cubic feet per pound. As it is frequently
necessary to distinguish between the specific volume, and the
volume of any given weight of the substance, wo shall use v to
denote the former and V the latter. Thus, in general, v will
denote the volume of one pound of the substance, l r the. volume
of M pounds ; hence
F= Mv.
This convention of small letters for symbols denoting quanti
ties per unit weight and capitals for quantities associated with
any other weight M will be followed throughout, the book.
Thus q will denote the heat applied to one pound of gas and Q
the heat applied to M pounds, u the energy of a unit weight of
substance, 7" the energy of M units, etc.
As regards the third coordinate, temperature, wo shall ac
cept for the present the scale of the air thermometer. Later
the absolute or thermodynamic scale will be introduced.
While the centigrade scale presents great advantages, tins com
mon use of the Fahrenheit scale in engineering practice, compels
the adoption of that scale in this book.
13. Characteristic Equation. In general, we may assume
the values of any two of the three coordinates p< v, T, and
then the value of the third will depend upon values of these
two. For example, let the system be one pound of air inclosed
in a cylinder with a movable piston. By loading the piston wo
may keep the pressure at any desired value ; then by the ad
dition of heat we may raise the temperature to any predeter
mined value. Thus we may fix p and T independently. Wo
cannot, however, at the same time give the volume v any value
we please ; the volume will be uniquely determined by the
assumed values of p and T, or in other words, v is a function
of the independent variables p and T. In a similar manner
we may take p and v as independent variables, in which ease T
will be the function, or we take v and T as independent and
p as the function depending on them.
or written in the explicit form
p .f ^ JT\ ^2\
The equation giving this relation is called the characteristic
equation of the substance. The form of the equation must be
determined by experiment.
For some substances more than one equation is required ; thus
for a mixture of saturated vapor and the liquid from which it
is formed, the pressure is a function of the temperature alone,
while the volume depends upon the temperature and a fourth
variable expressing the relative proportions of vapor and
liquid.
14. Equation of a Perfect Gas. Experiments on the socalled
permanent gases have given us the laws of Charles and Boyle.
Assuming these to be fol
lowed strictly, we may
readily derive the charac
teristic equation of a gas as
follows.
According to the law of
Charles, the increase of
pressure when the gas is
heated at constant volume is proportional to the increase of
temperature ; that is,
FIG.
This equation defines, in fact, the scale of the constant volume
gas thermometer. Charles' law is shown graphically in Fig. 2.
Point A represents the initial condition (p , ), point JB the
final condition (jp, ). Then
According to Charles' law, therefore, the points representing
the successive values of p and t, with v constant, lie on a straight
line through the initial point A, and the slope of this line is the
18 CHANGE OF STATE. TJUUKMALj UArAVi I ir,r> I IAI. u
constant k. Evidently k is independent of p and t, but it may
depend upoiifl; hence we write
A/00.
Substituting this value of k in (1), wo got
p ^ =C*V) /(.<>)
In this equation t and i are temperatures measured from Urn
Fahrenheit zero ; that is, from the origin (Fig. 2 ). Evidently
the difference tt Q is independent of the position of the as
sumed zero ; hence we may write
where ^and T^ denote temperatures measured from some. new
zero, assumed at pleasure. Let us choose this new xen> siicli
that T when p = 0. This is evidently equivalent to the
selection of a new origin 0' (Fig. 2) at the intersee.tion of the line.
AB with the iaxis. If we now take the, initial point A at 0' ,
we have p Q = 0, T Q = 0, and (2) takes the form
whence pv=Tvf(v~). (V, ;
By hypothesis, the substance follows Boyle's law; that is, the
product pv is constant when the temperature T is eonslaut.
From (3), therefore, the factor vf(v) is a constant ; and denot
ing this constant by B we have
which is the characteristic equation desired.
The name perfect gas is applied to a hypothetical ideal gas
which strictly obeys Boyle's law, and the internal energy of
which is all of the kinetic form, and, therefore, dependent on
the temperature only. No actual gas precisely fulfills these
conditions; but at ordinary temperatures, air, nitrogen, hydro
gen, and oxygen so nearly meet the requirements that 'they
may be considered approximately perfect.
15. Absolute Temperature. The zero of temperature defined
in the preceding article is called the absolute zero, ami tempera
tures measured from it are called absolute temperatures. The
AWT. 15J AJ3SUJLUTE TEMPERATURE 19
molecules on the containing walls. When this pressure is zero,
we infer that molecular motion of translation has entirely ceased,
and this is, therefore, the condition at absolute zero.
The position of the absolute zero relative to the centigrade
zero may be determined approximately by experiments on a
nearly perfect gas, such as air. From Eq. (4), Art. 14, we
have, assuming that the volume remains constant,
whence
Pi
~ Pi _ 2 ~"
t m \^J
Pi 'I
Let "us take 2j as the temperature of melting ice, T z that of
boiling water at atmospheric pressure. llegnault's experi
ments on the increase of pressure of air when heated at con
stant volume gave the relation
.)
)
Since for the C. scale
7^^
O.SOGGp, 100
we have  = m >
whence ^ =  = 272.85. (5)
O.oubo
That is, using air as the thermometric substance, the abso
lute zero is 272. 85 C. below the temperature of melting ice.
Other approximately perfect gases, as nitrogen, hydrogen, etc.,
give slightly different values for T r The experiments of
Joule and Thomson indicate that for an ideal perfect gas, one
strictly obeying the law expressed by the equation pv = BT, the
value of TI would be between 273.1 and 273.14. The corre
sponding value on the Fahrenheit scale may be taken as 491.6 ;
20 CHANGE OF STATE. THERMAL CAPACITIES [CHAP, u
denote ordinary temperatures by t and absolute temperatures
by T, we have
T 1 + 273.1, for the C. scale.
T=t + 459.0, for the K. .scale.
16. Other Characteristic Equations. The equation jn> = 7/7"
gives a close approximation to the changes of state of the. more
permanent gases. Other gases, as, for example, carbonic, acid,
which are in reality only slightly superheated vapors, show
marked deviations from the behavior of the ideally perfect gas,
and this equation does not give even a rough approximation to
the actual facts.
On the basis of the kinetic theory of gases, van der Wauls
has deduced a general characteristic equation applicable not
only to the gaseous but to the liquid state as well. It has the
following form :
BT a
r vl> v^
in which J9, a, and I are constants which depend it] ton the
nature of the substance.
An empirical equation for superheated steam is
' ~.l ' rjln V ~ '
It will be observed that for large values of T and ?>, that, is,
when the gas is extremely ratified, the hist term of both equa
tions becomes small and
the resulting equation ap
proaches more nearly the
equation of the perfect gas.
17. Characteristic Sur
faces. The characteristic.
V equation
ART. 18] THERMAL LINES 21
by its coordinates p^ v v T, and this state is therefore repre
sented by a point, on the surface. If the state changes, a
second point with coordinates p v i> 2 , 5^, will represent the new
state. The succession of states between the initial and final
states will be represented by a succession of points on the
surface. The point representing the state we will call the
statepoint. 'Hence, for any change of state there will be a
corresponding movement of the statepoint.
The surface representing the equation
is shown in Fig. 3. For other characteristic equations the sur
faces are of a less simple form.
18. Thermal Lines. If we impose the restriction that during
a change of state the temperature of the substance shall remain
constant, the statepoint will evidently move on the character
istic surface parallel to the jt?uplane. Such a change of state is
called isothermal, and the curve described by the statepoint is
an isothermal curve or, briefly, an. isotherm. By taking different
constant values for the temperature, we get a complete repre
sentation of the characteristic equation. For the perfect gas,
the isotherms consist of a system of equilateral hyperbolas hav
ing the general equation
pv const. (1)
The restriction may be imposed that the pressure of the sub
stance shall remain constant during the change of state. The
statepoint will in this case move parallel to the v2 7 plane, and
the projection of the path on the jpplane will be a straight line
parallel to 0V, as AB (Fig. 4). The relation between volume
and temperature is found l>y
combining the equation p D
Substituting this value of p in the characteristic, equation, wo
have
" ' '
If the substance changes its state at constant volume, the
statepoint moves parallel to the jp^Pphuie, and the projection
of the path on the pvplnno is a line parallel to the paxis, as
CD (Fig. 4). In the case of a perfect gas, the relation between
p and T for a change at constant volume is
. .
o
Lines of constant pressure are called isopiestic linns ; lines of
constant volume, isometric lines.
Besides the cases just given, others aro of frequent occur
rence, and will be taken up in detail later. Thus we may have
changes of state in which the energy of the system remains
constant; such changes are called isodynamic. Wo may also
have changes in which the system neither receives nor gives
out heat ; such are called adiabatic.
19. Heat absorbed during a Change of State. A change of
state of a system is generally accompanied by the absorption
of heut from external sources.
If we denote by q beat thus
absorbed pur unit, weight, we
may by giving r/ proper signs
cover all possible cases ; thus
+ q indicates heat absorbed, q
heat rejected; while if y 0,
we have the limiting adiabatle
change of state.
The heat absorbed may be
determined from the changes in
two of the three variables j>, t,, t that define the state of the
system. As we have seen, any pair may be selected as suits
our convenience. For example, let t and be talcen as the
independent variables, and let the curve AB f Ki. frt irnresent
FIG. 5.
UJtlAJN <JU
this curve to be replaced by the broken line PQR, then the
segment PQ represents an increment of volume Av with t
constant and the segment QR an increment of temperature
A with v constant. The rate of absorption of heat along PQ,
that is, the heat absorbed per unit increase of volume, is given
by the derivative (  j , the subscript t indicating that t is held
\dv/t
constant during the process. If the rate of absorption be mul
tiplied by the change of volume v, the product () Av is evi
\dvjt
dently the heat absorbed during the change of state represented
( \
) '
dtju
and the heat absorbed is the product (  ) A. The heat ab
1 \dtjv t
sorbed during the change PQR is, therefore,
(1)
^ J
Jvjt \dtjv
and the total heat absorbed along the broken path from A to
B is given by the summation
)At, + f??)A<. (2)
By taking the elements into which the curve is divided
smaller and smaller, the broken path may be made to approach 1
the actual path between A and B. Therefore, passing to the
limit, \ve have instead of (1)
, (3)
and for the heat absorbed during the change of state from A
By choosing other pairs of variables as independent, other
equations similar to (3) may be obtained. Thus, taking t and
or taking p and v as the independent variables, we have
dp
From (5) and (6) equations corresponding to (!) may bo
readily derived.
20. Thermal Capacity. Specific Heat. Of tho partial deriv
atives introduced in the preceding article, two are of special
importance, namely, (^] and ( ( ^) . In general, the heat
\vtjv \vtjp
required to raise the temperature of a body one, degree under
given external conditions is called the thermal capacity. of the.
body for these conditions. Hence, if Q denotes the boat, ab
sorbed by a body during a rise of temperature from t t lo / 2 , the
quotient  gives the mean thermal capacity of the, body ;
^~ tl O 7
and the quotient   =  , tho moan thermal capacity
1
of a unit weight. If the thermal capacity varies with tbo tem
perature, then the limiting value of the quotient , Miat
^ ~~ *i
is, the derivative J, gives the instantaneous value of (bo (her
Ctu
mal capacity. Accordingly, we recognixo in the', derivative
2] the thermal capacity per unit weight of the body under
Bt/v
the condition that the volume remains constant; and in the
derivative f 2 ) the thermal capacity with tho pressure, constant.
\ot /p
According to the definition of the thermal units (Art. 10),
the thermal capacity of 1 gram of water at 17.5 (1. is 1 calorie,
and that of one pound of water at 63.5 F. is 1 P>. t. u.
The specific heat of a substance at a given temperature t is
the ratio of the thermal capacity of the subsisting at this tem
perature to the thermal capacity of an equal muss of water at
some chosen standard temperature. If we take, 1T.;V ('.
j. JUUIUVJ.V.UJL.U
mal capacity per unit weight, then the specific heat c is given
by the relation
__ 7, (of subtance)
7ir.fi (of water) '
But for water y 176 = 1. cal. It follows that the specific heat at
the temperature t is numerically equal to the thermal capacity
of unit weight at the same temperature ; thus at 100 C. the
thermal capacity of a gram of water is found to be 1.005 cal.,
1.005. On account
7l7.fi
and the specific heat is 1 ^ = ? 05 1 '
.
of this numerical equality, we may consider that the derivative
~jf represents the specific heat, as well as the thermal capacity.
It is to bo noted, however, that a specific heat is merely a ratio,
an abstract number, and it is determined by a comparison of
quantities of heat. The deter
mination of thermal capacity, q
on the other hand, involves
energy measurements.
The specific heat of a sub
stance may be represented geo
metrically, as shown in Fig. (5.
Starting from some initial state,
let the rise of temperature be
taken as abscissa and the heat added to the substance as
ordinate. The resulting curve OMvfill represent the equation
and the slope of the curve at any point, as P, will give the de
rivative 2, or the specific heat at the temperature correspond
ttv
ing to P. With constant specific heat the curve OM is a
straight line ; if the specific heat increases with the tempera
ture, the curve is convex to the taxis.
The heat applied to a substance, as will be shown presently,
may have other effects than raising the temperature. The
tain temperatures the curve
temperature; hence, the value of the speeiho heal *i 1 dp,
upon the conditions under which the heat is absorbed I! tho
substance is in the solid or in the liquid form, the, HJ .or,! he i.'ut s are,
practically equal. For substances in the jy.isc.nus bnm however,
the specific heat may have any value from  cc to + oc, depen.lmg
upon the external conditions under which the heat is supplied.
21. Latent Heat. If the heat added to a substaneo and the
temperature be plotted as in Fig. 0, it may happen that, at, cer
tain tarrmflratures the curve has discontinuities. Knr example,
lut boat 1m applied to iei
at F. Tbe e.urve is
praetieally a st.raighl, lim
until tliu temptTaUuv. .'''
is reaclud, but. at, tbis
point considerable bea.t is
added witliout any cban.^e
in ti',inporal.ure. During
this addition of ben,t, rcp
resontod l>y Ibe. vi'i'tical
sogiuont AB (\ f \\r. 7), the
state oi r ji^Ljru^sil ion
changes from solid to liquid. As tlu; watcn. receives beat its
temperature rises, as indicated by BC, until tbc temperature
212 F. is reached (assuming atmospheric pressure), \vhere the
temperature again remains constant during the, addition of a
considerable quantity of heat, and the state of aggregation again
changes, this time from the liquid to the gaseous. The beat,
that is thus added to (or abstracted from) a substanee during
a change of state of aggregation is called latent heat. As
pointed out in Art. 4, substantially all of the. latent, heat is
stored in the system in the form of potential energy.
The specific heat =i becomes infinite during the changes
indicated by AB and CD, since t constant. The volume of
the substance changes, however, and the rate at which heat is
FIG. 7.
ART. 22J KUJJLjATlUNb iJ WTWliiJUJN TlllliKMAJj (JAl'AUiTJUUfc) Z/
is a thermal capacity called thu latent heat of expansion and
denoted by 1 . If tlie pressure also changes, we have in the
derivative ( ) the heat added per unit change of pressure.
Tli is thermal capacity is called the latent heat of pressure varia
tion, and is demoted by l p .
22. Relations between Thermal Capacities. Introducing the
symbols c v , c^, Z v , and l p in equations (8) and (5) of Art. 19, we
have
dq = l v dv + c v dT, (1)
dy = l lt dp + o v dT. (2)
liy means of the characteristic equation of the substance,
namely,
v =/(?, (3)
various relations between the thermal capacities may be de
rived. Some" of the most useful are the following.
From (3) we obtain by differentiation,
which substituted in (1) gives
.^=i,^+(, + z,)dr, (5)
Comparing (2) and (5), we have
dv "
In the same way, siibstituting
in (2), and comparing the resulting equation with (1), we
obtain
7 _ 7 5 , (8)
~
ing thermal capacities when any one is given }' dircd, <'.XJ><M'I
ment, provided the characteristic equation, of the substance is
known, so that the derivatives ^ ^ etc., can be determined.
For a perfect gas, as an example, ts p is known from experiment
and the ratio  has also been (letormimul. From the equation
c
of the gas pv = BT, we have the partial derivatives
8v _B dp _ It
Tf~~p* 52 r "V ;
hence from (7) and (9)
_ 7 ^ 7 ?' '  . "\
Vp C = (>v~~ 5 01> ^ ' It^'P '*"' '
and l v =V(c it c v }.
23. Interpretation of Diiferential Expressions. In thcnno
dynamics we frecjuently meet with exjtressions of tin; form
Mdx 4 Ntly
composed of two terms, of which eacli is tin? diffcnMitial of a
variable multiplied by a coefficient. The two c.o(>nic.ir.nis may
be constants or functions of the two vavitibh.s iuvolvud. Th
proper interpretation of differentials of this form is likely to
present difficulties to the student; we shall, lln'ivfori>, dnvott
this article to a discussion of such expressions, their projierties,
and their physical interpretations.
Let us consider first how such differential cxprossions may
arise. Suppose we have given the characteristic equation of a
substance in the form
jp=/0>, 0; (i)
by differentiation according to the wellknown methods of cal
cnlus, we obtain the relation
(3)
where M= ^, and JV=^.
aw' d
In Art. 19 we derived an equation of similar form, namely,
dq^dv+^dt, (4)
i dv dt ^ J
which may likewise be written in the form
dq = M'dv + N'dt. (5)
The second members of (3) and (5) are differential expressions
of the form Mdx + Ndy, which we have under consideration.
Kq. (3) was produced from a known functional relation be
tween p> V, and i, while Eq. (5) was derived directly from
physical considerations by assuming increments AV and A of
the independent variables and deducing from them the quantity
of heat A<7 that must necessarily be absorbed. No relation
between y, v, and t was given or assumed; in fact, it is known
that no such relation exists ; that is, q cannot be expressed as a
function of the variables v and t.
Let us see what is implied by the existence or nonexistence
of a functional relation between q, v, and t. Referring to
Fig. 5, let A and B denote the initial and final states of the
system. Since p is a function of v and t \_p=*f(y, t)~\, the
pressures at A and B are determined by the values of T and v
BT BT
at those points ; thus for a perfect gas, p = * and p z = a .
v i v z
Hence, the change of pressure p 2 p l in passing from A to B
is fixed by the points A and B alone and is independent of the
path between them. Similarly, if there is a functional rela
tion between q, v, and t, that is, if q = (v, ), we shall have at
A, ft = 0( v ii *i)' afc #' ( 72 = < ^( t; 2' f a) Therefore, the heat
absorbed in passing from A to B will be
<? 2  <li = $ ( V 2> ^2) ~ < 0>i *i)> ( 6 )
and this will be determined by the points A and B alone. On
the other hand, if the heat absorbed by the system depends
upon the path between A and B, there can be no relation
g = </>(v, ). As a matter of fact, the heat absorbed i different
for different paths between the same initial and iimil wtaU'H ;
hence it is not possible to express q in terms of v and C.
The conclusions just given may be stated in gonural toriiw a.s
follows. Given an expression of tho form
du =
(7)
where the coefficients M and TV are funotioiiH of x and //, there
mayor may not exist a functional relation between /. and the
variables x and y. If u is a function of x and //, say it . /''('% //),
then the change in u depends only on the initial and iinal
values of x and y and is independent of tho path. This
is found from (7) by integration ; thus
In this integration no relation between x and ?/ is required, for
since Mdx+'Ndy arises from differentiating the function
cf> (a, y), the integral must be (/> (#, ?/). In this ease ^l/f/.c f N<ltj
is said to be an exact differential.
As an example, consider the equation
du = ydx f inly.
Since ydx + xdy is produced by the differentiation of the prod
uct xy, we have the relation
u = xy + (7,
whence u z u^ = # 2 y 2 r.y.,.
The change of u is represented by the shaded area ( Fitf . K),
and is evid(uitly not. dep(Mident,
upon the path betwirn t.be points
If, however, no functional rela
tion exists between u and tin 1
variables a: and //, then J/r/.r +
TV}?// is said to be an inexact
differential. In this ease a value
sinned ; in other words, the value of u depends upon the path
between the initial and final points. For example, let
du = ydx 2 xdy
and let the initial and final points be respectively (0, 1) and
(2, 2). No function of x and y can be found which upon
differentiation will produce this differential. If we choose as
the path between the end points the straight line y = %x + l,
we have (since dy = \ d.ti),
u = j" [(J x 4 1 )dx xdx] = 1.
If we take as the path the parabola y =  x 2 4 1, we have
u = f ["(I x 2 + T)dx x z dx~] ~ 0.
The dependence of the value of u upon the path assumed is
evident.
The test for an exact differential is simple. If the differential
du = Mdx f Ndy is exact, then u must be a function of x and y,
say /(a;, v/). .By differentiation, we have
j du , . du ^
au = ax H ay.
dx dy y
Hence M and ^V" must be, respectively, the partial derivatives
O9/ O9/
__ and  By a wellknown theorem of calculus, we have
ox dy
d_(?u
dy\dx
thati S) y
dy dx
If relation (9) is satisfied, the differential is exact ; otherwise,
it is inexact.
As an example, we have from the differential ydx 2 xdy,
= 1, = 2 ; therefore, the differential is inexact, as was
dy dx
shown in the preceding discussion.
In thermodynamics we meet with certain functions that de
pend only upon the coordinates p, v, T of the substance under
consideration. From purely physical considerations the energy
(See Art. 26.) Hence if u is expressed in terms of two of
these coordinates as independent variables, thus,
we know at once that du is exact and we can write
f'A* = >! =/0 2 < zi) /O'n Zi ')
'i
Furthermore, from the test for an exact differential wo must
have the relation
T dv
By making use of this test when the differential IH known to
be exact, many useful relations are deduced.
We have also magnitudes that depend upon tho ootirdinatcs
and also upon the method of variation ; that in, upon tho path.
The heat q absorbed by a system in changing Htato is one of
these. If again we choose v and T as the independent variables,
we may write
but since dq is not exact, we cannot write
EXERCISES
1. Regnault's experiments on the heating of cmtain liquids are ex
pressed by the following equations :
Ether q = 0.529 t + 0.000200 * a ,  20 to + :)" ( '.
Chloroform q = 0.232 t + 0.0000507 t'\  '.W> to I (it) (!.
Carbon disulphide q= 0.235 t + 0.0000815 t",  :}()" to  !()" ('.
Alcohol q = 0.5476 1 + 0.001122 / a + 0.0000022 /", ~ 2:5" to + lili ' ( 1 .
From these equations derive expressions for tho Hpciniu; huat, and for
each liquid find the specific heat at 20 C.
2. From the data of Ex.1, find the mean heat oapaiily of i>iln>r IHIWIMMI
and 30 C. Also the mean heat capacity of alcohol bct/woon ()" and rIV ('.
3. If the thermal capacity of a substance at temp>ratun>. / is given by
the relation
y = a + U + ct~,
AWT. AJJ JUAEKCISES 33
4. In the investigation of the properties of gases, it is convenient to
draw the isothermal* (T = const.) on a plane having the pressure p as the
axis^ of abscissas and the product pv as the axis of ordinates. Show that
the isothermals of a perfect gas are straight lines parallel to the joaxis.
5. Show on the pop plane the general form of an. isothermal of super
heated steam, the characteristic equation being
As an approximate equation for superheated steam, the form
p(v + c)=BT,
has been suggested by Tumlirtz. Show the form of the isothermal when
this equation is used.
6. Derive relations between c in c m l p , and l v , similar to those given by
Eq. (10) and (11) of Art. 22, using van der Waal's equation
v b v 2
as the characteristic equation of the gas.
7. For a perfect gas, as will be shown subsequently, the thermal capacity
l v is Ap(A .J). Show that c p  c v = AB ; also that l }> =  Av.
B. Test the following differentials for exactness :
() vilp + npdv,
(J>) v n dp + n])v n  l dv.
(0 x +
9. Find the function u f{p, T) which produces the differential (c)
of Ex. 8.
10. The differential [c'(l  x)+ c"x] ^ + dx, which appears in the
discussion of vapors, is known to be exact, c' and c" may be taken as con
stants, while r is a function of T. Apply the test for exactness and thereby
deduce the relation c" c' = ^ ?
11. For perfect gases, dq  c v dT + Apdn. (See Ex. 7, and Art. 22.)
Making use of the characteristic equation pv = BT, show that while dq is
not an exact differential, is an exact differential.
REFERENCES
THERMAL CAPACITY. SPECIFIC HEAT
"VVeyraach : Grundriss der WiinneTheorie 1, (JO.
Chwolson : Lehrbuch der Physik 3, 172.
EXACT AND INEXACT DIFFKKENTIALH IN THERMODYNAMICS
Chwolson : Lehrbuch der Physik 3, d34.
Clausius: Mechanical Theory of Heat, Introduction.
Preston : Theory of Heat, 597.
Weyrauch : Gruudriss der WiirmeTheorie 1, 28.
Townsend and Goodenough: Essentials of Calculu.s, 245.
CHAPTER III
THE FIRST LAW OF THERMODYNAMICS
24. Statement of the First Law. The first law of Thermo
dynamics relates to the conversion of heat into work, and merely
applies the principle of conservation of energy to that process.
It may be formally stated as follows : When work is expended
in producing heat, the quantity of heat generated is proportional to
the work done, and conversely, when heat is employed to do work, a
quantity of heat precisely equivalent to the work done disappears.
If we denote by Q the heat converted into work and by "FT the
work thus obtained, we have, therefore, as symbolic statements
of the first law,
Tf= JQ, or Q = AW.
25. Effects of Heat. When a thermodynamic system, as a
given weight of gas or a mixture of saturated vapor and liquid,
undergoes a change of state, it in general receives or gives out
energy either in the form of heat or in the form of mechanical
work. These energy changes must, of course, conform to the
conservation law. Suppose in the first place that the system is
subjected to a uniform external pressure and that during the
change of state the volume is decreased. Mechanical work is
thereby done upon the system, or in other words, the system
receives energy in the form of work. At the same time heat
may be absorbed by the system from some external source.
Denoting by ATT the work received and by AQ the heat
absorbed, the increment AZ7 of the intrinsic energy of the
system is given by the relation
AZ7= J&Q + ATT. (1)
Ordinarily we take the work done by the system in expanding
as positive ; hence the work done on the system during com
_r .
36 THE FIRST LAW OF THERMODYNAMICS [DIAIMII
that is, the increase of energy of the system is equal to tho
energy received in the form of heat less tho energy tfivi'ii to
the surrounding systems in the form of work. Wo may also
write (2) in the form
and interpret the relation as follows. The heat absorbed by a
substance is expended in two ways : (1) in increasing the
intrinsic energy of the substance ; (2) iu tho performance of
external work.
Equation (3) is the energy equation in its most Amoral form.
Any one of the three terms may bo positivo or negative. Wo
consider A Q positive when the system absorbs boat., negative
when it gives out heat ; as before stated, A IK is positivo when
work is done by the system, negative whon work is done, on (lio
system; A U is positive when the internal energy is increased,
negative when the energy is decreased during tho change of
state.
26. The Intrinsic Energy. Tho increase A //"of I ho in( rinsio,
energy is, in general, separable into two parts: (1) Tho in
crease of kinetic energy indicated by a riso of temperature of
the system. As we have seen, this is duo to an increase in tho
velocity of the molecules of the system. (l2) Tin; increase of
potential energy arising from the inorcaso of volumo of tho,
system. To separate the molecules against their mutual at trac
tions, or to break up the molecular structure, as is dono in
changing the state of aggregation, requires work, and this
work is stored in the system as potential energy.
The energy U contained in a body depends' upon the state
of the body only, and the change of energy duo to a change
of state depends upon the initial and. final' states only. hi
Fig. 9, let A represent the initial, and ./>' tho tiual state'. The
point B indicates a definite state of the body as regards pres
sure, volume, and temperature. Now the, 'temperature indi
cated by B fixes the kinetic energy and Uo volume at P>
determines the potential em>r< rT,m,,,, +1,., i;,,.,i <..*..i ......... ^
ART. Z/J UA.TJttJtUNAjL WUJKJi.
to B. Whether we pass by the
path m or the path n, we have the
same volume and temperature at B
and therefore the same total energy.
Since V is thus a function of the
coordinates only, it follows that d II
is always an exact differential.
Choosing T and v as the hide FlG 9
pendent variables of the system,
we may express U as a function of these variables. We have,
therefore, ?7
whence dU= dT ' + dv.
dT dv
The term ~^,dT is the increment of energy due to the in
r) TT
crease of temperature d T. The factor is the rate at which
the energy changes with the temperature when the volume
n Try
remains constant. Hence ^dT is the change of energy due
merely to the rise of temperature, that is, it is the change
\ 7"7"
of kinetic energy. The term  dv is the change of energy
dv
due merely to the change of volume with the temperature
constant ; it is, therefore, the work done against molecular
attractions, the work that is stored as potential energy. For
a substance in which there are no internal forces between
the molecules, the energy is independent of the volume, that
is, ~~ = 0, and therefore the term dv is zero.
dv dv
27. The External Work. In nearly all cases dealt with in
applied thermodynamics, the external work ATT is the work
done by the system in expanding against a uniform normal
pressure. A general expression for the external work may
be deduced as follows. Let AJP denote an elementary area
on the surface inclosing the system and suppose that during
li JBJ..K.BT
normal pressure per unit area, the work donu against this
pressure is for this one element
p&Fs. )
When all the elements of tlie surface tiro lukim, HIM OXJMVS
sion for the work is
But evidently if 8 he taken sui'iiciontly small, A/'' is (In;
increase of volume AF"; hence we may writo
A If =^ A I 7 ; ()
from which we have
for a change of volume from V l to J T 2 .
The external work for a given change of staio is n^ircscnUMl
graphically by the area between the projuc.tinn of the. initli
of the statepoint on the j;Fplane and tin; F"uxis. Thus in
Fig. 10, let the variation of pressure and volume lx^ rcjircscnli'd
by the curve AJB; this is the projection on tlm p fplaiu. of tho
actual path of the statepoint on tho oharautnrislic, surl'ac.o.
The area A^BB^ under AB is clearly given by tho inU^nil
hence, it represents the work done by tho system in passing from
the initial to the final state according to the in von law.
n o
Tlio gonoral onorgy oqna
A tion (8), Art. U5, may now bo
written in tho form
or using tlio dilToroiitial nota
tion, in tlii! form
B
PIG. 10.
98
__ v For a unit weight of tho sub
fil stance, Ave havo
Jdq = du + pdt'. (V> a)
by the subscripts 1 and 2, respectively, we have
whence JQ = Z7 2  ^ + $p d F (1)
It should be noted carefully that since the energy U depends
only upon the state of the system and not upon the process of
passing from the initial to the final state, the change of energy
may be written at once as the difference U 2 U r The external
work
is evidently dependent upon the path of the statepoint between
the initial and final states. See Fig. 10. Hence the sum of
the change of energy and external work, that is, the heat added
to the system, must also depend upon the path. It follows
that dQ is not an exact differential, and we cannot write
In other words, we cannot properly speak of the heat in a
a body in the state 1 or the state 2 ; we can speak only of the
heat imparted to the body during the change of state with the
reservation, stated or implied, that the quantity thus imparted
depends upon the way in which the state is changed. For con
venience we shall denote by $ 12 the heat imparted to the sys
tem in passing from state 1 to state 2 ; and likewise by W lz the
corresponding external work done by the system.
29. Energy Equation applied to a Cycle Process. Let a sys
tem starting from an initial state pass through a series of pro
cesses and finally return to the initial state. The path of the
statepoint on the characteristic surface is a closed curve in
space and the projection of the path on the p Fplane is a closed
plane curve. See Fig. 11. Let A represent the initial state;
then in passing from A to B the external work done by the
system is
' p dV (along path m),
40 THE FIRST LAW OF THERMODYNAMICS [THAI. MI
which is represented by area A^AmSS^ while in passing from
B back to A along path n the external work is
f "0 dV= L p dV (along path ?<.)
J Pj J " ' O
and this is represented by area B^BnAA^ Hene.e tlie net
external work done by the system is represented by the area
inclosed by tho eurve of the
cycle.
Since tho energy I.' of tho
system depends upon the state
only, the change of energy for
the cycle is
/,/,= <>,
y and the energy equation ro
^ ,.," 01 duces to
FIG. 11.
That is, for a closed cycle, of processes, the heat imparted to thr.
system is the equivalent of the external work, and both are repre
sented graphically by the area of the cycle on the ^rplane.
30. Adiabatic Processes. When a system in changing its
state has no thermal communication with other bodies and
therefore neither absorbs nor gives out heat, tho change of
state is said to be adiabatic. In general, adiabatie. ehangos arc.
possible only when the system is inclosed in a nonoon<imaing
envelope. Rapid changes of state are approximately adiabatie,
since time is required for conduction or radiation of heat ; thus
the alternate expansion and contraction of air during the pas
sage of sound waves is nearly adiabatic; the flow of a gas or
vapor through can orifice is practically an adiabatic process.
Jor an adiabatic change, the term JQ of tho energy equation
reduces to zero, and we have, consequently,
During an adiabatic change, therefore, tho extern,! wnrt ,!,,
KT. 30]
ADIABATIC PROCESSES
41
B
The projection on the pT^plane of the path of the statepoint
.uring an adiabatic change gives the adiabatic curve. See Fig.
.2. The area A 1 ABB 1 represents the work TF 12 of the system
,nd from (1) it represents also the decrease of the intrinsic
nergy in passing from state 1
epresented by A to state 2
epresented by B. Making
ise of this principle, we can
rrive at a graphical represen
ation of the intrinsic energy
if a system. Suppose the
.diabatic expansion to be con
inued indefinitely; the adia
'atic curve AB will then FIG. 12.
pproach the F^axis as an
symptote, and the work of the expanding system will be
epresented by the area A^A oo between the ordinate AA, the
xis OF", and the curve extended indefinitely. The area A^Aca
epresents also the change of energy resulting from the expan
ion. Hence if we assume that the final energy is zero, we have
i = area A^A oo,
\ area A 1 A oo = ( y p d V.
It is instructive to compare
the adiabatic curve with the
isothermal. When the two
curves are projected on the
pF~plane, the adiabatic is the
steeper. See Fig. 13. This
follows from the fact that dur
ing adiabatic expansion the
nergy decreases and as a result the temperature falls ; hence
or
FIG. 13.
JblKST JUAW UJf
1U
On the other hand, the area under the indefinitely extended
isothermal is infinite.
31. Isodynamic Changes. If tho intrinsic, energy of tho
system remains unchanged during a change, of slate, the change
is called isodynamic or isoenergic. In this case the energy
equation reduces to the form
For perfect gases, the isodynamic curve is also tho isothermal,
but for other substances this is not tho case.
32. Graphical Representa
tions. The throe magnitudes
JQiv <r/ 2 ^r :U1( ^ '^12 l>11 ^ !1 '
ing into the energy equation
can bo represented graphically
by areas on the p\ 'plant!.
Suppose tho change of state
to bo represented by tho curve;
m between the initial point, A
Y and final point H (Kig. M ).
FIG. u.
,
'idiabatiu lines be drawn
through A and /i siinl ex
tended indefinitely; then from preceding considerations we have
F
12 = area
= area
<x>
Hence, JQ 12 = U z  ^ + F 12
= area A l ABB l + area B,B oo  area A,A
= area AB oo .
That is, the heat imparted is represented on the. p}"pl<m<> 1, t/,<>
area included between the path and two twfc/to/// m.,.,,,!,,?. ',H,
latics drawn through the initial and final pointy rmn^hchi
Ihrough the initial point A let an Lsodynamie be, drawn,
cutting BB,m the point 0, and through let tlio i,ldi,,it,ly
extended adiabafao (7=o be drawn. Then the energy r a of tho
system m state is equal to U and, therefore, '
ART. 32]
43
It should be noted that the p
area representing U z L\ is
not influenced by the path m.
A second graphical repre
sentation is shown in Fig. 15.
Through the initial point A
an isodynamic line is drawn,
and through the linal point B
an adiabatic is drawn, the two
lines intersecting at point (7.
We have then, denoting the
energy in the' state C by ?7 3 ,
A
B,
FIG. 15.
L y
2  ^ = z  z = area
W lz = area A^ABB V
JQ V1 = W lz + t/2  /! = area
As before, the change of energy is independent of the path w,
while botli the external work and the heat imparted depend
upon the form of m.
EXERCISES
1. Show that the energy equation may be written in the form
and that consequently the derivative ( ^ ) must be equal to Jc v .
2. If the energy of a substance is independent of the volume, show that
the energy equation reduces to the form
Jdq = Jc v dT+pdv.
3. Using the method of graphical representation, show by areas Qi,
U'2  Ui, and Ww () for a change at constant pressure, (b) for a change at
44 ltUtU JblltoT JLAW UJ. 1
7. Apply the general energy equation to the. procenn of changing ice ;
32 F. to water. What is the effect of greatly incroaidng the pro.s.snrc. o
the ice during the process 'I
REFERENCES
Preston : Theory of Heat, 590.
Zeuner: Technical Thermodynamics (Klein) 1, i28.
Planck: Treatise on Thermodynamics (Ogg), IkS.
THE SECOND LAW OF THERMODYNAMICS
33. Introductory Statement. While the first law of tliermo
ynamics gives a relation that must be satisfied during any
iiange of state of a system, and of itself leads to many useful
isults, it is not sufficient to set at rest all questions that may
rise in connection with energy transformations. It gives no
idications of the direction of a physical process ; it imposes no
mditions upon the transformations of energy from one form to
lother except that there shall be no loss, and thus gives no in
ication of the possibilities of complete transformation of dif
>rent forms; it furnishes no clue to the availability of energy
>r transformation under given circumstances. To settle these
uestioiis a second principle is required. This principle, called
le second law of thermodynamics, has been stated in many ways.
i effect, however, it is the principle of degradation of energy,
ist as the first law is the principle of the conservation of
There are conceivable processes which, while satisfying the
jqiurements of the first law, are declared to be impossible be
mse of the restrictions of the second law. As a single ex
mple, it is conceivable that an engine might be devised that
mild deliver work without the expenditure of fuel, merely by
sing the heat stored in the atmosphere; in fact, such a device
as been several times proposed. The first law would not be
iolated by such a process, for there would be transformation,
ot creation of energy; in other words, such an engine would
ot be a perpetual motion of the first class. Experience shows,
owever, that a process of this character, while not violating
le conservation law, is nevertheless impossible. The statement
34. Availability of Energy. In Art. 8 was noted the (list/mo
tion between various forms of energy with respect to the pos
sibility of complete conversion. Wo shall now consider the
point somewhat in detail.
Mechanical and electrical energy stand on the same footing
as regards possibility of conversion; either can be completely
transformed into the other in theory, and nearly so in practice.
Either mechanical or electrical energy can bo completely trans
formed into heat. On the other hand, experience shows that
heat energy is not capable of complete conversion into mechan
ical work, and to get even a part of heat energy transformed
into mechanical energy, certain conditions must bo satisfied.
As a first condition, there must bo two bodies of different, tem
perature; it is impossible to derive work from the heal of a body
unless there is available a second body of lower temperature.
Suppose we have then a source 8 at temperature T { and a re
frigerator R at lower temperature .7! 2 ; how is it possible to
derive mechanical work from a quantity of heat energy Q { stored
in SI If the bodies $ and R are placed in contact., the heat
Q will simply flow from S to R and no work will bo obtained.
Hence, as a second condition, the systems *S r and H must be kept
apart and a third system M must be nsed to convey energy.
This third system is the working fluid or medium. In the steam
plant, for example, the boiler furnace is the source /S Y , the con
denser is the refrigerator R at a lower temperature, and the
steam is the medium or working fluid M. The medium M
is placed in contact with S and receives from it heat Q^ it then
by an appropriate change of state (expansion) gives up energy in
the form .of work, and delivers to R a quantity of heat $ 2 ,
smaller than Q v the difference Q l  Q z being tho heat trans
formed into work. The details of this process will be given in
following articles, where it will be shown that in no other way
can a larger fraction of the heat be transformed into work.
The part of the heat Q l that can be thus transformed into work,
that is, Q l Q yt is the available part of Q^ and the purl $ 2 that
must be rejected to the refrigerator R, and which is of no further
.._. ,, ' ~ . . . a. ~ o.
cal work. In general, the term availability signifies the fraction
of the energy of a given system in a given state that can be
transformed into mechanical work.
In Art. 8 attention was called to the apparent tendency of
energy to degenerate into less available forms. We have now
to investigate this point somewhat closely in connection with
reversible and irreversible changes of state.
35. Reversibility. The processes described in thermo
dynamics are either reversible or irreversible. A process is
said to be reversible when the following conditions are fulfilled :
1. When the direction of the process is reversed, the system
taking part in the process can assume in inverse order the
states traversed in the direct process.
2. The external actions are the same for the direct and re
versed processes or differ by an infinitesimal amount only.
3. Not only the system undergoing the change but all con
nected systems can be restored to initial conditions.
A process which fails
to meet these require
ments in any particular
is an irreversible pro
cess. The following
examples illustrate the
above definitions.
(1) Suppose a con
fined gas to act on a
piston, as in the steam
or gas engine. See
Fig. 16. If A is the
piston area, the pres
sure acting on the face
of the piston is pA,
and for equilibrium
this pressure must be equal to the force F. If now we assume
the force pA slightly greater than F, the piston will move
slowly to the right and the confined gas will assume a succes
UbLJlJ SJiUJUlNJJ JUa.YV UJ?
sion of states indicated by the curve All, It at the slate .#
the motion is arrested and I 7 is made infinitesimally greater
than pA for all positions of the piston, the .scries of status from
B to A will be retraced and the system (tin 1 , confined gas in
this case) will be brought back to its original state without
leaving changes in outside bodies. The reversed process is
accomplished by an infinitely small modification of tins external
force F. The process is therefore reversible.
(2) Let the force F be removed entirely. Thou the piston
will move suddenly and the confined gas will bo thrown into
commotion. When the gas finally attains a stato of thermal
equilibrium with the volume F" 2 , that state will be represented
by some point as B 1 . No path can be drawn between A and Jl'
because during the passage from A to Ji' the gas is not in
thermal equilibrium, and its state at any instant cannot, there
fore, be determined. Evidently, therefore, the gas cannot be
returned to state A by reversing in all particulars the direct
change from A to B'. It can be returned to stato A, however,
in the following manner : A force F, slightly greater than ;;A,
is applied to the piston and the gas is thus compressed slowly,
the successive states being indicated by the enrve II' A', say.
Then the gas in the state A' is cooled at the, constant volnmo
V l until the original state A is attained. The restoration of
the^gas to its initial state has, however, left changes in other
bodies or systems. Thus the work of compression from tt 1 to
A' must be furnished from one external body, and the heat
given up by the cooling from A' to A must bo absorbed by
another external body. The free expansion of tho gas is,
therefore, an irreversible process.
It is easy to see that the flow of a fluid through ,,n online
trom a region of high pressure to a region of low pressure is
essentially equivalent to the irreversible expansion just de
scribed. Such cases are of frequent occurrence in t'eehnical
applications of thermodynamics. The flow of liquid aunnonia
through the expansion valve of the refrigerating machine may
be cited as an example.
AUT. ouj vjrJujiNJtujui/vju >x.t\..i. JIUV.I.JUUN x \jj}
and bearing due to the conversion into heat of the work of
overcoming friction. A complete reversal of this process would
involve turning the shaft in the opposite direction by cooling
the bearing.
(4) The conduction of heat from one body to another is an
irreversible process. There must be a temperature difference
to produce the flow of heat, and heat of itself will not flow in
the reverse direction ; that is, from the colder to the hotter
body. If, however, we take the temperature difference A T in
definitely small and let the transfer take place very slowly, the
process can be reversed by changing the sign of A 21 Hence
we can conceive of reversible flow as the ideal limiting condi
tion of the actual irreversible flow.
Strictly speaking, there are no reversible changes in nature.
We must consider reversibility as an ideal limiting condition
that may be approached but not actually attained when the
processes are conducted very slowly.
36. General Statement of the Second Law. According to
the first law, the total quantity of energy in a system of bodies
cannot be increased or decreased by any change, reversible or
irreversible, that may occur within the system. It is not, how
ever, the total energy, but the available energy of the system
that is of importance ; and experience shows that a change
within the system usually results in a change in the availability
of the energy of the system.
It may be considered as almost selfevident that no change
of a system which will take place of itself can increase the
available energy of the system. On the other hand, experience
teaches that all actual changes involve loss of availability. Con
sider, for example, the flow of heat from a body of temperature
T v to another at temperature T 2 . For the flow to occur of it
self we must have 5\ > T^ and as a result of the process there
is a loss of availability. To produce an increase of availability
would require T 2 to be greater than 2j ; in that case, however,
the process would not be possible. In the limiting reversible
of energy, are based 'entirely on experience:
I. No change in a system of bodies that nan take, plane of ifxe/f
can increase the available energy of the xi/nl.c.m.
II. An irreversible change causes a low of anai/a!>t'!itt/.
III. A reversible change doe's not affwt, the. ai>aitat>ility.
These statements may be regarded as fundamental natural
laws underlying all physical and chemical changes, The seeond
and third together constitute the law of degradation of ene.rgy.
The first may be taken as a general statement of the .second law
of thermodynamics.
By considering special processes the general statement of tin;
second law here given may be thrown into special forms. Tims
if heat could of itself pass from a body of lower to a body of
higher temperature, the result of the process would be. an in
crease of available energy, a result that is impossible according
to our first statement. We have, therefore, Clausius' form of
the second law, viz :
It is impossible for a self acting machine, unaided I if any e.rler
nal agency to convey heat from one body to another at hi<//ier
temperature.
Again, if we consider the increase of available, unorgy thai,
would result from deriving work directly from the heat of tbo
atmosphere, we are led to Kelvin's statement, namely :
It is impossible by means of inanimate 'material agency to derive
mechanical e/ect from any portion of matter bt/ e.ooUng it below th,:
temperature of surrounding objects.
In order to estimate the available energy ,,f a system in a
given state, or the loss of available energy when the system
undergoes an irreversible change, it is necessary to know tbo
most efficient means of transforming heat into mechanical work
under g lv en conditions. This knowledge is furnished by a
study of the ideal processes first described by Oarnot in mi.
37 Carnot's Cycle. Suppose that the conditions stilted in
Art. 34 are furnished ; that is, let there be a source of beat fi
at temperature ^, a refrigerator R at a lower temperature T v
ART. 37]
CARNOT'S CYCLE
A l
01
FIG. 17.
and an intermediate system, the working fluid or medium M.
The medium we may assume to be inclosed in a cylinder
provided with a piston (Fig. 18).
Let the medium initially in a state represented by B (Fig. 17),
at the temperature T of the reservoir $, expand adiabatically
until its temperature falls to T v
the temperature of body R.
By this expansion the second
state Q is reached, and the
work done by the medium is
represented by the area S 1 SOO r
The expansion is assumed to
proceed slowly so that the pres
sures on the two faces of the
piston are sensibly equal, and
the process is, therefore, re
versible. The cylinder is now
placed in contact with R so that heat can flow from Jf.to R,
and the medium is compressed. The work represented by the
area C l QDD l is done on the medium, and heat Q 2 passes from
the medium to the refriger
ator. The process is again
assumed to be so slow as to
be reversible. From the
state D the medium is now
compressed adiabatically,
the cylinder being removed
from R until its tempera
ture again becomes T v that
of the source 8. D uring this
third process work repre
sented by the area D^DAA^
is done on the fluid. Finally,
rjG 18 the cylinder is placed in
contact with S and the
fluid is allowed to expand at the constant temperature T
52 THE SECOND LAW OF THERMODYNAMICS [CIIAV. iv
temperature is kept constant by the flow of heat ^ from
StoM.
The area ABGD inclosed by tho four curvo.s of the, cycle
represents the mechanical work gained; that is, the excess of
work done by the medium over that done on tho medium.
Denoting this by W, we have from the first, law,
The efficiency of the cycle is the ratio of tho work gained to
the heat supplied from the source ti. Denoting thu elliciency
by 97, we have
QI ^i
Since all the processes of the Carnot cycle arc revorsiblo, it
is evident that they may be traversed in reverse order. Thus
starting from B, the fluid is compressed isothermal ly from Ji to
A and gives up heat Q 1 to S; from A to .D it expands udiabal
ically, from D to (7 it expands at the constant temperature 7!>
and in so doing receives heat Q 2 from Ji ; limilly it is com
pressed adiabatically from Q to the initial state. H. In this ease
the work TF represented by area ABCD is done nn tin; lluid ,17,
heat Q z is taken from the refrigerator 7, and the sum Q z \ A \V
= Q 1 is delivered to the source 8. This ideal reversed, engine
is the basis of our modern refrigerating machines.
38. Carnot's Principle. The efficiency of Carnot's ideal
engine evidently depends upon the temperatures 7 r , and r l\ of
the source and refrigerator, respectively. Thu question at once,
arises whether the efficiency depends also upon thu properties
of the substance M used as a working iluid. The answer is
contained in Carnot's principle, namely :
Of all engines working between the mme sour*: nnJ tlir n<inn>
refrigerator, no engine can have an efficiency </r<>at<'r than Unit of
a reversible engine.
In other words, all reversible engines working Ixtweei. tins
same temperature limits T, and 2!, have the same efficiency;
that is. the offimo, ^.3 7 , ",. ., , . . ._
emcient than, a reversible engine B working between the same
temperatures, then A and B can be coupled together in such a
way as to produce available energy without a compensating loss
of availability.
Suppose the two engines A and B (Fig. 19) to take equal
quantities of heat Q^ from the source when running direct.
Then, since by hypothesis A is the more efficient,
and
Now let engine B be run reversed. It will take heat Q Z B from
R and deliver Q 1 to S. If A and B are coupled together, A
will run B reversed and deliver
in addition the work W A W B .
The source is unaffected since it
simultaneously receives heat Q l
and gives up heat Q r The re
frigerator, however, loses the
heat Q z a Q Z A , which is the
equivalent of the work W A W B
gained. We have, therefore, an
arrangement by which unavail
able energy in the form of heat
in the reservoir is transformed
into mechanical work. In other
words, by a selfacting process the available energy of the
system of bodies $, R, A, and B is increased. According to
the second law (Art. 36), such a result is impossible ; if such
a result were possible, power in any quantity could be obtained
from the heat stored in the atmosphere without consumption of
fuel.
The assumption that engine A is more efficient than the
reversible engine B leads to a result that experience has shown
to be impossible. We conclude, therefore, that the assumption
is not admissible and that engine A cannot be more efficient
than engine B. But if engine A is also reversible, B cannot
he morfi p.ffimfint than A. and it follows that all reversible
FIG. 19.
54 THE SEUUJNU JUAVV vx AiAi^w. "*'" ........ i.
engines between the same source and the same refrigerator are
equally efficient.
39. Determination of the Efficiency. Since the omoieney of
the reversible Carnot engine is independent of the properties of
the medium and depends upon the temperatures of .source and
refrigerator only, we have
i_7"_? = / ( 7\, 7 ! 2 ) , )
I <i/i
whence = 1  iy == JF(2i, 2^) ; ()
that is, the quotient ~f is some function of iho temperatures
Vi
2\ and T^. The form of this function in required.
So far, we have considered temperatures as given by a mer
cury or air thermometer. The different temperatures of a
series of bodies are indicated by sets of numbers which may
denote (1) the different lengths of a column of mercury or
(2) the different pressures of a mass of confined gas. These
sets may or may not precisely agree. Now there are other
ways in which such a set of numbers may be chosen. Suppose
we take several sources of heat /S^ M v >S' 3 , , *S', ( , whoso tem
peratures are t^ 2 , 8 , , t M as defined by the mercury or gas
scale, and let
*i>*a>8 >>*,.
If we use S l as a source and S z as a refrigerator, a reversible
engine will take Q^ from S l and deliver ^ to /S' 2 . vSinuo the
bodies S 1 and ^ 2 have definite temperatures T L and 7!,, what
ever the scale adopted, the function .F(T V 2! 2 ) lias some defi
nite value; therefore, from (2) the fraction ^ must have a
Vi
definite value, and consequently @ a has one and only one value.
If ^ 2 is used as a source and S 9 as a refrigerator, a second
engine taking Q z from # 2 will give up Q z to N 3 , and so on.
Starting with Q v we thus obtain a determinate set of values
O n f} Qfn TrrVvi <,!> ,^,,.,4. JJICIl J.I 1 .
JIT. 39] KELVIN'S ABSOLUTE SCALE 55
rlere we have a set of numbers suitable to define a scale of
leruperature. Starting with the heat Q 1 taken from the source
1, to each source there corresponds a number indicating the
leat that would be rejected to it if it were used as a refrigerator
n connection with S v If we choose these numbers to define a
iew scale, then denoting the new temperatures by
T T 7 T T
*!' ty *& ' 'i Jni
ve have
T^kQv T 2 = kQ T 3 = kQ B , ..., T n = kQ n ,
vhence follows
<?i_02_ &L m
m r/j ' " rn ' \ )
Li Jz 'n
Returning now to the quotient ~, we have at once
lence, using this new scale, the efficiency of the Carnot engine
s
uid the form of the function is determined.
The scale of temperatures arrived at from the investigation
)f Caruot's cycle was first proposed by Lord Kelvin in 1848,
i/nd is known as the absolute scale because it is independent of
he property of any substance. The scale is simply such that
my two temperatures on it are proportional to the quantities
)f heat absorbed and rejected by a reversible Carnot engine
vorking between these temperatures.
If in (5) we make Q z = 0, tj = 1 and T z = 0. If we con
lei ve a temperature lower than the zero on the absolute scale,
T T
hat is, if we assume a negative value for jT 2 , then ~  > 1,
be shown subsequently that tins absolute zero is precisely the
same as that derived from the reduction in pressure, of ;i perfect
gas, and that the new scale coincides with Hint of a ther
mometer using a perfect gas as a iluid.
40. Available Energy and Waste. Caruot\s ideal eyele gives
us a means of measuring the available energy of a system and
the waste due to an irreversible change of state. Suppose, that,
a quantity of heat A$ is absorbed by the system at a. tempera
ture T, and that we wish to find the part of this heat, that can
possibly be transformed into work. As we have seen, no device.
can transform a larger portion of A Q into work than the ideal
Carnot engine. If T Q is the lowest temperature that, can be
T 7 r
obtained for a refrigerator, the fraction  '" of A^> can be
transformed into work by a Carnot engine, and this is, thenfore,
the availability of A$ under the given conditions. The avail
able part of A$ is, therefore,
T T / f f r
* " /
I 
T
and the waste is A Q ^ .
The temperature T Q cannot be lower than that of surrounding
objects, i.e. the atmosphere;* for even if a refrigerator could
be found with a temperature lower than that of the, atmosphere,
it could not be maintained in that state. Ilene.e, the tempera
ture of the atmosphere imposes a natural limitation on the avail
ability of heat in the performance of work.
EXAMPLE. If the absolute temperature of aourc is 1 ()<)()" F. and llisii, of
the atmosphere is 520, the available energy in
1000  520 n . ,
= 0.48 of the. total i>ncr'y.
Therefore, for every 1000 B. t. u. received from tlu> source i.>t, more Lluui
480B.t.u. can by any means whatever bo transformed into wurk, and at
least 520 B. t. u. must be rendered unavailable
* Possibly under special conditions a refrigerator whoso t.mpcratun'. is p,r
mnently Mow that of the atmosphere may exist; ,,/. the water of the. o,,an
or of one of the great lakes.
associated with certain important irreversible processes.
(1) Conduction of Heat, Suppose a quantity of heat Q to
pass by conduction from a source at a temperature T to
another at lower temperature T z . At the original temperature
the available energy was
The same quantity of heat in the second source has the avail
able energy
The available energy is, therefore, decreased by the quantity
and the unavailable energy is increased by an equal amount.
(2) Irreversible Conversion of Work into Heat. A common
irreversible process is the conversion of Avork into heat in the
interior of a system through the agency of friction. Examples
are found in the flow of steam through nozzles and blades, and
in the Motional losses due to internal whirls and eddies in
fluids. Heat thus produced we shall denote by the symbol H,
reserving Q to denote heat brought into the system from outside.
If now within the system the small quantity of heat A.H" is
generated while the system remains at the temperature T, the
part of AJ^Tthat is available is
rrr rn / rn
A TT J ~
where, as usual, T denotes the lowest available temperature.
Of the work Jb.II expended in producing the heat A//, the
part
may therefore be recovered in the form of work. The re
mainder
is rendered unavailable.
05
To obtain the total increase of unavailable energy, when tho
quantity of heat .fiTis generated, the temperature of the. system
varying in the meantime, we sum the element of tin 1 , type just
obtained. Thus if the temperature risen from T t to '1\ during
the process, we have for the total waste
(3) Free expansion of a {/as. The waste due, to free expan
sion, as described in Art. 85, may be determined by returning
the gas to its initial state and observing the changes left in
outside bodies.
The compression indicated, by B' A' (Fig. 1(5') requires that
work W, represented by area B' A' A^ v lie supplied from an
outside body $ 2 . Another outside body /V., must receive from
the gas heat Q equivalent to the work W. The gas, the
the system S v has the same available energy as at first,, being
restored to its initial condition; system A' 2 has lost available
energy W=JQ; and system ;S' 3 has received energy JQ of
which only part is available. On the whole, therefore, there is
an increase of unavailable energy. The loss of availability duo
to the original irreversible expansion of the ga.s (system A'j) is
repaired in this system, but an equal loss is brought about in
systems S z and S y It can be shown that the, waste thus in
curred is given by an expression of the form 7, f'//
/ /
41. Entropy. The expressions for the increase of unavail
able energy derived under various conditions are alike in hav
ing TV the lowest temperature available for a refrigerator, as a
factor. It appears, therefore, that the unavailable, energy
changes with T ; the lower T () can be taken, the, smaller the
waste and the larger the fraction of the heat supplied that can
be transformed into work.
The other factor in the expression must necessarily, for the
sake of consistent units, have the form Q or J "^. To this
measure of the change in the unavailable energy of the system ;
an increase of entropy involves an increase of unavailable
energy, and vice versa. We may formally define entropy as
follows :
If, from any cause whatever, the unavailable energy of a system
is increased and if the increase be divided by T^ the lowest tem
perature available for a cold body, the quotient is the increase of
entropy of the system.
This definition requires close examination to obviate possible
misconception. The " system " spoken of may be either a
single substance, as the medium employed in a heat motor, or
it may be all the bodies taking part in the process. Now, ac
cording as we take one or the other of these viewpoints we get
a particular notion of the significance of the term entropy.
To illustrate this point, let us consider a simple example.
Suppose we have a fluid medium M and a source of heat S, as
described in connection with the Carnot engine. We may
direct our attention either to the system M alone or to the sys
tem M+ 8 composed of the medium and source. Let both M
and S be at the temperature T and suppose that at this tem
perature heat Q is transferred from S to M. This is the ideal
reversible transfer assumed in the description of the Carnot
engine. In receiving Q the system M has its available energy
f T\
increased by Q 1 1 9 j and its unavailable energy increased by
T Q \ J
Q~m = ^oTjfr; hence by the definition just given the entropy of
.system M is increased by j," At the same time system S has
lost the energy Q and, therefore, the unavailable energy Q ~ ;
hence the entropy of S is decreased by ~ It follows that the
change of entropy of the system M+ S is zero. As the result
of the reversible transfer of heat from 8 to M there is no
change in the unavailable energy of the large system S + M and
no change in the entropy of this system. Suppose now that sys
tem M. is again at temperature T, but that system S has a higher
temperature T', as must be the case in any actual transfer
60 THE SECOND LAW OF THERMODYNAMICS ICUAI'. iv
of heat. If now heat Q passes from /S y to M, the unavail
able energy of M is increased by <??}, as before, and tho increase
of entropy of system M is ~ The system H has, however,
m
lost the unavailable energy (?J, and its entropy has decreased
by  The system tf + M has had its unavailable energy in
creased by tlie amount (?  (?j = ^  ^) The irre
versible transfer has therefore resulted in a not loss of available
energy of this amount, and this degradation is accompanied by
an increase of entropy ^ r ~ The result hero obtained for
two systems may be applied to any number of systems.
When we apply the notion of increase of entropy to tin; sys
tem composed of all the bodies involved in a process, in other
words, an isolated system, we are led to the conception (hat
the increase of entropy measures the degradation of energy in
cident to the process. If we combine this notion with that
expressed by the second law, we arrive at the following im
portant principles :
1. Any process that can proceed of itself IK cteeoinpanii'd hi/ an
increase of the entropy of the system of bodies involved in (he.
process.
2. The direction of a process, physianl or eJievu'ral, tJnit own)'*
of itself is such as will bring about an increase, of entropy in the
system.
These principles lie at the foundation of the application of
thermodynamics to chemistry.
42. Second Definition of Entropy. While the conception of
entronv as thp. ffl.nf.nr f.lmt w,n^,,.,,
ART. 42] SECOND DEFINITION OF ENTROPY 61
VA
unavailable energy of this single system^involves an increase in
the entropy of the system^ but, as we have seen, degradation
does not necessarily follow, for the increase of unavailable
energy of M may be compensated by an equal loss in some
other system taking part in the process.
We now inquire by what means the unavailable energy of
the single system under consideration can be increased. There
are at least three ways that are suggested from the previous
discussion of available energy (Art. 40).
(1) If energy is added to the system in the form of heat, the
total energy of the system is increased, and consequently the
unavailable energy is increased. If the heat A Q is thus added
when the temperature of the system is T, the resulting increase
of unavailable energy is
If, as is generally the case, the temperature rises as heat is
added, we shall have for the increase
*. /
r '*2
", T'
(2) The unavailable energy may be increased by the con
version of work into heat through internal friction. As shown
in Art. 40 (2), the increase of unavailable energy from this
cause is
(3) If the parts of the system are not at the same tempera
ture, there will be an irreversible flow of heat from one part of
the system to another, and this will increase the unavailable
energy. We may remove this source of unavailable energy by
assuming that the system is at all times of uniform temperature
throughout, an assumption that is usually justifiable.
Neglecting this third effect, we have for the increase of un
available energy from state 1 to state 2,
62
whence by definition, the increase of entropy ia
Now while the actual change of the system from state 1 to state
2 may, and usually does, involve Motional effects, wo can r.nn
ceive of a hypothetical change in which thesis internal irroversi
ble effects are entirely absent and in which the, increase, of
unavailable energy is due entirely to the addition to the system
of heat from some external source. Denoting by Q r tho heat
thus added, we have for the increase of entropy involved in
this particular process the integral
The important question now arises: Does tho increase of en
tropy of the single system under consideration depend only
upon the initial and final states or upon the path connecting
the states? It is easily shown that the increase of entropy,
like the increase of energy, depends upon the initial and final
states only. For the change of energy is independent of the
path; therefore, the change of the unavailable part of the en
ergy, as determined by the constant temperature 7 r and the
temperatures 2\ and T z at the initial and final states, is also
independent of the path; therefore the change of entropy,
which is the change of unavailable energy divided by .7 r , is
also independent of the path. It follows that the integral
T 3 r\
J ~ has the same value whether taken along the path r
(Fig. 20) or any other reversible path r' . We may write, there
fore,
where S denotes a function of the coordinates of tho system
which, may be termed the entropy of the system. We have,
then, the following definition :
The change of entropy of a system correspond!,/!*/ to a clianye
of the system from state. 1 tn st.nto 9 , f o *7, ,1,, /;,,.;/ ,;/,,,.,, 7 C'^'J^'
ART.
JU.MJliS5UAJLj.LTi: UJT UJUAUSIUS
According to this more restricted conception, the entropy of
a system, like the energy, pressure, or temperature, is a magni
tude determined by the state of the system, and change of en
tropy has no necessary connection with degradation of energy.
It should be noted that entropy as thus denned is like energy
purely relative. We are never concerned with the absolute
value of the entropy of a system in a given state ; what is
desired is the change of entropy associated with a given change
of state. For convenience of calculation we assume the zero
of entropy to be the entropy of a system in some specified state.
Thus, in dealing with vapors we assume the zero of entropy to
be the entropy of a unit weight of liquid at C.
43. The Inequality of Clausius. If an actual irreversible
change be represented by the path i, Fig. 20 (assuming it to
be possible to give such a repre
sentation), a correct value of the
change cannot be obtained from
.. y, .7 Q
the integral ( '77 taken along
the path i. For as we have seen
a f T * 'I*
"i=J ' 7,7
T. L
V
FIG. 20.
where 2 is the increase of en
tropy due to the internal irre
versible changes. For the actual irreversible change we have,
therefore,
This is the inequality of Clausius.
44. Summary. To present the important principles of this
chapter in concise form and in logical order the following sum
mary is added.
1. Experience shows that heat energy is not completely
transformable into mechanical work. The ratio of the energy
2. Experience further shows that an irreversible process
always decreases the availability of a system.
3. The second law of thermodynamics asserts that tho avail
able energy of an isolated system cannot be increased by any
process that takes place of itself.
4. To gain a means'of measuring availability the ideal ( arnot
engine is introduced. By the aid of the second law it is shown
that no engine working between the same temperature limits
can have an efficiency greater than tho Carnot engine, and as a
consequence, that the efficiency of this engine is a function of
the temperature limits only.
5. By the introduction of Kelvin's absolute scale of tempera
ture the efficiency of the Carnot engine is found to be given by
T T
the fraction  1 2 .
T T
6. Having the efficiency fraction 1 2, the available part
A
of a given quantity of heat Q at temperature T is found to bo
$(l o] an( i the unavailable part, Q^
7. By special examples of irreversible processes it is found
that the expression for the loss of available energy in such pro
cesses has the general form ^T) or 7\ } ( ' '^ .
8. The factor Vi or j"^. which multiplied by 7', gives tho
increase of unavailable energy is called the incrcttM of cut ><>]> >/
of the system.
9. Two conceptions of entropy are possible: (a) If atten
tion be directed to all the bodies involved in a process, the
increase of entropy of the whole system of bodies measures tho
degradation of energy resulting from the process. (/>) If at
tention be directed to a single body, as a medium used in a heat
motor, the entropy of this simple system is merely a function
of the coordinates of the system.
10. The change of entropy of a simple system is given by
/ J.N JL/ OJ^XVY
tlie initial and final states. The value of this integral is inde
pendent of the path.
11. For an irreversible change of state the change of entropy
r 2 ' 2 dO
is greater than \ %.
J y\ T
45 Boltzmann's Interpretation of the Second Law. A very clear insight
into the real physical meaning of natural irreversible processes and of the
second law of thermodynamics is afforded by the researches of Boltzmann
and Planck. In this article it is possible to give merely a brief outline of
Boltzmanu's contribution ; for a complete exposition the reader is referred
to Professor Klein's admirable book, The Physical Significance of Entropy.*
According to the molecular theory, the ultimate particles of matter are
in a state of incessant motion, the character of the motion depending upon
the state of aggregation, solid, liquid, or gaseous. In a gas it is assumed
that a particle has a free path and moves along a straight line until it col
lides with another particle or with a restraining surface, as the wall of the
containing vessel. To the motion of particles as to the motion of masses
we may apply the conception of constraint or control. Thus, in the wave
motions that characterize sound, the motion of the particles that constitute
the mediums is in some degree controlled or ordered. The molecular
motion that constitutes heat is, on the other hand, wholly uncontrolled and
disordered. For any given particle of a gas all directions of motion are
equally possible and, therefore, equally probable; and the direction of
motion and velocity of any particle is independent of the motions of other
particles. In a volume of gas particles will be moving in all directions
with all possible velocities. However, because of the great number of par
ticles even in a small volume, the values of magnitudes that depend upon
the molecular motion, such as pressure and temperature, remain constant
notwithstanding the haphazard character of the molecular motion.
According to Boltzmann, there is apparently a universal tendency
toward the disordered motion that characterizes heat. A motion that is
in any degree ordered or controlled tends to become disordered. Thus, as
sound waves die out the uniform motion of the particles in the wave
changes to disordered motion, and the energy of sound is transformed into
heat energy. The relative motion of two bodies in contact is retarded by
friction, and the work of overcoming friction is transformed into heat; that
is, the constrained motion of the particles in the mass gradually changes
to the disordered motion of heat. Since the energy of disordered molecular
motion is necessarily less available for direction into any required channel
than the energy of constrained or controlled motion, it follows that a change
from a less probable state of controlled motion to a more probable state of
/3ic.m..3ni.n>3 if.rvJi/i.. ; ,.!> r,,.,^ ,./.iv^ a nnnrli firm nf (TVOflt.PV El.Vni1il.hlft fillftrCW
to a condition of less available energy. II<mce, the statement of the. nal ural
tendency toward disordered motion is iu reality a broad statement <l the
second law of thermodynamics.
From the preceding considerations a physical interprelation of entropy
is readily deduced. A system of itself passes from a less probable, to a
more probable state ; that is, to a state of mure disordered moleeular motion.
The entropy of the system during the change must, inornaso. ^Therefore,
the entropy of the system may bo associated with the. probability of tins
state of the system. From the laws of probability, 1'lanek has shown that
the entropy is proportional to the logarithm of the probability of th .slate.
The following quotations from Prof. Klein's book indicate in some degree,
the significance of this conception of entropy.
" Growth of entropy is a passage from a somewhat regulated to a less
regulated state."
"Entropy is a universal measure of the disonli'r in the mass points of a
system."
"Entropy is a universal measure of tho spontaneity wit.li whie.h a system
acts when it is free to change."
"Growth of entropy is a passage from a concentrated (.<> a disl.ribul.cil
condition of energy; energy originally concentrated variously in t.lm system
is finally scattered uniformly in said system. In this aggregate aspect, it is
a passage from variety to uniformity."
EXERCISES
1. If a source of heat has an absolute tomporaturo of MOO" F. and tho
lowest available temperature is 525 F., what fraction of tho beat drawn
from the source is available ?
2. In a boiler 10,000 B. t. n. pass from the hot gases of the. fnrnat'e, tin;
temperature of which is 2500 F., through the boiler shell into water at a
temperature of 330 F. If the lowest available temperature is 80" F., iind
the loss of available energy.
3. Show how the result of Ex. 2 suggests tho superior dlhuency of the
gas engine compared with the steam engine.
4. Point out the loss of available energy when heat Hows from steam in a
radiator at a temperature of 225 into a room at 70". J)evisu a system of
heating that would obviate this loss.
5. A mass of water weighing 60 Ib. at a temperature of 70" F. is churned
by a paddle wheel until the temperature rises to 120. Find the increase, of
entropy, and the loss of available energy. Take the spec.itu heat of water
as 1.
6. In the demonstration of Garnet's principle, Art. 158, ;iKsnme the two
engines A and B to do the same work W. Then show that if. emrine A
ART. 45] LITERATURE ON THE SECOND LAW 67
REFERENCES
REVERSIBLE AND IRREVERSIBLE PROCESSES
Planck : Treatise on Thermodynamics, Ogg's trans., 82.
Bryan : Thermodynamics, 34, 40.
Klein : Physical Significance of Entropy, 29.
Chwolsou : Lehrbuch der Physik 3, 443.
Parker : Elementary Thermodynamics, 105.
THIS SECOND LAW. ENTROPY
Sudi Carnot: Reflections on the Motive Power of Heat. Translated by
Tlrarston.
Claxisius : Mechanical Theory of Heat.
Rankine: Phil. Mag. (4) 4. 1852.
Thomson : Phil. Mag. (4) 4. 1852.
Franklin : Phys. Rev. 30, 770. 1910.
Lorenz : Teehnische Warmelehre, 104.
Chwoison : Lehrbuch der Physik 3, 485, 497.
Bryan : Thermodynamics, 43, 57.
Preston : Theory of Heat, 025.
Klein : Physical Significance of Entropy.
Magie : The Second Law of Thermodynamics (contains Garnet's " Reflec
tions" and the discussions of Clausius and Thomson).
Planck: Treatise on Thermodynamics (Ogg), 86.
Parker : Elementary Thermodynamics, 104.
CHAPTER V
TEMPERATURE ENTROPY REPRESENTATION
46. Entropy as a Coordinate. It was shown in Art. 12 that
the entropy of a system measured from an arbitrary /ero is
dependent only upon the state of the system ; that, in, tho
entropy is a function of the coordinates of the system. It
follows that the entropy itself may bo included amon^ tho
coordinates used to define a system. We have, therefore, live
coordinates, namely, p, v, T, u, and 8, that may bo thus used.
From these five, ten pairs may be selected, and the change of
state of a system may be represented by ten different curves on
ten different planes. Of these possible graphical representa
tions two are of special importance : (1) representation on tho
jpFplane, because the area between the curve and /'axis repre
sents the external work done by the system; ( L 2) representa
tion on the T$plane, because with certain restrictions tho area
under the curve represents the heat absorbed, by the system
from external sources. Graphical representations on the, ^f
plane have been considered in Art. 82. This chapter will be
devoted chiefly to representations on the 2Wplano.
From the second definition of entropy, we have
%> CO
 I J^
from which relation we obtain at once the differential forms
and TdS=dQ.
Let the curve ATt
But from (3) this integral is the heat Q lz absorbed by the
system from external sources during the change of state. It
follows that the area between
T
the curve AB and the axis OS
represents graphically the heat
absorbed along the path AB.
One most important restriction
must, however, be observed. In
defining entropy by means of
equation (1) it was expressly
stated that the change of state ^ J?i
must not involve any internal
irreversible effects. If such effects are present, the equation
for the change of entropy is
where S denotes the increase of entropy due to internal
processes, conduction between the parts of the system, trans
C T tdO
formation of work into heat through friction, etc., and J r 
is the increase of entropy due to the absorption of heat from
external bodies. From (4) it follows that in this case
whence
* a _ a
~ * v
dQ<TdS,
(5)
or the heat absorbed from outside is less than the area between
the 2Wcurve and the #axis. This area therefore may be taken
as representing the heat absorbed by the system when, and only
when, the change of state involves no irreversible effects. Neglect
of this restriction has led to many errors.
47. Isothermals and Adiabatics. If the temperature of the
system remains constant during the change of state, the
M/V
**
~
D
A'
FIG. 22.
In this case we have merely to divide the heat added to the
system (assuming, of course, that the change of state is revers
ible) by the constant tempera
ture T, and the quotient is the
change of entropy.
If the state point passes from
B to A, that is, so as to de
crease the entropy, the area
A^ABB^ represents heat re
jected by the system to outside
bodies.
For an adiabatie change of
state, dQ = ; hence from (1) $, = h\ and tlio adiabatie line
on the 5Wplane, if the change of .state involves no irreversible
effects, is a straight line parallel to the .'/axis, as (.11) ( Kig. IW).
If the statepoint moves from Oio D, indicating a decrease of tem
perature, external work is done by the system, and tlio ehange
of state is an adiabatie expansion. If tlio point moves upward
from D to (7 the change of state is an adiabatie compression.
48. The Curve of Heating and Cooling. From the equation
do
G = * ,
which defines the specific heat of a substance, we have
(1)
Substituting this expression for ity in (1), Art. 4(1, we got for a
reversible process
.
T
If the specific heat c is constant during tlio change of state,
we have for the change of entropy of unit weight of the sub
stance
For the weight If,
(3 a)
If, however, c is variable, it can usually be expressed as a func
tion of the temperature ; that is, we can write
whence
T, . T
(4)
The integration can be effected when the function /(2 1 ) is
known.
EXAMPLK. Let tho specific heat o'f a substance be given by the relation
c = a + W = a + &(r450.G);
wo have then
r T <1 T rT
s a  8 L = (a  459.0 b) i  ~~  + I \ dT
J r t 1 J ?*[
= (  459.0 6) log, ^ +b(T s  J 1 ,).
^i
The general form of the curve that represents Eq. (3)
is shown in Fig. 28. This curve
represents the ordinary pro
cess of heating* a body or sub
stance, as the healing of water
iu a boiler or metal in a furnace.
It is called by some writers the
polytropic curve. The subtan
gent of the curve is constant
and numerically equal to the
specific heat. Thus from the
F E
FIG. 23.
figure we have
~~ ~d^~ dT
It follows that the smaller the value of c, the greater the slope
of the curve.
The isothermal and adiabatic curves (Fig. 22) ms?y be con
. ^f 4T>^ V,/>n'finn. oiirl nn nil nor miTVP.. T^OT
72 TEMPERATUJRJfi
niAi>. v
Cases may arise in which tluj
slope of the 2/S'ourvo is nega
tive, as sliown in Fig. 24. In
such cases abstraction of lioat i.s
accompanied by a rise in tem
perature or vine verxa. Evidently
the speciJic heat ff1 niu.st bo
it ,L
negative, as is indicated geo
metrically by tho negative sub
tangent. Examples will be shown in the compression of air
in the ordinary air compressor, and in the expansion of dry
saturated steam with the provision that it remains dry during
the expansion.
:iy bo
series
49. Cycle Processes. Since any reversible process m
sliown by a curve in ^coordinates, it follows that a
of such processes forming a
closed cycle may be repre
sented by a closed figure on
the 2Eplane. In Fig. 25 is
shown such a cycle composed
of two polytropics AB and
DE, an isothermal J3C, and
two adiabatics CD and HA.
In any such cycle the area
included by the cycle repre
sents the net heat added to
(or abstracted from) the work
ing fluid during the cycle process. Assuming the cycle to bo
traversed in the clockwise sense, we have
ab = area
Q bc = area 1 BOO V
= ABODE.
the cycle is traversed in the counterclockwise sense, we have
it from the first law, Q is the heat transformed into work;
nee for the direct cycle
area AS ODE = Q = AW,
d for the reversed cycle
area AB ODE = Q**AW.
This reasoning evidently holds for any number of processes,
d therefore for a reversible
>sed cycle of any form. Thus
? the cycle shown in Fig. 26,
> have
area F= Q = AW,
area F= =
3ording as the cycle is traversed
the clockwise or counter clock
se sense.  FlG _ 2(1>
tn later developments it will
quently be necessary to show cycle processes on the iZWplane.
)0. The Rectangular Cycle. When the curves representing
s four processes of the Carnot cycle are transferred to the
2%'plane, the cycle becomes the
simple rectangle ABQD, Fig. 27.
The area A^ABB^ represents the
heat Q 1 absorbed by the medium
from the source during the iso
thermal expansion AB, and the area
B^CDAy the heat Q z rejected to the
refrigerator during the isothermal
compression CD. The lines BO
and DA represent, respectively, the
adiabatic expansion and the adia
FIG. 27. batic compression.
Tn
B
74 TEMPERATURE ENTROPY RKPUKSKNTATION [CHAP, v
From the geometry of the figure, we have
A IV T, T,,
whence f] = ^ ~ ~rn "
as already deduced in Art. 89.
When the cycle is traversed in the counterclockwise souse,
the heat Q 2 is received by the medium from the cold body during
the isothermal expansion J9(7, and the larger amount of beat Q t
is rejected to the hot body during isothermal compression JiA.
The difference $ 2 ^ == J. T7 represented K v ^ 10 t! y^ u ari!Jl
is the work that must be done on the medium, and must there
fore be furnished from external sources.
The reversed heat engine may be used either as a rof rigerating
machine or as a warming machine. In the lirst case Uie space,
to be cooled acts as the source and delivers then heat Q z = area
A 1 DCB 1 to the medium. In the second case the space, to bo
warmed receives the heat Q 1 = area B l BAA l from the medium.
51. Internal Frictional Processes. Referring to Art. 4U, the
increase of entropy when heat is generated in the interior of a
system is seen to be
2 1 ~ ^ j\ ~T J r ~T r '
If $=0, that is, if no heat enters the system from outside
sources, the increase of entropy is
and is due entirely to the generation of beat in the interior of
the system. If it be assumed that this process is steady, so that
the system at every instant is approximately in thermal equi
librium, the usual graphical representation may be applied to
(2), and the area under the 2^curve will in this ease repre
sent not the hfifl.t bvnnoht. infn +!IQ c.irfi 1^,^ 4.1,., 1 4. 77
A
FIG. 28.
int A (Fig. 28) lias its pressure decreased in passing along the
zzle, and as a result the temperature likewise falls. The
Dceas is adiabatic, that is, no heat
received from external bodies;
nee, if there were no internal
ction, the drop in temperature
iuld be indicated by a motion of
3 statepoint along AA r But
irk is expended in overcoming
3 friction between the fluid and
rale wall. This work is neces
:ily transformed into heat, which
retained by the fluid. It follows
it there is an increase of entropy, as indicated by the curve AB.
om (2) the heat generated is represented by the area A 1 AS r
52. Cycles with Irreversible Adiabatics. In certain cases the
>sed cycle of operations of a heat motor may contain an adia
tic irreversible process, the irreversibility arising either from
:ernal generation of heat or from the free expansion or wire
awing of the working fluid. Even if it is possible to draw
a T&curve representing such
a process, the area under that
curve does not represent the
heat entering the system from
an external source. Hence
some care is required to inter
pret properly the graphical
representations of cycles with
such irreversible parts.
In the cycle shown in Fig. 29,
suppose the process SO to be
FlG> 29 ' an irreversible adiabatic, the
ler parts of the cycle being reversible. Since AB is revers
.e, the heat absorbed in passing from A to B is represented by
3 area A 1 ABB r Likewise area C 1 ODA 1 represents the heat
lected by the system in changing state from to D. The
. v
76 TEMPERATURE ENTROPY REPRESENTATION I
process DA is adiabatic, hence $ llB Oj and by hypothesis
= 0. The value of 2# for the cycle is, therein,
V((6 ~i~ VJ " ^' 1 ' 1 r T7 r/r
= area ABKD  area /^M ( r
The energy equation applies to any process, reversible or
irreversible. Therefore for this
cycle, as for those previously
considered, we have
FIG. 30.
It appears, therefore, that, the
work derived is less by tho area
B l KOO i than il, would have
been il: tho reversible adiabalie
.3 BE had been followed.
For the reversed, cycle
(Fig. r'50) we have as the
work required from external sources
W=J(Qaa+ Q^ = ~ aron D V DAA 1 + area ./^/iOf^
Comparing this cycle with the cycle A.E<JD having the re.vers
ible adiabatic AZ7, it is seen that the heat absorbed from the
cold body is smaller by the heat represented by the area
A^EBBy while the work required to drive the machine is
greater by an equal amount. In every case tho irreversible
process results in a reduction of the useful effect.
53. Heat Content. Since the quantities p, 7', .7 r , H, and s are,
function of the state of a system only, it follows that any com
bination of these quantities is likewise a function of the state
only. For example, let
(V)
r. 53]
HEAT CONTENT
77
tentials, and are used in certain, applications of thermo
namics to physics and chemistry. The function I has use
L applications in technical thermodynamics.
To gain a physical meaning for the function I, let us consider
3 process of heating a substance at constant pressure. If t/p
, and p l denote the initial energy, volume, and pressure,
jpectively, and 7 2 , V y and p z the final values of the same
jrdinates, we have from the energy equation
since p z = p 1
= A[U z ~C7 l
tat is, the change in I is equal to the heat added to the sys
n during a change of state at constant pressure. For this
ison I is called the heat con
it of the system at constant I
essure, or, more briefly, the
ieat content."
In some subsequent investiga
ns, especially those relating to
3 How of fluids, it will be con
:iient to use / and S as the in
pendent variables and to repre
it changes of state by curves on Q .
i /xSplane. The great advantage
the /^representation over the
'representation lies in the fact that in the former quantities
heat are represented by linear segments, while in the latter,
we have seen, they are represented by areas. A reversible
.abatic on the J&plane is a vertical line, as BQ (Fig. 31).
t in this diagram segment BO represents a quantity of heat
tead of a change of temperature,,
FIG. 31.
2. Assuming that the specific heat of water is constant, c 1, plot uu
crosssection paper the rScuwo reproHonting the heating tif water from
32 to 212.
3. Langen's formula for the .specific heat of CO., ut constant pressure in
c = 0.195 4 0.000066 t. Find tlm increase, of entropy when CO., is healed
a^t constant pressure from 500 to 2000 F. ; aim) tlm heat, absorbed.
4. A direct motor operates on a rectangular cycle between temperature,
limits ^=840 and T z = 000 and reeeivos from the. source 'J(K) 15. t,. u. per
minute. Find the efficiency, and the work don<>, per ininuti 1 .
5. A reversed motor, rectangular cycle., operates between temperature
limits of 10 and 130, and receives liOO It. t. u. per minnln from the cold
body. Find the heat rejected to the hot body, and the. horsepower required
to drive the motor.
6. A direct motor, rectangular cycle, operating between temperatures
2\ = 900 and T 2 = 080, takes 1000 B. t. u. from a boiler. The heat rejected
is delivered to a building for heating purposes. This direct, motor driven
a reversed motor which operates on a ra'.tangnlar cycle between tempera
tures r 4 = 460 (temperature of outside, atmosphere) and 7'., : <>00. The
reversed motor takes heat from tho atmosphere and rejects heat io the.
building. Find the total heat delivered to tho building pur 1000 It. t. u.
taken from the boiler.
7. In the vaporization of water at atmospheric pressure, the. temperature
remains constant at 212 F., and 970.d B. t. vt. arc required for the process.
Find the increase of entropy.
8. The expression for the energy U for a given weight of a permanent
pV
gas is _  + U m where k and U are constants. Derive an expression for
the heat content I of the gas.
9. Combine the energy equation dQ = AdU + AjxlV \\illi tho deiining
equation I = A ( U + p 7) and show that d I = d Q + A } 'dp.
REFERENCES
USE OF TEMPEUATUUEENTHOPY COOUPINATKS
Berry : The TemperatureEntropy Diagram.
Sankey : The Energy Diagram.
Boulvin : The Entropy Diagram.
Swinburne : Entropy.
USE OF HEAT CONTENT AND ENTROPY AS COOUDINATKS
Berry : The TemperatureEntropy Diagram, 127.
Mollier: Zeit. des Verein. deutscher Ing. 48 271.
CHAPTER YI
GENERAL EQUATIONS OF THERMODYNAMICS
54. Fundamental Differentials. The introduction of the
entropy s and the functions i, F, and $ (Art. 52) permits the
derivation of a large number of relations between various
thermodynamic magnitudes. While the number of formulas
that can be thus derived is almost unlimited, we shall intro
duce in the present chapter only those that will prove useful
in the subsequent study of the properties of various heat media.
In this article we shall by simple transformations express the
differentials of u, i, F, and <J> in terms of the differentials of the
variables jp, v, T, and s.
We have to start with the fundamental energy equation
dq = A(du + pdv), (1)
and for a reversible process the relation
dq=Tds. (2)
Combining (1) and (2), we obtain
T
du = ds~pdv, (3)
A
an equation that gives u as a function of the independent varia
bles s and v.
From the defining equation
we have
di = Adu + Ad (pv)
= Adu + Apdv + Avdp.
Introducing the expression for Adu given by (8), we get
80 GENERAL EQUATIONS OF THERMODYNAMICS [CHAP, vi
Here i is given as a function of s and^> as independent
variables.
Likewise, from the relation
I=Au Ts,
dF '= Adu  Tds  sdT ';
whence from (3)
 dF = sdT + Apdv. (5)
Finally, from the defining relation
<E> = Au + Apv Ts,
d$> = Adu + Ad(pv)  d(Ts^)
= Tds Apdv + Apdv + Avdp  Tds sdT;
or d = Avdp  sdT. (6)
Now since the functions w, i, JF, and <& depend on the state
only, their differentials are exact ; hence the second members
of (3), (4), (5), and (6) are all exact differentials.
Certain results can be deduced at once from the differential
equations (3)(6). For example, from (6), if a system changes
state reversibly under constant pressure and at constant tem
perature, the function $ remains constant. Again from (5), if
a change of state occurs at constant temperature, the external
work clone is equal to the decrease of the function F. These
results are important in the application of thermodynamics to
chemistry.
55. The Thermodynamic Relations. The fact that the dif
ferentials in (3), (4), (5), and (6) of the last article are exact
gives a means of deriving four important relations. In (3)
we have u expressed as a function of the variables s and v;
that is,
M =/(*> v),
whence du = ds+~dv.
ds 3v
Comparing this symbolic equation with (3), it appears that
dv\dsj Bs\dv)'
that is,
Adv^ J r)8
(If) =*() (A)
The subscripts denote the variables held constant during the
differentiations indicated.
Relation (A) may be expressed in words as follows : The
rate of increase of temperature with respect to the volume
along an isentropic is equal to A times the rate of decrease of
the pressure with respect to the entropy along a constant vol
ume curve. That is, if the reversible change of state be repre
sented by curves, one on the 2Vplane, another on the jpsplane,
the slope of the second curve at a point representing a given
state is A times the slope of the first curve at the point that
represents the same state.
In (4) we have s and p as the independent variables ; and
since di is exact, the necessary condition of exactness gives
dp
dp. \dsp
That is, the rate of increase of temperature with respect to the
pressure in adiabatic change is A times the rate of increase of
volume with respect to the entropy in a constantpressure
change.
Since in (5) dF is an exact differential, we have
From (6), likewise, we obtain
The relations given by (A), (B), (C), and (D) are known
as Maxwell's thermodynamic relations. They hold for all
(C) and (D) by means of the relation Us = ~ \, aro usolul :
(CV)
dpT \<Y/'v
56. General Differential Equations. From tho thenno
dynamic relations certain useful general equations arc at oneo
deduced. As in Art. 19, we may write
according as T and v or 27 and ^ are taken a.s tho indopondont
variables. Now replacing (^\ and ("^ ^)} r '' ml. ( ' ; ,i r 
\o yj, yd /. y^ ;
spectively, and (~2 ) and (2 ) by tho exprcission.s i^ivon in
VSu/y \dpjy
((7') and (i>') 5 these equations become, rospectivoly,
\dv, (I)
Eliminating dT between (I) and (II), a third equation having
p and v as the variables is obtained. Thus
Two other important equations may be derived from (I) and
(II). Since from the energy equation
du = Jdq pdv,
we have from (I)
di = c p dT A  tjp. (V)
The general equations (I)(V) hold for reversible changes
I state. The partial derivatives involved may be found from
he characteristic equation of the substance under investi
ation.
As an application of (IV), we may derive expressions for the
lange of energy (a) of a gas that follows the law pv = BT ;
b") of a gas that obeys van der Waals' equation
ence
(a) From the characteristic equation pv J3T, we have
*\ =*.
dT) v v '
/"ftrji
du = Jc v dT+( ^
\ v
= Jc v dT,
rid u z u 1 =
=
ssuming c v to be a constant.
(5) From van der Waals' equation, we have
B
dTj v vb'
, r^rf d P\
'henoe ^^
'rom (IV), we have, therefore,
du = Jc,dT+dv,
v 2
rhence, assuming again that c v is constant,
It appears, therefore, that if a gas follows the law jw = IW\ the
energy is a function of the temperature only, while ii il. follows
van der Waals' law, the energy depends upon the temperature
and volume; in other words, the gun possesses 1)uth kinetie
and potential energy. .
57. Additional Thermodynamic Formulas. For certain in
vestigations of imperfect gases, especially the superheated
vapors, certain formulas involving the specific. heals <> and
c v are useful. The most important of these urn (VI;, (VII),
and (VIII) following.
Since du is an exact differential, wo obtain, upon applying
the criterion of exactness to (IV),
whence = A (VI)
\dvJ T \dT*J v
In a similar manner, since di is exact, we have from (V)
Equations (VI) and (VII) may be used to show tho depend
ence of the specific heats c v and c v upon tho pressure and vol
ume. For example, if a gas follows the equation pv BT we
find  = 0, whence from (VII) <1 \ = 0, and it follows that
dpji'
c p does not depend upon the pressure, though it may vary with
the temperature. Also _JL=: 0, whence it follows that e v does
9T*
not vary with the volume. The student may show that the
second result follows from van der Waals' equation or from any
equation in which p and T appear in the first degree only.
If, however, we take the characteristic equation
hich applies to superheated steam, we obtain
hence (?**}  A <n + V)(l +
\ap/r T n+1
itegrating this with 27 constant, we have
here 0(27), an arbitrary function of T, is the constant of inte
ation. In this case it is seen that c p is a function of both T
Lcl p.
An expression for c p c v is obtained as follows : Writing the
itropy s as a function of p and v, we have
d8 = dp + ~dv.
dp dv
bis, combined with the familiar equation
Adu = Tds Apdv,
3o flo
ves the equation Adu = T dp + (T  Ap~)dv.
nee du is an exact differential, we have
L( T ?\=*(T&
dv\ dp} dp\ dv
at is,
dv dp dvdp dp dv dpdv
, dTds dTds A f ^
tience  = A. (1)
dv dp dp dv
:om the definition of specific heat, we have
G= ^=T^
dT dT"
.d if we express both s and T as functions of p and v, this re
fcion becomes
^dp+^dv
Q ~T dp dV _ (2)
** dT 7 , dT J ' ^ }
If p is constant, c= c v and dp = ; lunico wo huvu from (^'2 i
1
c v ~TlJL.
~dt>
Likewise, when v is constant we; have
00
c ~
(4)
Combining (3) and (4), we obtain
dv dp t
Making use of (1), we get finally
c n o
EXAMPLE. For the character! stio uquatiou;> = 7J7', \vn huvo
= ~ dlL = Ji
dT p' dT~'v'
Therefore, from (8),
c p  Cv =A^r ==AB .BT =
pv p v
That is the difference Cp  Cv is constant (JV( , U jf an(1 ith t ,
temperature.
Taking Zeuuer's equation for superlieatod atoani, vi/,:
pv = BT~ Cp",
we have j?l _ ^ Jg. .B_
32 1 j' 32 7 nC>''HV
whence c p  c v = ^5 ^H__ _ y ^ JIT __
n Cp n + p v ( n _ 1 )c yl ^. 7/7"
In this case, therefore, the difference c ,  c v varies with 7' and p.
By varies substitutions and transformations wo c,mld add
Sr md " finitel y to this "at of thermodynamic fc.rnmlas.
However the eight formulas (I)(VIII) arc suflioiont fop the
mvestifirationof nnnriwon ^.i... ,,
.apter T must necessarily denote the temperature defined by
e Kelvin absolute scale. The coincidence of this scale with
e perfect gas scale will be shown in the next chapter.
58 Equilibrium. For irreversible processes the equations of Art. 54
ist be replaced by inequalities. Since for an irreversible process,
dq<Tds, (])
[. (3), (4), (5), and (6) of Art. 54 become, respectively,
Adu<Tds  Apdv, (2)
di < Tds + Avdp, (3)
dF>sdT + Apdu, (4)
d$<AvdpsdT. (5)
From the inequalities (4), (5), and (1) the following conclusions are at
ce apparent :
1. If the temperature and volume of a system remain constant, then from
), rZZ' T <0. That is, tJF must be negative, and any change in the system,
ist result in a decrease of the function F.
2. If the temperature and pressure remain constant, as in fusion, and
porization, theii from (5), d$ < 0. Hence any change in the system must
such as to decrease the function cfr,
3. If the system be isolated, q = 0, and from (1), tfs>0. Hence in an
dated system any change must result in an increase of entropy.
The conditions of equilibrium are readily deduced from these conclusions,
ider the condition of constant T and v, change is possible so long as F
a decrease. When F becomes a minimum, no further change is possible
d the system is in stable equilibrium. Likewise, with T and p constant,
ible equilibrium is attained when the function $ is a minimum.
The fiinctions F and $ are evidently analogous to the potential function
in mechanics. A mechanical system is in a state of equilibrium when
3 potential energy is a minimum, and similarly a thermodynamic system
in equilibrium when either the function F or the function $ is a minimum.
>r this reason F and <$ are called thermodynamic potentials.
By the use of thermodynamic potentials, problems relating to fusion,
porization, solution, chemical equilibrium, etc., are attacked and solved.
EXERCISES
1. From (V) derive an expression for the change of the heat content i
len a gas following the law/w = BT changes state.
2. If the gas obeys van der Waal's law, find an expression for the
ange of the heat content i.
3. Apply equations (II), (IV), and (V) to the characteristic equation
superheated steam,
GENERAL EQUATIONS OF THERMODYNAMICS [CHAP, vi
4. Callendar has proposed for superheated steam the equation
Apply (VII) to this equation and show that c is a function of p and T.
5. Give geometrical interpretations of the thermodynamic relations
(C) and (D).
6. From (I) and (II) derive expressions for dq and also for y for a
gas following the law pv  BT. Show that the expressions for ^ are
iutegrable, while those for dq are not.
7. Derive (VI) and (VII) by the following method: Divide both mem
bers of (I) and (II) by T, and knowing that ^ = ds is exact, apply the
criterion of exactness to the resulting differentials.
8. Deduce the following relation between the specific heats and the
functions F and
r v rr& F n\
(a) c.= r_; (6) c,=
9. Using temperatureentropy coordinates, deduce a system of graphical
representation for the three magnitudes Q, U 2 U v and W that appear in
the energy equation.
Suggestion. Through the point representing one state draw an iso
dynamic, through the other point a constant volume curve.
REFERENCES
GENERAL EQUATIONS OF THERMODYNAMICS
Bryan: Thermodynamics, 107.
Preston : Theory of Heat, 637.
Chwolson : Lehrbuch der Physik 3, 466, 505.
Buckingham : Theory of Thermodynamics, 117.
Parker: Elementary Thermodynamics, 239.
EQUILIBRIUM. THERMODYNAMIC POTENTIALS
Planck: Treatise on Thermodynamics (Ogg), 115.
Gibbs : Equilibrium of Heterogeneous Substances.
CHAPTER VII
PROPERTIES OF GASES
59. The Permanent Gases. The term "permanent gas"
survives from an earlier period, when it was applied to a series
of gaseous substances which supposedly could not by any
means be changed into the liquid or solid state. The recent
experimental researches of Pictet and Cailletet, of Wroblewski,
Olszowski, and others have shown that, in this sense of the
term, there are no permanent gases. At sufficiently low tem
peratures all known gases can be reduced to the liquid state.
The following are the temperatures of liquefaction of the more
common gases at atmospheric pressure :
Atmospheric air  192.2 C.
Nitrogen  193.1 C.
Oxygen  182.5 C.
Hydrogen  252.5 C.
Helium 263.9C.
It appears, therefore, that the socalled permanent gases are
in reality superheated vapors far removed from temperature of
condensation. We shall understand the term " permanent gas "
to mean, therefore, a gas that is liquefied with difficulty and
that obeys very closely the BoyleGay Lussac law. Gases that
show considerable deviations from this law because they lie
relatively near the condensation limit will be known as super
heated vapors.
60. Experimental Laws. The permanent gases, at the pres
sures usually employed, obey quite exactly the laws of Boyle
and Charles, namely :
1. Boyle's Law. At constant temperature, the volume of a
given weiaht of aas varies inversely as the pressure.
z. v/nanes
sure
, of a gas is proportional to the change of temperature.
By the combination of these laws the characteristic equation
pv = BT is deduced. (See Art. 14.) In this equation T
denotes absolute temperature on the scale 'of the gas ^ther
mometer, and not necessarily temperature on the Kelvin
absolute scale.
The classic experiment of Joule showed that permanent gases
obey very nearly a third law, namely :
3. Joule's Law. The intrinsic energy of a permanent gas is
independent of the volume of the gas and depends upon the temper
ature only. In other words, the intrinsic energy of a gas is all
the kinetic form.
Joule established this law by the following experiment. Two
vessels, a and 6, Fig. 32, connected by a tube were immersed in
a bath of water. In one vessel air was compressed to a pres
sure of 22 atmospheres, the other
vessel was exhausted. The tem
perature of the water was taken
by a very sensitive thermometer.
A stopcock G in the connecting
tube was then opened, permit
ting the air to rush from a to
J, and after equilibrium was es
tablished the temperature of the
No change of temperature could be
FIG. 32.
water was again read,
detected.
From the conditions of the experiment no work external to
the vessels a and 6 was done by the gas ; and since the water
remained at the same temperature, no heat passed into the gas
from the water. Consequently, the internal energy of the air
was the same after the expansion into the vessel 5 as before.
Now if the increase of volume had required the expenditure of
internal work, i.e. work to force the molecules apart against
their mutual attractions, that work must necessarily have come
from the internal kinetic energy of the gas, and as a result the
temperature would have been lowered. As the temperature
remained constant, it is to be inferred that no such internal
was required. .a. gas nas, uiereiore, no appreciaoie inter
nal potential energy ; its energy is entirely kinetic and depends
upon the temperature only.
Joule's law may be expressed symbolically by the relations :
The more accurate porousplug experiments of Joule and
Lord Kelvin showed that all gases deviate more or less from
Joule's law. In the case of the socalled permanent gases, air,
hydrogen, etc., the deviation was slight though measurable ; but
with the gases more easily liquefied, the deviations were more
marked. The explanation of these deviations is not difficult
when the true nature of a gas is considered. Presumably
the molecules of a gas act on each other with certain forces, the
magnitudes of which depend upon the distances between the
molecules. When the gas is highly rarefied, that is, when it is
far removed from the liquid state, the molecular forces are van
ish.in.gly small ; but when the gas is brought nearer the liquid
state by increasing the pressure and lowering the temperature,
the molecules are brought closer together and the molecular
forces are no longer negligible. The gas in this state possesses
appreciable potential energy and the deviation from Joule's
law is considerable.
61 . Comparison of Temperature Scales. Joule's law furnishes
a means of comparing the two temperature scales that have
been introduced: the scale of the gas thermometer and the
Kelvin absolute scale.
Since the intrinsic energy u is, in general, a function of T and
v, we may write the symbolic equation
CD
But from the general equation (IV), Art. 56,
&\ p~}dv (2)
dJ J
paring (1) and (2), we obtain
For a gas that obeys Joule's law  = 0, wlienco from (H)
& <3u
=. (4)
'A y ^ ;
Equation (4) is, however, precisely the equation that expresses
Charles' law when T is taken as the absolute temperature on
the scale of the constant volume gas thermometer. Thus, if
the change of pressure is proportional to the change of tem
perature when the volume remains constant, we have, taking jw
as the pressure at C.,
1=1
It follows that the value of T is the same whether taken
on the Kelvin absolute scale or on the scale of a constant
volume gas thermometer, provided the gas strictly obeys the
laws of Boyle and Joule. The fact that any actual gas, as
air or nitrogen, does not obey these laws exactly makes
the scale of the actual gas thermometer deviate slightly from
the scale of the ideal Kelvin thermometer. From the porous
plug experiments of Joule and Kelvin, Rowland has made a
comparison between the Kelvin scale and the scale of the air
thermometer.
62. Numerical Value of B. The value of the constant B for
a given gas can be determined from the values of p, v, and T be
longing to some definite state. The specific weights of various
gases at atmospheric pressure and at a temperature of C.
are given as follows :
Atmospheric air ...... 0.08071 Ib. per cubic foot.
Nitrogen ....... O.OT829 Ib. per cubic foot.
Oxygen ........ 0.08922 Ib. per cubic foot.
Hydrogen ....... 0.00561 Ib. per cubic foot.
Carbonic acid ...... 0.12268 Ib. per cubic foot.
A pressure of one atmosphere, 760 mm. of mercury, is 10,333 kg.
per square meter = 14.6967 Ib. per square inch =2116. 32 Ib.
per square foot. Taking as 491.6 the value of T on the F.
scale corresponding to C., we have for air
2116  32 =5334
T <yT 0.08071x491.6
In metric units the corresponding calculation gives
7? = 10333 __ 9 Q 9 g
273.1 x 1.293 ' '
The values of B for other gases may be found in the same way
by inserting the proper values of the specific weight 7.
63. Forms of the Characteristic Equation. In the character
istic equation as usually written,
(1)
v denotes the volume of unit weight of gas. It is convenient
to extend the equation to apply to any weight. Letting M
denote the weight of the gas, we have for the volume F~of M
Ib. (or kg.), V= Mv, whence instead of (1) we may write :
pV=MBT. (2)
This equation is useful in the solution of problems in which
three of the four quantities, p, v, T, and M, are given and the
fourth is required.
EXAMPLE. Find the pressure when 0.6 Ib. of air at a temperature of
70 F. occupies a volume of 3.5 cu. ft.
From (2)
p = MBI = 0.6 x 58.34 x (70 + 469.6) = 484g>7 ^ per square feot
V o.o
= 33.63 Ib. r>er sauare inch.
advantageous in the solution of problems that involve tw
states of the gas. If ( p v F r T and (> 3 , F^, T^) are the tw
states in question, then
~~m~~ ~~T~
With this equation any consistent, system of units may be usec
EXAMPLE. Air at a pressure of 14.7 Ib. per square inch and having
temperature of 60 F. is compressed from a volume of 4 cu. ft. to a volun
of 1.35 cu. ft. and the final pressure is 55 Ib. per square inch. The fun
temperature is to be found.
From (3) we have
14.7 x 4 _55x 1.35
60 + 459.6 t 2 + 459.6'
whence t 2 = 196.5 F.
EXERCISES
1. Find values of B for nitrogen, oxygen, and hydrogen.
2. Establish a relation between the density of a gas and the value of tl
constant B for that gas.
3. Find the volume of 13 Ib. of air at a pressure of 85 Ib. per square inc
and a temperature of 72 C.
4. If the air in Ex. 3 expands to a volume of 30 cu. ft. and the fin
pressure is 20 Ib. per square inch, what is the final temperature ?
5. What weight of hydrogen at atmospheric pressure and a teinperatu
of 70 F. will be required to fill a balloon having a capacity of 12,000 cu. ft
6. A gas tank contains 2.1 Ib. of oxygen at a pressure of 120 Ib. p
square inch and at a temperature of 60 F. The pressure in the tank shou
not exceed 300 Ib. per square inch and the temperature may rise to 100 !
Find the weight of oxygen that may safely be added to the contents of tl
tank.
64. General Equations for Gases. The general equatioi
deduced in Chapter VI take simple forms when applied 1
perfect gases. From the characteristic equation
we obtain by differentiation
JTh
introducing tnese values or tue derivatives in the general
equations (I)(V) and (VIII), the following equations are
obtained :
da = c v dT + AS  dv, (Id)
< v y
dq = c p d TAB dp, (II a)
P
, AB ( T 7 , T , ^ , TTT ,
dq = G P dv + G V dp , (III a)
c p o v \ v p * J
du = J<j v dT, (IV a)
di = C] ,dT, (Yd)
c p ~c v = AB. (VIII a)
The first two equations may be still further reduced by
means of the characteristic equation to the forms
dq = c v dT 4 Apdv, (I 5)
dq=c p dTAvdp (115)
The ratio ^ of tlie two specific heats is usually denoted by
c v
k. The introduction of this ratio reduces (III a) to the sim
pler form,
d q = A. [kpdv + vdp] . (Ill 5)
K JL
Equation (IV a) simply expresses symbolically Joule's law
that the change of energy of a gas is proportional to the change
in temperature. Equation (I 5) follows independently from
(IV a) and the energy equation ; thus
dq = Adu + Apdv
= c v dT+ Apdv, since AJ 1.
EXERCISES
1. Deduce (VIII a) from (I 6), (II 5), and the characteristic equation.
2. Derive (V ) from (IV a) and the equation jw = BT.
3. From (I ), (II a), and (III n) derive expressions for
If
4i From (III&) deduce the equation of the adiabatic curve in _pucoordi
nates.
93 PROPERTIES OF GASES [CHAP, vn
5. From (I a) derive the equation of an adiabatic in TVcoordinates.
6. Using the method of graphical representation explained in Art. 32,
show a graphical representation of equation (I fc).
65. Specific Heat of Gases. If a gas obeys the law pv = BT,
the specific heat of the gas must be independent of the pressure
and also independent of the volume. This principle was shown
in Art. 57. The specific heat (c p or c w ) may, however, vary
with the temperature, and the results of recent accurate experi
ments over a wide range of temperature show that such a vari
ation exists. As a general rule, the law of variation is
expressed by a linear equation ; thus
c v = a + bt,
Cp = a' + bt.
When the range of temperature is large, as in the internal
combustion motor, the variation of specific heat with tempera
ture must be taken into account. In the greater number of
problems that arise in the technical applications of gaseous
media it may be assumed with sufficient accuracy that the
specific heat has a mean constant value.
For air the value of c p , as determined by Regnault, is 0.2375
from to 200 C. Recent experiments by Swann give the
following values :
0.24173 at 20 C.
0.24301 at 100 C.
In ordinary calculations we may take c p = 0.24.
The value of e p for carbon dioxide (CO 2 ) is usually given as
0.2012. Swann found the values
0.20202 at 20 C.,
0.22121 at 100 C.
The value of e p for other gases for temperatures between
and 200 C. may be taken as follows:
Hydrogen .... 3.4240
Nitrogen .... 0.2438
Values of the ratio k =  have been determined by various
c*
experimental methods. For air the results obtained range from
k = 1.39 'to Tc = 1.42. From the experimental evidence it seems
probable that the true value lies between 1.40 and 1.405. In
calculations that involve this constant, we shall take the value
1.4 as convenient and sufficiently accurate. For air, there
fore, ^=0.241.4 = 0.171.
The values of k and of <? for other gases may be taken as
follows :
k Co
Hydrogen ..... 1.4 2.446
Nitrogen ..... 1.4 0.174
Oxygen ..... 1.4 0.155
Carbon dioxide ... 1.3 0.162
Carbon monoxide . . 1.4 0,173
Ammonia ..... 1.32 0.387
If in equation (VIII #), c v is replaced by A the result is the
k
relation
,,*=! A3.
Each of the four magnitudes <? p , &, A, and B have been deter
mined experimentally, and this equation serves as a check.
66. Intrinsic Energy. An expression for the intrinsic
energy of a gas is obtained by integrating (IV a). Thus
O , (1)
if c v is assumed to be constant. The constant of integration
U Q is evidently the energy of a unit weight of gas at absolute
zero. Since, however, we are not concerned with the absolute
value of the energy, but the change of energy for a given
change of state, the constant M O drops out of consideration
when differences are taken, and we need make no assumption
as to its value. Hence, if (^ v v T^) and (jt? 2 , v v T^) are the
coordinate of the initial and final states, we have
ttM^Jb.CTaZi). (2)
98
PROPERTIES OF GASES
[CHAP, vn
This formula gives the change of energy per unit weight of
gas. For a weight M the formula becomes
UzU^JMc^TJ. (3)
A clear understanding of the physical meaning of formula
(2) is of such importance that it is desirable to give a second
method of derivation, one based directly upon Joule's law.
According to Joule's law the energy of a unit weight of gas
is dependent on the temperature only. Hence, if T v Fig. 33,
is an isothermal, the energy
of the gas in the state A is
the same as in the state D;
likewise, the energy of the
gas at all points on the iso
thermal T a is the
FIG. 33.
same. It
follows that the change of
energy in passing from tem
perature jPj to temperature T^
is the same, whether the path
is AJB, AC, orDJS.
Since the energy is directly proportional to the temperature,
the change of energy is directly proportional to the change of
temperature. Hence
Uzu^a^TJ, (4)
in which a denotes a proportionalityfactor. To determine the
factor a, we choose some particular path between the isother
rnals T and T z (Fig. 33). As we have seen, if this constant
is established for one path it holds good for every other path.
The most convenient path for this purpose is a constant volume
line, as A 0. The heat required for a rise in temperature from
V'Ziis fca^CZiZi)
Since in the constant volume change, the external work is zero,
we have from the general energy equation
Comparing these equations, we have
A loimuia lor tne cnange ot energy in terms of p and Fmay
be derived from (3). Multiplying and dividing the second
member by JB,
kl ' W
In (5) Kj and V l denote the final and initial volumes, respec
tively, of the weight of gas under consideration; consequently
it is not necessary to find the weight M in order to calculate the
change of energy. It is to be noted, however, that in using
(5) pressures must be taken in pounds per square foot.
EXAMPLK. Find the change of energy when 8.2 cu. ft. of air having a
pressure of 20 Ib. per square inch is compressed to a pressure of 55 Ib. per
square inch and a volume of 3.72 cu. ft.
Using the value k = 1.40,
55x3.7220x8.2
U = = 144 x
67. Heat Content. The change in heat content correspond
ing to change of state of a gas is readily derived from the
general equation (Va).
Thus, i = c v dT= c p T+ i , (1)
and 2 j = c p (T z T^) . (2)
Introducing the factor AB in the second member of (2),
For a weight of gas M, (2) and (3) become, respectively,
IZI^MC^TJ, (4)
and ^Ji^^r
68. Entropy. Expressions for the change of entropy are
easily derived from the general equations (la), (II a), and
(Ilia). Dividing both members of these equations by T, we
have
dq dT
^^
(2)
' ^ }
dv dp ,QN
ds = c P~+ c f' < 3 )
Hence for a change of state from (p r v v T{) to (p 2 , v a , T 2 ),
s 2  j = e, log e ^ + J.5 log, ^ (4)
g c (5)
^i Pi
8=8 c, log. ^+ c.log e s. (6)
These formulae give the change of entropy per unit weight
of gas. For any other weight M, the change of entropy is
M (s 2 Sj). Equations (4), (5), and (6) are in reality identi
cal. Each can be derived from either of the other two by
means of the relations pv = BT, c p c v = AB. In. the solution
of a problem, the equation should be chosen that leads most
directly to the desired result.
EXERCISES
1. From (4), (o), and (6) deduce expressions for the change of entropy
corresponding to the following changes of state : (a) isothermal, (b) tit con
stant volume, (c) at constant pressure.
2. By making s 2 s l = in (4), (5), and (6), deduce relations between
Tand v, T and;;, and p and v for an adiabatic change of state.
69. Constant Volume and Constant Pressure Changes. In
heating a gas at constant volume the external work is zero.
Hence,
Q = A( U,  Uj = Mo v (T 2  zy. (1)
(2)
\i fin" g,4'5 is hi'utttl at constant prewsure, the external
si
iruf ul<lrt in,
nf rniTjiy is, us in all canon, given by the
, given by the relation
C r >)
ha\r IHMU writtmi tlin
f entropy in
limiM c~> nntl (7) may
In* ilrnvril ilirt't'lly from
^jKM'ul fipiutiuiiH for *n
, Art. iJS.
f I'ljitnijf.s uf state just
Irrnl art npn'sruti'il on
fWjilaiu' by curvi'S of tbe
al form sbmvn in Ki.u r . &l*
rurvi' ,4/^. wliicli rcp
Is tbe const ant, volume (>
,n\ is MrquT ihsin the
'
(6)
FIG. 34.
that is, area A 1 ABB l < area
70. Isothermal Change of State. If T is made constant in
the equation p V=MB T, the resulting equation
P ^' r Pi Vi = constant (1)
is the equation of the isothermal curve in p F'coordinates. This
curve is an equilateral hyperbola. The external work for a
change from state 1 to state 2 is given by the general formula
(2)
Using (1) to eliminate p, we have
(3)
For the change of energy,
Z7 2  U^JMc^T^ TJ = 0; (4)
hence Tr
^12=^^12 = ^1^^^, (5)
and l 'i
Since in isothermal expansion the work done is wholly sup
plied by the heat absorbed from external sources, it follows that
if the expansion is continued indefinitely, the work that may be
obtained is infinite. This is also shown by (3), thus :
JL
71. Adiabatic Change of State. To derive the ^equation of
an adiabatic change of state, we may use the general differen
nun o^ucnuui.1. uujuumumg p uuu v as variaoies. me most con
venient form of this equation is (III a),
j_
dq = j T (ydp + /qpcfo) . (1)
During an adiabatic process no heat is supplied to or ab
stracted from the system ; hence in (1) dq 0, and therefore
vdp + kpdv = 0. (2)
Separating the variables,
<3/p_ Jcdv __ ^
P v ~ '
whence log e jt? + Tc log e v = log (7,
or jpy* = (7. (3)
The relation between temperature and volume or between
temperature and pressure is readily derived by combining (3)
and the general equation pv = BT. Thus from
pv* = C,
pv = BT,
we get by the elimination of p,
*!_ .
that is, 2V 1 = const. (4)
Similarly, by the elimination of v, we obtain
TDk
,*! _  mk .
p gJ. ,
xi
7} k
that is,  = const. (5)
If we choose some initial state, p v v v T r the constants in
(4) and (5) are determined, and the equations may be written
in the homogeneous forms
37
kl
Since in an adiabatic change the heat Q is zero, the energy
equation gives
whence using the general expression for the change of energy,
By means of the equation
the final volume V z may be eliminated from (8). The result
ing equation is
EXAMPLE. An air compressor compresses adiabatically 1.2 cu. ft. of free
air (i.e. air at atmospheric pressure, 14.7 Ib. per square inch) to a pressure
of 70 Ib. per square inch. Find the work of compression; also the final
temperature if the initial temperature is 60 F. K . .
For the final volume, we have
F 2 = 1.2 j~ = 0.3936 cu. ft.
The work of compression is
piVip,V 2 _ 144(14.7 x 1.2  70 x 0.3036) _ o ri , Q *.
kl 04 ~ ~
The initial temperature being 60 + 459.6 = 519.6 absolute,, we have for
the final temperature
0.4
T 2 = 519.6 (~Y= 811.6 abs.,
whence z 2 = 352 F.
72. Poiytropic Change of State. The changes of state con
sidered in the preceding sections are special cases of the more
general change of state defined by the equation
By giving n special values we get the constant volume, constant
pressure, and other familiar changes of state. Thus :
for n = 0, pv = const., i.e. p = const,
for n = co, p^v = const., v = const.
for n = 1, pv = const., isothermal,
for n = 7c, pv h = const., adiabatic.
The curve on the p F~plane that represents Eq. (1) is called
by Zeuner the polytropic curve.
By combining (1) with the characteristic equation^? VMBT,
as in Art. 71, the following relations are readily derived
For the external work done by a gas expanding according to
the law p V n p l V{ 1 const.,
from the volume FJ to the volume V v we have
tf
Pl *
The change of energy, as in every change of state, is
77  IT  P* V<i ~~ Pi T/ i (^
Uz Ul k^T~ w
Hence, from the energy equation, we have for the heat absorbed
by the gas during expansion
JO UU + W
</Vi 2 u z 1/1+ WM
K 1 L n
or J0 1n =  
Comparing (3), (4), and (5) we note that the common factor
(poV* p,V,} occurs in the second member of each expression.
useful relations :
W = ^1 (6)
CZT 1n ^
These may be combined in the one expression
W:U Z  U l :JQ=7cl:ln:Jcn. (9)
EXAMPLE. Let a gas expand according to the law
j) F 1>2 = const.
Taking k 1.4, we have
that is, the external work is double the equivalent of the beat absorbed by
the gas and also double the decrease of energy.
73. Specific Heat in Polytropic Changes. From Eq. (5),
Art. 72, an expression for Q lz in terms of the initial and final
temperatures of the gas may be readily derived. Since
(5) becomes
hence, Q^ = McJ^(T 2  T^. (1)
J  7i
We have, in general,
Qu=Me(T,T 1 '). (2)
where c denotes the specific heat for the change of state under
consideration. Comparing (1) and (2), it appears that
Hence, for the polytropic change of state, the specific heat is con
ART. 73] SPECIFIC HEAT IN POLYTROPIC CHANGES 107
stant (assuming c v to be constant) and its value depends on the
value of n in the equation p V n const.
It is instructive to observe from (3) the variation of c as n
For ra = 0, c = kc v =
and the
P
FIG. 35.
is given different values.
change of state is repre
sented by the constant
pressure line a, Fig. 35,
36. For n = 1, c = oo, and
the change of state is iso
thermal (line 5). If n = 7c,
then c = 0, and the ex
pansion is adiabatic (line
cT). For values of n lying
between 1 and 7c, the value
of c as given by (3) is
evidently negative ; that
is, for any curve lying
between the isothermal b
and adiabatic d, rise of temperature accompanies abstraction of
heat, and vice versa. This is shown by the curve c.
It will be observed that
in passing through the
region between curves a
and 5, n increases from
to 1 and c increases from c p
to oo ; then as n keeps on
increasing from 1 to k, c
changes sign at curve b by
passing through oo, and
increases from oo to 0.
As n increases from n = k
to n = + co, c increases
from c = to <?=<?; and
for n oo, the constant
volume case, c becomes c v .
T
O
S
FIG. 36.
108 PROPERTIES UF UAbHitt
74. Determination of n. It is frequently desirable in experi
mental investigation to fit a curve determined experimentally
_ aSj for example, the compression curve of the indicator
diagram of the air compressor by a theoretical curve having
the general equation pV n =c. To find the value of the
exponent n we assume two points on the curve and measure to
any convenient scale p v p v V v and V v Then since
wehave =*r g . CO
log F!  log F 2
EXAMPLE. In a test of an air compressor the following data were
determined from the indicator diagram :
At the beginning of compression, p = 14.5 Ib. per square inch.
^ = 2.50 cu. ft.
At the end of compression, jo 2 = G8.7 Ib. per square inch.
F a = 0.77 cu. ft.
Assuming that the compression follows the law p V n const., we have
for the value of the exponent
ff= log 68.7 log 1*.5 = 132
log 2.56  log 0.77
The work of compression is
0.7715.5x2.56)
1 n
The increase of intrinsic energy is
" 14 5 x 2.56) f b
k 1 0.4
and the heat absorbed is
56807100, .
^12  ^g  =  l.od B. t. u.
The negative sign indicates that heat is given up by the air during com
pression ; this is always the case with a waterjacketed cylinder.
If the initial temperature of the air is 00 F., or 519.6 absolute, the final
temperature is
EXERCISES
1. A curve whose equation is pV n = C is passed through the points
Pi = 40, F! = 6 and jt? 2 = 16, F 2 = 12.5. Find the value of n.
2. Air changes state according to the law pV n C. Find the value of
n for which the decrease of energy is one half of the external work; also the
value of n for which the heat abstracted is one third of the increase of energy.
3. If 32,000 ft.lb. are expended in compressing air according to the
law^F 1  28 = const., find the heat abstracted, and the change of energy.
4. In heating air at constant pressure 35 B. t. u. are absorbed. Find
the increase of energy and the external work.
5. A mass of air at a pressure of 60 Ib. per square inch absolute has a
volume of 12 cu. ft. The air expands to a volume of 20 cu. ft. Find the
external work and change of energy : (a) when the expansion is isothermal ;
(&) when the expansion is adiabatic ; (c) when the air expands at constant
pressure.
6. If the initial temperature of the air in Ex. 5 is 62 F., what is the
weight? Find the heat added and the change of entropy for each of the
three cases.
7. Find the specific heat of air when expanding according to the law
p v i.25 = const. If during the expansion the temperature falls from 90 F. to
10 F., what is the change of entropy?
8. Find the latent heat of expansion of air at atmospheric pressure and
at a temperature of 32 F.
9. The volume of a fire balloon is 120 cu. ft. The air inside has a
temperature of 280 F., and the temperature of the surrounding air is 70 F.
Find the weight required to prevent the balloon from ascending, including
the weight of the balloon itself.
10. A tank having a volume of 35 cu. ft. contains air compressed to
90 Ib. per square inch absolute. The temperature is 70 F. Some of the
air is permitted to escape, and the pressure in the tank is then found to be
63 Ib. per square inch and the temperature 67 F. What volume will be
occupied by the air removed from the tank at atmospheric pressure and at
70 F. V
11. Air in expanding isothermally at a temperature of 130 F. absorbs
35 B. t. u. Find the heat that must be absorbed by the same weight of air
at constant pressure to give the same change of entropy.
12. Air in the initial state has a volume of 8 cu. ft. at a pressure of
75 Ib. per square inch. In the final state the volume is 18 cu. ft. and the
pressure is 38 Ib. per square inch. Find: () the change of energy; (b) the
change in the heat content ; (c) the change of entropy.
13. Find the work required to compress 30 cu. ft. of free air to a pressure
of 65 Ib. per square inch, gauge according to the lawpw 1  3 = const. Find the
heat n.lisf.ra.fltprl rlnvino nnmnrfissinri.
recourse to general equations.
SUGGESTION. Let one pound of gas be heated through the temperature
range T 2  T l (a) at constant volume, (/;) at constant pressure. Find an
expression for the excess of heat required for the second case and then
make use of the energy equation.
15. Suppose the specific heat of a gas to be given by the linear relation
c v = a + bt. Deduce relations between p, v, and T for an adiabatic change.
SUGGESTION. Use the general equation dq = c v dT + Ajxlv and the char
acteristic equation pv = BT.
REFERENCES
CHARACTERISTIC EQUATION OF GASES. DEVIATION FROM TIIK
BOYLEGAY LUSSAC LAW
Zeuner: Technical Thermodynamics (Klein) 1, 03.
Preston : Theory of Heat, 403.
Barus: The Laws of Gases. N.Y. 1809. (Contains the researches of
Boyle and Amagat.)
Regnault : Relation des Experiences 1.
Weyrauch : Grundriss der WiirmeT heorie 1, 124, 127.
THE POROUSPLUG EXPERIMENT. THE ABSOLUTE SCALK OF
TEMPERATURE
Thomson and Joule: Phil. Trans. 143, 357 (1853) ; 144, 321 (1851) ; 152,
579 (1862).
Roselimes: Phil. Mag. (0) 15, 301. 1908.
Callendar: Phil. Mag. (0) 5, 48. 1903.
Olszewski : Phil. Mag. (6) 3, 535. 1902.
Buckingham : Bui. of the Bureau of Standards 3, 237. 1908.
Preston : Theory of Heat, 695.
Bryan : Thermodynamics, 128.
Chwolsou : Lehrbuch der Physik 3, 546.
SPECIFIC HEAT OF GASES
Regnault: Relation des Experiences 2, 303.
Swann : Proc. Royal Soc. 82 A, 147. 1909.
Zeuner : Technical Thermodynamics (Klein) 1, 116.
Chwolson : Lehrbuch der Physik 3, 22G.
Preston : Theory of Heat, 339, 243.
Weyrauch : Grundriss der WsinneTheorie 1, 146.
GENERAL EQUATIONS
Zeuner : Technical Thermodynamics 1, 122.
Weyrauch : Grundriss der WarmeTheorie 1, 152.
Bryan : Thermodynamics, 116.
CHAPTER VIII
GASEOUS MIXTURES AND COMPOUNDS. COMBUSTION
75. Preliminary Statement. In the preceding chapter we
discussed the properties of simple gases with the implied
assumption that chemical action was excluded. For many
technical applications a knowledge of such properties is suffi
cient for the consideration of all questions that arise. On the
other hand, investigations of the greatest importance, those
relating to internal combustion motor, have to deal with
gaseous substances that enter into chemical combination and
(after combustion) with mixtures of inert gases. In the
present chapter, therefore, we shall consider some of the pro
perties of gaseous compounds as dependent on chemical com
position, and also the properties of mixtures of gases.
76. Atomic and Molecular Weights. Let U v E^, etc. denote
different chemical elements and a r a 2 , etc. their corresponding
atomic weights. Then if n^ w 2 , etc. denote the number of atoms
of JE r jKj, etc. entering into a molecule of a given combination,
the molecular weight of the compound is
m = n^ + n z a z + etc. = "^na. (1)
For the elements that enter into subsequent discussions the
atomic weights (referred to the value 16 for oxygen) are as
follows :
APPROXIMATE
EXACT VALUE INTEGRAL VALUE
Hydrogen 1.008 1
Oxygen 16.000 16
Nitrogen . 14.040 14
Carbon 12.000 12
Sulphur 32.060 32
Tlie approximate uj.tegj.tn vo.uuo ,^ **** *j ~ 
practical purposes, in view of unavoidable errors in experi
mental results.
Using these values, we have as the molecular weights of cer
tain important substances the following :
Water H 2 m=2x 1 + 1x16 = 18
Carbon monoxide CO 1 x 12 + 1 x 16 = 28
Carbon dioxide C0 2 1 x 12 + 2 x 16 = 44
Ammonia NH 8 1 x 14 + 3 x 1 = 17
Methane CH 4 Ixl2 + 4x 1 = 16
Nitrogen N 2 2x14 = 28
Hydrogen H 2 2x1 = 2
The composition by weight of a compound is readily deter
mined from the value of , a, and m. Thus in a unit weight
(pound) of compound there is
Ml Ib. of element _EL
m
lb. of element E v etc.
m
For example, C0 2 is composed by weight of  carbon anct
If oxygen, NH 3 is composed by weight of \$ nitrogen and ^
hydrogen.
77. Relations between Gas Constants. If in the character
istic equation pv = BT, which holds approximately for any
gaseous substance (mixture or compound), we replace v by 
we have ^
Here 7 denotes the weight of unit volume of the gas. From
this relation it is seen that for a chosen standard pressure and
temperature, for example, atmospheric pressure and 0C., the
product By is the same for all gases. But since the specific
weight 7 of a gas is directly proportional to the molecular
weight m, it follows that the product Bm is likewise the same
tor all gases. Denoting this product Bm by JK, we have for
the characteristic equation of any gas
pv = ~T. (2)
f m ^ '
From (1) we obtain the relation
&; (3)
hence the numerical value of R can be found when the values
of m and 7 are accurately known for any one gas. From Mor
ley's accurate experiments, we have for oxygen 7 = 0.089222 Ib.
per cubic foot at atmospheric pressure and 32 F. ; and for
oxygen m = 32. Inserting these numerical values in (3), we
obtain
2116.3x32
0.089222x491.6
The constant R is called the universal gas constant. From it
ffche characteristic constant B of any gas can be determined at
pnce from the molecular weight. Thus for carbonic acid we
Ijiave
= 1^ = 35.09.
j*i 4:4
^ From the general formula
[ CV C. = AB (4)
for the difference between the specific heats of a gas, we have
AR 1544 1 1.9855 ,^
/ __ i ft ^^_^^. r  _, J _  .  I > I
p v ~~ m ~ 777.64m m ' W
This relation gives a ready method of calculating one specific
heat from the other when the molecular weight m is known.
Thus for C0 2 , Cp  c ,= i^^ = 0.0451, and if e, = 0.2020, we
have c v = 0.2020  0.0451 = 0.1569.
It is convenient to express the specific weight 7 and the
specific volume v of a gas in terms of the molecular weight m.
These constants are referred to standard conditions, namely,
atmospheric pressure and a temperature of 32 F. From (3)
we have 7 = JLw, (6)
whence inserting tne numerical viuuea, jj **
!F=491 6
7 = 0.002788 w.
For the normal specific volume, we have
v== l = OTl.
7 p m
358.65
or
v =
in
(7)
(8)
(9)
From the preceding relations, the following values are readily
found for the constants of certain gases.
Volume per
Character
niH'oronco
Weight per
pound
Gas
Chemical
Symbol
Molecular
Weight
istic
Constant
of Specific
Heats
cubic foot nt
!!'2 1<\ and
Atmo.sphoric
at !i V.
and Atmos
pheric
m
B
CpC v
Pressure
Treasure
Oxygen ....
2
32
48.249
0.0021
0.089222
11.208
Hydrogen . . .
H 2
2.016
765.86
0.9849
0.005621
177.9
Nitrogen . . .
N 2
28.08
54.985
0.0707
0.07829
12.773
Carbon dioxide .
C0 2
44
35.09
0.0451
0.12268
8.151
Carbon monoxide
CO
28
55.142
0.0709
0.078028
12.81
Methane . . .
CH 4
16.032
96.314
0.1238
0.04470
22.37
Ethylene . . .
CJI 4
28.032
55.079
. 0.0708
0.078036
12.794
Air
53.34
0.0086
0.08071
12.39
78. Mixtures of Gases. Dalton's Law. A mixture of several
gases that have no chemical action on each other obeys very
closely the following law first stated by Dalton :
The pressiire of the gaseous mixture upon the walls of the con
taining vessel is the sum of the pressures that the constituent gases
would exert if each occupied the vessel separately.
Like Boyle's law, Dalton's law is obeyed strictly by mix
tures of ideal perfect gases only. Mixtures of actual gases show
deviations from the law, these being greater with gases most
easily liquefied. For the purpose of technical thermodynamics,
however, it is permissible to assume the validity of Dalton's
law even in the case of a mixture of vapors.
Let F denote the volume of a given mixture, M v M z , M& . .
the weights of the constituent gases, and
J9 2 , J5 3 ,
the
ART. 78] MIXTURES OF GASES. DALTON'S LAW 115
constants for those constituents ; then the partial pressures of
the constituents, that is, the pressures they would exert sepa
rately if occupying the volume V, are :
B8
m l i IY\ __ & " m _ " " /I 1
Pi y iPi ' ~Y '"3 Y ' '" ^  1
According to Dalton's law the pressure p of the mixture is
P=Pi+Pi + PB + ' = ~ r (.M 1 B 1 +M 2 2 +M 3 S 3 + ). (2)
Furthermore, if Mis the weight of the mixture,
(3)
Let us now introduce a magnitude B m defined by the equation
MB m = M& + M,B Z + Jf 8 J5 8 +; (4)
then (2) takes the form
pir=MB m F. (5)
The constant B m may be regarded as the characteristic con
stant of the mixture. It is obtained from (4), which may be
written in the more convenient form
5m= (6)
The partial pressures may readily be expressed in terms of
the pressure of the mixture. Thus combining (1) and (5),
we obtain
, etc. (7)
' ^
EXAMPLE. A fuel gas has the composition by weight given below. The
value of the constant B m for this gas is found as indicated by the following
arrangement :
CONSTITUENTS WEIGHT
r,n n ....... 0.04
Since M= 1 and 2MB = 103.24, we have
B m = 103.24.
The apparent molecular weight of the mixture is
1544 1/( nr
m= ios^r 1U)G>
and the weight per cubic foot under standard conditions is, therefore,
y = 0.002788 x 14.90 = 0.0417 Ib.
79. Volume Relations. Let V v F 2 , F 3 , , denote the vol
ume that would be occupied at pressure p and temperature T
by several gaseous constituents; then if B^ B^ J9 3 , , denote
the characteristic constants of these gases, we have
pV,= M&T, pV, = M Z B Z T, pV s = M Z B Z T, .... (1)
If now the gases be mixed, keeping the same pressure and
temperature, the mixture will occupy the volume
FF 1 +F a +F 8 +, (2)
and its weight will be necessarily
M=Mi + Mt + M % +.~ (3)
Taking B m as the characteristic constant of the mixture, we
have
pV=MB m T. (4)
Comparing (1) and (4), we obtain the relations
V 1 _M 1 B, 3_J^ _
V~ MBj V~MB m ' 1 " ^ }
It will be seen that the volume ratios given by (5) are equal
to the pressure ratios given by (7) of Art. 78.
If 7 denotes the weight of a unit volume (1 cu. ft.) of gas,
then
1 M xnx
7 =  = y (6)
For the several constituents of a mixture, we have, therefore,
M l = 7l V v M, = 72 F 2 , M z = 73 F 3 , . .., (7)
and for the mixture
Similarly, we have for the specific volume of the mixture
Since 7 = 0.002788 m = km (see Art. 77), we have from (7)
and M M + M,+ .
m j;  m i I ., W 9 ,
Therefore, ^ = =*_*= , _1 = 2_A (10)
J(K? STO^ jf Sm^JY
If further we denote by w ro the quotient ^, we have from (8)
/c
i m m = y2.m i V i . (11)
The constant w m we maj 7 " regard as the apparent molecular
weight of the mixture, and from it we may determine the con
stants B m , c p <?, 7, and v of the mixture.
Equations (10) and (11) are useful in the investigation of a
mixture when the composition by volume is given. The follow
ing example shows the method of procedure.
EXAMPLE. A producer gas has the composition by volume given below.
.Required the composition by weight and the constants of the mixture.
EU . . .
.... 0.08
2
0.16
2m f F"{
0.006
CO ....
.... 0.22
28
6.16
0.2308
CH 4 ....
.... 0.024
16
0.384
0.0144
CO 2 ....
.... 0.066
44
2.904
0.1088
N 2 ....
.... 0.61
28
17.08
0.64
1.000 26.688 1.000
According to (10) the last column gives the composition by weight. The
constant m m is 26.688; hence we have
= 57.85. v = 0.002788 x 26.688 = 0.07441.
'
26.688
1.9855
pv 2688
80. Combustion : Fuels. The elements that chiefly combine
with oxygen to produce reactions characterized by the evolution
rvf Tioof QVQ TnrrJy^rrcm onrl n.a.rlnrm . rinmnrmnrls that are made
up largely ot these elements are lueis; ror example, metnane
CH 4 , benzol C 6 H 6 , alcohol C 2 H 6 0. The product of complete
combustion of hydrogen is H 2 0, water ; that of complete com
bustion of carbon is C0 2 , carbon dioxide. Sulphur is a possible
constituent of fuels, and the product of combustion is SO 2 , sul
phur dioxide.
Chemical reactions are, in general, characterized by the evo
lution or absorption of heat. The union of a combustible with
oxygen is accompanied by the evolution of a considerable
quantity of heat, and the heat evolved by the combustion of a
unit weight of the combustible is called the heating value of
the combustible. The heating value of hydrogen alone or car
bon alone must be determined by experiment, but the heating
value of a compound of C and H may be calculated, at least
approximately.
Hydrogen and compounds containing hydrogen have two
heating values, called respectively the higher and the lower.
This arises from the fact that the product H 2 O may be either
water or steam. If the temperature after combustion is above
212, the product exists as vapor, and the heat necessary to
keep it in the vapor form is not set free ; hence, we have the
lower heating value. If, however, the vapor condenses, the
heat of vaporization is recovered, and we have the higher heating
value.
The heating values of various fuels are given in the follow
ing table.
B. T. U. 1
KK POUND
15. T. u. VKH Ounic
High "
Low
Low
Hydrogen ....
H,
02100
52230
2!M
Carbon
c
14(500
1 '1 (if)O
Carbon monoxide .
^2
CO
4380
4380
312
Methane ....
CH 4
23842
21385
951)
Ethyleue ....
C 2 II 4
2142.9
20025
15(i3
Acetylene ....
C 2 H 2
21429
20073
14!)!)
The heating value of a fuel mixture is determined from the
heating values of the separate constituents. Denoting bv M,.
ART. 81] AIR REQUIRED FOR COMBUSTION 119
My , the weights of the constituents, by H v jET 2 , H^ , the
corresponding heating values per pound, and by H m the heat
ing value of the mixture, we have
whence H m = . (1)
By a similar procedure the heating value per cubic foot may
be obtained when the composition by volume is given.
EXAMPLE. Required the lower heating value of the producer gas de
scribed in the example of Art. 79.
For the heating value per pound we have
Jf IT MIT
H a ........ 0.006 52230 313.38
CO ....... 0.2308 4380 1010.9
CH 4 ....... 0.0144 21385 307.94
2 MH = 1632.2
Since M = 1, we have // = 5 MH = 1632.2 B. t. u. per Ib.
The heating value per cubic foot (at 32 F. and atmospheric pressure) is
evidently the product
II m j = 1632.2 x 0.07441 = 121.5 B. t. u.
Or from the composition, we have
Ho ....
r
. . . . 0.08
IT
294
r/7
23.52
CO ...
"
342
75.24
CH 4 . . .
. . . . 0.024
956
22.94
121.7 B. t. u. per cu. ft.
The difference in the two results is due to approximations in the calculation,
and is of no importance.
81. Air required for Combustion and Products of Combustion.
The oxygen required for the complete combustion of a given
fuel is determined from the equation of the reaction. For
example, the combustion of methane, CH 4 , is given by the
equation
CH 4 + 2 2 = C0 2 + 2 H 2 ;
120 GASEOUS MIXTURES AND COMPOUNDS [CHAP, vin
and two molecules of H 2 O. Since by Avogaclro's law the
volumes are proportional to the numbers of molecules enter
ing into the equation, we may also read the preceding chemical
equation as follows : two volumes of oxygen combine with one
volume of CH 4 , producing one volume of C0 2 and two volumes
of H 2 0.
Taking the molecular weights of the four gases into con
sideration, we may write the equation
16 + 2x32 = 44 + 2x18.
From this it appears that one pound of CH 4 requires for com
plete combustion f f = 4 Ib. of oxygen and the products are
if = 2.75 Ib. of CO 2 and ff = 2.25 Ib. of H 2 O.
Since oxygen is 23 per cent of air by weight, the weight of
air required for the complete combustion of one pound of CJT 4
4.
is = IT. 4 Ib. The volume of air required for the burning
2
of one cubic foot of CBL is ^ = 9.52 cu. ft.
4 0.21
We may generalize the process illustrated by the preceding
example as follows :
Let the gaseous fuel have the composition C ni H Ba O n ,, and let
!, 2 , 3 denote the atomic weights of C, H, and 0, respectively.
Then the molecular weight of the fuel in question is
m = a 1 ?i 1 + a z n z + a 3 n 3 .
The equation of the reaction may be written
where $, ?/, and z indicate the number of molecules of the
respective substances. Comparing the two members of the
equation, we find
whfinp.fi
ART. 81] AIR REQUIRED FOR COMBUSTION 121
for alcohol C 2 H 6 O, x = 2 +  = 3, y = 2, and g = 3, showing
that for the combustion of one cubic foot of alcohol vapor,
3 cu. ft. of oxygen are required, and the resulting products are
2 cu. ft. of CO 2 and 3 cu. ft. of H 2 0.
To get the relations between the weights of the substances
under consideration we must introduce the molecular weights
in the reaction equation. Thus we obtain
m + 2 a B x = y(ai + 2 3 ) + z (2 a 2 + a 3 ),
from which follow the ratios :
, weight of oxvgen 2 ax , , , N
x   1+111 = = 3(2 n 1 +l~n z n B ) ;
weight of fuel m m
i weight of OCX, yCa, + 2 a^) n, , . .
y = .^ a = &^LZ: az _ _i ( a + 2 a ) ;
weight of fuel m m
g ; _ weight of H 2 O _ g(2 a 2 + 8 ) _ M 2 XQ g + a ).
weight of fuel m 2m 2 3
If we make use of the integral values of the atomic weights,
namely, ^ = 12, a 2 = 1, a s = 16, we have for the complete com
bustion of one pound of the combustible :
w /*
x' = oxygen required = (2 n^ + w 2 n s) lb. ;
y' = C0 2 produced = 44 ^1 Ib. ;
wi
' = HoO produced = 9^ Ib.
2 ^ m
Taking alcohol, C 2 H 6 0, for example, we have
! = 2, Wjj = 6, TC S = 1, w = 2 x 12 + 6 x 1 + 16 = 46, whence
x' = (2 x 2 + i x 6  1) = 2.08T;
, 44 x 2 1 Q1 o .
= =1.913;
= 1.174.
46
The weight of air required per pound of alcohol is
and the weight of nitrogen appearing among the products of
combustion is, therefore, 9.075  2.087 = 6.988 Ib.
If a gaseous fuel is a mixture of several combustible con
stituents, the values of x\ y ! , and z' may be found for the indi
vidual constituents separately. Then if M^ M v M z , , are the
weights of the constituents respectively, we have
, y ,
__ _ . at' __ */__,_ fy' 
' y ~ '
__
M ' ~ M ' M
EXAMPLE. For the producer gas heretofore investigated, we have the
following values :
jir
Q)'
y'
a'
Ufa'
Mi/'
Ms,'
H,
0.006
8
9
0.048
0.051
CO
0.2308
0.571
1.571
0,1 '5 18
().3(!2G
CH 4
0.0144
4
2.75
2.25
0.0570
O.OSOG
0.0324
C0 2
0.1088
1
0.1088
N 2
0.64
1.00
0.2871
0.511
0.08(31
One pound of the gas requires 0.2374 Ib. of oxygen for complete combustion.
The weight of air required is, therefore, 0.2374 f 0.23 = 1.032 Ib., and this
air brings with it 1.032  0.2374 = 0.7940 Ib. of nitrogen. We have then the
following balance :
CONSTITUENTS PRODUCTS
Fuel gas 1.00 Ib. C0 2 0.511 Ib.
A ir 1.0^2 H,Q 0.08(54
2.032 Ib. No 0.64 + 0.7040 = l.ljWn.
(W2 Ib.
Taking the composition by volume, the following results are found :
V
te
y
z
Fin
Vff
r~
H Q
0.08
0.5
1
0.04
0.08
CO
0.22
0.5
1
0.11
0.22
CH 4
0.024
o
1
o
0.048
0.024
0.048
C0 2
0.066
1
O.OOli
N 2
0.61
1.00
0.198
0.31
0.128
Since 0.198 cu. ft. of oxygen is required per cubic foot of gas, the volume of
air required is 0.198 * 0.21 = 0.943 cu. ft., and the volume of nitrogen corre
sponding is 0.943  0.198 = 0.745 cu. ft., which is added to the O.(il cu. ft. in
the fuel gas. The volume of gas and air before combustion is 1 + 0.943 =
1.943 cu. ft., and the volume of the products is 0.31 + 0.128 + 0.01 + 0.745
82. Specific Heat of Gaseous Products. In deducing the
special equations for gases we assumed that the specific heat
of any gas remains constant at all pressures and temperatures.
In many technical applications this assumption is sufficiently
near the truth and is justified by the simplicity of the analysis
based upon it ; but when a very wide range of temperature is
encountered, as in the case of the internal combustion motor, the
assumption of constant specific heat is no longer permissible.
The gaseous products that come under consideration may be
separated into two classes. (1) The simple or diatomic gases,
as nitrogen, oxygen, air, etc. ; (2) the compounds, like carbon
dioxide (CO 2 ) and steam (H 2 0), which may be regarded as
superheated vapors rather than as gases. For the products in
the first group, the law pv = B T holds quite exactly, and, there
fore (see Art. 57), the specific heat must be independent of the
pressure, but may vary with the temperature. The substances
in the second group, which are comparatively near the liquid
state, do not follow the gas law closely, and for these the
specific heat may vary with the pressure as well as with the
temperature. The character of the variation of the specific heat
of steam is shown in Fig. 71, Art. 133. At the lower tempera
tures the specific heat increases with the pressure, but as the tem
perature rises the influence of the pressure becomes negligible
and the specific heat rises with the temperature. It is probable
that the specific heat of CO 2 varies in somewhat the same manner.
Experiments on the specific heats of various gases show that
in general the specific heat rises with the temperature, and that
the law governing the variation is expressed sufficiently well
by the simple linear equation
G = a f bt.
The formulas, as usually stated, give molecular specific heats,
the molecular specific heat being numerically equal to the
thermal capacity of a weight of the substance expressed by the
molecular weight. Thus, since the molecular weight of carbon
monoxide (CO) is 28, the molecular specific heat of CO is
numerically equal to the thermal capacity of 28 pounds of CO.
We mav denote molecular specific heat bv the product me. It
gases are quite different, the molecular specific heats are sub
stantially identical.
The results of Langen's experiments are given by the follow
ing formulas, in which * denotes temperature in degrees C.
For all simple gases
me,** 4. 8 + 0.0012*. (1)
For carbon dioxide
we, = 6. 7 + 0.0052*. (2)
For water vapor
we, = 5.9 + 0.0043*. (3)
Dividing by the appropriate value of the molecular weight m,
the heat capacity of a gas per unit weight is readily found.
Thus for oxygen m = 32, and from (1) we have
c,= 0.15 + 0.0000375*;
for C0 2 , m = 44, and from (2) we obtain
c w 0.1523 + 0.0001182*.
Formulas (1), (2), and (3) give molecular specific heats at
constant volume. From the relation m(c p <?)= 1. 1)855 (see
Art. 77), we have approximately mc p = mc v + 2. Therefore,
from the preceding equations we obtain corresponding equa
tions for Op, namely :
mc p = 6.8 + 0.0012 *; (4)
; (5)
CO
For temperatures F. the preceding formulas become respec
tively:
1. For simple gases
<? =  (4. 77 + 0.000667*)
= (4.48 + 0.00066720
m
< = 1(6.75 + 0.0006670
m '
= 1(6.46 + 0.000667 T\
m J
2. 1< or carbon dioxide
c,= 0.15 + 0.000066*
= 0.12 + 0.000066 I 7 .
c p = 0.195 + 0.000066
= 0.165 + 0.000066.^1' ^ ^
3. For superheated water vapor
c v = 0.324 + 0.000133S
= 0.263 + 0.000133 T
c p = 0.435 + 0.000133S
= 0.374 + 0.000133^]
83. Specific Heat of a Gaseous Mixture. Let M^ M z ,
respectively, denote the weights of the constituents of a mix
ture and e Vi , <? V2 , , the corresponding specific heats. It is
assumed that for a given temperature rise each constituent
requires the same quantity of heat when mixed with other
constituents as it would if separated from them. Hence, the
heat Q required for a temperature change T z T^ is
But we have also
where M=M 1 {M z + , and c v denotes the specific heat of
the mixture. Combining these expressions, we obtain
__
or c v
TM
Likewise, c p = .,
EXAMPLE. Find the specific heat c v of a mixture of 1 Ib. of the pro
ducer gas described in the example of Art. 79 and 1.25 Ib. of air, which, is
about 20 per cent in excess of the air required for complete combustion.
Find also the specific heat c v of the products of combustion.
Of the 1.25 Ib. of air furnished 0.2875 Ib. is oxygen and 0.9625 Ib. is
nit'.i'ii. .Mintti;: ntt nut. ,:> n i> MM IM.I m. jjj tJu ^UH, y u , ,
t.lKKJ.'. Hi. \Vi ]iu\" UKII
v  ^ .v...
....... 'HJ ' "'""'(l.iH + O.OOCHJOT 7')
n 'J'U m
;"' '7^ ( l.'JH  O.OOOliliT 7')
rn ...... , n.in^ u,j OHM (0.12 j o.ooiKKm 7')
N, ....... l.Uii;;, ','' tjjll ""''( 1,1S . I).()(){)()li7 7')
ys
? 7')
7'
o (i . 1;i y
Fur the jirniluri nf ,.tu!.nf;,.n, lu,v' t * Art. SI
V.,,
tj.r.ll (O.lL*  O.ODOOlili 7')
]:j;j 7')
t!  n: i '" 1 ( I.1S 1 O.OIHHiiiT 7')
n.:U7 l.'i i II.ODOOS.I.M 7'
i> i., . .,<(.. ..;';..:' 7V
84. Adiabutic Chan^rs with VitryinK Specific Heats. AVhon
tlu i sjiriifir IniU ni ,i ; ^ s , 5 , ul^n ,i .1 fniu'tion of Uunporaiurc,
tints
,.  .1 / 'A
, s ./' * /?*
lh siuipli' ivlaiJMiis .lrri\ t ,l m V:t. 71 i lt>n^i!r apply. We
have as lii'fntv, ltu\vr\rj,
For an adiabatic change dq = ; hence from (T), we have
c v dT= Apdv,
or
v
From (4) we obtain upon integration
. (5)
M z
From the characteristic equation pv=BT, we have 2 =^2,
^i
therefore (5) becomes
or alog. + J(2',y 1 ) = (^5 + a)log.. (6)
jrl "2
Finally, if in (5) we substitute for 1 its equivalent ^ 2 , we
, . v z
obtain
. a .  .
Pl T
whence
For the external work of adiabatic expansion, we have
TF=.Z7 _ TJ
. (8)
Equations (5), (6), and (7) are readily applied when the
initial and final temperatures are given. When, however,
the final temperature is required, the equation in T is tran
scendental and its solution requires a process of successive
approximations. The illustrative example of the following
article shows the method of procedure.
85. Temperature of Combustion. A close analysis of' the pro
cess of burning a fuel gas under given conditions involves com
plicated equations, especially when the specific heat is taken as
variable. The temperature and pressure at the end of the pro
cess are the results usually uesireu,
least approximately, by a simple method.
Let ^ denote the temperature of the gaseous mixture at the
beginning of combustion and T z the desired final temperature ;
H the lower heating value of the fuel per pound, and M the
combined weight of one pound of fuel and of the air furnished
for combustion (M is evidently also the weight of the products
of combustion). It is assumed that the combustion is complete,
and that the heat His all expended in raising the temperature
of the products from ^ to T v As a matter of fact, the com
position of the mixture during the combustion process is con
tinually changing, but as the specific heat changes but little, it
is considered permissible to base the calculation on the final
products alone. We have then
H=M( T \a + bT)dT, (1)
Tl
where a + bT is the expression for the variable specific heat of
the products. From (1) we obtain upon integration
from which T z may be calculated.
EXAMPLE. The mixture of producer gas and air in the example of
Art. 83 is compressed adiabatically from an initial pressure of 14.7 Ib. pel
square inch to a pressure of 150 Ib. per square inch absolute. The initial
temperature is 530 absolute. The mixture is then burned at constant
volume and the products of combustion expand adiabatically to the initial
volume. Required the temperature and pressure after compression, after
combustion, and after expansion. Also the work of compression, and the
work of expansion.
The characteristic constants of the fuel mixture and of the mixture of
the products, respectively, are first required. Tor the fuel mixture we have
M i) j//;
H 2 ....... 0.006 765.86 4.5!);)! (5
CO ...... 0.2308 55.142 12.72077
CH 4 ...... 0.0144 90.311 1.3861)2
C0 2 ...... 0.1088 85.00 3.81770
N 2 ....... 1.6025 54.985 88.11340
2 ....... 0.2875 48.249 13.8715!)
2.25 124,512
B = 124.512 s. 2.25 = 55.34 ; AB = 0.07116.
.r or me mixture 01 products, we ODtain JJ 51.50; AB 0.06621.
For the fuel mixture, the expression for the specific heat is
c v = 0.1618 + 0.00002643 T.
We have, therefore, from (7), Art. 84
0.23296 log e ^= 0.07116 log^  0.00002643 (T z  Ti).
i  JLur. I
To solve this equation for T 2 let us assume as a first approximation
T 2  I\ = 500. Then
l T* = 0.16529 O.ni3215 =0
h 2\ 0.2329(3
and ^=1.921.
Therefore, T 2 = 1.921 x 530 = 1018.1,
and r 2  7 7 1 = 488.1.
As a second approximation, we assume Tz T l = 490. We obtain
T, 0.165290.012951 = Ot6539
^^ 0.23296
^ = 1.9231, T z = 1.9231 x 530 = 1019.2,
TI
T 2  T v = 489.2.
As the assumed value of T 2  T\ is so nearly attained, we may take the
value Tz = 1020 as sufficiently exact.
The ratio of initial and final volumes is now readily found from the
relation
Thus,
V l p 2 Ti 150 530
For the external work required to compress one pound of the mixture, we
have
W= J . 1 (0.1618 + 0.00002643 T)dT  69460 ft.lb.
If T s denotes the temperature after combustion, we have from (2), taking
c v = 0.1544 + 0.00003753 T for the products of combustion,
8 . 1020") =
whence T z = 3949.
To find the pressure j> 3 , we must take account of the change of composi
tion during combustion. For the initial state, p 2 V = 55.34 T 2 , at the end of
combustion p s V = 51.50 T 3 . Hence, we have
130
GASEOUS MIXTURES AND COMPOUNDS [CHAP.VIII
For the adiabatic expansion, the ratio of volumes is the same as for the
adiabatic compression. That is, r =0.1887.
From (5) Art. 84, we have
which may be written in the form
Inserting the known values AB = 0.06021, a = 0.1544, 6 = 0.00003753,
T s = 3949, ^ = 0.1887, we get
log 7*4 = 3.7028  0.000105(5 T v
This equation may be .solved
graphically, as shown in Fig. 37. As
the value of !T, evidently lies between
2500 and 3000 we plot the curves
3.45
3.44
8.42
3.41
25X)
3.40 /
2600 2700
FIG. 37.
2800
?/ = ^g T,
and y = 3.7028  0.0001056 T^
between these limits. The intersec
tion gives the desired value,
T = 2049.
The external work of expansion is
/ row
W = J\ (0.1544 + 0.00003753 T)dT
Jwa ^ '
=287,940 ft.lb.
EXERCISES
The following are the compositions by volume of two gases, one a rich
natural gas, the other a blast furnace gas :
NATURAL GAS (Indiana)
H 2 0.02
CO 0.007
CH, 0.931
BLAST FURNAOM GAS
II Z 0.05
CO 0.27
CH 0.015
Work the following examples for each of these gases :
1. Find the composition, by weight.
2. Find the heating value :
(a) per cubic foot under standard conditions;
(6) per pound.
3. Calculate the constants B m , y, v, and c p c v .
4. Find the volume of air required for the combustion of one cubic
foot.
5. Find the weight of air required for the combustion of one pound.
6. Find the products of combustion, by weight.
7. Find the specific heat c,, of a mixture of the gas with air, the weight
of air being 15 per cent in excess of that required for complete combustion.
8. Find c v for the products of combustion, assuming that 15 per cent
excess of air is used.
9. Find the constants B m , y, and v of the mixture of Ex. 7; also of the
products of combustion.
10. The mixtiire of Ex. 7 is compressed adiabatically from a pressure of
14.7 Ib. per square inch to a pressure of 120 Ib. per square inch in the
case of the natural gas and to a piessure of 175 Ib. per square inch in the
case of the blast furnace gas. The initial temperature in each case is 80 F.
Find the temperature at the end of compression in each case.
11. Find the work of adiabatic compression.
12. Find the ratio of initial to final volume.
13. If at the end of adiabatic compression the mixture is ignited and
burns at constant volume, find the temperature at the end of the process,
assuming that no heat is lost by radiation.
14. After combustion the products expand adiabatically to the initial
volume. Calculate the final temperatures.
15. Find the work of adiabatic expansion.
16. Assume that the adiabatic compression follows the law p 7" = const.
Find the values of n. Find also the values of n for the adiabatic expansion.
REFERENCES
GAS MIXTURES
Preston : Theory of Heat, 350.
Bryan : Thermodynamics, 121.
Zeuner: Technical Thermodynamics (Klein) 1, 107.
Wevrannh : Grnndriss cler WanneTheorie 1, 137, 140.
FUELS. COMBUSTION. HEATING VALUES
Levin : Modern Gas Engine and Gas Producer, 80.
Carpenter and Diederichs : Internal Combustion Engines, 129.
Zeuner : Technical Thermodynamics 1, 405, 410.
Weyrauch : Grundriss der WiirmeTheorie, 216, 255.
Jones : The Gas Engine, 293.
Poole : The Calorific Power of Fuels.
In the field of thermochemistry reference may be made to the extei
sive researches of Favre and Silbermann, Berthelot, and J. Thomson. Fc
tables of heating values see Landolt and Bornstein : Physik.chemiscb
TabeUen.
VARIABLE SPECIFIC HEAT OF GASES
Mallard and Le Chatelier : Annales des Mines 4. 1883.
Vieille: Comptes rendus 96, 1358. 1883.
Langen : Zeit. des Verein. deutsch. Ing. 47, 022. 1903.
Haber : Thermodynamics of Technical Gas Reactions, 208.
Clerk: Gas, Petrol, and Oil Engines, 341, 301.
Zeuner: Technical Thermodynamics 1, 146.
Carpenter and Diederichs : Internal Combustion Engines, 220.
THERMODYNAMICS OF COMUUSTION
Zeuner: Technical Thermodynamics 1, 423, 428.
Lorenz : Technische Wiirmelehre, 392.
Stodola: Zeit. des Verein. deutsch. Ing. 42, 1045, 1086. 1898.
CHAPTER IX
TECHNICAL APPLICATIONS. GASEOUS MEDIA
86. Cycle Processes. In any heat motor, heat is conveyed
from the source of supply to the motor by some medium, which
thus simply acts as a vehicle or carrier. In practically all
cases the medium is in the liquid or gaseous state, though a
motor with a solid medium is easily conceivable. The perform
ance of work is brought about by a change in the specific
volume of the medium due to the heat received from the source.
By a proper arrangement of working cylinder and movable pis
ton this change of volume is utilized in overcoming external
resistances. (In the steam turbine another principle is em
ployed.) The medium must pass through a series of changes
of state and return eventually to its initial state, the series of
changes thus forming a closed cycle. To use a crude illustra
tion, the medium taking its load of heat from the source at high
temperature, delivering that heat to the working cylinder and
to the cold body (condenser) and returning to the source for
another supply may be compared with an elevator taking
freight from an upper story to a lower level and returning
empty for another load.
Where the medium is expensive it is used over and over, and
thus passes through a true closed cycle. Examples are seen in
the ammonia refrigerating machine and in the engines and
boilers of ocean steamers, in which fresh water must be used.
In such cases we may speak of the motor as a closed motor.
If the medium, on the other hand, is inexpensive or available in
large quantities, as air or water, open motors are quite generally
used. In these the working fluid is discharged into the atmos
phere and a fresh supply is taken from the source of supply.
Even in this case the medium mav pass through a closed cycle,
but all the changes of state are not completed in the organs of
the motor.
In this chapter we shall take up the analysis of several cycles
that are of importance in the technical applications of gaseous
media. In general, we shall assume ideal conditions, which
cannot be attained in actual heat motors. However, the con
clusions deduced from the analysis of such ideal cycles are
usually valid for the modified actual cycles ; furthermore, the
ideal cycle furnishes a standard by which to measure the effi
ciency of the actual cycle.
87. The Carnot Cycle. Although the Carnot cycle is of no
practical importance, it possesses the greatest interest from a
theoretical point of view. Hence an analysis of it is included.
Referring to Fig. 18, the heat absorbed from the source dur
ing the isothermal expansion AB is given by the equation
a log e , (1)
'a
and the heat rejected to the refrigerator is
77"
, = Av V loo LA (v\
Vcd "jfc ' c lu be rr " \^}
' c
The heat transformed into work is, therefore,
A W= Q a(i + Q cd = A( PU V a log fi p  p. V c log. IT). (3)
\ ' a I <;/
Since in the state A the temperature is T v we have
p a r a = MBT v (4)
and likewise p c V c = MB T 2 . (."> )
Furthermore, for the adiabatic BO we have the relation
and for the adiabatic DA the relation
~~ T'
K K
ART. 88] CONDITIONS OF MAXIMUM EFFICIENCY
135
Introducing in (3) the results given by (4), (5), and (8), we
obtain
whence
AW
Q t
ab
(9)
(10)
f n
This expression for the efficiency is identical with that deduced
from the Kelvin absolute scale of temperature. We have in
Eq. (10) a proof, therefore, that the Kelvin absolute scale coin
cides with the perfect gas scale.
E
D
r,
88. Conditions of Maximum Efficiency. On the SWplane
the Carnot cycle is the simple
rectangle ABCD (Fig. 38), hav
ing the isothermals AB and CD
at the temperatures T^ and T 2 of
the source and refrigerator, respec
tively. This geometrical rep
resentation affords an intuitive
insight into the property of maxi
mum efficiency. Between the
same isothermals let us assume
some other form of cycle, as the
trapezoidal cycle EB CD, For the
rectangular cycle the efficiency is
heat transformed into work_ area ABCD
heat supplied area A 1 ABB 1
For the trapezoidal cycle, likewise, the efficiency is
area DEBC
A,
#1
FIG. 38.
But
DEBC
area A l DEBB l
ABCDAED
~~ A^ABB^  AED
ABCD
that any cycle lying wholly within the rectangular cyle AB CD
has a smaller efficiency than the rectangular cycle.
With a given source and refrigerator, the conditions of maxi
mum efficiency, which may be approached but never actually
attained, are the following :
1. The medium must receive heat from the source at the
temperature of the source. No heat must be received at lower
temperature.
2. The medium must reject heat to the refrigerator at the
temperature of the refrigerator.
3. Provided the medium, source, and refrigerator are the
only bodies involved in the transfer of heat, it follows from 1
and 2 that the intermediate processes must be adiabatic, as any
departure from the adiabatic would mean passage of heat to
or from some body at a tem
perature different from either
the source or refrigerator.
89 . Isoadiabatic Cycles . Let
a cycle be formed with the iso
thermals AB and CD as in the
Carnot cycle, but with the
adiabatics replaced by similar
curves BC and AD (Fig. 39) ;
that is, curve BC is simply
^r ^ g s curve DA shifted horizontally
FIG. 39. a distance AB. Then AB =
DC, as in the Carnot cycle. If
the cycle is traversed in the clockwise sense, the heat entering
the medium is
Qda +Qab = area D 1 DAA 1 + area
while the heat rejected by the medium is
Qbc + Qcd = area B^B CC l + area 1 CDD r
The heat transformed into work is the same as in the Carnot
cycle, for the area of the figure ABCD is equal to that of the
r>. j. j. i ^T . ,. ., _ "
D 1 DAA 1 is taken from the source of heat, the efficiency of the
cycle is
_ heat transformed __ area ABQD
heat taken from source ~~ area D l DABB l '
and this is manifestly smaller than the efficiency of the Carnot
cycle. Let it be observed, however, that
V&c Qdal
that is, area B l BOO l = area D^AA^
If the heat rejected by the medium during the process BO
could be stored instead of thrown away, then this heat might
be used again during the process DA, thus saving the source
the heat Q da . In this case we should have the following series
of steps :
1. Medium absorbs heat Q^ from source.
2. Medium rejects heat Q be , which is stored.
3. Medium rejects heat Q cd to refrigerator.
4. Medium absorbs the heat Q da (= Q b J) stored during
step 2.
Since in this case the source furnishes only the heat Q&, the
efficiency is
area ABCD
77
area
which is the same as that of the Carnot cycle. A cycle in
which the adiabatics of the Carnot cycle are replaced by similar
curves, along which the interchanges of heat are balanced, is
called an isoadiabatic cycle. Any such cycle has the same ideal
efficiency as the Carnot cycle.
90. Classification of Air Engines. Heat motors that employ
air or some other practically perfect gas as a working fluid may
be divided into two chief classes : (1) Motors in which the fur
nace is exterior to the working cylinder, so that the medium is
heated by conduction through metal walls. (2) Motors in which
the medium is heated directly in the working cylinder by the
combustion of some gaseous or liquid fuel. These are called
internal combustion motors.
We mav make a, second division based on the manner in
which the working fluid is used. In the closedcycle type of
motor, the same mass of air is used over and over again, fresh
air being supplied merely to replace leakage losses. In the
opencycle type a fresh charge of air is drawn in at each stroke,
and after passing through its cycle is discharged again into the
atmosphere.
Air motors of the first class, namely, those with the furnace
exterior to the working cylinder, are usually designated as hot
air engines. Motors of this class are no longer constructed
except in small sizes for pumping and domestic purposes ; they
are, however, of historical interest, and besides they furnish in
structive illustrations of the application of the regenerative
principle. We shall, therefore, describe briefly the two leading
types of hotair engines and give an analysis of the cycles.
91. Stirling's Engine. The engine designed by Robert
Stirling in 1816, and bearing his name, is of the external fur
nace closedcycle type.
The general features of
the engine are shown in
IP Fig. 40. A displacer
piston Q works in a cyl
inder Q. Between and
an outer cylinder D is
placed a regenerator RR,
made of thin metal plates
or wire gauze. At the
upper end of the cylinder
is a refrigerator M, com
posed of a pipe coil through
which cold water is made
to circulate. At the lower
FIG. 40. end is the lire F. The
piston Q is filled with some
nonconducting material. The working cylinder A has free
communication with the displacer cylinder. In the actual
piston P to be at the beginning of its upward stroke and the
displacer piston at the bottom of its cylinder. The air is,
therefore, all in the upper part of the cylinder in contact with
the refrigerator, and its state may be represented by the point
D (Fig. 39). Now let the displacer piston be moved suddenly to
the upper end of its cylinder. The air is forced through JK
and the perforations in O into the lower end of the cylinder.
The air remains at constant volume, since the piston P has not
yet moved, and has received heat in passing through R. Hence
the change of state is a heating at constant volume represented
by DA in the diagram. The air now receives heat from the
furnace and expands at constant temperature during the up
ward working stroke of piston P, This process is represented
by AB. When the piston P reaches the upper end of its
stroke, the displacer piston Q is suddenly moved to the bottom
of the cylinder, thus forcing the air back through R into the
refrigerator M. This again is a constant volume change and is
represented by BO. Lastly, during the return stroke the air is
compressed isothermally, as represented by (7Z>, and heat is re
jected to the refrigerator.
The ideal cycle is seen to be an isoadiabatic cycle with
the adiabatics of the Carnot cycle replaced by constantvolume
curves. The cycle given by the actual engine deviates consid
erably from the ideal cycle on account of the large clearance
necessary between the two cylinders.
A double acting Stirling engine of 50 i. h. p. was used for
some years at the Dundee foundry, but was eventually aban
d.oned because of the failure of the regenerators. This
engine had an efficiency of 0.3 and consumed 1.7 Ib. of coal
per i. h. p.
92. Ericsson's Air Engine. The Swedish engineer Ericsson
made several attempts to design hotair engines of considerable
power. His large engines proved failures, however, because of
their enormous bulk and the rapid deterioration of the regener
ators. The engines for" the 2200ton vessel Ericsson had four
singleacting working cylinders 14 ft. in diameter and 6 ft.
stroke and ran at 9 r.p.m. They developed 300 li.p. with a
fuel consumption of 1.87 Ib. of coal per h.p.hour.
The working of the Ericsson engine was substantially as fol
lows : A pump compressed air at atmospheric temperature into
a receiver, whence it passed through the regenerator into a
working cylinder. The pump was waterjacketed so as to act
as a refrigerator. During the passage through the regenerator
the air was heated at constant pressure. After the air was cut
off in the working cylinder, it expanded isotherimilly, the nec
essary heat being furnished
by a furnace external to the
working cylinder. On the
return stroke the air was dis
charged through the regener
ator at constant pressure.
The p /^diagram is shown
in Fig. 41. The pump cycle
is DCJ?E, the motor cycle
JEAJBF. The operations are
as follows:
(1) Compression in pump from to D, heat abstracted by
pump waterjacket. (2) Discharge from pump to regenerator,
represented by DE. (3) Suction of air into working cylin
der represented by EA. (4) Isothermal expansion from A to
J9, during which air receives heat from furnace. (5) Dis
charge of air, represented by BF. (G) Suction of air into pump,
represented by FO.
Deducting the work of the pump from that of the motor, the
effective work is given by the diagram AB CD composed of the
two isothermals and two constantpressure lines.
93. Analysis of Cycles. The ideal cycles of the Stirling and
Ericsson engines are isoadiabatic cycles. In the Stirling cycle
the constantvolume lines DA and BO (Fig. 39) replace the
adiabatics of the Carnot cycle. Using the iWplane we have
Q* = Ap a V a log e ^ = ABTJf log. !J
' o I' a
n 7I/T, / m m \
ART. 94] HEATING BY INTERNAL COMBUSTION ' 141
Q cd = Ap c V c log, p =  A MBT Z log, f
But since F a = V d and F c = F 6 ,
The heat ^ is taken from a regenerator, and therefore the
heat Qa alone is supplied from the source ; hence the efficiency
s
" ft* " *i '
For the Ericsson cycle Z>J. and .#(7 are constantpressure
lines and the analysis is essentially the same except that c v is
replaced by c p .
94. Heating by Internal Combustion.* While the hotair
engine with exterior furnace should apparently be an efficient
heat motor, experience has proved the contrary. The difficulty
lies in the slow rate of absorption of heat by any gas. Even
with high furnace temperatures and comparatively large heat
ing surfaces it has been found impossible to get a high tempera
ture in the working medium. Furthermore, if the air could be
effectively heated, the metal surfaces separating the furnace from
the hot medium would be destroyed; hence, while high tempera
ture of air is necessary for high efficiency, low temperature is
necessary to secure the durability of the metal.
These contradictory conditions are completely obviated by
the method of heating by internal combustion. The rapid
chemical action supported by the medium itself makes possible
the rapid heating of large quantities of air to a very high
temperature. The medium and the furnace being within the
t.>io nntop onvFunA rvf t.Tip. mftta.l walls can be keT)t at
low temperature by a water jacket, and consequently the inner
surface can be exposed to the high temperature desired without
danger of destruction. .Furthermore, the low conductivity
of gases becomes here an advantage as it prevents a rapid flow
of heat from the medium to the cylinder walls. The low gas
temperature of the hotair engine results in a small effective
pressure and makes the engine very bulky for the power
obtained. The high temperature possible in the internal
combustion motor, on the other hand, permits high effective
pressures, and therefore gives a relatively small bulk per
horsepower.
95. The Otto Cycle. The cycle of the wellknown Otto
gas engine has five operations as follows :
1. The explosive mixture
is drawn into tho cylinder.
Represented by HI), Fig. 42.
2. The mixture is com
pressed, as represented by
DA.
3. The charge is ignited,
causing a rise of temperature
and pressure, as shown by AB.
4. The gases in the cyl
FIG. 42. inder expand adiabatically as
shown by BQ.
5. The burned gases are expelled in part. Represented
by DE.
In the case of the fourcycle Otto engine, each of the opera
tions 1, 2, 4, and 5 occupies one stroke of the piston, while
operation 3 occurs at the beginning of a stroke. The cycle
is completed in four strokes, whence the term fourcycle.
It is customary in the analysis of gasengine cycles to
assume in the first instance that the medium is pure air
throughout the cycle and that the air receives during the
process AB an amount of heat equal to that developed by the
combustion of the fuel in the actual cycle. This assumed ideal
CVfilfi is rpfprrorl f
VT, , _._ , _
ART. 95]
THE OTTO CYCLE
143
On the Titfplane, the ideal cycle has the form shown in
Fig. 43, AS and CD being constant volume curves. The
medium in the state repre
sented by point A is heated at
constant volume, as shown by
the curve AB. For this pro
cess we have (assuming that c v
is constant)
For the adiabatic expansion
represented by BO,
W 
~
For the cooling at constant volume, represented by (7.Z), we
have Q cd = Jfc,( T d  T c ) = 
Finally the medium is compressed adiabatically from D to J.,
and for this change of state
W 
rr
7 
K 1
The heat changed into work is
(1)
The work of the cycle is
W bc + W cd + W^
It is easily shown that these results are identical.
The efficiency is
/
TJBJUJlJNUJAJj Arr JUJLOA A
This expression for rj may be simplified as follows : From
Fig. 43 we have
S b  S a =S. S d = Mc v log. ? = Mc v log. ',
,
hence,
Therefore,
rn T T
Jt c ^v, L c _ *rf
 or
_
,
J. a *& x a
or
It appears, therefore, that the Otto cycle has the same efficiency
as a Carnot cycle having the extreme temperatures T a and T d
or the extreme temperatures T b and T of the adiabatics, but a
smaller efficiency than a Carnot cycle having T b and T d as
extreme temperature limits.
The expression for the ideal efficiency may be written in
another convenient form. Since the curve DA represents an
adiabatic process, we have
whence
11,
or
(5)
It appears from the last expression that the higher the com
pression pressure^, the greater the ideal efficiency.
If the ratio of volumes ~ be denoted by r* we have for the
T'
* a
ideal efficiency the expression
11 1 (6)
EXAMPLE. If the air is compressed from 14.7 Ib. to 45 ll>., the ideal
The temperature and pressure represented by the point B
are readily calculated for this ideal case. Let q l denote the
heat absorbed per pound of air during the process AB; then
whence ^A + l. (7)
* a Cv*a
Since F.= F 6 ,
The value of q l for a given fuel depends upon the heating
value of the fuel and the weight of air required for the com
bustion of a unit weight of the fuel.
96. The Joule or Brayton Cycle. In the Otto type of motor,
the fuel gas is mixed with air previous to compression, and
when the mixture is ignited the combustion is so rapid as
to produce an explosion; the heat is supplied, therefore, at
practically constant volume. Another type of motor was first
suggested by Joule and was developed in working form by
Brayton (1872). In the Bray ton engine the mixture of air
and gas was compressed into a reservoir to a pressure of per
haps 60 Ib. per square inch and from the reservoir flowed into
the working cylinder, where it was ignited by a flame. A wire
gauze diaphragm was used to prevent the flame from striking
back into the reservoir. The mixture was thus burned quietly
in the working cylinder during about one half the stroke of
the piston, and by proper regulation of the admission valve the
rate of combustion was so regulated as to give practically con
stant pressure during the period of admission. The ideal
cycle of operations is as follows:
1. Charge drawn, into compressor cylinder, ED (Fig. 44).
2. Adiabatic compression, DA.
146 TECHNICAL APPLICATIONS. GASEOUS MEDIA [CHAP, ix
3. Expulsion at constant pressure from compressor, AF;
simultaneous admission to motor cylinder, FB, The charge
during the passage from
compressor to motor is
heated at constant pres
sure and the volume is
thereby increased as in
dicated by AB,
4. Adiabatic expansion,
BC, after cut off.
5. Expulsion of burned
T , A , gases, OE.
FIG. 44. b '
The area JEDAF repre
sents the negative work of the compressor, the area FBQJH
the work obtained from the motor ; hence, area ABCD repre
sents the net available work.
On the T/S'plane, the ideal Joule cycle has the same form as
the Otto cycle (Fig. 43). The curves AB and (72), however,
represent, respectively, heating and cooling at constant pressure.
We have, therefore,
= Q ab + Q cd =
.,__?''
\ T:
Also, ~ = .
0)
(2)
(3)
(5)
97. The Diesel Cycle. The principle of gradual and quiet
combustion as opposed to explosion was seized upon by Diesel
in the design of the Diesel motor. In this motor air without
fuel is compressed in the working cylinder to a pressure ap
proximating 500 Ib. per square inch. The temperature at the end
of compression is consequently higher than the ignition tempera
FIG. 45.
expand at practically constant
pressure, or if desired, with
falling pressure and nearly
constant temperature. As in
the Brayton engine, govern
ing is effected by cutting off
the fuel injection earlier or
later.
The ideal cycle of the Diesel o
engine is shown in Fig. 45. It
resembles the Otto cycle except
that the process AS in this case represents a constant pressure
rather than a constant volume combustion. It was the original
aim of Diesel so to regulate the
injection of fuel that a short
period of combustion AM
would be followed by isother
mal expansion JfZV, the fuel
being cut off at the point N.
On the 2!$plane the ideal
Diesel cycle is shown in
Fig. 46, in which AB is a
constantpressure curve and
CD a constantvolume curve.
We have then
(1)
(2)
' (3)
FIG. 46.
c p (T b T a )
If the cycle includes an isothermal process, as MN, we have
Q am = Mc p ( T m  T a }, (5)
V,
f~\ A ]\/rj2 fT* Inrr . _
*Vmn == jti.JLrj.JL> J m. J U ^P tr 1
<,*, o* Y^
?n+ <?,
and 77 = ^
:=1
(6)
a)
FIG. 47.
98. Comparison of Cycles. The three principal cycles are
shown superimposed in Fig. 47. The minimum temperature
at J) and maximum temperature at B are the same for all
three. With this assumption
it is seen that the Bray ton
cycle A'BC'D has the largest
area, the Otto cycle ABGD,
the smallest. Hence, between
the same temperature limits
and with the same maximum
pressure jp 6 , the Bray ton cycle
is the most efficient, the Otto
cycle the least efficient. Com
$ paring the maximum volumes,
it is seen that the Otto and
Diesel cycles have the same
maximum volumes V& while the Bray ton cycle requires a
greater volume, as indicated by the point O 1 '. The Diesel
cycle, therefore, combines the advantages of the high efficiency
of the Brayton cycle due to the high compression pressure
and the smaller cylinder volume of the Otto cycle.
99. Closer Analysis of the Otto Cycle. In the preceding
analysis of gasengine cycles two assumptions have been made :
(1) That the medium employed has throughout the cycle the
properties of air. (2) That the specific heat of the medium is
constant. While the approximate analyses based on these
assumptions are of value in giving the essential characteristics
of the various cycles, and an idea of their relative efficiencies,
they give misleading notions regarding the absolute magnitudes
of those efficiencies. To obtain the true value of the maximum
possible efficiency of a gasengine cycle, it is necessary to take
account of the properties of the fuel mixture entering the cylin
der and of the mixture of the products of combustion after the
fuel is burned. Making use of the principles and methods
laid down in Chapter VIII, we may thus make an accurate
nislied by the example of Art. 85, shows such an analysis for
the Otto cycle.
EXAMPLE. Determine the ideal efficiency of an Otto cycle in which the
compression, combustion, and expansion follow the course described in the
example of Art. 85. Compare this efficiency with the "air standard"
efficiency under the same conditions.
In the example quoted, the work of adiabatic compression was found to
be 69,460 ft.lb., the work of expansion 287,940 ft.lb. These results refer
to 1 Ib. of the fuel mixture. The heating value of the fuel per pound was
found to be 1632.2 B. t. u. ; hence the heating value per pound of fuel mix
ture is 1632.2  2.25 = 725.4 B. t. u. The net work derived from the cycle
per pound of mixture is 287,940  69,460 = 218,480 ft.lb. Therefore, the
efficiency is
Q.387.
J x 72o.4
The "air standard" efficiency depends upon the ratio of initial and final
Vz
volumes, which ratio was found to be = = 0.1887. Hence, for this efficiency
1 i
we have rj = 1  0.1887* = 0.487.
The discrepancy between the two efficiencies is in a large measure due to
the assumption of constant specific heat in. the analysis of Art. 95.
100. Air Refrigeration. The term refrigeration is applied
to the process of keeping a body permanently at a temperature
lower than that of surrounding bodies. Since heat naturally
flows from the surroundings to the body at lower temperature,
this heat must be continually removed if the body is to remain
permanently at its lower temperature. Hence a refrigerating
machine has the office of removing heat from a body of low
temperature and depositing it in some other convenient body
of higher temperature.
The operation of a refrigerating machine is thus precisely
the reverse of the operation of the directheat motor ; and if
the cycle of a heat motor be traversed in the reverse direc
tion, it will give a possible cycle for a refrigerating machine.
When air is used as a medium for refrigeration, the reversed
Joule cycle is employed. Fig. 48 shows diagrammatically the
arrangement of the refrigerating machine, Fig. 49 the ideal
j? ^diagram, and Fig. 50 the ^diagram. Air in the state A
in the cold room is drawn into the compressor c and is com
150 TECHNICAL APPLICATIONS. GASEOUS MEDIA [CHAP, ix
pressed adiabatically as indicated by AB. It then passes into
the cooling coils, about which cold water circulates, and is
cooled at constant pressure, as indicated by BO. In the state
the
>
ff
f
U ' <
'Jill
IB
3
f
It
C Cooling 1
Cold
J
w
3
s~
N
/
>
C Coils
Room
4
1"
 /
1
J
f
III!
Illl
Illllllll
1
J ,
Ij, ..^
U.
J_
_Li
j
FIG. 48.
air passes
into the expansion
cylinder e and is
permitted to ex
pand adiabatically
down to the pres
sure in the cold
room, i.e. atmos
pheric pressure.
The final state is
represented by
point D. Finally the air absorbs heat from the cold room, and
its temperature rises to the original value T a . Referring to
Fig. 49, the actual compression diagram is ABFE, while the
diagram JFCDJE taken clockwise is the diagram of the expan
sion cylinder. The net work done on the air is, therefore,
given by the diagram ABOD.
The Allen denseair machine has a closed cycle and the air
is always under a pressure much higher than that of the atmos
phere. Thus the pressure DA (Fig. 49) is perhaps 40 to t>0,
and the upper pressure, say
200 Ib. per square inch. The
air, after expanding to the
lower pressure, is led through
coils immersed in brine and
absorbs heat from the brine.
In the following analysis of
the airrefrigerating machine E
we shall assume ideal condi
tions. In the actual machine " JT IO . 49.
these conditions are to some
ART. 100]
AIR REFRIGERATION
151
minute, and M the weight of air
circulated per minute. Then
since in passing through the cold
body the temperature of the air is
raised from T d to T a (Fig. 50), we
have
Q = Mc p (T a T^. (1)
Let pL denote the suction pres
sure of the compressor cycle
(atmospheric pressure, in the
case of the open cycle) and p z
the pressure at the end of com
pression ; then, assuming adiabatic compression, we have
FIG. 50.
~T ~~ ( ~rT I ' (Q
a \ J f'l /
and if the pressure at cutoff in the expansion cylinder is also
p z (as in the ideal case), we have also
(3)
(4)
(5)
$\)
whence _ = .
The work required per minute is
rn m
rn J b~ La
" ~ " v " area O l DAB l * T a '
and the heat rejected to the cooling water, represented by the
area B l BGO l (Fig. 50), is
W T/,
The compressor cylinder draws in per minute M pounds of air
having the pressure p l and temperature T a , Denoting by N
the number of working strokes per minute and by V c the volume
displaced bv the comnressor mston. we have for the ideal case
152 TECHNICAL APPLICATIONS. GASEOUS MJSJUJLA ICHAP.IX
or
Likewise, the volume V e of the expansion cylinder is given by
the relation
EXAMPLE. An airrefrigerating machine is to abstract GOO B. t. u. per
minute from a cold chamber. The pressure in the cold room is 14.7 Ib. per
square inch, and the air is compressed acliabatically to 05 Ib. per square inch
absolute. The temperature in the cold room is 30 F. and the air leaves the
cooling coils at 80 F. The machine makes 120 working strokes per minute.
Kequired the ideal horsepower required to drive the machine, and the volumes
of the compression and expansion cylinders.
The first step is the determination of the temperature T d at the end of
expansion. From the relation
0.4
we have T d = 539.0 (iM ) " = 352.9.
From (1) we obtain for the weight of air that must be circulated per minute
M = Q _ ^ __ = 17r,'Ml,
c P (T a T d ) 0.24(405.0  352.9)
The work required per minute is
W = JQ T "~ Td = 778 x GOO x 5:j9  6 ~ 352  <} = 240,950 ft. Ib. ,
T d 352.9 ' '
and the horsepower under these ideal conditions is therefore
246950 _ 7 P
33000 ''
For the volume of the compressor cylitider, we have
v 17.52 x 53.34 x 495.0 . U0 ..
Fc= 120x14.7x144 = Lb2c " ft >
and for the volume of the expansion cylinder
ny in mining, tunneling, ana metallurgical processes,
impression of air may be effected by rotary fans and
s or by piston compressors. In the piston compressor,
itmospheric pressure is drawn into a cj'linder through, in
ves and is then compressed upon the return stroke of the
When the desired pressure is attained, the outlet valves
sued and the air is discharged into a receiver. The ideal
or diagram of an air
p
;ssor has, therefore, the
hown in Fig. 51. The c
4. represents the drawing
le air ; the curve AB rep
i the compression from
wer pressure p 1 to the
sr pressure jt? 2 ; and BO D
jnts the expulsion of the _
the higher pressure. It FlG> 51 _
be noted that the curve
ipresents a change of state, while lines DA and BO
nit merely change of locality ; thus BQ represents the
D of the air (in the same state} from the compressor
31* to the receiver.
V^ denote the volume denoted by point A, and V 2 the
} after compression ; then the work of compression (area
B is
'"* nl
ng that the compression curve follows the law pV n = const.
)rk of expulsion (represented by area B^BOO) is evidently
3 work done by the air during the intake (area ODAA^) is
the total work represented by the area of the diagram
n
(1)
1
V V I I
2 ~ x W '
whence combining (1) and (2) we get
(2)
(3)
a formula that does not contain the final volume Y v
For the temperature at the
end of compression we have the
usual formula
(4)
The action of the air com
pressor may be studied advanta
geously by means of the T8
diagram. Let the point A (Fig.
52) represent the state of the
air at the beginning of com
pression, and suppose that AB
represents the compression pro
cess. Through B a line repre
senting the constant pressure
p z is drawn, intersecting at F an isothermal through A. It
can be shown that the area A l ABFF l represents the work W
given by (1). Denoting by T 2 the final temperature corre
sponding to point B, we have
area A l ABS l = Mo v ^ (T 2  TJ,
FJG> 52>
area
Mc
area A l ABFF l =
n c ~
n 1
1 n
B
102. Water jacketing. Unless some provision is made for
withdrawing heat during the compression, the temperature will
rise according to the adiabatic law. Ordinarily the energy
stored in the air due to its increase of temperature, that is, the
energy U 2  U,= Mc^T.T^
is never utilized because during the transmission of the air
through the mains heat is lost by radiation and the temperature
falls to the initial value. Hence
a rise in the temperature during
compression indicates a useless
expenditure of work. The water
jacket prevents in some degree
this rise in temperature and
decreases the work required for
compression. The curve AE
(Fig. 53) represents adiabatic
. TP ,
compression. If the compres
sion could be made isothermal, the curve would be AF, less
steep than AE, and the work of the engine would be reduced
per stroke by the area AEF. The water jacket gives the curve
AS lying between AE and AF, and the shaded area represents
the saving in work. Because of the water jacket the value of
the exponent n in the equation p V n = const, lies somewhere
between 1 and 1.40. Under usual working conditions, n is
about 1.8.
For any value of n the relation between the heat abstracted,
work done, and change of energy is given by the proportion
JQ:(U 2  ZTj) : TF= (k  n) : (1  n) : (k  1).
This applies only to the compression AB not to the expulsion
of the air represented by B 0.
The influence of the water jacket is shown more clearly by
the ^diagram, Fig. 52. The vertical line AE indicates adia
batic compression from p l to jp 2 , the horizontal line AF, isother
mal compression, and the intermediate curve .&., compression
according to the law p V n const., with n between 1 and 1.4.
The area A^ABB^ represents the heat abstracted from the air
during compression, and the area AEB represents the work
saved by the use of the jacket. A more efficient jacket would
give a compression curve with its extremity lying nearer the
point F. In the case of the isothermal compression represented
by AF, the area A l AFF l represents the heat absorbed from the
air and also the work done on the air. These must necessarily
be equivalent, since there is no change in the internal energy.
103. Compound Compression. The excess of work required
by the increase of temperature during compression may be obvi
ated in some measure by
dividing the compression
into two or more stages.
Air is compressed from
the initial pressure p 1 to
an intermediate pressure
p', it is then passed
through a cooler where
the temperature (and con
sequently the volume) is
FlGt 54 ' reduced, and finally it is
compressed from p' to the desired pressure p T In Fig. 54,
DA represents the entrance of air into the cylinder, and A 6r,
which lies between the adiabatic AE and the isothermal AF,
the compression in the first cylinder. From Gr to Jt the air is
cooled at constant pressure in the intercooler. The curve HL
shows the compression in the second cylinder, and the line
LQ the expulsion into the receiver. In a single cylinder the
diagram would be ABQD ; hence compounding saves the work
indicated by the area B CrHL.
The saving is shown even more clearly if we use the TS
plane (Fig. 55). During the first compression AGr the heat
represented by the area A 1 AGrGr 1 is absorbed by the water
jacket. Then the heat G^GrHH^ is abstracted by the inter
cooler. During the second compression the heat HMLL, is
ART. 103]
COMPOUND COMPRESSION
157
abstracted by the water jacket,
and finally the heat
is radiated from the receiver
and main. As shown in the
preceding article, the area
A l AGHLFF l gives the work
of the compressor. Evidently
area BGrSL represents the
work saved by compounding.
If we take (3) of Art. 101,
we find for the work done in
the first cylinder
FIG. 55.
and for the work done in the second cylinder
nl
n
p
i S
where V is the volume indicated by point H (Fig. 54).
But since point .ffis on the isothermal AF> we have
and, therefore,
nl
P
The total work is, consequently,
nl
l\ n
(1)
The work required is numerically a minimum when the
is variable. Using the ordinary method of the calculus, we
find that this expression is a maximum when
P 1 =
Equation (2) is useful in proportioning the cylinders of a com
pound compressor.
Referring to Fig. 55, we have
With the condition expressed by (2) we have
nl n1
and likewise,
nl n~l
\PiPzJ \Pi,
Hence, T l = T a ;
that is, for a minimum work of compression the points G and L
should lie on the same temperature level. The same statement
applies to threestage compression.
104. Compressedair Engines. Compressed air may be used
as a working fluid in a motor in substantially the same way
as steam. In fact, compressed air
has in some instances been used
in ordinary steam engines. The
indicator diagram for the motor
should approach the form shown
in Fig. 50. With clearance and
compression, A. 1 12' will replace
FlG 56 AE. The work per stroke is
readily calculated in either case.
The expansion curve BO may be taken as an adiabatic.
105. TSdiagram of Combined Compressor and Engine. The
2Wdiagram shows clearly the losses in a compressedair system
losses. In the following discussion we shall take up first an
ideal case and afterwards several modifications that may be
made.
In Fig. 57, m represents the compressor diagram, n the
motor diagram, both without clearance. Air in the state repre
sented by point A is
taken into the com
pressor at atmos
pheric pressure and
temperature. The
compression, a s
sumed here to be
adiabatic, is repre
sented on the TS
plane by the vertical line AB (Fig. 58). The expulsion of
the air into the receiver and thence into the main is merely a
change of locality and does not itself involve any change of
state ; hence, it is not represented on the ^fWplane. However,
the passage of the air along the main is usually accompanied
by a cooling, and this is represented by BQ (Fig. 58), the final
point representing the state of the air at the beginning of
expansion in the motor. The adiabatic expansion to atmos
pheric pressure in the motor is
represented by CD. This is
accompanied by a drop in tem
perature which is given by the
equation
kl
A
T
FIG. 58.
The air discharged from the motor
in the state D is now heated at
the constant pressure of the atmos
phere until it regains its original temperature T a . This heating
is represented by DA.
The complete process is a cycle of four distinct operations,
L ^ ,,,, ,, . (l.r.f
what does the area AJUJJU ot tne cycle represent sometnmg
useful or something wasteful ? To answer this question let us
recur to the original energy equation
JQ = Z7 2  Z/i + W,
and apply it to the air which passes through the cycle process
just described. We have
Work done on air = area of diagram m = W m .
Work done by air = area of diagram n + W n .
Total work = W n  W m .
Heat absorbed by air = area under DA.
Heat rejected by air = area under SO.
Total heat put into system = area ABQD.
Change of energy = U a U a 0.
Hence, j x ^ AB Q]) = ^ _ ^
that is, the area ABQD represents the difference between the
work done by the compressor and the work delivered by the
motor. Consequently it
represents a waste, which
is to be avoided as far as
possible.
Various modifications
of the simple cycle of
Fig. 58 are shown in
Fig. 59. The effect of
using a water jacket is
shown at (a). The
shaded area represents
the saving.
Figure 59 (7>) shows
the effect of reheating
the air before it enters the motor. In the main the air cools,
as indicated by BO, but in passing through the reheater it is
heated again at constant pressure, and the state point retraces
its path, say to D. Then follows adiabatic expansion DE, and
constantpressure heatins EA. This rpVmntincr RH.VPR work
B
(d)
FIG. 59.
vy LUC area \JJJJUM. xo wuuiu ue pussiiuie TO carry
D to the right of B, in which case the waste would "become
zero or even negative. The area CDJ3]? does not, however,
represent clear gain, as account must be taken of the heat
expended in the process CD.
In Fig. 59 (c) is shown the effect of compound compression,
and in Fig. 59 (c?) the effect of compound compression with
a compound motor. In each case the shaded area represents
the saving.
It would not be difficult to represent also the loss of pressure
in the main due to friction.
EXERCISES
1. Find the efficiency of a Stirling hotair engine "working under ideal
conditions between the temperatures 1340 F. and 140 F. Find the weight
of air that must be circulated per minute per horsepower.
2. An air compressor with 18 in. by 24 in. cylinder makes 140 working
strokes per minute and compresses the air to a pressure of 52 Ib. per square
inch, gauge. Assuming that there is no clearance, find the net horsepower
required to drive the compressor. Take the equation of the compression
curve as p V 1  3 = const.
3. If 200 cu. ft. of air at 14.7 Ib. is compressed to a pressure of 90 Ib. per
square inch, gauge, find the saving in the work of compression and expulsion
by the use of a water jacket that reduces the exponent n from 1.4 to 1.27.
4. Find the efficiency of the ideal Otto cycle (air standard) when the
compression is carried to 120 Ib. per square inch absolute.
5. Draw a cxirve showing the relation between the efficiency of the Otto
cycle and the compression pressure. Take values of p from 40 to 200 Ib.
per square inch.
6. An airrefrigerating machine takes air from the cold chamber at a
pressure of 40 Ib. per square inch and a temperature of 20 F., and com
presses it adiabatically to a pressure of 200 Ib. per square inch. The air
is then cooled at this pressure to 80 F. and expanded adiabatically to
40 Ib. per square inch, whence it passes into the coils in the cold chamber.
The machine is required to abstract 45,000 B. t. u. per hour from the cold
room, (a) Find the net horsepower required to drive the machine. () If
the machine makes 80 working strokes per minute, find the necessary
cylinder volumes.
7. Air is to be compressed from 14.7 Ib. per square inch to 300 Ib. per
square inch absolute. If a compound compressor is used, find the interme
diate pressure that should be chosen.
8. In Ex. 7, the compression in each cylinder follows the law p F 1  3 =
;onst. Find the saving in work effected by compounding, expressed in per
uent of the work required of a single cylinder.
9. Using the results of Ex. 1015 of Chapter VIII, find the efficiencies of
the Otto cycle with the natural gas and the blast furnace gas, respectively,
under the conditions stated. Compare these efficiencies with corresponding
air standard efficiencies.
10. On the TSplane draw accurately an ideal Diesel cycle from the fol
lowing data: Adiabatic compression of air from 14.7 to 500 Ib. per square
inch absolute ; heating at constant pressure to a temperature of 2200 F. ;
idiabatic expansion to initial volume ; cooling at constant volume to initial
state. Calculate the ideal efficiency of the cycle.
11. Modify the Diesel cycle of the preceding example by stopping the
jonstantpressure heating at 1600 F. and continuing with an isothermal
jxpansion (as shown by MN, Fig. 40). Calculate the efficiency of this
modified cycle.
12. The ideal Lenoir cycle has three operations, as follows : heating of air
it constant volume, adiabatic expansion to initial pressure (atmospheric), and
jooling at constant pressure. Show the cycle on pV and TSplanes, and
lerive an expression for its efficiency.
13. Let the expansion in the Otto cycle be continued to atmospheric
pressure. Show the resulting cycle on pV and T'Splanes and derive an
jxpression for the efficiency.
REFERENCES
HOTAIR ENGINES
Snnis : Applied Thermodynamics for Engineers, 129.
5euner : Technical Thermodynamics (Klein) 1, 340.
Stankine: The Steam Engine (1897), 370.
Swing : The Steam Engine, 402.
GASENGINE CYCLES
"lerk : Gas, Petrol, and Oil Engines, 67.
Carpenter and Diederichs : Internal Combustion Engines, 65.
^evin : The Modern Gas Engine, 43.
terry: The Temperature Entropy Diagram, 107.
Sum's : Applied Thermodynamics, 154.
'eabody: Thermodynamics of the Steam Engine, 5th ed., 304.
jorenz : Technische Warmelehre, 421.
Veyrauch : Grundriss der WarmpTViPmMa 077
AIR REFRIGERATION
Ennis : Applied Thermodynamics, 396.
Ewing : The Mechanical Production of Cold, 38.
Peabody : Thermodynamics of the Steam Engine, 5th ed., 397.
Zeunev: Technical Thermodynamics, 384.
AIR COMPRESSION
Peabody : Thermodynamics, 5th ed., 358.
Ennis : Applied Thermodynamics, 96.
CHAPTER X
SATURATED VAPORS
106. The Process of Vaporization. The term vaporization
may refer either (1) to the slow and quiet formation of vapor
at the free surface of a liquid or (2) to the formation of vapor
by ebullition. In the latter case, heat being applied to the
liquid, the temperature rises until at a definite point vapor
bubbles begin to form on the walls of the containing vessel and
within the liquid itself. These rise to the liquid surface, and
breaking, discharge the vapor contained in them. The liquid,
meanwhile, is in a state of violent agitation. If this process
takes place in an inclosed space as a cylinder fitted with a
movable piston so arranged that the pressure maybe kept
constant while the inclosed volume may change, the following
phenomena are observed:
1. With a given constant pressure, the temperature remains
constant during the process ; and the greater the assumed pres
sure, the higher the temperature of vaporization. The tempera
ture here referred to is that of the vapor above the liquid. As a
matter of fact, the temperature of the liquid itself is slightly
greater than that of the vapor, but the difference is small and
negligible.
2. At a given pressure a unit weight of vapor assumes a
definite volume, that is, the vapor has a definite density;
and if the pressure is changed, the density of the vapor changes
correspondingly. The density (or the specific volume) of a
vapor is, therefore, a function of the pressure.
3. If the process of vaporization is continued at constant
pressure until all the liquid has been changed to vapor, then if
heat be still added to the vapor alone, the temperature will rise
and the specific volume will increase ; that is, the density will
decrease.
164
uaoo 10 ociiv^. i/u uo oenuiaicu. emu. line uuJJ.UttiJU lit;m]Jt;ra/UUI.e
sponding to the pressure at which the process is carried on is
the saturation temperature. If no liquid is present, and through
absorption of heat the temperature of the vapor rises above the
saturation temperature, the vapor is said to be superheated.
The difference between the temperature of the vapor and the
saturation temperature is called the degree of superheat.
The process just described may be represented graphically
on the jt?F"plane. See Fig. 60. Consider a unit weight of
liquid subjected to a pressure p represented by the ordinate of
the line A' A 1 ' ; and let the
volume of the liquid (de p
noted by ') be represented
by A'. As vaporization
proceeds at this constant
pressure, the volume of
the mixture of liquid and
vapor increases, and the
point representing the
state of the mixture moves
along the line A' A" . The
point A' r represents the
volume v" of the saturated
vapor at the completion
segment A' A" represents
FIG. 60.
of vaporization ; therefore, the
the increase of volume v" v'.
Any point between A' and A'\ as M, represents the state
of a mixture of liquid and vapor, and the position of the
point depends on the ratio of the weight of the vapor to
the weight of the mixture. Denoting this ratio by x, we have
x f n , whence it appears that at A', 3 = 0, while at A",
3 = 1. This ratio x is often called the quality of mixture.
If the mixture is subjected to higher pressure during vapor
ization, the statepoint will move along some other line, as B'B".
The specific volume indicated by B" is smaller than that indicated
by A". The curve v", giving the specific volumes of the satu
rated vapor for different pressures, is called the saturation curve ;
while the curve v\ giving the corresponding liquid volume, is
the liquid curve. These curves v', v" are in a sense boundary
curves. Between them lies the region of liquid and vapor
mixtures, and to the right of v" is the region of superheated
vapor. Any point in this latter region, as Z7, represents a state
of the superheated vapor.
107. Functional Relations. Characteristic Surfaces. For a
mixture of liquid and saturated vapor, the functional relations
connecting the coordinates jp, v, and t are essentially different
from the relation for a permanent gas. As explained in the
preceding article, the temperature of the mixture depends
upon the pressure only, and we cannot, as in the case of a
gas, give p and t any values we choose. The volume of a unit
weight of the mixture depends (1) upon the specific volume
of the vapor for the given pressure and (2) upon the quality
x. Hence we have for a mixture the following functional
relations :
* = /GO, orp = ^(0, (1)
v= ( f}(p,x'). (2)
With superheated steam, as with gases, p and t may be
varied independently, and consequently the functional relation
between p, v, and t has the general form
, , = 0. (3)
The characteristic surface of a
saturated vapor is shown in Fig. 61.
It is a cylindrical surface iS whose
generating elements cut the piplane
in the curve p = F(t')> These ele
t ments are limited by the two space
curves v f and v", which when pro
jected on the jwplane give the
curves v', v" of Fig. 60. The space
curve v" is the intersection of the
iuo. Relation oetween Pressure and Temperature. The rela
tion p = ]?() between the pressure p and temperature t of a
saturated vapor must be determined by experiment. To Reg
nault are due the experimental data for a large number of
vapors. Further experiments on water vapor have been made
by Ramsey and Young, by Battelli, and very recently by Hoi
born and Henning. These lastmentioned experiments were
made with the greatest accuracy and with all the refinements
of modern apparatus; they may, therefore, be regarded as
furnishing the most reliable data at present available on the
pressure and temperature of saturated water vapor. Experi
ments on other saturated vapors of technical importance, carbon
dioxide, sulphur dioxide, ammonia, etc., have been made by
Amagat, Pictet, Cailletet, Dieterici, and others. It is likely,
however, that further experiments must be made before the
data for these vapors are as reliable as those for water vapor.
If the experimentally determined values of p and t be plotted,
they will give the curve whose equation is p = f(t) (Fig. 61),
To express this relation many formulas have been proposed,
some purely empirical, some having a more or less rational
basis. A few of these formulas are the following :
1. Siot's Formula. As used by Regnault, Biot's equation
has the form
log p = a ba n + c/3 n , (1)
where n = t C.
This formula is purely empirical. Having five constants, the curve
may be made to pass through five experimentally determined
points.; hence, the formula may be made to fit the experimental
values very closely throughout a considerable range. The follow
ing are the values of the constants as given by Prof. Peabody :
FOR STEAM FROM 32 TO 212 F., p FOR STEAM FROM 212" TO 428 F., p
IN POUNDS PER SQUARE INCH. IN POUNDS PER S<JUAUE Iscu.
a =3.125906 a = 3.743976
log b == 0.611740 log 5 = 0.412002
log c = 8.13204  10 log c = 7.74168  10
log a = 9.998181  10 log a = 9.998562  10
log /3 = 0.0038134 log/3= 0.0042454
n = t  32 n = t212
2. Rankings Formula. Rankine proposed an equation of
the form 7? ,7
log^=JL + +^ 2 , (2)
in which T denotes the absolute temperature. This formula
has been much used in calculating steam tables, especially in
England. Having but three constants, it is not as accurate
as the Biot formula. The following are the values for the
constants, when p is taken in pounds per square inch, and
.4 = 6.1007; B = 2719.8; (7=400125.
3. The DuprHertz formula has the form
ablogT~ (3)
This equation has been derived rationally by Gibbs, Bertrand,
and others, and gives, with a proper choice of constants, results
that agree well with experiment. Using the results of Reg
nault's experiments, Bertrand found the following values of the
constant for various vapors (metric units).
ale
Water 17.44324 3.8682 2795.0
Ether 13.42311 1.9787 1729.97
Alcohol 21.44687 4.2248 2734.8
Chloroform 19.29793 3.9158 2179.1
Sulphur dioxide .... 16.99036 3.2198 1604.8
Ammonia 13.37156 1.8726 1449.8
Carbon dioxide .... 6.41443  0.4186 819.77
Sulphur 19.1074 3.4048 4684.5
4. Bertrand 's Formulas. Bertrand has suggested two equa
tions, namely : ,
* Wo.
 7  ^ eo
and p^k^Tiy. (5)
The latter may be written in the more convenient form
log p = log k n log (6)
Bertrand's second formula (6) has the advantage over the
others suggested of lending itself to quick and easy computa
tion. Furthermore, although it has but three constants, it
gives results that agree remarkably well with the experiments
of Holborn and Henning on water vapor. The constants are
as follows (English units) :
T=t + 459.6
n = 50.
FROM 32  90 F.
FROM 00  23T F.
FROM 238 420 F.
6 = 140.1 6 = 141.43 6 = 140.8
log&= 6.23167 log& = 6.30217 log 7c = 6.27756
The agreement between observed and calculated values is
shown in the following table. The maximum difference is
one tenth of one per cent.
TEMPERATURE, C.
PRESSURE IN MM. OF MERCURY
Bertrand's Formula
tWflj
// tsf^e
Experiments of
Holborn and Kenning
'LjLS^et^s.fft**.
4.577
4.579
10
9.208
9.205
20
17.511
17.51
30
31.682
31.71
40
55.121
55.13
50
92.325
92.30
60
149.21
149.19
70
233.55
233.53
80
354.97
355.1
90
525.64
525.8
100
760
760
110
1075.2
1074.5
120
1489.7
1488.9
130
2025.2
2025.6
140
2708.3
2709.5
150
3566.7
3568.7
160
4631.1
4633
170
5935.2
5937
180
7515
7514
190
9409.1
9404
200
11658
11647
5. Marks' Equation. Professor Marks nas deduced an
equation that gives with remarkable accuracy the relation
between ? and ^throughout the range 32 F. to 706.1 F., the
latter temperature being the critical temperature, as established
by the recent experiments of Holbom and Baumami. The
form, of the equation is
log p = a tcT+eT*. (7)
The constants have the following values: a = 10.515354, 1
4873.71, c = 0.00405096, e = 0.000001392964.
109. Expression for ^ In the ClapeyronClausius formula
dt
/Jin
for the specific volume of a saturated vapor, the derivative _
dt
is required. An expression for this derivative is obtained by
differentiating any one of the equations (1) to (7) of Art. 108.
Thus from (6),
dp _ ( 1 1 \ _ nip . , i N
dt ~ np \T^b ~ Tj ~ T(T b)' ^ )
whence
log & = log nb + logp  log T  log (S 7  5).
dt
Values of ^ are readily calculated since the terms log T,
dt
log (_T 5), and log p appear in the calculation of p from (6).
110. Energy Equation applied to the Vaporization Process.
It is customary in estimating the energy, entropy, heat content,
etc., of a saturated vapor to assume liquid at 32 F. (0C.) as a
datum from which to start. Thus the energy of a pound of
steam is assumed to be the energy above that of a pound of
"water at 32 F.
Suppose that a pound of liquid at 32 is heated until its
temperature reaches the boiling point corresponding to the
pressure to which the liquid is subjected. The heat required
is given by the equation
where c' denotes the specific heat of the liauid. This process
LRT. 110]
VAPORIZATION PROCESS
171
s represented on the ^fitfplane by a curve AA' (Fig. 62).
Che ordinate OA represents the initial absolute temperature
52 + 459.6 = 491.6, the ordinate A^A! the temperature of va
)orization given by the relation = /(j?), and the area OAA'A l
,he heat q' absorbed by the liquid. This heat q' is called the
ieat of the liquid.*
When the temperature of vaporization is reached, the liquid
)egins to change to vapor, the temperature remaining constant
luring the process. A definite quantity of heat, dependent
ipon the pressure, is required to change the liquid completely
nto vapor. This is called the
ieat of vaporization and is de
loted by the symbol r. In Fig.
52, the passage of the state
)oint from A' to A" represents
;he vaporization, and the heat
is represented by the area
^A'A'Ay For a higher pres
iure the curve AB' represents
.he heating of the liquid and
ihe line B' B" the vaporization.
During the heating of the
iquid the change in volume is
small and may be neg
A'f
M
A 1 B 1
MT
FIG. 62.
ected ; hence, the external work done is negligible also, and
ubstantially all of the heat q f goes to increase the energy of
he liquid. During the vaporization, however, the volume
ihanges from v' (volume of 1 Ib. of liquid) to v" (volume of
. Ib. of saturated vapor). Since the pressure remains constant,
he external work that must be done to provide for the increase
>f volume is I f = p (v"  v') ( 2 )
According to the energy equation, the heat r added during
vaporization is used in increasing the energy of the system and
is the heat required to increase the energy of the unit weight
of substance when it changes from liquid to vapor. This heat
is denoted by p and is called the internal latent heat. Since
during the vaporization the temperature is constant, there is no
change of kinetic energy ; it follows that p is expended in in
creasing the potential energy of the system. The heat equiva
lent of the external work, namely, Ap (y" v'), is called the
external latent heat, and for convenience may be denoted by ^.
We have then . . ^.N
r = p + ^. (4)
The total heat of the saturated vapor is evidently the sum of
the heat of the liquid and the heat of vaporization. Thus,
q" = q' + r,
or q" = q' + p + "f (5)
Comparing (5) with the general energy equation, it is evident
that the sum q' + p gives the increase of energy of the saturated
vapor over the energy of the liquid at 32 F. Denoting this
by w", we have , , , ,.
J Au" = q' +p. (6)
If the vaporization is not completed, the result is a mixture
/ A'M\
of saturated vapor and liquid of qviality x f x = ), as indi
\ A A j
cated by the point M (Fig. 60 and 62). In this case the heat
required to vaporize the part x is xr heat units and the total
heat of the mixture, which may be denoted by q x , is given by
q x = q' + xr
= q f + xp + x^r. (7)
The energy of the mixture (per unit weight) above the energy
of water at 32 F. is, therefore, given by the relation
Au x = q' + zp, (8)
and the external work done is
L x = Jx^. (9)
If heat is added at constant pressure, after the vaporization is
completed, the vapor will be superheated. The statepoint will
thft P.nrvP 4"7fJ Cff\rr d.8"\ nnrl +.1.0 liocif. /> ft f. n ~\
epresented by the area A 2 A" ' EE^ will be added. Here c p de
lotes the mean specific heat of the superheated vapor, t e the
inal temperature, and t" the saturation temperature correspond
ng to the pressure p. The total heat corresponding to the
)oini E and represented by the area OAAA'EE^ is, therefore,
q e =q'+r + c p (t e t"). (10)
f v e denotes the final volume, and u e the energy above liquid
,t 32 F., then the external work for the entire process is
L=p(y e v<), (11)
,nd, therefore,
Au>. = q e Ap (y e v'). (12)
111. Heat Content of a Saturated Vapor. By definition we
iave for the heat content of a unit weight of saturated vapor
i" = A(u" +pv"~) = q' + p + Apv". (1)
iince the total heat is
<? = q' + p + Ap(v"^ (2)
; appears that i" is larger than q" by the value of the term
\.pv'. As v', the specific volume of water, is small compared
,dth v", the term Apv' may be neglected except for very high
ressures, and q" and i" may be considered equal.
In most of the older steam tables values of q" were given ;
i the more recent tables, the values of i" instead of q" are
.sually tabulated.
112. Thermal Properties of Water Vapor. From the relation
q" = q' + r,
> appears that if any two of the three magnitudes q", q', r are de
srmined by experiment, the third may be found by a combina
ion of those two. Various experiments have been made to
etermine each of these magnitudes for the range of temperature
rdinarily employed, and as a result several empirical formulas
ave been deduced. Naturally the greatest amount of attention
as been given to water vapor, and we may consider the proper
ies of this medium as quite accurately known at the present
ime. Ammonia, sulphur dioxide, and other vapors have not
UtitJUL BUUUJ.CU Wltii UHC aclillC
cm AM. UJJL&J.J.
are as yet only imperfectly known.
In the sections immediately following we shall give briefly the
results of the latest and most accurate experiments on water
vapor.
113. Heat of the Liquid. Denoting c r the specific heat of
water, the heat of the liquid above 32 F. is given by the re
IatioQ j /*. CD
If the specific heat c' were constant at all temperatures, this
equation would reduce to the simple form q' = c'(t 32). As
a matter of fact, however, c' is not constant, and its variation
with the temperature must be known before (1) can be used to
calculate q'. Between C. and 100 C. (32212 F.) the
experiments of Dr. Barnes may be regarded as the most trust
worthy. Taking c' 1 at a temperature of 17.5 C., the fol
lowing values are given by Griffiths as representing the results
obtained by Barnes.
TEMPERATURE
SPECIFIC HEAT
TKMl'KKATUUE
Si'KOinc HKAT
C.
F.
C.
F.
32
1.0083
55
131
0.9981
5
41
1.0054
60
140
0.9987
10
50
1.0027
65
149
0.9993
15
59
1.0007
70
158
1.0000
20
68
0.9992
75
1C7
1.0007
25
77
0.9978
80
176
1.0015
30
86
0.9975
85
185
1.0023
35
95
0.9974
90
194
1.0031
40
104
0.9973
95
203
1.0040
45
113
0.9974
100
212
1.0051
50
122
0.9977
These values are shown graphically in Fig. 63. From them
values of q' may be obtained by means of relation (1).
In the actual calculation of the tabular values of q', the fol
lowing method may be used advantageously. Since the specific
heat c' does not differ greatly from 1, let
c'= 1 + &,
ART. 114]
LATENT HEAT OF VAPORIZATION
175
1.008
1.000
1.004
1.002
1.000
0.908
0.096
\
\
\
/
\
f
"7
\
,
/
\
/
\
/
U u
^
iO L
4
X
8
w
\
S
'
N.
*~
FIG. 63.
where k is a small correction term. Then for q' we have
q' = f c'dt = t  32 + f kdt.
1 JM J32
If now values of Jc are plotted as ordinates with correspond
ing temperatures as abscissas, the values of the integral (kdt
may easily be determined by graphical integration.
For temperatures above 212 F. the only available experi
ments giving the heat of
the liquid are those of
Regnault and Dieterici.
The results of these ex
periments are somewhat
discordant and unsatis
factory. Fortunately,
we have for the range
212 to 400 F. reliable
formulas for the total
heat q" and the latent
heat r, and we may therefore determine q' from the relation
q' = q" r.
114. Latent Heat of Vaporization. The latent heat of water
vapor for the range to 180 C. (32356 F.) has been accu
rately determined by direct experiment. The results of the
experiments of Dieterici at C., Griffiths at 30 and 40 C.,
Smith over the range 1440 C., and Henning over the range
30180 C. show a remarkable agreement, all of the values
lying on, or very near, a smooth curve. The observed values
are given in the third column of the following table. As the
thermal units employed by the different investigators were not
precisely the same, all values have been reduced to a common
unit, the joule.
It is readily found that a seconddegree equation satis
factorily represents the relation between r and t. Taking r in
joules, the following equation gives the values in the fourth
176
SATURATED VAPORS [CHAP, x
LATENT HEAT OF WATER, IN JOULES
IiATENl
llKAT
TEMPKKA
DlFFKKHNOK
PUB CUNT
Observed
Calculated
2493.8
2495.8
0.08
30.00
2429.3
2430.8
 0.06
Griffiths
40.15
2403.6
2407.5
0.16
13.95
2467.6
2406.3
+ 0.05
21.17
2451.2
2450.5
+ 0.03
.
28.06
2435.0
2435.2
U.U1
39.80
2405.8
2408.3
0.10
30.12
2424,8
2430.6
 0.24
Henning,
First Series ....
40.14
64.85
77.34
2385.3
2343.0
2313.7
2386.2
2347.7
2316.0
0.04
 0.20
0.10
89.29
2285.6
2284,6
+ 0.05
100.59
2254.2
2254,0
+ 0.01
102.34
2248.7
2249.2
 0.02
Henning,
120.78
2200.2
2197.2
+ 0.14
Second Series . . .
140.97
2134.2
2137.6
0.10
100.56
2077.0
2077.2
0.01
180.72
2018.6
2012.3
+ 0.31
The differences between the observed values and those calcu
lated from this formula are shown in the last column.
The mean calorie is equivalent to 4.184 joules ; hence, divid
ing the constants of Eq. (1) by 4.184, the resulting equation
gives r in calories. This equation is readily changed to give
r in B. t.u. with t in degrees F. We thus obtain finally
r = 970.4  0.655 (*  212)  0.00045 (t  212) 2 . (2)
This formula may be accepted as giving quite accurately the
latent heat from 32 F. to perhaps 400 F.*
115. Total Heat. Heat Content. For the temperature range
32 to 212 F. the total heat q" is obtained from the relation
q" = q' + r. As has been shown, values of q' and of r can be
accurately determined for this range. For temperatures be
tween 212 and 400, we are indebted to Dr. H. N. Davis for
the derivation of a formula for the heat content of saturated
vapor of water. The earlier experiments of Regnault led to
the formula q n ^ 1091 . 7 + 0>305 ^ __ 33^
which has been extensively used in the calculation of tabu
lar values. By making use of the throttling experiments of
Grindley, Griessmann, and Peake, Dr. Davis* has shown that
Regnault's linear equation is incorrect, and that a seconddegree
equation of the form
q" = a + b (t  212) + e (t  212) 2
may be adopted. Dr. Davis obtains for the heat content i"
the formula
i" = 1150.4 + 0.3745(15 212)  0. 00055 (t  212) 2 . (3)
From this formula the total heat q" is readily determined from
the relation q" = i" Apv' . It is found, however, that slight
changes in the constants are desirable in view of Henning's sub
sequent experiments on latent heat. The modified formula
i" = 1150.4 + 0.35 (t  212)  0.000333 (t  212) 2 (4)
may be accepted as giving with reasonable accuracy values of
i" for the range 212 to 400 F.
116. Specific Volume of Steam. The specific volumes" of
a saturated vapor at various pressures may be determined
experimentally. For water vapor accurate measurements of
v" for temperatures between 100 and 180 C. have been made
by Knoblauch, Linde, and Klebe. It is possible, however, to
calculate the volume v" from the general equations of thermo
dynamics ; and the agreement between the calculated values
and those determined by experiment serves as a valuable check
critical temperature, 689 F. At the higher temperatures it doubtless gives more
accurate values than the seconddegree formula. See Proceedings of the Amer.
Acad. of Arts and Sciences 45, 284.
* Trans. Am. Soc. of Mecli. Engs. 30, 1419, 1908. See Art. 104 for a dis
cussion of thfi inRf.bnrl mrmlnvfirl in t.hfi rlflviva.tion of formula (3}.
on the accuracy with which the factors entering into the theo
retical formula have been determined.
The general equation (Art. 56)
do c v dT\ AT(  ) dv (1)
\dtjv
applies to any reversible process. Let us apply it to the pro
cess of changing a liquid to saturated vapor at a given constant
temperature. For a saturated vapor, the partial derivative
is simply the derivative , and this is a constant for any
dtjv " " dt
given temperature (Art. 10T). Hence, for the process in ques
tion, we have (since dT 0)
(2)
But in this case q is the heat of vaporization r ; hence we have
.. , r 1 Jr 1 .ox
1)" v' =  .. {&)
dt dt
This is the ClapeyronClausius formula for the increase of vol
ume during vaporization.
Having for any temperature the derivative  (Art. 109)
and the latent heat r, the change of volume v" v' is readily
calculated. The following table shows a comparison between
the values of v" determined experimentally by Knoblauch,
Linde, and Klebe, and those calculated by Henning from the
Clapeyron equation, using the values of r determined from his
own experiments. The third line gives values of v" calculated
from the characteristic equation of superheated steam. (See
Art. 132.)
SPECIFIC VOI.UMBB, Cu. MKTBKB I'KII Ko.
100
120
140
160
180 C.
Experimental ....
Hennin r
1.674
1.073
1.073
0.8922
0.8912
0.8915
0.5001
0.5078
0.5084
0.3073
0.3071
0.3071
0.1043
0.1947
0.1945
o
From the equation for
superheated steam . .
ART. 117] ENTROPY OF LIQUID AND OF VAPOR 179
The relation between the pressure and specific volume v" of
saturated steam may be represented approximately by an equa
tion of the form // _. 0 ,^
Zeuner, from the values of v" given in the older steam tables,
deduced the value n = 1.0646. Taking the more accurate
values of v" given in the later steam tables, we find
91 = 1.0631, (7=484.2.
117. Entropy of Liquid and of Vapor. During the process
of heating the liquid from its initial temperature to the tem
perature of vaporization the entropy of the. liquid increases.
Thus, referring to Fig. 62, if the initial temperature be 32 F.,
denoted by point 4, and if the temperature be raised to that
denoted by A', the increase of entropy of the liquid is repre
sented by OA}, the heat of the liquid by area OAA'A V
Since dq' = c'dT, we have as a general expression for the
entropy s 1 of the liquid corresponding to a temperature T,
 c T ^L C T m
J 491.6 T ~ J 4916 T
If the specific heat c' is given as a function of T, the inte
gration is readily effected. In the case of water, where the
specific heat varies somewhat irregularly, as shown by the
table of Art. 115, the following expedient may be used. Put
c' = 1 + k ; then k is a small correction term that is negative
between 63 and 150 F. and positive elsewhere. From (1) we
have, therefore,
The first term is readily calculated and the small correction
term may be found by graphical integration. This method was
used in calculating the values of s' in table I.
The increase of entropy during vaporization, represented by
v
A' A!' (Fig. 62), is evidently the quotient = Hence the en
tropy of the saturated vapor in the state A" is
For a mixture of quality x, as represented by tlie point M, the
entropy is
= *' + f. (4)
118. Steam Tables. The various properties of saturated
steam considered in the preceding articles are tabulated for
the range of pressure and temperature used in ordinary tech
nical applications. Many such tabulations have appeared.
The older tables based largely upon Regnault's data are now
known to be inaccurate to a degree that renders them value
less. The recent tables of Marks and Davis * and of Peabody, f
however, embody the latest and most accurate researches on
saturated steam.
Table I at the end of the book has been calculated from
the formulas derived in Arts. 108116. The values differ but
little from those obtained by Marks and Davis. The first col
umn gives the pressures in inches of mercury up to atmospheric
pressure, and in pounds per square inch above atmospheric
pressure ; the second column contains the corresponding
temperatures. Columns 3 and 4 give the heat content of the
liquid and saturated vapor, respectively. The values in col
umn 3 may be taken also as the heat of the liquid q' ; similarly,
column 4 may be considered as giving the total heat q" of the
saturated vapor. As we have seen, the difference between i"
and q" is negligible except at high pressures.
119. Properties of Saturated Ammonia. Several tables of
the properties of saturated vapor of ammonia have been pub
lished. Among these may be mentioned those of Wood, Pea
body, Zeuner, and Dieterici. The values given by the different
tables are very discordant, as they are for the most part obtained
by theoretical deductions based on meager experimental data.
For temperatures above 32 F. the values obtained by Dieterici
as the result of direct experiment are most worthy of confidence.
Dieterici determined experimentally the specific volume v"
of the saturated vapor for the temperature range to 40 C.
* Marks and Davis, Steam Tables and Diagrams, Longmans, 1908.
(32 to 104 F.) and also for the same range the specific heat c f
of the liquid ammonia. The formula deduced by Dieterici for
specific heat is, for the Fahrenheit scale,
c' = 1.118 + 0.001156 (t  32). (1)
From this formula, the heat of the liquid q r and the entropy of
the liquid s' are readily calculated by means of the relations
/ 1 / 7 r /* / 0> X
q' = I c at, s' = ( c'
J 32 Jfi>l.G T
The relation between pressure and temperature is given by
the experiments of Regnault. The results of these experiments
are expressed quite accurately by Bertrand's formula
log p = 5.87395  50 log m T QA . (2)
Above 32, having Dieterici's experimental values of v" and
from (2) the derivative . we
dT
the ClapeyronClausius formula
from (2) the derivative JL we may find the latent heat r from
Ct ~L
r = A(v"v'~)T. (See Art. 116.) (3)
Gv JL
For temperatures below 32 we have neither v" nor r given
experimentally; hence for this region values of various prop
erties can only be determined by extrapolation, and the ac
curacy of the results thus obtained is by no means assured. In
calculating the values of table III the following method was
used. The values of r for temperatures above 32 were calcu
lated by means of (3). It was found that these values may be
represented quite accurately by the equation
log r = 1.7920 + 0.4 log (266  ), ( 4 )
in which 266 is the critical temperature of ammonia. (See p.
176, footnote.) Formula (4) was assumed to hold for the range
32 to 30 ; and from the values of r thus obtained values of
v" were calculated by means of the Clapeyron relation (3).
120. Other Saturated Vapors. Several saturated vapors in
addition to the vapors of water and ammonia have important
technical aDplications. Sulphur dioxide and carbon dioxide in
particular are used as media for refrigerating machines. The
properties of the former fluid have been investigated by Cailletet
and Mathias, those of the latter by Amagat and M oilier. The
results of these investigations are embodied in tables.*
The properties of several vapors of minor importance have
also been tabulated, the data being furnished for the most part
by Regnault. These include ether, chloroform, carbon bisul
phide, carbon tetrachloride, aceton, and vapor of alcohol, f
121. Liquid and Saturation Curves. If for various tem
peratures the corresponding values of s', the entropy of the
liquid, be laid off as abscisses, the result is a curve s', Fig. 62.
This is called the liquid curve. If, likewise, values of
be laid off as abscissa;, a second curve s' f is obtained. This
is called the saturation curve.
As already stated (Art. 106), any point between the curves s'
and s" represents a mixture of liquid and vapor, the ratio x de
pending upon the position of the point. It is possible, there
fore, to draw between the curves s' and s" a series of constanta;
lines. Each of the horizontal segments A' A", B'B", etc., is
divided into a convenient number (say 10) of equal parts and
corresponding points are joined by curves. The successive
curves, therefore, are the loci of points for which x = 0.1,
#=0.2, etc.
The form of the saturation curve has an important relation
to the behavior of a saturated vapor. For nearly all vapors,
the curve has the general form shown in Fig. 62 ; that is, the
entropy s" decreases with rising temperature. In the case of
ether vapor, however, the entropy increases with rising tem
perature and the curve has, therefore, the same general direc
tion as the liquid curve s'.
122. Specific Heat of a Saturated Vapor. Kef erring to the
saturation curve of Fig. 62, suppose the statepoint to move
* For tables of the properties of saturated vapor of S0 2 and C0 2 in English
units, see Zeuner's Technical Thermodynamics, Klein's translation, Part II.
t See Peabody's Steam and Entropy Tables, or Zeuner's Technical Thermo
"
ART. 122] SPECIFIC HEAT OF A SATURATED VAPOR 183
from A" to B" '. This represents a rise of temperature of the
saturated vapor during which the vapor remains in the satu
rated condition. The process must evidently be accompanied
by the withdrawal of heat represented by the area A^Al'IP'S^ ;
and the reverse process, fall in temperature from B" to A", is
accompanied by the addition of heat represented by the same
area. It appears, therefore, that along the saturation curve
the ratio ^ is negative (except in the case of ether) ; that is,
ZA
the specific heat of a saturated vapor is, in general, negative.
An expression for the specific heat c" of the saturated vapor
may be obtained as follows. The entropy of the saturated
vapor is given by the equation
hence the change of entropy corresponding to a change of
temperature is obtained by differentiating (1), thus
(2)
But <fo' = ^fr < 3 )
and similarly for the saturation curve,
*" = ^. (4)
Substituting these values ds' and ds" in (2), the result is
'
m "> j_
1 dT\T ] '
But since c' = TJ, (5) may be written
^d(q'+r*) r
c ~ dT T
a IS Known. JLIIUS lur wauor vtipui ciuuvo *JJA , wo JU.O/VD
2 " = a + 6(<  212)  c(t  212) 2 ;
whence
where 5 = 0.35 and c = 0.000333.
At 212, we have, for example,
r 9704
T 212 + 459.6
123. General Equation for Vapor Mixtures. Let heat be
added to a unit weight of mixture of liquid and saturated
vapor, of which the part x is vapor and the part 1 x is
liquid. In general, the temperature T and quality x will
change ; hence the heat added is the sum of two quantities :
(1) the heat required to increase the temperature with x
remaining constant; (2) the heat required to increase x with
the temperature constant. The first is evidently c'(l x~)dT
\c"xdT; and the second is rdx ; hence we have
dq = c'(l  x)dT+ c"xdT + rdx (1)
as the general differential equation for the heat added to a
mixture.
From (1) the general expression for the change of entropy
of a mixture is given by
7 dq c'fl x^ 4 c"x im , rj /o\
rts=x! = _A t 2 T f_ (&. C2)
M/J. ( _ \^ J
The fact that ds is an exact differential leads at once to the
rekti n arV(i*) + tf
dx\_ T
whence
c = c _.,
dT T
the relation that was obtained in Art. 122.
F
A"
124. Variation of x during Adiabatic Changes. Let the point
A" (Fig. 64) represent the state of saturated vapor as regards
pressure and temperature. Adiabatic expansion will then be
represented by a vertical line A" E, the final point H being at
lower temperature. Adiabatic compression will be shown by a
vertical line A" Gr. With a saturation curve of the form
shown, it appears that during adiabatic expansion some of the
vapor .condenses, while adiabatic compression results in super
heating. If the statepoint is originally at M so that x is some
what less than 1 (say O.T or 0.8),
then adiabatic expansion is ac
companied by a decrease in #,
adiabatic compression by an in
crease of x.
If the saturation curve slopes
in the other direction, as in the
case of ether, the conditions just
stated will, of course, be reversed.
Adiabatic expansion of the
liquid is represented by the line
A'F ; evidently some of the
liquid is vaporized during the
process. If the mixture is originally mostly liquid, as indicated
by a point .ZVnear the curve *', then adiabatic expansion results
in an increase of #, adiabatic compression in a decrease of x.
For a given pressure there is some value of x for which an
indefinitely small adiabatic change produces no change in x ;
in other words, at this point the constanta? curve has a vertical
tangent. For this point we have evidently dq = and dx 0,
and the general equation (1), Art. 123, becomes
FIG. 64.
whence
or
x =
c'c"'
(1)
(2)
(3)
The locus of the points determined by (3) is a curve n (Fig. 64),
dq = rax ; (4)
that is, all the heat entering the mixture is expended in vapor
izing the liquid. The zero curve is of little practical importance.
The change of the quality x during the adiabatic expansion
of a' mixture is readily calculated by means of the entropy
equation. In the initial state, the entropy of the mixture is
and in the final state it is
z
But for an adiabatic change s 2 = s 1 ; therefore, we have the
relation s/ + ^ = s 2 ' + 'j^ 2 , (5)
in which # 2 is the only unknown quantity.
125. Special Curves on the TSplane. The region between
the liquid and saturation curves may be covered with series of
curves in such a way that the position of the point represent
ing a mixture indicates at once the various properties of the
mixture.
In the first place, horizontal lines intercepted between the
curves s' and s" are lines of constant temperature, also lines of
constant pressure ; while vertical lines are lines of constant
entropy.
Lines of constant quality, z v # 2 , # 3 , . . . may be drawn as
explained in Art. 121.
Curves of constant volume may be drawn as follows : The
volume of a unit weight of mixture whose quality is x is given
by the equation
v = x(v" v'} + v', (1)
whence
x
V ~ v
V V
Suppose that the curve for some definite volume (say v 5 cu.
ft.) is to be located. For different pressures p^ p v p y . . .
the saturation volumes v/', v/, v a ", . . . are known from the
tables, substituting successively these values of v" in (2),
values of #, as x v x v a; 8 , . . . corresponding to the pressures
Pv> Pv> P& ' w ^ be f un d. The value of v' may be taken
as constant for all pressures. The value of x l locates a definite
point on the p l line, that of x 2 a point on the p z line, etc. The
locus of these points is evidently a curve, any point of which
represents a mixture having the given volume v ; hence it is a
constant volume curve.
In a similar manner curves of constant energy u may be
located. Since u = q'+xp, (3)
u q'
we have x = . (4)
P
For given pressures p v p 2 , . . .
f f
ry *! fv* _ *% 0,4p
JU\ ' 3 *Vn ^ uUV_/
Pi Pz
Values of q' and p for different pressure are given in the table,
and therefore for a given w, values of x v # 2 , . . . are readily
calculated. These locate points on the corresponding ^>lii
.and the locus of the points is
the desired constantw curve. T
By the same process may be
drawn curves of constant total
heat,
q = q' f xr const.
or curves of constant heat
content
i = i' \xr const.
In Fig. 65, the various curves
are shown drawn through the ~ FlG G5
same point P. From the general
course of the curves the behavior of the mixture during a
given change of state may be traced. Thus : (1) If a mixture
expands adiabatically, v increases but p, T, u, and i decrease.
The quality x decreases as long as the statepoint lies to the
right of the zero curve. ' (2) If a mixture expands isody
namicallv (u= const.), v, s, and x increase, p, T, and i decrease.
for water vapor, taking values of s' and s" from the steam table. Then
draw the curves v = 2, v = 10, v  40 cu. ft. Also draw the curves u = 600
B. t. u., M = 800 B. t. u.
126. Special Changes of State. Certain of the curves de
scribed in preceding articles represent important changes of
state of the mixture of saturated vapor and liquid. The prin
cipal relations governing some of these changes will be de
veloped in this article. It is assumed that the system remains
a mixture during the change, that is, that the path of the state
point is limited by the curves s' and s".
(a) Isothermal, or Constant Pressure, Change of State. Let
x 1 denote the initial quality, x z the final quality. Then the
initial volume is
and the final volume is
/ n t~\ i t
1)n = Xn(V V ) + V .
The change in volume is therefore
v v = (x x'}(v"v'') 00
and the external work is
The change of energy is
and the heat absorbed is
q^rtxtxj. (4)
These equations refer to a unit weight of mixture.
EXAMPLE. At a pressure of 140 lb., absolute, the volume of one pound
of a mixture of steam and water is increased by 0.8 cu. ft. The change of
quality is 2L = = 0.2514. The external work is
140 x 144 x 0.8 = 16,128 ft.lb.
The increase of energy is Jp(x z  xj = 778 x 786.1 x 0.2514 = 153850 ft.lb. ;
and the heat absorbed is r (x z  xj = 869 x 0.2514 = 218.5 B. t. u.
(6) (Jliange oj /state at Constant Volume. Since the volumes
> l and v% are equal, we have
*i(V'"0=z 2 (< <;'), (5)
ivhere v^" and v z " are the saturation volumes corresponding to
ihe pressures p and p^ respectively. From (5) the quality x z
.n the final state may be determined. The external work TTis
Hero ; hence we have for the heat absorbed
 ( ? /  x lPl ) . (6)
' EXAMPLE. A pound of a mixture of steain and water at 120 Ib. pressure,
quality 0.8, is cooled at constant volume to a pressure of 4 in. of mercury.
Required the final quality and the heat taken from the mixture.
From (5)
^ = *,* > ) = 0.8(3.7240.017) = ^
v 2 "  v' 176.6
rherefore
q = 311.9 + 0.8 x 795.8  (93.4 + 0.0167 x 959.5) = 839.2 B. t. u.
(c) Adiabatio Change of State. For a reversible adiabatic
change the entropy of the mixture remains constant ; hence we
have
'i' + % L = '*' + *> CO
L\ J2,
from which equation the final quality z 2 can be found. Having
z; 2 , the final volume v 2 per unit weight is
v = z 2 <> 2 "^)+^. (8)
Since the heat added is zero, the external work is equal to the
decrease in the intrinsic energy of the mixture. That is,
(9)
EXAMPLE. Three cubic feet of a mixture of steam and water, quality
0.89, and having a pressure of 80 Ib. per square inch, absolute, expands
adiabatically to a pressure of 5 in. Hg. The final quality, final volume,
and the external work are required.
From the steam tables we find the following values :
fl P . T
For^SOlb. 281.8 819.6 0.4533 1.1667 5.464
Forp = 5in. Hg. 101.7 953.7 0.1SSO 1.7170 143.2
The weight of the mixture is
M  3 = = 0.6167 Ib.
m ~ Xi ( v _ ') + ' 0.89(5.464  0.017) + 0.017
From (7), the quality x 2 in the second state is given by the relation
0.4533 + 0.89 x 1.1667 = 0.1880 + 1.7170 x a ,
whence x % = 0.759.
The volume in the second state, neglecting the insignificant volume of the
liquid, is
V 2 = 0.6167 x 0.759 x 143.2 = 67.02 cu. ft.
Finally, the external work is
W = 778 x 0.6167 [(281.8 + 0.89 x 819.6)  (101.7 + 0.759. x 953.7)] = 89,080
ft.lb.
(dT) Isodynamic Change of State. If the energy of the mix
ture remains constant, we have
Wj = Up
or ft' + x lPl = qj + x z p 2 . (10)
From (10) the final value of x is determined, and the final
volume is then found from (8).
For the isodynamic change, the heat added to the mixture is
evidently equal to the external work. There is no simple way
of finding the work. As an approximation, an exponential
curve
p 1 v 1 n =pv n (11)
may be passed through the points p^ v^ and j? 2 , v 2 , and the
value of n can be found. This curve will approximate to the
true isodynamic on the j?vplane, and the external work will
then be approximately
ytri v i~P2 v <2 (12^)
nl ' ^ J
In practice the isodynamic of vapor mixtures is of little
importance.
127. Approximate Equation for the Adiabatic of a Vapor Mix
ture. In certain investigations, especially those relating to the
flow of steam, it is convenient to represent the relation between
p and v during an adiabatic change by an equation of the form
RT. 127] APPROXIMATE EQUATION OF ADIABATIC
191
?he value of the exponent n is not constant, but varies with the
litial pressure, the initial quality, and also with the final
ressure ; and at best the equation is an approximation,
tankine assumed for n the value ! for all initial conditions.
a
ieuner, neglecting the influence of initial pressure, gave the
ormula
n = 1.035 + 0.1 x.
(2)
Ir. E. H. Stone,* using the tables of Marks and Davis, has
.erived the relation
n = 1.059  0.000315 p + (0.0706 + 0.000376^>. (3)
The following table gives values of n calculated from (3).
nitial
iuuli
ty
INITIAL PRESSURE IN POUNDS PUR SQUAKE INCH, ABSOLUTE
20
40
60
80
100
120
140
160
180
200
220
240
1.00
1.131
1.132
1.133
1.134
1.136
1.137
1.138
1.139
1.141
1.142
1.143
1.145
0.95
1.127
1.128
1.128
1.130
1.131
1.131
1.132
1.133
1.134
1.135
1.136
1.137
0.90
1.123
1.123
1.124
1.124
1.125
1.125
1.126
1.126
1.127
1.127
1.128
1.129
0.85
1.119
1.119
1.119
1.119
1.120
1.120
1.120
1.120
1.120
1.120
1.120
1.121
0.80
1.115
1.115
1.114
1.114
1.114
1.114
1.113
1.113
1.113
1.113
1.112
1.112
0.75
1.111
1.110
1.110
1.109
1.109
1.108
1.107
1.106
1.106
1.105
1.104
1.104
0.70
1.108
1.106
1.105
1.104
1.103
1.102
1.101
1.100
1.099
LOSS
1.097
1.096
0.65
1.104
1.102
1.101
1.099
1.098
1.096
1.095
1.093
1.092
1.091
1.089
1.088
0.60
1.100
1.098
1.096
1.094
1.093
1.091
1.089
1.087
1.085
1.083
1.081
1.080
0.55
1.096
1.093
1.092
1.089
1.087
1.085
1.083
1.080
1.078
1.076
1.074
1.072
0.50
1.092
1.089
1.087
1.084
1.082
1.079
1.077
1.074
1.071
1.069
1.066
1.064
Having the initial values p v V v and x# and the final pressure
> 2 , the final volume V 2 is found approximately from (1), the
ppropriate value of n being taken from the table. The exter
lal work is found approximately by the usual formula for the
hange represented by (1), namely,
w=
(4)
n
EXAMPLE. Taking the data of the example of Art. 126 (c), we have
_ on T7 a . n so lionpp 1.193. The final tiressure is 5 in. Hff.
and W = 144 x = 88 OT4 b " lb '
Comparing these results with the results obtained by the exact method,
it appears that the volume F 2 is about 0.36 per cent smaller and the work
W about 0.13 per cent smaller. Hence the approximation is sufficiently
close for all practical purposes.
EXERCISES
1. From Bertrand's equation calculate the pressure of steam corre
sponding to the following temperatures : 60, 250, 400 F.
2. Find the values of the derivative ( P for the same temperatures.
at
3. Using the results of Ex. 1 and 2, find the specific volumes for the
given temperatures.
4. Find (a) the latent heat, (&) the total heat of saturated steam, at a
temperature of 324 F.
5. Calculate the latent heat of steam, (a) by the quadratic formula (2),
Art. 114; (b) by the exponential formula (see footnote, p. 170) for the tem
peratures 220 F. and 380 F. Compare the results.
In the following examples take required values from the steam table,
p. 315.
6. Find the entropy, energy, heat content, and volume of 4.5 Ib. of a
mixture of steam and water at a pressure of 120 11). per square inch, quality
0.87.
7. Find the quality and volume of the mixture after adiabatic expan
sion to a pressure of 16 Ib. per square inch.
8. Find the external work of the expansion.
9. Using the data of the preceding examples, calculate the volume and
work by means of the approximate exponential equation p V n = C.
10. A mixture, initial quality 0.97, expands adiabatically in a 12 in. by
12 in. cylinder from a pressure of 100 Ib. per square inch, gauge, to a pressure
of 10 Ib. per square inch, gauge. Find the point of cutoff.
11. The volume of 6.3 Ib. of mixture at a pressure of 140 Ib. per square
inch is 17.2 cu. ft. Find the quality of the mixture ; also the entropy
and energy of the mixture.
12. The mixture in Ex. 11 is cooled at constant volume to a pressure of
20 Ib. per square inch. Find the final value of x and the heat abstracted.
13. At a pressure of 180 Ib. per square inch the volume of 2 Ib. of a
mixture of steam and water is increased by 0.9 cu. ft. Find the increase of
quality, increase of energy, heat added, and external work.
14. A mixture of steam and water, quality 0.85, at a pressure of 18 Ib.
per square inch, is compressed adiabatically. Find the pressure at which
tne water is completely vaporized, .eina aiso tne woric 01 compression,
per pound of mixture.
15. Steam at a pressure of 80 Ib. per square inch expands, remaining sat
urated until the pressure drops to 50 Ib. per square inch. Find approxi
mately the heat that must be added to keep the steam in the saturated
condition.
16. Water at a temperature of 352 F. and under the corresponding
pressure expands adiabatically until the pressure drops to 30 Ib. per square
inch. Find the per cent of water vaporized during the process. Find the
work of expansion per pound of water.
17. Two vessels, one containing M t Ib. of mixture at a pressure p 1 and
quality x\, the other M 2 Ib. at a pressure p and quality x 2 , are placed in
communication. No heat enters or leaves while the contents of the vessels
are mixing. Derive equations by means of which the final pressure ps and
final quality x s may be calculated.
18. Let 1 Ib. of mixture at a pressure of 20 Ib. per square inch, quality
0.96, enter a condenser which contains 20 Ib. of mixture at a pressure of 3 in.
Hg., quality 0.05. Assuming that no heat leaves the condenser during the
process, find the pressure and quality after mixing.
REFERENCES
PRESSURE AND TEMPERATURE OF SATURATED VAPORS
Kegnault: Mem. de 1'Inst. de France 21, 465. 1847. Rel. des exper. 2.
Henning : Wied. Ann. (4) 22, 609. 1907.
Holborn and Henning : Wied. Ann. (4) 25, 833. 1908.
Holborn and Baumann: Wied. Ann. (4) 31, 945. 1910.
Risteen : The Locomotive 26, 85, 183, 246 ; 27, 54 ; 28, 88.
These articles contain a very complete account of the experiments
of Regnault, Holborn and Henning, and Thiesen.
Chwolson : Lehrbuch der Physik 3, 730.
Gives comprehensive discussion of the many formulas proposed for the
relation between the pressure and temperature of various vapors.
Preston : Theory of Pleat, 330.
Marks and Davis : Steam Tables and Diagrams; 93.
Peabody: Steam and Entropy Tables, 8th ed., 8.
Marks : Jour. Am. Soc. Mech. Engrs. 33, 563. 1911.
PROPERTIES OF SATURATED STEAM
(a) Specific Heat of Water. Heat of Liquid
Regnault : Mem. de 1'Inst. de France 21, 729. 1847.
Dieterici : Wied. Ann. (4) 16, 593. 1905.
Barnes : Phil. Trans. 199 A, 149. 1902.
Rowland : Proc. Amer. Acaa. oi Arcs ana sciences **, < u ; jo, oo. j.oov
1881.
Day: Phil. Mag. 46, 1. 1898.
Griffiths : Thermal Measurement of Energy.
Marks and Davis : Steam Tables and Diagrams, 88.
(6) Latent Heat
Regnault: Mem. de 1'Inst. de France 21, 635. 1847.
Griffiths : Phil. Trans. 186 A, 261. 1895.
Henning: Wied. Ann. (4) 21, 849, 1906; (4) 29, 441, 1909.
Dieterici: Wied. Ann. (4) 16, 593. 1905.
Smith : Phys. Rev. 25 145. 1907.
(c) Total Heat
Davis: Proc. Am. Soc. of Mech. Engrs. 30, 1419. 1908. Proc. Amer.
Acad. 45, 265.
Marks and Davis : Steam Tables and Diagrams, 98.
(!) Specific Volume
Fairbairn and Tate : Phil. Trans. (I860), 185.
Knoblauch, Linde, and Klebe : Mitteil. liber Forschungsarbeit. 21, 33. 1905
Peabody : Proc. Am. Soc. Mech. Engrs. 31, 595. 1909.
Peabody : Steam and Entropy Tables, 8th ed., 12.
Marks and Davis : Steam Tables and Diagrams, 102.
Davis : Proc. Am. Soc. Mech. Engrs. 30, 1429.
PROPERTIES OF REFRIGRATING FLUIDS
(a) Ammonia
Dieterici : Zeitschrift fur Kalteindustrie. 1904.
Jacobus: Trans. Am. Soc. Mech. Engrs. 12, 307.
Wood : Trans. Am. Soc. Mech. Engrs. 10, 627.
Peabody : Steam and Entropy Tables, 8th ed., 27.
Zeuner: Technical Thermodynamics (Klein) 2, 252.
Lorenz : Technische W'armelehre, 333.
(I) Sulphur Dioxide
Cailletet and Mathias : Comptes rendus 104, 1563. 1887.
Lange: Zeitschrift fur Kalteindustrie 1899, 82.
Mathias : Comptes rendus 119, 404. 1894.
Miller : Trans. Am. Soc. Mech. Engrs. 25, 176.
Wood: Trans. Am. Soc. Mech. Engrs. 12. 137.
Zeuner : Technical Thermodynamics 2, 256.
(c) Carbon Dioxide
iagat: Comptes rendus 114, 1093. 1892.
llier : Zeit. fur Kalteindustrie 1895, 66, 85.
Liner: Technical Thermodynamics 2, 262.
GENERAL EQUATIONS FOR VAPORS. CHANGES OF STATE
.iner: Technical Thermodynamics 2, 53.
>,yrauch : Grundriss der WarmeTheorie 2, 33.
*Hlon : Theory of Heat, 650.
L*ry : Temperature Entropy Diagram, 43.
CHAPTER XI
SUPERHEATED VAPORS
128. General Characteristics of Superheated Vapors. The
nature of a superheated vapor has been indicated in Art. 106,
describing the process of vaporization. So long as a vapor is
in immediate contact with the liquid from which it is formed it
remains saturated, and its temperature is fixed by the pressure
according to the relation t = /"(#>). When vaporization is com
pleted, or when the saturated vapor is removed from contact
with the liquid, further addition of heat at constant pressure
results in a rise in temperature. If t s denotes the saturation
temperature given by t t =/Q?) and t the temperature after su
perheating, the difference t t s is the degree of superheat. Thus
for steam at a pressure of 120 Ib. per square inch, t s = 341.3^;
hence if at this pressure the steam has a temperature of 460,
the degree of superheat is 460  341.3 = 118.7.
As soon, therefore, as a vapor passes into the superheated
state, the character of the relation between the coordinates p, v,
and t changes. The temperature is freed from the rigid con
nection with the pressure that obtains in the saturated state,
and p and t may be varied independent!}' . The volume v of
the superheated vapor depends upon both p and t thus taken as
independent variables ; that is,
as in the case of a perfect gas. The form of the characteristic
equation (1) for a superheated vapor is, however, less simple
than that of the gas equation pv = BT.
The state described by the term " superheated vapor " lies
between two limiting states ; the saturated vapor on the one
hand, and the perfect gas, obeying the laws of Boyle and Joule,
on the other. The characteristic equation therefore should
196
be of such form as to reduce to the equation of the perfect
gas, as the upper limit is approached and to give the proper
values of p, v, and t of saturated vapor when the lower limit
is reached. In the case of compound substances like water
or ammonia, however, one disturbing element is introduced
at very high temperatures. The vapor may to some extent
dissociate ; thus steam may in part split up into its components
hydrogen and oxygen, ammonia into nitrogen and hydrogen.
Nernst has found for example that at a pressure of one atmos
phere 3.4 per cent of water vapor is dissociated at a temperature
of 2500 C. Manifestly the existence of dissociation must in
fluence the relation between the variables p, i>, and t. However,
at the temperatures and pressures with which we are concerned
in the technical applications of thermodynamics, the amount of
dissociation is entirely negligible, and the characteristic equation
may be assumed to hold for all temperatures within the range
of ordinary practice.
129. Critical States. The region between the limit curves
v', v" (Fig. 60) or s', s" (Fig. 62) is the region of mixtures of
saturated vapor and liquid.
The fact that these two curves
approach each other as the tem
perature is increased suggests
that a temperature may be
reached above which it is im
possible for a mixture of liquid
and vapor to exist. Let it be
assumed that the two limit
curves merge into each other
at the point S (Fig. 66), and 0'
thus constitute a single curve,
of which the liquid and saturation curves, as we have previously
called them, are merely two branches. The significance of this
assumption may be gathered from the following considerations.
Let superheated vapor in the initial state represented by
point A (Fig. 66 and 67) be compressed isothermally. Under
usual conditions, the pressure will rise until it reaches the pres
FIG. 6(3.
sure of saturated vapor corresponding to the given constant
temperature *, and the state of the vapor will then be represented
by point B on the saturation curve. Further compression at
constant temperature results in condensation of the saturated
vapor, as indicated by the line B 0. If the liquid be compressed
isothennally, the volume will be
decreased slightly as the pres
sure rises, and the process will
, / \B' A > be represented by curve CD.
' \ The isothermal has therefore
three distinct parts : along AB
the. fluid is superheated vapor,
along BO a mixture, and along
QD a liquid. If the initial tem
perature be taken at a higher
I 1 s value ', the result will be similar
FIQ 67 l except that the segment B' O' will
be shorter. If the limit curves
meet at point IT, it is evident that the temperature may be
chosen so high that this horizontal segment of the isothermal
disappears ; in other words, the isothermal lies entirely outside
of the single limit curve.
In Fig. 66 the segment BO represents the difference v" v'
between the volume v" of saturated vapor and the volume v 1 of
the liquid; and in Fig. 67, the area B 1 B00 1 represents the la
tent heat r of vaporization. For the isothermal t a that passes
through J?, the segment BO reduces to zero; hence, for this
temperature and all higher temperatures, we have
v" v' = 0, or v" = i>',
and r = 0.
The second result also follows from the first when we consider
the Clapeyron equation
v  v ' = Jr ^L
Tdp.
dT
The experiments of Andrews show that the condition just
dioxide as determined oy Andrews are snown in Jfig. t>o. Jb or
t= 13.1 and 21. 5 C. the horizontal segments corresponding
to condensation are
clearly marked. For
*= 31.1 the horizontal
segment disappears and
there is merely a point
of inflexion in the
curve. At 48.1 the
point of inflexion dis
appeared, and the iso
thermal has the general
form of the isothermal
for a perfect gas.
The temperature t c
was called by Andrews
the critical tempera
ture. It has a definite
value for any liquid.
The pressure p c and
volume v c indicated by the point S are called respectively the
critical pressure and critical volume. Values of t c and p c for
various substances are given in the following table:
50
FIG. 68.
SUBSTANCE
t c , DEOUEES C.
PC, ATMOSPHERES
Water . . . ...
365.0*
200.5
Ammonia ... ....
130.0
115.0
Ether
197.0
35.77
155.4
78.9
30.92
77.0
277.7
78.1
146.0
35.0
Oxygen
118.0
50.0
Hydrogen
220.0
20.0
Air
140.0
30.0
* According to the recent experiments of Holborn and Baumann, the critical
temperature of water is 706.1 F (374.5 C) and the critical pressure is 3200 11).
per square inch. See article by Prof. Marks, Jour. A. S. M. E., Vol. 33, p. 563.
Although at sufficiently high pressure the fluid may be in the
liquid state, the closest observation fails to show where the
gaseous state ceases and the liquid state begins. As stated by
Andrews, the gaseous and liquid states are to be regarded as
widely separated forms of the same state of aggregation.
It has been proposed to make the critical temperature the
basis of a distinction between gases and vapors. Thus, air,
nitrogen, oxygen, nitric oxide, etc., whose critical temperatures
are far below ordinary temperature, are designated as gases,
while steam, chloroform, ether, etc., whose critical temperatures
are above ordinary temperature are designated as vapors.
The determination of the critical values c , p c , and v c by ther
modynamic principles is a problem of great theoretical interest,
but lies beyond the scope of this book.
130. Equations of van der Waals and Clausius. Many
attempts have been made to deduce rationally a single charac
teristic equation, which with appropriate change of constants
will represent the properties of various fluids in all states from
the gaseous condition above the critical temperature to the
liquid condition. Such a general equation is that of van der
Waals, namely,
v  a v
which was deduced from certain considerations derived from
the kinetic theory of gases. As van der Waals' equation does
not accurately represent the results of Andrew's experiments
on carbon dioxide, Clausius suggested a modification of the
last term of the equation and ultimately arrived at an equation
of the form
where /( 2") is a function of the absolute temperature that takes
the value 1 at the critical temperature.
The equations of van der Waals and Clausius are constructed
with special reference to the behavior of fluids in the vicinity
>f the critical state ; hence they apply more particularly to
such fluids as carbon dioxide, the critical temperature of which
.s within the range of temperature encountered in the practical
implications of heat media. The critical temperatures of most
mportant fluids, as water, ammonia, and sulphur dioxide are,
lowever, far above the ordinary range, and for these media
ihe general equations do not give as good results as certain
purely empirical equations deduced from experiments covering
i relatively small region. For some fluids, notably ammonia,
:here is unfortunately a lack of experimental data; for the
.nost important fluid, water, we have, however, reliable data
tarnished by the recent experiments at Munich.
131. Experiments of Knoblauch, Linde, and Klebe. The
sxperiments made at the Munich laboratory were so con
iucted that three important
relations could be obtained
simultaneously. These
were :
1. Relation between pres
sure and temperature of
saturated steam.
2. Relation between spe
sific volume and temperature
of saturated steam.
3. Relation between pres
sure and temperature of
superheated steam with the
volume remaining constant.
The experiment covered
the range 100 to 180 C.
The apparatus employed is
shown diagrammatically in
Fig. 69. An iron vessel a contains a smaller glass vessel 5 to
which is attached a glass tube c. A similar glass tube d leads
B
a tube/ leading to a mercury manometer, oteam is mi/ruuuoeu.
into vessel a from a boiler, and suitable provision is made for
returning the condensed steam to the boiler.
A given weight of water is put into the glass vessel b and
is evaporated gradually by the heat absorbed from the steam
surrounding it. As long as vessel b contains a saturated mix
ture, the pressure within b must be the same as that within a,
since the temperature is the same throughout. Hence the
mercury levels m, m in tubes o and d will be at the same height.
When the water in b is all vaporized and the pressure and
temperature of the steam in a is further increased, the steam
in b becomes superheated. While
the temperature is still the same in
vessels a and 5, the pressures in the
two vessels are not equal. This
may be shown by the ^diagram
(Fig. 70). Let point A on the
saturation curve s" denote the state
of the steam in vessel b just at the
end of vaporization ; it also repre
sents the state of the saturated
steam in the outer vessel a. As
the temperature rises from ^ to t z the state of the steam
in a changes as represented by the curve A , that is, the
steam in a is saturated at the pressure p v The apparatus
is so manipulated, however, that the mercury level m in tube o
is held constant, thus keeping a constant volume of steam in
vessel b. The point representing the state of the steam in b
moves along the constant volume curve AS in the superheated
region, and the final pressure p 3 given by the point JS is smaller
than the pressure p 2 of the saturated steam in a. As a result
the mercury level in the tube d will be depressed to the
level n. A comparison of the mercury level in the manometer
with the level m gives the relation between the pressure and
temperature of superheated steam at the given constant
volume v\ and a comparison witli the level n gives the
relation between the pressure and temperature of saturated
steam.
L FiG. 70.
.RT. 132] EQUATIONS FOR SUPERHEATED STEAM 203
132. Equations for Superheated Steam. To represent the
esults of the Munich experiments, Linde deduced the empiri
al equation
 JZ>. (1)
n metric units with p in kilogram per square meter, the con
tants have the following values :
J5 = 47.10 tf= 0.031 =3.
a = 0.0000002 D= 0.0052
English units and pressures in pounds per square inch, the
iquation becomes :
pv = 0.5962 Tp(l + 0.0014^?) A 5030 ^ 000 _ o.0833\ (2)
Fhe form of Eq. (1) is such as to make it inconvenient for
he purpose of computation ; and the constant D in the last
,erm leads to complication in the working out of a general
heory. A modified form of the equation, namely,
* + ' = (l+*)fi ( 3 )
s free from these objections and with constants properly chosen
epresents the results of the Munich experiments as accurately
is Linde's equation. The constants are as follows :
METRIC UNITS ENGLISH UNITS
B = 47.113 B 85.87, p in pounds per square foot
= 0.5963, p in pounds per square inch
,ogm = 11.19839 log TO = 13.67938
n = 5 n = 5
c = 0.0055 . c 0.088
a = 0.00000085 a = 0.0006, p in pounds per square inch.
Fhe final equation with constants inserted is therefore
T fi . A nnnfl m \ 47795 x 10 _ SA*.
An equation o tne simple lorm
v + c=^ (5)
P
has been proposed by Tumlirz on the strength of Battelli's
experiments. Lincle has shown that this equation may be made
to represent with fair accuracy the results of the Munich ex
periments. For English units and with p in pounds per square
inch, the equation becomes
v + 0.256 = 0.5962. (G)
For moderate pressure this formula is quite accurate, but at
high pressures and superheat the volumes given by it are con
siderably smaller than those indicated by the experiments.
Two other characteristic equations deserve mention. For
many years Zeuner's empirical equation
pv = BT Cp n (7)
has been extensively used. The results of the Munich experi
ments have shown that the form of this equation is defective,
and that it cannot accurately represent the behavior of super
heated steam over a wide range. Callendar, from certain theo
retical considerations, has deduced the equation,
which in form resembles Eq. (3), but lacks the factor p in the
last term. While this equation is somewhat simpler than
Eq. (3), it is less accurate.
133. Specific Heat of Superheated Steam. The experimental
evidence on the specific heat of superheated steam may be clas
sified as follows :
1. The early experiments of Regnault at a pressure of one
atmosphere and at temperatures relatively close to
saturation.
2. The experiments of Mallard and Le Chatelier, Langen,
and others at very high temperatures.
3. The experiments of Holborn and Henning at atmospheric
pressure and at temperatures varying from. 110 to
1400 0.
4. Recent experiments with steam at various pressures and
with temperatures close to the saturation limit. Of
these, the experiments of Knoblauch and Jakob are
considered the most reliable.
Regnault concluded from his experiments that at a pressure
f one atmosphere the specific heat of superheated steam has
he constant value 0.48 for all temperatures. This value has
een largely used for all temperatures and for all pressures as
rail.
Experiments by Mallard and Le Chatelier and by Langen at
igh temperatures agree in making the specific heat a linear
auction of the temperature. Thus, according to Langen,
c p = 0.439 + 0. 000239 t, (1)
rhere t is the temperature on the C. scale.
The earlier experiments of Holborn and Henuing at much
Dwer temperatures than those of Langen lead to the formula
c p = 0.446 + 0.0000856 t. (2)
?his is again a linear relation, but the coefficient of t is smaller
han that in Langen's formula. Equations (1) and (2) show
hat the specific heat varies with the temperature at least, and
hat the convenient assumption of the constant value 0.48 is
Lot permissible.
Finally, the experiments of Knoblauch and Mollier show con
lusively that c p depends also upon the pressure. In these
experiments, steam was run through a first superheater in
diich all traces of moisture were removed. It was then run
hrough a second superheater consisting of coils immersed in
m oil bath. The heat was applied by means of an electric
lurrent and could be measured quite accurately, and a com
>arison of the heat supplied with the rise of the temperature of
lie steam gave a means of calculating the mean specific heat over
.he temperature range involved. Experiments were conducted
ii, mAssmAs of 9, 4. fi. a.nrl 8 ICQ. TtBT sauare centimeter. The
206 SUPERHEATED VAPORS [CHAP, xi
results are shown by the points in Fig. 71. From these
results the following conclusions may be drawn : (1) The
specific heat varies with the pressure, being higher the higher
the pressure at the same temperature. (2) With the pressure
constant, the specific heat falls gradually from the saturation
limit, reaches a minimum value, and then rises again.
Starting with the characteristic equation (3), Art. 132, it is
possible to deduce a general equation for the specific heat c p
that will give results substantially in accord with the experi
mental results of Knoblauch and Mollier. For this purpose we
make use of the general relation
From the characteristic equation,
BT ^
00
in x ' * s 'J.
we obtain by successive differentiation
dv B mn ,
.
C 1 + op). (6)
Substituting in (3), the result is
dc p \ Amn(n
""
Talcing T as constant and integrating (7) with p as the in
dependent variable, the result is
Amn(n + 1) / a \ , , , . J
C P =  jrs+i p(^ + nP )+ const, of integration.
Now since T was taken as constant, the constant of integration
may be some function of T; hence we may write
(8)
60
160
240
480
560
3SO 400
Temperature
le groups of points represent the results of experiments at 2, 4, 6, and 8 kg. per sq. cm.
respectively, beginning with the lowest group.
FIG. 71.
increased. From JLangen s experiments, it is seen tnat at very
high, temperatures c p is given by an equation of the form
hence we are justified in assuming that
where and /3 are constants to be determined from experi
mental evidence. Equation (8) thus becomes
0)
This is the general equation for the specific heat of superheated
steam at constant pressure.
It may be seen at once that this equation gives results agree
ing in a general way with those of Knoblauch and Mollier. At
a given temperature T the specific heat increases with the pres
sure ; furthermore for a given pressure, c p has a minimum value
as appears by equating to zero the derivative
a rn ' Wn+Z
The following values of the constants have been found to
make Eq. (9) fit fairly well the experimental results of Knob
lauch and Mollier :
a = 0.367
/3 = 0.00018 for the C. scale.
/3 = 0.0001 for the F. scale
Replacing the product Amn(n + 1) by a single constant (7,
we have as the final formula for the specific heat
c p = 0.367 + 0.0001 T+p(l + 0.0003 j?) ~, (10)
where log (7=14.42408 (pressure in pounds per square inch).
Figure 71 shows the curves representing this formula for the
pressures of the Knoblauch and Mollier experiments. The
agreement between the points and curves is satisfactory, con
sidering the difficulty of the experiments. In Fig. 72 the
<?pcurves for various pressures in pounds per square inch are
UJb' SUPERHEATED STEAM 209
134. Mean Specific Heat. Formula (10), Art. 133, gives
he specific heat at a given pressure and temperature. For
ome purposes it is desirable to have the mean specific heat be
ween two temperatures, the pressure remaining constant.
?his is readily calculated by the mean value theorem ; thus
L enoting by (c p ~) m the mean specific heat, we have
\. c p)m~~7jn rfi" ^ J
J 2 M
Jsing the general expression for c p , we have, therefore,
/ N 1 f^f , 0/77, Amn(n}V) ( ., , a
'
(2)
The calculation, while straightforward is rather long, and if
^curves are available, it is usually preferable to determine
he mean c p by Simpson's rule or by the planimeter.
Curves of mean specific heat are shown in Fig. 73. For any
degree of superheat the mean specific heat between the satura
ion state and the given state is given by the ordinate corre
ponding to the given degree of superheat and the given
iressure. For example, at a pressure of 150 Ib. per square
ach the mean specific heat for 240 superheat is 0.529.
135. Heat Content. Total Heat. Having a formula for the
pecific heat at constant pressure, equations for the heat con
ent and the intrinsic energy of a unit weight of superheated
team at a given pressure and temperature are readily derived.
for this purpose the general equation
dq = c p dT AT dp (see Art. 54) (1)
^
300 400
Superheat, Deg. F.
500
GOO
i=A(u+pv^
we have di = A [du +
or di = dq + Avdp. (2)
Hence, making use of (1),
From the characteristic equation we have
dv _ B n 4 ^ TO ^
___ + rc( a P)7jwi
whence T 7^ v = (w + 1) (1 + ap') 7 ~~ + c.
Introducing in (3) this expression for T  v and the general
expression for c p , the result is
Since z depends upon the state of the subtance only, the second
member of (4) must be an exact differential. The integral is
readily found to be
i Q . (5)
The constant of integration i Q is determined by applying
Eq. (5) to the saturation state. For a given pressure and cor
responding saturation temperature the second member of (5)
exclusive of can be calculated. The first member is the
value of i for the assumed pressure as given in the steam table.
Hence i Q is found by subtraction. By this method the mean
value i =886.7 is obtained.
Introducing known constants, Eq. (5) becomes
i = ^(0.367 + 0.00005 T)  p (1 + 0.0003^)^
0.0163^ + 886.7. ' (6)
Here log (7= 13.72511 when p is taken in pounds per square
inch.
The total heat of a unit weight of superheated vapor is the
heat required to raise the tem
perature of the liquid to the
boiling point at the given con
stant pressure, evaporate it, and
then superheat it, still at con
stant pressure, to the tempera
ture under consideration. On
the ^ZWplane, the process is
shown by the line ABCD (Fig.
74). The area OABCO l rep
resents the total heat of the
saturated vapor, which has
been denoted by q" , The area
A
FIG. 74.
represents the heat added to superheat the vapor.
This heat is evidently given by the integral
taken between the saturation temperature T 8 at point and
the final temperature T at point D. This integral is, in fact,
the product (c p ) OT (2 7 T a ~), where (e p ) w is the mean specific
heat for the temperature range T T t . The total heat of a
unit weight of superheated steam is given therefore by the
expression q= <? +(c,^T  Tj. (7)
The term (c p ^) m (T T s ) is easily found from the mean
specific heat curves (Fig. 73), and gr"(=i") is given in the
steam table. Hence with the aid of the curves, an approxi
mate value for the heat content may be calculated.
EXAMPLE. Find the heat content of one pound of steam at a pressure
of 150 Ib. per square inch superheated 200.
From the steam table t"(= <?") for this pressure is 1194.6 B.tu.; and
from Fig. 73 the mean specific heat from saturation to 200 superheat is
0.534.
Hence i = 1194.6 + 200 x 0.534 = 1301.4 B. t. u.
Thp roclllf. rnirar. K,T frvTTYiTila. f(\\ il 1 2f)1 .7 "R. t. 11.
214 SUPERHEATED VAPORS [CHAP, xi
136. Intrinsic Energy. For the intrinsic energy we have
from the defining equation i = A(u + pv),
Au = i Apv. (1)
Using the expressions for i and v heretofore derived, we obtain
the equation
* . (2)
This expression gives the intrinsic energy in B. t. u. of a unit
weight of superheated steam. Introducing the proper constants,
we have, when p is taken in pounds per square inch,
AM = 2 T (0.2566 + 0.00005 T^~ (1 + 0.00024 p) + 886. 7, (3)
where log (7=13.64593.
The intrinsic energy may also be found quite exactly by
the following method. For the given pressure p the energy
of one pound of saturated steam is
Au" = q' + p,
and the increase of energy due to the superheat is
where (c^) m denotes the mean specific heat at constant volume.
The difference (c^) m (c v ) m varies somewhat with the pressure
and superheat, but 0.13 may be taken as a mean value. Hence
the energy of one pound of superheated steam is given by the
equation
A U =q'+p + [(*,)  0.13](^ r a ). (4)
Values of q' and p are given in the steam table and the
proper value of (c p ) m may be found from the curves of Fig. 73.
EXAMPLE. Find the intrinsic energy of one pound of steam at a pres
137. Entropy. From the general equation
ntroducing in this equation the expressions previously derived
or c p and ( ^) (see Art. 133), the result is
\dJTJp
ds = + dT+ Amnp(  n +
["his is necessarily an exact differential since s is a function of
he state only. The integral is found to be
+ir (3)
nserting the known constants and passing to common loga
ithms, (3) becomes
s = 0.8451 log T+ 0.0001 T 0.2542 logp
0.0003^)  6  0.3964. (4)
11 using (4), p is taken in pounds per square inch, and
og (7=13.64593. The constant 0.3964 is determined by
>assing to the saturation limit, as was done in finding the
ralue of .
Equation (4) gives the entropy of one pound of superheated
iteam at any given pressure and temperature.
The entropy may also be found as follows. Let the point D
'Fig. 74) represent the state of the fluid and assume CD
;o be a constant pressure line cutting the saturation curve
it 0. Then OO 1 gives the entropy s" of saturated steam
it the same pressure as the superheated steam, and
en! ropy is
> the iiiii
,." !
thi' MIMW in ih' '*;!*' /'
fit , i Tf .4 * ?'' t . *
?' ! /if f _, /,. 1 .,
ulili*, h\vr\T, .i.' if 'l"'
rr. )
1. Pillil llli rtitj'i'p.^, 'IU;>. as.l )i'' >' !
Hlcain ul ;i jurotuv { >1 t! ISi, J<i vn,r. :..! .'( '!'' ;t'. ; I ; t";^  ' ;ifs "
tif I in F.
2. S.tiiirat''l f.l":nit ;il 11 j.jr..m. .< Hull; ]t la^. ,;...!; V*".. 'nf< ' f
Nlli('r!l';t(f'il (u U fria"i :s!n' if .'>' I a "" '*.*.' j..v.j: J rS:'l
(,i) h:vt iwlili.l; (A) rh.ur,:" { >u^:\ f. l .!..; .;  i' s.'i^j . J J ' 5 ^ S
el' fili'aJn.
3. A'.HUIMI i.U;i ;uiS l 'Uij4'.." !l. n. n"i. .M.v,s.I 1. v^,:.,; M ai. t'/}
nf Art. i:i;.
4. Av,uitH' il.ii.i tuil iskuhit* !?:.)> 1;. ?"!: tM .."! * *. ^ !t ' i '*'*
Coiuj'iUo ti'.nU 1 '.
138. Special Change of State. Uv ju..ti ( > !,.u.i. !;> i*'
I'tpUtttUll (o) % Art. l:'L', UUll llir i'rjtrj.l! r,,'i.s!l..:r. jL.if L.t^*"
"iicrn (li'iltti'i'il for tin* li.it < intiul, !;,"* , .UP! : j IMJ.*, , iu"'t
pnilili'ins (luit arise in fujun'fijMU \%j?h lit' *h.iH .** "f '".t *
stiiti'rhcali'tl sti.un may } Ml\nl *,Mfis ..jir, .&:;'' .. { * !l
Ht'l'iiUnl, li>W(M'r, [ the f.l.'I ii,U fhr ',j., ( :!: L. ,if '! ?. S H
given by a snnu\\hat fiiupli.tt'l f.isjr.t'.;, it > .un;"! i'*  *'^"
ju'rti'tl lliat tin rclatiuns Is.'r. .irn ! will h.r, ? ; H :?in*^ < Iiin
nf tlmsr fur pcrfrrl ;j;t^s. In fhr }.;!.!!:.' '! :' U'<'4' >H' *'*
spi'i'ial chanins ttf ;taf\ v> c sh.t'.l ',;IM* ini;.'., .tu in!an "i <^*'
jiroct'ssi'.s involved, Iraviii:,^ the ilfMil'' I" ) I;'/.'l in ^'> ^*''
slutU'iit.
1. Constant Pressure. Let superheated stearn change state
i constant pressure from an initial temperature ^ to a final
mperature t z . For the heat added we have
 Amp (n + 1) (l + ^VL _ 1 \ (1)
\ it j \jt 2 Ji J
he external work is given by the relation
W=p(v z vJ=B(T,TJmp (I + ap) [~L _ 1 1 (2 )
Lfz  L l J
he change of energy may be found from the energy equation
u z uj, = Jq W,
1 independently by calculating from the general formula the
lergies in the initial and final states.
The change of entropy may be obtained, likewise, from the
3neral equation for entropy or from the relation
/QN
(3)
The preceding equations apply to a unit weight of the
lid.
2. Constant Volume. If T and T 2 denote, as before, the
itial and final temperatures, respectively, we have from the
laracteristic equation
l + (4)
om which p z may be found. Having T v p v and T z , p v the
itial and final values of the energy and entropy may be de
rmined from the general formulas. Since the external work
zero, the heat added is equal to the increase of energy.
3. Isothermal Expansion. Let the initial and final pressures
characteristic equation. For the change of entropy per unit
weight we have from the general equation for entropy
(6)
The heat added during the expansion per unit weight is
therefore
For the external work, taking dv from the characteristic equa
tion, we have
fl + S!L(f l *pf). (7)
Pz *
The change of energy may be found by combining (6) and (7)
or from the general equation of energy. It is found to be
%  % = {0>i JPa) + f (  1) Oi 2  ?)] (8)
It should be noted that in the case of superheated steam con
stant temperature does not, as with perfect gases, indicate con
stant intrinsic energy.
4. Adiabatic Change of State. For an adiabatic change the
entropy remains constant ; hence, for the relation between the
final pressure p z and temperature T z , we have from the general
equation for entropy
where is a constant determined from the initial state. The
pressure p 2 is generally given ; therefore, we have the tran
scendental equation
z =C', (9)
Having the initial and final values of p and T, the initial and
nal values u^ and w 2 of the intrinsic energy may be calculated,
'he external work per unit weight is then
W=u l u z . (10)
In problems connected with the flow of steam the change of
.eat content resulting from an adiabatic expansion is required.
?his difference is found by calculating from the general equation
or the heat content the initial and final values i t and z 2 .
If the adiabatic expansion is carried far enough, the expansion
Ine, as >JE (Fig. 74), will cross the saturation curve s", and the
tatepoint will enter the region between the curves s' and s".
lliis means that at the end of the expansion the fluid is a mix
ure of liquid and vapor. The investigation of this case presents
to difficulties. The entropy and energy at the initial point D
,re calculated from the general equation. Knowing the pressure
or the final state JS, the quality x is readily determined from
he equation
xr n
vhere s denotes the entropy in the initial state. Having x, the
snergy in the final state is calculated from the equation
u 2 = J^' + z/> 2 ). (12)
Flien the external work per unit weight is given by the equation
(13)
EXAMPLE. Steam at a pressure of 150 Ib. per square inch absolute and
superheated 100 F. expands adiabatically to a pressure of 5 in. of mercury.
Required the final condition of the fluid and the external work per pound;
ilso the pressure at which the steam becomes saturated.
From the general equation the entropy in the initial state is found to be
L.6346. From the steam table we obtain for the final pressure s' = 0.1880,
= 1.7170; hence
T 1.6346 = 0.1880 + 1.7170 x,
)r x = 0.8425.
[n the initial state the energy in B. t. u. is
4wi = 918.1(0.2566 + 0.00005 x 918.1)  JfjrrgC 1 + 000024 x 150) + 880.7
= 1153.9 B. t. u.
In tne janal state tne energy is
Au z  q z ' + x 2 p 2 = 101.7 + 0.8425 x 953.7 = 905.2.
Hence, the external work per pound of steam is
W= MI  Ma = 778(1153.9  905.2) = 193,490 ffc.lb.
The initial entropy 1.6346 is the entropy of saturated steam at a pressure of
66.6 Ib. per square inch. Hence the steam becomes saturated at this pressure.
139. Approximate Equations for Adiabatic Change of State.
Exact calculations that involve adiabatic changes of superheated
steam are tedious on account of the transcendental form of the
.equation for entropy ; and it is therefore desirable to introduce
simplifying approximations, provided the results obtained by
them are sufficiently accurate. An investigation of a number
of cases covering the range of values ordinarily used in the
technical applications of superheated steam shows that a set of
equations similar in form to the equations for a perfect gas
may be obtained, and that the error involved in using these
approximate equations does not in general exceed one or two
per cent.
The relation between pressure and volume during an adiabatic
change may be represented approximately by the equation
p (v + o) n = const. (1)
The value of c is taken the same as in formula (4), Art. 131,
namely, c = 0.088.
The value of n probably varies slightly with the initial pres
sure and with the degree of superheat ; however, it appears that
the value n 1.31 gives quite accurate results for the range of
pressure and superheat found in practice. If now we take the
approximate characteristic equation
p(y + c) = BT, (Art. 132) (2)
we get by combining (1) and (2),
or
Given the initial state of the fluid, the volume in the final
;ate may be found from (1), the final temperature from (4),
ad the external work from (5).
EXAMPLE. A pound of superheated steam at a pressure of 200 Ib. per
[uare inch and superheated 200 expands adiabatically to a pressure of
) Ib. per square inch. Kequired the final condition and the external work.
The initial volume is found to be 2.973 cu. ft., and the initial entropy
6657. Using the formula for s (Art. 137), the final temperature is found
r trial to be 752.5 absolute ; and taking this value of T, the exact value
; the final volume is found to be 8.6S1 cu. ft.
From (3), Art. 136, the energy in the initial state is found to be 1200.57
. t. u., that in the final state 1098.82 B. t. u. ; hence the external work is
'8 (1200.57  1098.82) = 79,262 ft.lb.
Taking the approximate formulas, we have
i _i_
v 2 + c = ( Vl + c) ( iV= (2.973 + 0.088) f?2V a = 8.819:
\2 } 'i' \50 J
hence v 2 = 8.819  0.088 = 8.731 cu. ft.
H4
It will be seen that for practical purposes the results obtained from the
iproximate equations are satisfactory as regards accuracy.
140. Tables and Diagrams for Superheated Steam. The lead
g properties of superheated steam volume, entropy, and
tal heat for various pressures and degrees of superheat
ive been calculated and tabulated by Marks and Davis and
r Peabody. The values in the Marks and Davis tables are
srived from specific heat curves that differ somewhat from the
irves of Fig. 72, and they therefore differ from the values
itained from the equations of Arts. 135137. However,
.roughout the range of ordinary practice, the difference does
t exceed one half of one per cent.
The Marks and Davis tables are accompanied by graphical
.arts that may be used to great advantage in the approximate
22 SUPERHEATED VAPORS
jlution of numerical problems. The principal chart has the
eat content i as ordinate and the entropy s as abscissa. The
turated steam at various pressures. The region above this
irve is the region of superheat, and the lines running approxi
ately parallel to the saturation curve are lines of constant
igree of superheat. Below the saturation curve is the region
wet steam, and the lines running parallel to the saturation
irve are lines of constant quality. The lines that cross the
ituration curve obliquely are lines of constant pressure.
The first conception of the heat contententropy chart is
.ie to Dr. E. Mollier of Dresden, hence we shall refer to it as
ie Mollier chart. In addition to the chart published by
"arks and Davis, one is contained in Stodola's Steam Turbines
id one in Thomas' Steam Turbines. In the light of the
icently acquired knowledge of the properties of saturated and
iperheated steam, the Marks and Davis chart must be regarded
j the most accurate.
The Mollier chart may be used for the approximate solution
: many problems that involve the properties of saturated and
iperheated steam, and it is specially valuable in problems on
le flow of steam. The following examples illustrate some of
le uses of the chart :
Ex. 1. Steam at a pressure of 150 Ib. per square inch superheated 200 F.
rpands adiabatically to a pressure of 3 Ib. per square inch.
The point representing the initial condition lies at the intersection of the
instantpressure line marked 150 and the line of 200 superheat. Locating
ds point on the chart, it is found at the intersection of the lines i 1300
id s = 1.087. The heat content and entropy in the initial state are thus
itermined. The line 5 = 1.687 intersects the constantpressure curve p = 3
i the line i = 1002 ; hence the heat content after adiabatic expansion is
)02 B. t. u. The quality in the final state is found to be 0.88.
Ex. 2. When steam is wiredrawn by flowing through a valve from a
igiou of higher pressure j t to a region of lower pressure p, the heat content
mains constant. Steam at a pressure of 200 Ib. per square inch and
lality 0.95 flows into the atmosphere ; required the final condition of the
earn.
Drawing a line of constantheat content from the initial point to the
irve p = 14.7, it is found that the final point lies above the saturation curve
id that the steam is superheated about 12 at exit. The entropy increases
om s = 1.498 to s = 1.766.
141. Sunftrheated Ammonia and Sulphur Dioxide. Experi
224 SUPERHEATED VAPORS [CHAP, xi
other than that of water is very scant, and our knowledge of
such properties is accordingly imperfect. For superheated
ammonia Ledoux has proposed the characteristic equation
pv = BT Qp m , (1)
and this form has been accepted by Peabody, who derives the
following values of the constants (English units) :
=99, (7=710, m = .
For sulphur dioxide Peabody uses the same equation with the
constants :
5 = 26.4, (7=184, TO = 0.22.
According to Regnault the specific heat of superheated ammo
nia has the constant value 0.52. It is very likely that this
specific heat is no more constant than that of superheated
steam and that it varies with pressure and temperature. How
ever, experimental evidence on this point is lacking. Lorenz
finds that for superheated sulphur dioxide c v = 0.329.
The problem that most frequently arises in connection with
the use of these fluids as refrigerating media is the determi
nation of the state of the superheated vapor after adiabatic
compression. It may be assumed that the relation between
pressures and temperatures for an adiabatic change follows
approximately the law for perfect gases, namely:
. (2)
_ r
Zeuner found that for superheated steam the exponent
in (2) is equal to the exponent m in the characteristic equation
(1). Hence, using the values of m assumed by Peabody, we
have:
For ammonia n =  = = 1.333.
l_ m 1Q.25
For sulphur dioxide n = = 1.282.
BT.
vapor, juet A. ^rig. 10; represent tne initial state,
id B the final state after adiabatic compression. EA and
'B are constantpressure curves. Denoting by TJ the satura
on temperature correspond
ig to the pressure p r the
icrease of entropy from E
T
) A is Cploge^, and the
>tal entropy in the jstate A is
s/'fCplog.^l.
ikewise, the entropy in the
;ate B is
T.
II
FIG. 76.
ince AB is an adiabatic, the entropies at A and B are equal,
id therefore
i this equation s/', s 2 ", 2V, and 3Y' ^ re tabular values corre
jonding to the given pressures p 1 and
[ence, !T 2 is the only unknown quantity.
and 2j is given.
EXERCISES
1. Calculate by Eq. (2), (4), and (6), respectively, of Art. 132 the vol
ne of one pound of superheated steam at a pressure of 180 Ib. per square
.ch and a temperature of 430 F. Compare the results.
2. If the products pv are plotted as ordinates with the pressures p as
>scissas, show the general form of the isothermals T = C when Eq. (3),
rt. 132 is used ; when Eq. (6) is used.
3. For ammonia, Peabody gives the following equations for the latent
?at of vaporization : r = 540 0.8 (t 32) . If at the critical temperature
= 0, find t c for ammonia by means of this formula and compare with the
ilue of t c given in Art. 129. Explain the discrepancy.
4. Following the method of Art. 133, deduce an equation for c p , using
le approximate equation (5), Art. 132; also using Calendar's equation (8).
5. By means of Eq. (3), Art. 132, calculate the specific volume of satu
,ted steam at the following pressures : 5 in. Hg., 20, 50, 150 Ib. per square
men. USB
pare the results with the values of v" given in the table.
6. Calculate the mean specific heat of superheated steam at a pressure
of 140 Ib. per square inch between saturation and 250 superheat. Compare
the result with the curves of Fig. 73.
7. Using the mean specific heat curves, Fig. 73, find the heat content
and energy of one pound of superheated steam at a pressure of 85 Ib. per
square inch and a temperature of 430 F.
8. A pound of saturated steam at a pressure of 120 Ib. per square inch
is superheated at constant pressure to a temperature of 386 F. Find the
heat added, the external work, and the increase of energy.
9. The steam after superheating expands adiabatically tmtil it again be
comes saturated. Find the pressure at the end of expansion and the
external work.
10. The following empirical equation has been proposed for the value
of c p very close to the saturation limit :
^=.,
(Jc la)
in which t c is the critical temperature, 689 F., and t a is the saturation tem
perature corresponding to an assumed pressure. Using the curves of
Fig. 72, calculate the value C for several assumed pressures, and thus test
the validity of the formula for these curves.
11. The following equation has also been proposed for the value of c p
at saturation : (c p ) Ba t = a + bt s . Test this equation, and if it holds good
within reasonable limits determine the constants o and &.
12. In the initial state 6.4 cu. ft. of superheated steam has a temperature
of 420 F. and is at a pressure of 160 Ib. per square inch. By the approxi
mate equations of Art. 139 find the temperature and volume after adiabatic
expansion to a pressure of 80 Ib. per square inch ; also the work of expansion.
13. Assume for the initial state of superheated steam p^ = 80 Ib. per
square inch, v : = 20 cu. ft., ^ = 350 F. Plot the successive pressures and
volumes for an isothermal expansion to a pressure of 30 Ib. per square inch.
Compare the expansion curve with the isothermal of air under the same
conditions.
14. With the data of Ex. 13 find the external work, heat added, and
change of energy (a) for the superheated steam ; (fc) for air.
REFERENCES
THE CRITICAL STATE. EQUATIONS OF VAN DER WAALS AND CLAUSIUS
The literature on these subjects is very extensive. For comprehensive
discussions, reference may be made to the following works :
Preston: Theory of Heat, Chap. V, Sections 6 and 7.
euner: Technical Thermodynamics (Klein) 2, 202229.
hwolson : Lehrbuch de Physik 3, 791841.
CHARACTERISTIC EQUATIONS
allendar : Proc. of the Royal Soc. 67, 266. 1900.
inde : Mitteilungen iiber Forschungsarbeiten 21, 20, 35. 1905.
euner : Technical Thermodynamics 2, 223.
/"eyrauch : Grundriss der WarmeTheorie 2, 70, 87.
SPECIFIC HEAT OF SUPERHEATED STEAM
[allard and Le Chatelier : Annales des Mines 4, 528. 1883.
angen : Zeit. d. Ver. deutsch. Ing., 622. 1903.
olborn and Henning : Wied. Annalen 18, 739. 1905. 23, 809. 1907.
egnault: Mem. Inst. de France 26, 167. 1862.
noblauch and Jakob : Mitteilungungen iiber Forschungsarbeiten 35, 109.
noblauch and Mollier : Zeit. des Ver. deutsch. Ing. 55, 665. 1911.
horaas : Proc. Am. Soc. Mech. Engrs. 29, 633. 1907.
A most complete discussion of the work of various investigators is given
f Dr. II. N. Davis, Proc. Am. Acad. of Arts aud Sciences 45, 267. 1910.
GENERAL THEORY OF SUPERHEATED VAPORS
allendar : Proc. of the Royal Soc. 67, 266.
r eyrauch : Grundriss der "WarmeTheorie 2, 117.
3uner : Technical Thermodynamics 2, 243.
CHAPTER XII
MIXTURES OF GASES AND VAPORS
142. Moisture in the Atmosphere. Because of evaporation
of Welter from the earth's surface, atmospheric air always con
tains a certain amount of water vapor mixed with it. The
weight of the vapor relative to the weight of the air is slight
even when the vapor is saturated. Nevertheless, the moisture
in air influences in a considerable degree the performance of
air compressors, air refrigerating machines, and internal com
bustion motors ; and in an accurate investigation of these ma
chines the medium must be considered not dry air but rather a
mixture of air and vapor. The study of air and vapor mixtures
is also important in meteorology and especially in problems
relating to heating and ventilation. Finally, it has been pro
posed to use a mixture of air with highpressure steam as the
working medium for heat engines, and the analysis of the action
of an engine working under this condition demands a special
investigation of air and steam mixtures.
Experiment has shown that Dalton's law holds good within
permissible limits for a mixture of gas and vapor. The gas has
the pressure p' that it would have if the vapor were not present,
and the vapor has the pressure p" that it would have if the gas
were not present. The pressure of the mixture is
P=p'+p". (1)
If the vapor is saturated, the temperature t of the mixture must
be the saturation temperature corresponding to the pressure
p". If the temperature is higher than this, the vapor must be
superheated.
The water vapor in the atmosphere is usually superheated.
Let point A, Fig. 77, represent the state of the vapor, and let
A.B be a Constant DresSlirH P.nrvo nnt.f.inrr f.lm unfn vrHirm p.nrvft
RT. 142]
MOISTURE IN THE ATMOSPHERE
229
b B. Further, let m denote the weight per cubic foot of the
apor in the state A, and m^ the weight per cubic foot of satu
xted vapor at the same temperature, that is, in the state 0.
'he ratio is called the humidity of the air under the given
onditions. If the mixture of air and vapor is cooled at constant
ressure, the vapor will follow the
ath AB and at B it will become
iturated. Upon further cooling
3me of the vapor will condense.
?he temperature T Q at which con
ensation begins is called the dew
oint corresponding to the state A.
The humidity may be expressed
pproximately in terms of pressures.
<et p a " denote the pressure of the
apor in the state A and p c n the
ressure of saturated vapor at the
nine temperature, hence in the state represented by 0. At the
)w pressures under consideration we may assume that the vapor
allows the gas law p V MET. Hence, taking V 1, we have
FIG. 77.
'herefore, denoting the humidity by <, we have
m ">"
p
(2)
'hat is, the humidity is the ratio of the pressure corresponding
D the dew point to the saturation pressure corresponding to
le temperature of the mixture.
For investigations that involve hygrometric conditions, the
ata ordinarily required may be found in table II, page 319.
At 70 the saturation pressure is, irom taoie JLI, U.MO incnes 01 ug,
while at 52 the saturation pressure is 0.13905 inches of Hg. The humidity
is therefore
03905 =
v 0.738
If the air were saturated at 70, it would contain 8.017 grains of vapor per
cuhic foot. Hence with 52.9 per cent humidity the weight of vapor per
cubic foot is
8.017 x 0.529 = 4,211 grains.
EXAMPLE 2. Atmospheric air has a temperature of 90 F. and a humidity
of 80 per cent. It is required that air be furnished to a building at 70 F.
and with 40 per cent humidity.
From table II, the pressure of saturated vapor at 70 is 0.738 inches
of Hg; hence from (2) the pressure corresponding to the dew point is
0.40 x 0.738 = 0.2952 inches of Hg, and the dew point is 44.5. In the initial
state one cubic foot of air contains 0.80 x 14.85 = 11.88 grains of vapor.
The air is cooled to 44.5 by proper refrigerating apparatus and in this state
contains 3.39 x 459 ' 6 + M ' 5 = 3.11 grains, the difference 11.88  3.11 = 8.77
459.6 + 90
grains being condensed. The air freed from the condensed vapor is now
heated to the required temperature, 70.
143. Constants for Moist Air. The constants B, c v , <?,
etc., given in Chapter VII apply only to dry air. For ail
containing water vapor the constants must bo changed some
what, the magnitude of the change depending, of course, upon
the relative weight of vapor present.
An expression for the constant B of the mixture may be
obtained by the following method. Let the volume V contain
M 1 Ib. of air at the pressure p' and M z Ib. of water vapor at
the pressure p". Then assuming that the gas law may be
applied to the vapor, we have
, (1)
(2)
Let Ta = 3, an d ^2 = e ; then from (3)
(4)
hence *>"=.Pff^ ^f'
dding the members of (1) and (2), we obtain
'he constant m of the mixture is, however, given by the
^nation
pV=(M^M^B m T. (7)
[ence, comparing (6) and (7), we have
Taking the molecular weight of water vapor as 18, we have
=85.72,
. j jD 9 85. 1 2 1 PI
ld '"^so* 1  81 
EXAMPLE. Find the value of B for air at 90 F. completely saturated
Lth water vapor. The pressure of the mixture is 14.7 Ib. per square inch.
From the table the pressure p" of the vapor is 0.691 Ib. per square inch ;
erefore the pressure p 1 of the air is 14.7 0.691 = 14.009 Ib. per square
ch. From (5), 1 + ez = = ^ = 1.0493, ez = 0.0493, and z =
'/) J.'x.UUty
1 D4.Q3
0.0306. Therefore, B m = 53.34 x ~^ = 54.31.
l.OoOo
The specific heat of the mixture is found by applying the
w deduced in Art. 83. If cj and c p " denote respectively
.e specific heats of the air and steam, then the specific heat of
.e mixture is given by the equation
EXAMPLE. Taking c p for air as 0.24, and for steam at 90 as 0.43, the
seine heat of the mixture given in the preceding example is
0.24 + 0.0306 x 0.43 _
1 4 0.0306
144. Mixture of Wet Steam and Air. In a given volume V
let there be M Ib. of air and M z Ib. of saturated vapor mixture
of quality x. The absolute temperature of the entire mixture
is T, and the total pressure p. The pressure p is the sum of
the partial pressures p' and p" of the air and steam, respec
tively. This follows from Dalton's law, which whithin reason
able limits holds good for the case under consideration. We
have then
p' + p" = p, (1)
p'V^MJBT, C2)
F= Jf 2 [>(*/'  ') + <], (3)
where, as usual, v' r and v' denote, respectively, the specific
volumes of steam and water at the saturation temperature T.
The energy of the mixture is the sum of the energies of the
two constituents ; hence, we have
AU= M lCv T+ M z (j + xp) + Z7 . (4)
Likewise, the entropy of the mixture is
S = M l [> log, T+ (c 9  O log e F] + M z + + ff . (5)
By means of these equations various changes of state may be
investigated.
145. Isothermal Change of State. Since ^remains constant,
we have from (4)
A( U z  Z7i) = Mf(zt  xj, (1)
and from (5)
%S^ M,AB log e ^ + M 2 Lfa  xj. (2)
Hence, the heat added is given by the equation
Q = T(S Z  SJ = MtABTlog. p + Jf 2 r(^ 2  ^. (3)
The external work is
(4)
1 neglecting the small water volume v',
VZ =
hile in the initial state
(6)
ence, combining (5) and (6),
_ JF
From (7) it appears that isothermal expansion is accompanied
r an increase of the quality #, that is, by evaporation, while
^thermal compression involves condensation.
146. Adiabatic Change of State. In the case of an adiabatic
ange the final total pressure j9 2 is usually given. Assuming
at the steam in the mixture does not become superheated,
e final temperature T z of the mixture must be the saturation
cnperature corresponding to the partial pressure p 2 " of the
jam. The determination of the final state of the mixture
solves the determination of two unknown quantities ; namely,
3 partial pressure p 2 " and the quality # 2 of the saturated
por. Hence two relations are required. One is given by
3 condition that the entropy of the mixture shall remain
astant during the change, the other by the condition that
3 final volume V z may be considered as occupied by each
istituent of the mixture independently of the other.
En the application of the first condition it is convenient to
3 an expression for the entropy of the mixture of a form
ferent from that given by (5), Art. 144. In terms of the
nperature and pressure, the entropy of a unit weight of air
^iven by the expression
s = c p log, TAB log e p + s ;
ice for the mixture we have
S=M, (c Joer, TAB loer. '") + A
IVJLJL^VJ. \JJ
As the constant $ disappears when the difference of entropy
between two states is taken, it may be ignored in the calculation.
Let 8} denote the entropy in the initial state. Then since
the entropy remains constant, we have
^ = M l (c v log fl T z  AB log a K) + JfV + (2)
\ j.% /
In Eq. (2), S r M v M. 2 , and the coefficients c v and AB are
known, as is the final total pressure p z . The partial pressures
p 2 r and p z ", the quality a; 2 , and temperature T z are unknown.
However, T z depends upon ? 2 ", and p z ' is found from the
relation p z + p z " = p 2 when p z ' f is determined. Denoting the
final volume by V v we have
whence W
Inserting this expression for x z in (2), we have finally
SMIo TABlo Mfs
+  (4)
\ M z Pz V 2 /
In this equation p z is the only unknown. The solution is
most easily effected by assuming several values of p z " and
calculating for these the values of the second member. These
calculated values are then plotted as ordinates with the corre
sponding values of p z " as abscissas and the intersection of the
curve thus obtained with the line ^ = const, gives the desired
value oip z "'
The external work of expansion or compression is equal to
the change of energy. Hence, using the general expression
for the energy of the mixture, we have
(5)
EXAMPLE. In a compressor cylinder suppose water to be injected at the
beginning of compression in such a manner that the weight of water and
water vapor is just equal to the weight of the air. Let the pressure of the
mixture be normal atmospheric pressure 20.92 in. of mercury, and let the
temperature be 79.1 F. The mixture is compressed to a pressure of 120 Ib.
ing to 79.1 is 1 in. Pig, hence the partial pressure of the air is 28.92 in. Hg.
The initial quality x is found from the relation
whence
53 ' 34 x 538 ' 7
= 0.0214.
M 2 pi'vi" 28.92 x 0.4912 x 144 x 656.7
The factor 0.4912 x 144 is used to reduce pressure in inches of mercury to
pounds per square foot.
For l.he entropy of the mixture we obtain from (1) (neglecting the con
stant So)
Si = 0.24 log e 538.7  0.0686 log e (28.92 x 0.4912) +0.0916 + 0.0214 x 1.9482
= 1.4587.
Since the ratio of the final to the initial pressure of the mixture is =1 = 8.2,
we assume that the pressure pz" of the vapor after compression will be
approximately 8 times the initial pressure pi". Hence we assume p 2 " 7,
8, and 9 in. of mercury, respectively, and calculate the corresponding values
of the second member of (2). Some of the details of the calculation are
given.
FROM STEAM TABLE
(in. I
!")"
' n>
I
T z
**'
j o
V*'
Pz
& 2 sq.
in.
7
3.43
116.57
146,
.0
606.5
0.2007
1011.1
104.4 ]
8
3.92
116.08
152.
3
611.9
0.2186
1007.9
92.18
\ Data
9
4.41
115.59
157.
1
616.7
0.2265
1005.0
82.57 J
I
p a
ABlog
1.5378
1.5400
1.5418
0.3264
0.3262
0.3259
0.0308
0.0349
0.0390
1.4519 1
1.4673 I Results
1.4814 J
The pressure p 2 " that gives the value S = 1.4587 lies between 7 and 8 in. Hg
and by the graphical method or by interpolation we find p 2 " = 7.44 in. Hg,
or 2>a" = 3.65 Ib. per square inch. Therefore p 2 ' = 120  3.65 = 116.35 Ib.
per square inch. From the steam table the following values are found for
the pressure p a " = 7.44 in. Hg : t = 149.3, T 2 = 608.9. qj = 117.3, r 2 = 1000.4,
p 2 = 942.8, v 2 " = 99. The final quality is
53.34 x 608.9
p 2 'v 2 " 116.35 x 144 x 99
The external work per pound of air is
W= ,7[0.17(149.3  79.1) + 117.3  47.2 + 0.0214 x 989.8  0.01958 x 942.8]
= 61566 ft. Ib.
236 MIXTURES OF GASES AND VAPORS [CHAP, xn
The volume of the mixture at the end of compression is
V = * = 6*8421608.0 = 1<08ao cu< f t
p a ' 110.85 x 144
and the work of expulsion is therefore
1.9380 x 120 x 144 = 33408 ft. Ib.
Hence, the work of compression and expulsion is 950(5'! ft. Ib.
The effect of injecting water into a compressor cylinder may be shown
by a comparison of the result just obtained with the work of compressing
and expelling 1 Ib. of dry air under the same conditions.
The initial volume of 1 Ib of air is f = la  C74 cu  fti 
The final volume after adiabatic compression to 120 Ib. per squaro inch is
JL
13.574 (^pY' 4 = 3.0290 cu. ft.
\ .l^U /
The work of compression is
,7 x 13.574  120 x 3.0290) = 59044 ft. Ib.,
the work of expulsion is 3.0290 x 120 x 144 = 52350 ft. Ib., and the sum is
111394 ft. Ib. The effect of water injection is therefore to reduce the
volume and temperature at the end of compression and the work of com
pression and expulsion. The reduction of work in this case is about 17
per cent.
147. Mixture of Air with Highpressure Steam. In the pre
ceding articles, we have dealt with mixtures of steam and air
in which the pressure of the vapor content was small. The
suggestion has been made that a mixture of air at relatively
high temperature and pressure mixed with steam either super
heated, saturated, or with a slight amount of moisture be used
as a medium for heat engines. An analysis of the action of
such a medium in a motor demands in the first place a discussion
of the process of mixing, afterwards a discussion of the change
of state of the mixture.
Let M^ Ib. of air compressed to a pressure p l and having a
temperature T^ be mixed with M z Ib. of wet steam having a
pressure w~ and mialihv 1 Tlio fomT\ovof.iivo T 1 nf
g the air into a receiver which contains steam, or vice versa.
nee under these conditions the pressure of the mixture can
it be raised above the pressure of the constituents, the volume
the mixture cannot be taken as the original volume V l of
e air. We assume, on the other hand, that the conditions
e such that the volume of the resulting mixture is the sum
the volumes of the constituents ; that is,
F=F 1+ F 2 . (1)
s a second condition, the internal energy of the mixture is
Tial to the sum of the energies of the constitutents ; hence
e have the equation of condition
U=U l+ U r (2)
Let T denote the temperature after mixing p' the partial pres
.re of the air and p" the partial pressure of the steam. Then,
ovided the steam does not become superheated, the tempera
.re T must be the saturation temperature corresponding to the
essurep".
The following relations are readily obtained.
(3)
1 since the quality x 2 is nearly 1,
(4)
(5)
here x denotes the quality after mixing, and v" is the specific
)lume of steam corresponding to the pressure//'.
(6)
z (q ! + xp). (8)
com (2) we have
= M,e n T, + MM + a; 2 P 2 ). (9)
(10)
Pi
Having V calculated from (10), we obtain from (5)
and this expression for x substituted in (0) gives finally
(12)
In (12) the second member is known from the initial condi
tions. In the first member q', p, and v" are dependent on T \
hence T is the one unknown. As usual, tho solution is ob
tained by taking various values of T and plotting the resulting
values of the first member of (12).
EXAMPLE. Let 1 Ib. of wefc steam, quality 0.85, at a pressure of 200 Ib.
per square inch, be mixed with 2 Ib. of air at a pressure of 220 Ib. per
square inch and a temperature of 400. Required the condition of the
mixture.
From the data given, the following values are readily found :
Vi = 2.895 cu. ft. ; V z = 1.948 cu. ft. ; V = 2.81)5 + 1.948 = 4.843 cu. ft.
U=Ui+U>2 = 1273.8 B. t. u.
Equation (12) becomes
0.34 T + q' + 4.843 ^ = 1273.8.
v"
We now assume for p" the values 50, 75, and 100 Ib. per square inch;
from the tables we find the corresponding values of q', p, v", and T, and
calculate the values of the first member. The results are :
For /'= 50, OSlB.tu.
p"= 75, 1 222.3 B. t.u.
p" = 100, 1451 B. t. u.
Plotting these results, we find p" = 81 Ib. per square inch very nearly. The
temperature of the mixture is therefore 313 F. and the quality of the
4 843
steam is x = ~ = 0.897. (5.4 is the specific volume v" corresponding to
a pressure of 81 Ib.) The partial pressure p' of the air is found from
(5) to be 130 Ib. per square inch. Hence the pressure of the mixture is
130 + 81 = 211 Ib. per square inch.
JLU is seen mia,\j, tis L.UW IBSUIL UL mixing, trie temperature is considerably
vered, the pressure takes a value between pi and jo 2 , and the quality of
3 steam is increased.
If the steam is initially superheated, the preceding equations
list be modified by inserting for V 2 arid V z the appropriate
:pressions for the volume and energy, respectively, of super
ated steam. To reduce as far as possible the complication
the formulas we shall take the approximate equation (5),
Lt. 132, for the volume. We have then
F 2 = M z v z = M 2 (^  A (13)
ae constant B is written with a prime merely to distinguish
from the constant for air. The intrinsic energy of the steam
given by Eq. (2), Art. 136. This equation can be simplified
ith a small sacrifice of accuracy by dropping the term con
ining a. The modified equation then takes the form
Au = !F(e + fT) & + 886.7, (14)
which e = 0.2566, /= 0.00005, and log 0= 13.64593.
From (6) and (14) the energies of the constituents before
ixing can be calculated, and the sum of these gives the
Lergy If of the mixture. We have then as one equation of
>nditioii
+ fT)  + 886.7] =AU. (15)
nee p" and T are here independent, there are two unknowns
id a second condition is required. From (3) and (13) the
itial volumes V^ and V 2 are found and the sum gives the
>lume V of the mixture. Then
.P'
P" = ~V
Jf 2
:om (15) and (16) the unknowns p" and STcan be found.
240 MIXTURES OF GASES AND VAPORS [CHAP, xn
EXAMPLE. Let 5 Ib. of air at 60 F. be compressed adiabatically from
atmospheric pressure to a pressure of 200 Ib. per square inch and mixed
with 1 Ib. of steam at 200 Ib. per square inch superheated 100. The con
dition of the mixture is required.
The temperature of the air after compression
^=519.6(^)^=1095.
The saturation temperature of steam at 200 Ib. per square inch is 381.8 F ;
hence T z = 381.8 + 100 + 459.0 = 941.4. The energy of the air is
5 x 0.17 x 1095 = 930.75 B. t. u.
and that of the steam is, from (14),
941.4 (0.2566 + 0.00005 x 941.4) C^ + 880.7 = 1100.0 B. t. u.
v ' 941. 4 5
Hence A ( U l + Z7 2 ) = A U = 030.75 + 11GO.O = 2001.35. B. t. u.
We have then from (15).
0.85 T + 7X0.2566 + 0.00005 T)Q~ = 1204.65.
To derive an expression for the partial pressure p" the total volume V must
be found. Before mixing, the volume of the air is
v MZ. = 5x63.34x1095 =
1 p l 144x200 '
and the volume of the steam is
05062 x 941.4
 _
pz 200
Hence V= 10.14 + 2.55 = 12.09 cu. ft.
After mixing the superheated steam at the partial pressure p" and tem
perature T occupies this volume ; hence, we have (since M% = 1)
_ B'T 0.5062 T
V+c 12.69 + 0.256
Introducing this expression for p" in the term ~~, that term becomes
C 1
, where log C'  12.30919. The equation in T then becomes
1.1066 T + 0.00005 T z   = 1204.65.
T 1.1060 T 0.00005 T*  Sum
JT 4 J"' 1 "
1000 1106.6 50. 2.04 1154.56
1050 1161.93 55.13 1.68 1215.38
1100 1217.26 60.5 1.39 1276.37
By interpolation it is readily found that T= 1041. The pressure of the
:am is
0.5962 x 1041 Afr nA ,, . ,
p  TTTai  = 47.94 Ib. per square inch,
lile the pressure of the air is
. 53.34x1041x5 ,_, no ..
P' 777 77 = 151.93 Ib. per square inch.
terefore p =p' + p" = 199.9 Ib. per square inch.
The total pressure p should evidently be 200 Ib. per square inch ; hence
} result may be regarded as a check on the calculation.
Having now the initial condition of the mixture, the condi
>n after adiabatic expansion to any assumed lower pressure
d the work of expansion may be found by the methods of
?t. 146.
The discussions of Arts. 146 and 14T furnish the necessary
uations for the analysis of the action of a motor that uses a
.xture of air and steam as its working fluid.
EXERCISES
1. Find the humidity and the weight of vapor per cubic foot when the
nperature is 85 and the dew point is 70.
2. The humidity is 0.60 when the atmospheric temperature is 74 F.
id the dew point.
3. Find the value of B for air at 80 with 70 per cent humidity. Fiud
the specified heat c p of the mixture.
4. A mixture of air and wet steam has a volume of 3 cu. ft. and the
nperature is 240 F. The weight of the air present is 1 Ib., that of the
am and water 0.4 Ib. Find the partial pressures of the air and vapor, the
al pressure of the mixture, and the quality of the steam.
5. Let the mixture in Ex. 4 expand isothermally to a volume of 5 cu. ft.
id the external work, the heat added, the change of entropy, and the
inge of energy.
6. Let the mixture expand adiabatically to a volume of 5 cu. ft. Find
1 condition of the mixture after expansion, and the external work.
7. Let 1 Ib. of steam, quality 0.87, at a pressure of 150 Ib. per square
inch, be mixed with 4 Ib. of air at a pressure of 100 Ib. per square inch and
a temperature of 340 F. Find the condition of the mixture.
8. Let the mixture in Ex. 7 expand adiabatically to a pressure of 40 Ib.
per square inch. Determine the final state of tho mixture and calculate the
work of expansion.
9. Let 1 Ib. of steam at a pressure of 150 Ib. per squaro inch and super
heated 140 be mixed with 6 Ib. of air at a pressure of 100 11). per square
inch and a temperature of 340 F. Find the condition o the mixture.
10. Let the mixture in Ex. 9 expand adiabatically until the pressure
drops to 14.7 Ib. per square inch. Required the final state of the mixture
and the work of expansion.
REFERENCES
Berry: The Temperature Entropy Diagram, 130.
Zeuner : Technical Thermodynamics, 3'20.
Lorenz : Technische Warmelehre, 306.
' ' CHAPTER XIII
THE FLOW OF FLUIDS f> " '
148. Preliminary Statement. Under the title " flow of
fluids " are included all motions of fluids that progress continu
ously in one direction, as distinguished from the oscillating
motions that characterize waves of various kinds. Important
examples of the flow of elastic fluids are the following : (1) The
flow in long pipes or mains, as in the transmission of illuminat
ing gas or of compressed air. (2) The flow through moving
channels, as in the centrifugal fan. (3) The flow through
orifices and tubes or nozzles. The recent development of the
steam turbine has made especially important a study of the last
case, namely, the flow of steam through orifices and nozzles,
and it is with this problem that we shall be chiefly concerned
in the present chapter.
Of the early investigators in the field under discussion,
mention may be made of Daniel Bernoulli (1738), Navier (1829),
and of de Saint Venant and Wantzel (1839). The latter de
duced the rational formulas that today lie at the foundation of
the theory of flow ; they further stated correctly conditions for
maximum discharge, and advanced certain hypotheses regard
ing the pressure in the flowing jet which were at the time dis
puted but which have since been proved valid.
Extensive and important experiments on the flow of air were
made by Weisbach (1855), Zeuiier (1871), Fleigner (1874 and
1877), and Him (1844). These served to verify theory and
afforded data for the determination of friction coefficients. In
1897 Zeuner made another series of experiments on the flow of
air through wellrounded orifices.
Experiments on the flow of steam were made by Napier
(1866), Zeuner (1870), Rosenhain (1900), Rateau (1900),
Gutermuth and Blaess (1902, 1904).
243
Most of the experimental worK nere notcu .^,  ^
flow of fluids through simple orifices or through short con
vergent tubes. The more complicated relations between veloc
ity, pressure, and sectional area that obtain for How through
relatively long diverging nozzles have been investigated experi
mentally by Stodola, while the theory has been developed by
H. Lorenz and Prandtl. The flow of steam through turbine
nozzles has also been discussed by Zeuner.
149 Assumptions. In order to simplify tlio analysis of
fluid flow and render possible the derivation of fundamental
equations, certain assumptions and hypotheses must necessarily
be made.
1. It is assumed that the fluid particles move in noninter
secting curves stream lines which in the case of a prismatic
channel may be considered paral
^^rp^rrrrr:p^&r3? lul to tlxo tlxi ' S f th clianne1 '
"i^ir^^^^^tt We mav imagine surfaces
'^^^ ' r '" '^^^C^ stretched across the channel, as
FlG> 78 ' JF, .F', I", etc., Fig. 78, to which
the stream lines are normal. These are the cross sections
of the channel. They are not necessarily plane surfaces, but
they may usually be so assumed with sufficient accuracy.
2. The fluid, being elastic, is assumed to fill the channel
completely. From this assumption follows the equation of con
tinuity, namely :
in which I denotes the area of cross section, w the mean veloc
ity of flow across the section, M the weight of fluid passing in
a unit of time, and v the specific volume.
3. It is assumed that the motion is steady. The variables
p, v, T giving the state of the fluid and also the mean velocity
w remain constant at any cross section J? ; in other words, these
variables are independent of the time and depend only upon the
position of the cross section.
150. Fundamental Equations. The general theory of flow of
elastic fluids is based upon two fundamental equations^which
MTP. dp.rived hv aDDlvinsr the mlneii)le of conservation of
energy to an elementary mass of fluid moving in the tube or
channel.
Let w l denote the velocity with which the fluid crosses a
section F of a horizontal tube, Fig. 79, and w the velocity at
some second section F. A unit weight of the fluid at section
an 2
F l has the kinetic energy of motion ^ due to the velocity w^ ;
hence if u is the intrinsic energy of the fluid at this section, the
nn 2
total energy is w x + ~. Likewise, the energy of a unit weight
w 2
of fluid at section F is u f . In general, the total energy at
^9
section F is different from that at section F l and the change of
energy between the sections must arise : (1) from energy
entering or leaving the fluid in . .
the form of heat during the 
passage from F to F \ (2) from \
work done on or by the fluid.
The heat entering the fluid per
unit of weight between the two
sections we will denote by q. Evidently work must be done
against the frictional resistance between the fluid and tube ; let
this work per unit weight of fluid be denoted by z. The heat
equivalent Az necessarily enters the flowing fluid along with
the heat q from the outside. Aside from the friction work, the
only source of external work is at the sections F and F. As
a unit weight of fluid passes section F v a unit weight also passes
section F. Denoting by p l and v 1 the pressure and specific
volume, respectively, at F^ the work done on a unit weight of
fluid in forcing it across section J^ is the product p^ ; simi
larly, the product pv gives the work done ~by a unit weight of
fluid at section F on the fluid preceding it. For each unit
weight flowing the net work received at the section F l and F is,
therefore,
Equating the change of energy between F l and F to the energy
received from external sources, we obtain
7T " U
2 n
2\
^ )=
z a J
or
G)
This is the first fundamental equation.
It will be observed that the friction work z drops out of the
equation; the effect of friction is to alter the distribution
7/1**
between internal energy u and kinetic energy  at section jP,
.tj f/
leaving the sum total unchanged.
Differentiation of (1) gives
^ + du + d(pv) = Jdq, (2)
u/
a form of the fundamental equation that is useful in subsequent
analysis.
Equation (1), as is apparent, takes account only of initial
conditions at section jf\ and final conditions at section _F, and
gives no information of anything that occurs between these
sections. A second fundamental equation taking account of
internal phenomena between the two sections is derived as fol
lows. Consider a lamina of the fluid moving along the channel.
This element receives from external sources the heat dq and
also the heat Adz, the equivalent of the work done against
frictional resistances. Independently of its motion, the lamina
of fluid may increase in volume and thereby do external work
against the surrounding fluid, and its internal energy may
increase. According to the first law we have, therefore,
J(dq + Adz) = du + p dv . (3)
The first member represents the energy entering the lamina
during the passage from J^ to F, du is the increase of energy,
and pdv the external work done. Combining (3) with (2), we
get
wdw 7 , , A si\
 + vdp + dz = 0, (i)
9
whence by integration we obtain
ART. 151] FORMS OF THE FUNDAMENTAL EQUATION 247
The fundamental equations (1) and (5), or the equivalent dif
ferential equations (2) and (4), are perfectly general and hold
equally well for gases, vapors, and liquids.
151. Special Forms of the Fundamental Equation. In nearly
all cases of flow the heat entering or leaving the fluid is so
small as to be negligible, and we may, therefore, assume that
q = 0. The sum u+pv will be recognized as the work equiva
lent of the heat content i ; that is,
u +pv = Ji. (See Art. 52.)
Equation (1) of Art. 150 may, therefore, be written in the form
For a perfect gas
pv, (2)
K 1
whence,
.TON
(3)
If the fluid is a mixture of liquid and saturated vapor, the
heat content i is practically equal to the total heat. (See
Art. 86.) Hence we may put
i = q' + xr, (4)
and (1) becomes
^JW + WW + xrX. (5)
%g
For a superheated vapor, the general form (1) is used, the
values of ^ and i being calculated from formula (6), Art. 135.
Equations (3) and (5) being derived from the first funda
mental equation hold equally well for frictionless flow and for
flow with friction.
152. Graphical Representation. A consideration of the
fundamental equations developed in Art. 150 leads to several
convenient and instructive graphical representations, in which
jpaxis is given by
In the case of frictionless
flow, however, the second
FlGr 80> fundamental equation [(5),
Art. 150] becomes
(i)
Hence for frictionless flow, the increase of kinetic energy is
given by the area between the jpaxis and the. curve representing
the expansion.
2. If the flowing fluid is a saturated vapor of given quality,
the representation just given applies but the equation of the
expansion line AS must be
expressed in the form pv n
const. It is, therefore, more
convenient to use the tem
perature T and entropy S as
coordinates. If the flow is
frictionless and adiabatic,
the expansion curve AB is
the vertical isentropic, Fig.
81. The area OHCAA^ rep
resents the total heat of the
mixture in the initial state A,
FIG. 81.
A*
and the area OHDBA l the
total heat in the final state B ;
hence the difference of these areas, namely, the area ABDC,
represents the difference q^ + x^  (</ + ar), and from (5),
Art. 151, this area, therefore, represents the increase of energy
If the initial point is at A' in the superheated region, we
have
^ = area OJETOAA'A^
i = area OSDB'A^
\  i = area A'B'D OAA'.
3. The work z expended in overcoming friction may be
shown on either the pv or the ^ZWplane. When friction is
taken into account, the heat Az, the equivalent of the friction
work z, reenters the fluid, and consequently. the heat content i
and the volume v are both greater at the lower pressure p
than they would be were there no friction. Hence the expan
sion curve AB' , Fig. 82 and 83, for flow with friction must
lie to the right of the curve AS for flow without friction.
This statement applies to both figures.
Let ij denote the heat content in the initial state JL, i the
heat content in the state B, and i' the heat content in the final
state B' when friction enters into consideration. Then
i' > *,
whence
^ i' < *! i'
It follows from (1) Art. 151, that the change of kinetic energy
01) ^2 _ an 2
 1 for flow with friction is less than the change
2<7 *
in the case of frictionless flow. Friction, therefore, causes a
loss of kinetic energy given by the relation
2 w' 2 Tr ., . N /ON
~ = J (* 0 ( 2 )
On the ^Wplane, Fig. 83, this loss is represented by the area
A.BB'B'; for
i' = area OHDB' /,
i = area OHDBA V
;'&'= area A.BB'BJ.
fluid ; hence as explained in Art. 50, the increase of entropy
is jf '*:, and the area A^B'BJ under the curve AB repre
sents (in heat units) the friction work z.
On the jpfplane let a constant i line be drawn from point B',
Fig. 82, cutting the frictionless expansion line in the point Cr.
Then since the heat content
i' is the same at Gr as at B f ,
the difference i 1 i' in pass
ing from A to B' along the
actual curve is the same as
in passing from A to Gr along
the ideal frictionless expan
sion curve AB. .Hut the
change of i between the
mo. 82. states represented by points
A and Gr, which in work
units represents the increase of kinetic energy between A
and Cr, is given by the area AGf.EO. Hence we have :
For frictionless flow,
For the actual flow,
= area ARDO.
. n ffsy
1 = area A GrJEC.
Hence the loss of kinetic energy due to friction is given by the
area BDEG.
From the fundamental equation (5), Art. 150, we have
C Pl j w 2 Wi 2 /QN
z=\ vdp L, (3)
J P 2(/
in which the integral refers to the actual expansion curve.
Referring to Fig. 82, V \dp is given by the area AB'DC
ART. 152]
GRAPHICAL REPRESENTATION
251
T
,Z
\
FIG. 83.
while the change of energy for the actual flow is, as just shown,
given by the area AGrECi hence the difference, the area
AB'DEGA, represents the work
of friction z.
The friction work z (area
AiAB'Bl, Fig. 83) is greater
than the loss of kinetic energy
(area A 1 BS'B 1 1 '). The reason
for this lies in the fact that
part of the heat Az entering
the moving fluid is capable of
being transformed back into
mechanical energy. As shown or
in Chapter IY, the loss of
available energy, represented
by area A^B'B^, is the increase of entropy multiplied by the
lower temperature. The triangular area ABB' represents,
therefore, the part of the friction work that is recovered.
4. The most convenient graphical representation for practi
cal purposes is obtained by taking the heat content i and entropy
s as coordinates. On this is
plane a series of constant pres
sure lines are drawn, Fig. 84;
then a vertical segment AB
represents a Motionless adia
batic change from pressure p l
to a lower pressure jp, while a
curve AB' between the same
pressure limits represents an
expansion with increasing en
tropy, that is, one with fric
tion. The segment AB, there
of 2 w, z ..i ,
without
o'
FIG. 84.
fore, represents the increase of jet energy
friction, the segment AG, the smaller increase
with
which the pressure is p^ through an orifice or short tube, Jbig.
85, into a region in which exists a pressure p 2 lower than p v
If we take the section F l in the reservoir, the velocity w l will
be small and may be assumed to be zero. The second section
F will be taken at the end of the tube, and
the pressure at this section will be denoted
by p. Assuming the flow to be frictionless
and adiabatic, we have, since w^ = 0,
FIG. 85.
The law of the expansion is given by the
equation
m V n = pyn C2^\
where for air n = k, while for saturated or
superheated vapor it has a value depending on the conditions
existing. In any case, n can be determined, at least approxi
mately. Making use of (2) to evaluate the definite integral
of (1), we get n _j
w
n 
p\ I
'
p l j J
If F denotes the area of the orifice or tube, and M the
weight of fluid discharged per second, the law of continuity is
expressed by the equation
Mo = Fw, (4)
whence eliminating w between (3) and (4), we obtain
711
From (2), we have
w r / n \ n n
\ 1  (^
_l ^ \p^J J
(5)
\P
which substituted in (5) gives
ART. 153]
SAINT VENANT'S HYPOTHESIS
253
If now various values be assigned to the lower pressure p
and the values of w and M be found from (3) and (6), respec
tively, the relations be
tween p, w, and M will be
as shown in Fig. 86. The
*^11P!WS\
^^I^P^
initial pressure p 1 is rep
resented by the ordinate n
OQ, the lower pressure p
by the ordinate OH, and
the curve AS represents
the change of state of the
moving fluid starting from
the initial state A. The
shaded area GABff rep * FlG  86<
/PI
resents the integral J vdp and, therefore, the kinetic energy
w
of the jet at the section J?. The abscissa HE represents
*ff
the velocity w found from the equation
w = V2 g x area &ABH (in ft. lb.),
while the abscissa HD represents to some chosen scale the
weight of fluid discharged per second, as found from (4) or
directly from (6). Inspection of (6) shows that the discharge
M reduces to zero when p=p^ and also when p = 0. It fol
lows that the curve CrDO must have the general form shown
in the figure and that the discharge JjTinust have a maximum
value for some value of p between p = and p p z . Let
this value of p be denoted by p m . Evidently from (6), M is a
maximum when
2 nl
is a maximum. Placing the first derivative of this expression
r>
This ratio is called the critical ratio, and^ is called the critical
value of the lower pressure p. For air, taking n = Tc 1.4, this
ratio is 0.5283 or approximately 0.53 ; for saturated or slightly
wet steam, taking n= 1.135, the ratio is 0.5744.
The question now arises as to the relation between the pres
sure p in the jet at section F and the pressure p z of the region
into which the jet discharges. If it be assumed that p and p 2
are always equal, then p = when p z = 0, and from (6) M = 0.
This can only mean that no fluid can be discharged into a perfect
vacuum, a result manifestly absurd. It follows that under
certain conditions, p must be different from p z . Saint Venant
o.i
0.3 0.5
FIG. 87.
0.9
and Wantzel, to whom equations (3) and (6) are due, asserted
that the discharge into a vacuum must be a maximum and
advanced the hypothesis that for all values of p% lower than the
critical pressure p m the discharge is the same. We have, there
fore, two distinct cases : (1) If p% is greater than p m , the pres
sure p in the jet takes the value p 2 , and w and M are found from.
(3) and (6), respectively. (2) If p 2 is equal to or less than
p m the pressure p assumes the constant value p m given by (7),
and the velocity and discharge remain the same for all values
of p z between p 2 =p m and p z = 0.
The hypothesis of Saint Venant has been fully confirmed by
the experiments of Fleigner, Zeuner, and Gutermuth. Figure 87
shows the results of Gutermuth's experiments on the flow of
steam through a short tube with rounded entrance, usinp dif
f erent initial pressures p r In each case the discharge becomes
constant when the lower pressure reaches a definite value p m .
154. Formulas for Discharge. Since for all values of p z less
than p m the discharge remains constant and the pressure at the
plane of the orifice or tube takes the value p m , we may obtain
P
the maximum velocity and discharge by substituting for in
_n_ Pi
(3) and (6) of Art. 153 the critical value (  ^f 1 . The
U 4.' \+ V
resulting equations are :
and *_ ,. (2)
\n+\J * n + 1 VL ^ '
These equations give w and Jf for p z <p m i if jp 2 >p m the ratio
must be substituted for in the original equations.
Pi Pi
By easy transformations (1) and (2) may be given simpler
forms. The following are some of the wellknown formulas
that have been thus derived.
1. Fliegners Equation for Air. From the general equation
which applies to the air in the reservoir from which the flow
proceeds, we have
Substituting this expression for ^J in (2), and taking n*=lc,
v i
the result is
Inserting the numerical values of k and B for air, we get in
English units
J!f= 0.53 ^^i=. (4)
v y
x'^vv VX ' JJX.U11JS [CHAP. Xlli
This is the equation given by Fliegner as representing the re
suits of his experiments on the flow of air from a reservoir into
the atmosphere. It holds good when the pressure in the reser
voir is greater than twice the pressure of the atmosphere
When the pressure in the reservoir is less than twice the at
* f 110Wing em P irical equation is given by
J.
2. Grashofs Equation for Steam. In formula (2), Pl and v
refer to the fluid in the reservoir. If this fluidSs seated
steam, then^ and v, are connected by an approximate relation
in which for English units, m = 1.0631 and (7= 144 x 484 2
From (6) we readily obtain b4 ' 2 '
m+i
Pl =
Q~bn
and substituting this in (2), the resulting equation is
If now we take for steam the value n = 1.135, (7) reduces to
the simple form ^ w auuces to
1 m ^=0.01911^0.97.
In this formula, F is taken in square feet and in pound, npr
square foot. When the area is taken in square^ iX a td the
pressure in pounds per square inch, (8) becomes
^"=0.0165^097.
This formula is applicable for values of p 9 below the criH^T
backpressure^. ^nwoai
3. Rateau^K T^nvmiiJ^ r>_j. i TOI
modified the Grashof
the results of his experiments :
ABT. 155] ACOUSTIC VELOCITY 257
4. Napier's equations. The following simple, though some
what inaccurate, equations based upon the experiments of
Mr. R. D. Napier, are due to Rankine.
When the pressure in the reservoir exceeds of the back
pressure
when it is less than of the back pressure
~ 42
EXAMPLE. Find the discharge in pounds per minute of saturated steam
at 100 Ib. pressure (absolute) through, an orifice having an area of 0.4 sq. in.
The back pressure is less than the critical pressure, 57 Ib. per square inch.
1. By Grashof s formula
M = 60 x 0.0165 x 0.4 x lOO ^ = 34.493 Ib.
2. By Kateau's formula
6 X X 10
(16367  0.96 x 2) = 34.673 Ib.
3. By Napier's formula
M = ' 4 * 10 x 60 = 34.286 Ib.
4. The discharge may be found from the two fundamental formulas
to = VSflr/ (ij  f a ) = 223.7 V^  i g ,
and M=^.
v
The critical pressure p m is 57.44 Ib. per square inch. From the steam table
(or more conveniently, and with sufficient accuracy, from the ischart)
we find :
ii (for 100 Ib.) = 1186.5 B. t. u.
i m (for 57.44 Ib.) = 1142.7 B. t. u.
x m = 0.964.
t'm = arm Om"  v') + / = 7.07 cu. ft.
Then w = 223.7 VI 186.5  1142.7 = 1480 ft. per second,
0.4 1480
144 7.07
and M = 60 x M x ^2  34.89 Ib.
K tne ratio ? 01 tne specmc neats. JLJUBJU.
c v
in the medium is given by the relation
w = \Jgkpv. (See any textbook in Physics). (1)
If we denote by p m the critical back pressure, we have
k
(2)
p m __f 2 Vi
which combined with the adiabatic equation
a
ft VW* C8)
gives
w / 2 \*=I
^1=(__) . (4)
Vm \k + 1J ^ J
Combining (2) and (4), we have
W," 2
The velocity through the orifice is
and by the use of (5) this becomes
w = Vgkp m v m . (6)
Comparing (6) with (1), it appears that the maximum velocity
of flow from a short convergent tube is the same as the velocity of
sound in the fluid in the state it has at the critical section.
This result is due to Holtzmann (1861).
156. The de Laval Nozzle. The character of the flow through
a simple orifice depends largely upon the pressure jt? 2 in the
region into which the jet passes. There are two cases to be
discussed :
1. When^? 2 is equal to or greater than the critical pressure
p m given by the ratio
p m _f 2 Y&
Pi \k I 1
2, When p 2 is less than p m .
In the first case the pressure a f
as wo have seen, takes'
and therefore, t he jet
the
for
Fig. 88,
jet, the
, ,,
constant cross section. Furthermore
8
.
e> . dAio u.
remains practically con
stant at successive cross sections.
Ihis velocity is gi veu by m
Art. 151. W '
In the second case the pres
sure at section a takes the critical v
is greater than the pressure of the sun
As a result of the pressure difference 
jet will expand laterally, as shown Fig
furthermore, along the axis of the jet the
the
89.
i successive sections are passed.
itial velocity at section a is
FIG. 90.
at is, the acoustic velocity.
The lateral spreading of the jet may
prevented by adding to the orifice
)roperly proportional tube, as shown
Fig. 90. The orifice and tube to
^er constitute a de Laval nozzle.
Pressurr/ 6XPanSi0n f the
pressuie from^ at section to ^ 2 at section 3. The area
tne end section 5 depends upon the final pressure Pv At
aon a the jet has the acoustic velocity w 3 as if the added
The tube must diverge
its velocity increases and at the end section b takes the value
w 2 given by the relation
2
W. T/ N xi N
 = J(l m ^ z ). (1)
The general character of the flow through the de Laval
nozzle may be seen from the following analysis.
Assuming frictionless adiabatic flow, the fundamental equa
tions (6) and (7), Art. 150, become, respectively,
du+pdv = Q, (2)
wdw j ^ON
.  = vdp. (3)
&
We have also the equation of continuity
Fw = Mv, (4)
from which by differentiation we obtain
dw^ dF__dv _
+ IT . W
For perfect gases,
pv
ni f _,
% ~A;1'
while for superheated or saturated vapors,
nl
Therefore, (2) becomes
or Jcpdv + vdp = 0, (6)
, dv dp
whence ___..
v Icp
Combining this relation with (5), we obtain
dw dF dp __
^r + T + ^ u  w
Now from (3),
W
hence (7) becomes
p w/ F
By introducing the equation for the acoustic velocity
w? = kffpv,
(8) may be readily reduced to the form
ld p== kw 2 I dF
pdx~~ w 2 w 2 F Ax
(8)
(9)
(10)
The variable x may be used to denote the distance of a nozzle
section from some fixed origin, Fig. 90. For vapors, k may be
replaced by n.
The nozzle has two distinct parts: the rounded orifice ex
tending from to A, Fig. 91, and the diverging tube extend
ing from A to B. As the
cross sections decrease in
area from to A, the deriva
tive is negative for this
dx
part ; for the diverging part
(JF
from A to J5, is positive;
dx
for the throat A it has the
value zero. The pressure
drops continuously from
to B as shown by the curve _:
, dp . FIG. 91.
of pressure ; hence  is
negative throughout. Referring to (10) we have the following
results :
For orifice OA, is ;  is ;
dx dx
For tube AB, ~ is + ; ^ is  ;
dx dx
kw z
is ; w < iv a .
is f \ w>w a .
rl V
For throat A, ~ = ;
kw 2
= co ; w =='
Hence the velocity steadily increases until at the throat it
attains the acoustic velocity; then in the diverging tube it
further increases. Inspection of (10) shows that divergence is
necessary if the velocity w is to exceed the acoustic velocity w s .
157. Friction in nozzles. In the case of flow through a simple
orifice or through a short convergent tube with rounded en
trance, the friction between the jet and orifice, or tube, is small
and scarcely demands attention. With the divergent de Laval
nozzle, on the contrary, the friction may be considerable and
must be taken into account. As explained in Art. 152, the
*iJ)
effect of friction is to produce a decrease in the jet energy
^9
at the end section. Referring back to Fig. 83, suppose A to
denote the initial state of the fluid entering the nozzle, B' the
final state at exit, and B the final state that would have been
attained with frictionless flow ; then the area AjBB'B^' repre
sents the increase in the final heat content i z due to friction and
it likewise represents the decrease in jet energy at exit.
Let ij, z 2 , and i z ' denote, respectively, the heat content of the
fluid in the states represented by the points A, B, and B'.
Without friction, we have
2
y*2 ....
while with friction
The loss of kinetic energy due to friction is, therefore,
It is customary to take as a friction coefficient the ratio of the
loss of energy to the kinetic energy without friction. Denoting
this ratio by y we have, therefore,
whence
'
 (2)
The experiments that have been made on the flow of steam
through nozzles indicate that the value of y may lie between
0.08 and 0.20.
EXAMPLE. Steam in the dry saturated state flows from a boiler in which
the pressure is 120 Ib. per square inch absolute into a turbine cell in which
the pressure is 35 Ib. absolute. A de Laval nozzle is used, and the value of
y is 0.12. Find the velocity of the jet, and the loss of kinetic energy ; also
the final quality of the steam.
For the given initial state, i 1190.1 B. t. u. At the end of adiabatic ex
pansion to the lower pressure, xz is found to be 0.925, and i a is found to be
1095.8 B. t. u. The exit velocity on the assumption of frictionless flow is,
therefore,
w = 223.7 V1190.1  1095.8 = 2172 ft. per second,
while the actual velocity is
w' = 223.7 V(l  0.12) (1190.1  1095.8) = 2038 ft. per second.
The loss of kinetic energy is,
0.12 x 778 x 94.3 = 8804 ft.lb.,
or in B. t. u.,
0.12 x 94.3 = ll.SB.t.u.
This heat is represented by the rectangle A\BB'Bi!, Fig. 83. Hence, for the
quality x>z in the actual final condition J3', we have
x j  X2 = ?/0'i  *'Q = HA = 0>012
r z 938.4
and, therefore, xj = 0.925 + 0.012 = 0.937.
The effects of friction are : (1) to decrease the velocity of
flow at a given section ; (2) to increase the specific volume v
of the fluid passing the section. The latter effect is seen in
the case of steam in the increased quality or increased degree
of superheat due to the heat generated through friction reenter
ing the moving fluid. From the equation of continuity
F=M~, (3)
w
it appears that the effect of friction is to increase the numera
tor v and decrease the denominator w of the fraction of the
second member ; hence for a given discharge M> the cross sec
tion F must be larger the greater the friction, that is, for the
same lower pressure p 2 .
The effect of friction may be viewed from another aspect.
In Fig. 92, let the curve OMAE represent the pressures along
the axis of a de Laval nozzle on the assumption of no friction.
This curve is readily found for a given value of p 1 by finding
for various lower pressures^?
%_ J?> the proper cross section F by
means of the two equations,
T Mo
FIG. 92.
Ag w
Let A be a point on the pres
sure curve obtained in this
manner. If now friction is
taken into account, the sec
tion I" associated with the
lower pressure p has a larger
area than the section F calcu
lated on the assumption of
no friction ; therefore, the
point A is shifted by friction
to a new position A' underneath the new section F' . Similarly
the end section F e must be increased in area to JFJ, and the
point E on the frictionless pressure curve is shifted to a new
position U 1 . The effect of friction, therefore, is to raise the
pressure curve as a whole, that is, to increase the pressure at
any point in the axis of the nozzle.
158. Design of Nozzles. The data required in the design of
a nozzle are the initial and final pressures, the weight of steam
that must be delivered per hour or per minute, and the coef
ficient y. Two cross section areas must be calculated, that at
the throat, and that at the end of the nozzle. The following
example illustrates the method.
EXAMPLE. Kequired the dimension of a nozzle to deliver 450 Ib. of
steam per hour, initially dry and saturated, with an initial pressure of 175
TU nVici/^1 4n nnA fi v n 1 rvwcic< n l*k /xp 1ft 1V\ Q V\D<*1 Tt + l T" !* II f} 1Q
The critical pressure in the throat is 175 x 0.57 = 100 Ib. approx. Then
r frictionless adiabatic flow
t, = 1196.4 B. t. Ti.,
i m (at throat) = q m ' + x m r m = 298.1 + 0.962 x 888.4 = 1152.9,
a = ^2' + Wz = 181.1 + 0.863 x 909.7 = 1017.5,
^  i m  43.5 ; i,  12 = 178.9.
nee the throat is near the entrance, the effect of friction between entrance
id throat is practically negligible ; hence the velocity at the throat is
w m = 223.7 V41T5 = 1475 ft. per second,
iking account of the loss of energy (37 = 0.13), the velocity of exit is
w 2 = 223.7 V0.87 x 178.9 = 2791 ft. per second.
ae quality of steam at the throat was found to be 0.962, and that at exit,
ithout friction, 0.863. Because of friction, the quality at exit is increased
' the amount 178.9 x 0.13 * 969.7 = 0.024, thus giving a final quality
363 + 0.024 = 0.887. Neglecting the volume v' of a unit weight of water,
ice x is large, the specific volumes at throat and exit are respectively
4.42 x 0.962 = 4.252 cu. ft.
:d 26.23 x 0.887 = 23.26 cu. ft.
om the equation of continuity Fw = Mv, we have, since
M = 45 = 0.125 Ib. per second,
60 x 60 i '
F m =  125 >< 4.252 _ aoo036 sq> ftf
= 0.0519 sq. in.
the area of the cross section at the throat. The area at exit is
F z = 0125 x 23.26 _ 0>001042 sq . f t _ alg sq> in>
w i J J.
the cross section of the nozzle is made circular, the diameters at throat
d exit are respectively
d m = 0.251 in., d a = 0.437 in. ;
d taking the taper of the nozzle as 1 to 10, the length, of the conical part
10(0.437  0.257) = 1.8 in.
EXERCISES
1. .Find the weight of air discharged per minute through an orifice
inch in diameter from a reservoir in which the pressure is maintained at
orifice having an area of 0.4 sq. in. into a region in which the pressure is
55 Ib. per square inch. Find (a) the velocity; (b) the weight discharged
per minute. Compare the results obtained by using Grashof's, Napier's,
and Rateau's formulas, respectively.
3. If in Ex. 2 the back pressure is 80 Ib. per square inch, what in the
weight discharged? Assume the steam to be initially dry and saturated.
4. If for superheated steam the exponent n in the adiabatic equation
Pm
pv n = const, is taken as 1.30, find the critical ratio
5. A de Laval nozzle is required to deliver 080 Ib. of steam per hour.
The steam is initially dry and saturated at a pressure of 110 Ib. per square
inch and the final pressure is 8 in. of mercury. Find the necessary areas of
the throat section and end section of the nozzle, assuming frictionless ilow.
6. In Ex. 5 find the areas of the two sections when the loss of kinetic
energy is 0.15 of the available energy.
7. Find the area of an orifice that will discharge 1000 Ib. of dry steam
per hour, the initial pressure being 150 Ib. per square inch and the back
pressure 105 Ib. per square inch.
8. In an injector, steam flows through a diverging nozzle into a combin
ing chamber in which a partial vacuum is maintained, due to the condensa
tion of the steam in a jet of water. If the initial pressure is 80 Ib. per
square inch and the pressure in the combining chamber is 8 Ib. per square
inch, find the velocity of the steam jet. Assume y = 0.08.
9. Steam at 160 Ib. pressure superheated 100 flows through a nozzle
into a turbine cell in which the pressure is 70 Ib per .square inch. Find the
area of the throat of the nozzle for a discharge of 36 Ib. per minute.
10. Let steam at 1GO Ib. pressure, superheated 100, expand adiabatically
without friction. Take values of the back pressure p% as abscissas, and plot
curves showing (a) the available drop in heat content i\ i ; (?) the veloc
ity of the jet ; (c) the area of cross section required for a discharge of one
pound per second.
SUGGESTION. Find i z for the following pressures: 140, 120, 100, 80,
60, 40, 20, 10, 5 Ib. per square inch. Then find w from the formula
V} 223.7 Vt'i  i z , and the cross section from the equation of continuity.
11. Steam at 160 Ib. pressure superheated 100 is discharged into a
region in which the pressure is p through an orifice having an area of
0.25 sq. in. Take the values of p 2 given in Ex. 10 and plot a curve showing
the weight discharged for different values of p.
12. Show that if the loss of kinetic energy is y per cent of the available
energy, the decrease in the velocity of the jet is approximately \y per cent
of the ideal velocity.
Zeuner : Technical Thermodynamics 1, 225 ; 2, 153.
Lorenz : Teclmische Warmelehre, 99, 122.
Weyrauch : Grundriss der WarmeTheorie 2, 303.
Peabody: Thermodynamics, 5th ed., 423.
Stodola : Steam Turbines, 4, 45.
Rateau : Flow of Steam through Nozzles.
ORIGINAL PAPERS GIVING EXPERIMENTAL RESULTS OR DISCUSSIONS
Weisbach : Civilingeineur 12, 1, 77. 1866.
Eliegner : Civilmgenieur 20, 13 (1874) ; 23, 443 (1877).
De Saint Venant and Wantzel : Journal de I'E'cole polytechnique 16. 183
Comptes rendus 8, 294 (1839) ; 17, 140 (1843) ; 21, 366 (1845).
Gutermuth : Zeit. des Verein. deutsch. Ing. 48, 75. 1904.
Emden : Wied. Annallen 69, 433. 1899.
Lorenz : Zeit. des Verein. deutsch. Ing. 47, 1600. 1903.
Prandtl and Proell : Zeit. des Verein. deutsch. Ing. 48, 348. 1904.
Biichner : Zeit. des Verein. deutsch. Ing. 49, 1024. 1904.
Rateau: Ann ales des Mines, 1. 1902.
Rosenhain : Proc. Inst. C. E. 140. 1899.
Wilson : London. Engineering 13. 1872.
CHAPTER XIV
THROTTLING PROCESSES
159. Wiredrawing. The flow of a fluid from a region of
higher pressure into a region of much lower pressure through
a valve or constricted passage gives rise to the phenomenon
known as wiredrawing or throttling. ' Examples are seen in the
passage of steam through pressurereducing valves, in the
throttling calorimeter, in the passage of ammonia through the
expansion valve in a refrigerating
machine, and in the flow through
ports and valves in the ordinary
' steam engine. Wiredrawing is
FIG. 93. . , & . ., . &
evidently an irreversible process,
and as such, is always accompanied by a loss of available
energy.
The fluid in the region of higher pressure is moving with a
velocity w^ Fig. 93. As it passes through the orifice into the
region of lower pressure jt? 2 , the velocity increases to w 2 ac
cording to the general equation for flow, viz :
!lfi!..JXhS> (1)
The increased velocity is not maintained, however, because the
energy of the jet is dissipated as the fluid passing through the
orifice enters and mixes with the fluid in the second region.
9/J 2 ___ nn 2
Eddies are produced, and the increase of energy ^  is re
turned to the fluid in the form of heat generated through in
ternal friction. Utimately, the velocity w z is sensibly equal to
the original velocity w 1 ; therefore from (1), we obtain
L = L, (2)
ART. 160]
LOSS DUE TO THROTTLING
269
as the general equation for a wiredrawing process. The
initial and final points lie, therefore, on a curve of constant heat
content.
160. Loss due to Throttling. Let steam in the initial state
denoted by point A, Fig. 94, be throttled to a lower pressure,
the final state being denoted by
point B on the constantz curve
AB. Also let TQ denote the
lowest available temperature.
The increase of entropy during
the change AB is represented
by AiBp and this increase multi
plied by the lowest available
temperature 2* gives the loss of
available energy. Evidently this
loss is represented by the area
EXAMPLE. In a steam engine the pressure is reduced by a throttling
"valve from 160 Ib. per square inch to 90 Ib. per square inch absolute. The
initial quality is x 0.99 and the absolute back pressure is 4 in. of mercury
Required the loss of available energy per pound of steam.
From the steam table the initial heat content is 1187.2 B. t. u. At a pres
sure of 90 Ib. the heat content of saturated steam is 1184.5 B. t. u., therefor*
in the second state the steam is superheated. As the degree of superheat ii
evidently small, it may be determined with sufficient accuracy from th(
curves of mean specific heat. At a pressure of 90 Ib. the mean specific heai
near saturation is 0.55 ; hence the superheat is
1187.2  1184.5
0.55
= 5, nearly
The entropy in the second state is the sum of the entropy at saturation
1.6107 for a pressure of 90 Ib., and the entropy due to superheat, "which is
approximately.
0.55 log e LJ? = 0.55 log, f = 0.0035.
Hence, .93 = 1.6107 + 0.0035 = 1.6142. The entropy in the initial state
replaced by a corresponding reversible change with the condi
tion that the heat content i remains constant. The general
equation
di = Tds + Avdp,
then becomes,
= Tds + Avdp,
and approximately we have, therefore,
A S = ^, (1)
in which As is the increase of entropy corresponding to the
change of pressure Ap. Since Ajp is intrinsically negative, it
follows that As must be positive. Equation (1) may be
written in the more convenient form
For perfect gases (2) reduces to the simple form
For steam having the quality x, we have
i} = x(v" v'~) f t/,
and Apv Apx(v" v'~) f Apv' ;
or neglecting the small specific volume v r of the water,
Apv = xfy.
Eq. (2) therefore takes the form
Aff (4)
Mean values for p, T, and ^ should be taken.
EXAMPLE. If in passing into the engine cylinder the pressure of steam
is reduced by wii'edrawing from 125 Ib. to 120 Ib. per square inch, what is
the loss of available energy ? The initial value of x is 0.98 and the pressure
at exhaust is 16 Ib. per square inch.
Taking the two pressures 125 and 120, the following mean values are
fouud from the table :
p = 122.5, T = 802.4, if, = 82.5. Also, A p =  5
Hence, A 5 = x = 0.00398.
For T we take the temperature corresponding to the 16 Ib., namely, 675.9.
Therefore the loss of available energy is
6.75.9 x 0.00398 = 2.7 B. t. u. approx.
161. The Throttling Calorimeter. A valuable application
of the throttling process is seen in the calorimeter devised by
Professor Peabody for determining the quality of steam. In
the operation of the calorimeter steam from the main is led
into a small vessel in which the pressure is maintained at a
value slightly above atmospheric pressure. The steam is thus
wiredrawn in passing through the valve in the pipe that con
nects the main and the vessel. For successful operation the
amount of moisture in the steam must be small so that, as the
result of throttling, the steam in the vessel is superheated.
In Fig. 94, let point A represent the state of the steam in
the main and point B the observed state of the steam, in the
calorimeter ; then
i A = i s . (1)
But
i A = z y + xr v (2)
where ij and r^ refer to the pressure p 1 in the main ;
and ijB = h" + c P (t' Z '~t z ), (3)
where t 2 f is the observed temperature of the steam in the
calorimeter, t z is the saturation temperature corresponding to
the pressure p z in the calorimeter, i z " is the saturation heat
content corresponding to the pressure p z , and c p is the mean
specific heat of superheated steam for the temperature range
t 2 ' 2 . Combining the preceding equations, we obtain
EXAMPLE. The initial pressure of the steam is 140 Ib. per square inch,
the observed pressure in the calorimeter 17 Ib. per square inch, and the
temperature in the calorimeter 258 F. Required the initial quality.
The temperature of saturated steam at 17 Ib. pressure is 219.4 F. ; hence
the steam in the calorimeter is superheated 258 219.4 = 38.6. From the
curves of mean specific heat the value 0.477 is found for the pressure 17 Ib.
and the degree of surperheat in question ; and from the steam table we have
t," = 1153, ijf = 324.2, n = 869. Hence,
1153 + 0.477 x 38.6  324.2 n Q7 
x = __ _ 0.975.
The Mollier chart, Fig. 75, may be used conveniently in the
solution of problems that involve the throttling of steam.
Since the heat content remains constant during a throttling
process, the points representing the initial and final states lie
on the same horizontal line. In the preceding example the
final point is located from the observed superheat 38.6 and
the observed pressure 17 Ib. in the calorimeter. A horizontal
line drawn through this point intersects the constant pressure
line p = 140 Ib., and from this point of intersection the quality
x = 0.975 is read directly.
162. The Expansion Valve. In the compression refrigerat
ing machine the working fluid after compression is condensed
and the liquid under the higher pres
sure p is permitted to flow through
the socalled expansion valve into coils
in which exists a much lower pressure
p v Let point A, Fig. 95, on the liquid
curve represent the initial state of the
liquid. The point that represents the
final state must lie at the intersection
of a constant i curve through A and
S line of constant pressure p z . Evidently
we have
and i b = i 2 ' + # 2 r 2 ,
where x% denotes the quality of the mixture in the final state.
Therefore r _ t ,
1 2 ^ t 2'2'
li la
or
(2)
The increase of entropy (represented by
s
(3)
and the loss of refrigerating effect due to the expansion valve,
which is represented by the area A t GBB V is
h'h'W ,') (4)
The following equalities between the areas of Fig. 95 are
evident :
area ^ = area
area F6rA = area
163. Throttling Curves. If steam initially dry and saturated
be wiredrawn by passing it through a small orifice into a region
of lower pressure, then, as has been shown, it will be super
heated in its final state.
If the lower pressure p z
is varied, the tempera
400,
350
soo
S50
200
50 100 150
Pressure, Ib. per sq. in.
FIG. 96.
ture t z will also vary,
and the successive values
of p% and t 2 will be rep
resented by a series of
points lying on a curve.
By taking various initial
pressures a series of such
curves may be obtained.
Sets of throttling curves
for water vapor have
been obtained by Grind
ley, Griessmann, Peake,
and Dodge. The curves deduced from Peake's experiments
are shown in Fig. 96. Abscissas represent pressures, ordinates,
temperatures. The curve from which the throttling curves
start is the curve t=f(p~) that represents the relation between
the pressure and temperature of saturated steam.
It was the original purpose of Grindley, Griessmann, and
Peake to make use of the throttling curves in finding the
specific heat of superheated steam. The theory upon which
this determination rests is simple. From Eq. (4), Art. 161, we
readily obtain . / . ,/
*i ~r ^i^i ? 2 /IN
The temperature difference t z ' 2 for any lower pressure p z is
the vertical segment between the throttling curve and the satu
ration curve and is given directly by the experiment. Hence if
the initial quality x is known, and if i^' and z* 2 " are accurately
given by the steam tables, the mean value of c p is readily cal
culated. The results obtained were, however, discordant and
of no value. The form of Eq. (1) is such that a slight error
in any of the terms of the numerator of the fraction produces
a large error in the calculated value of c p .
The impossibility of deriving consistent values of o p by the
method just described led to the belief that Regnault's formula
for the total heat of saturated steam, hitherto regarded as
authoritative, must be incorrect. The experiments of Kno
blauch and Jakob on the specific heat having appeared,
Dr. H. N. Davis of Harvard University discerned the possibility
of reversing the method and deriving by it a new formula for
total heat.
164. The Davis Formula for Heat Content. The method
employed by Dr. Davis in deriving from the throttling curves
a formula for the heat content of
steam may be described as follows :
Let AD, Fig. 97, be one of the
series of throttling curves, and
AD' the saturation curve. The
heat content is constant along the
throttling curve, that is
p
FIG. 97. Let p 2 be the lower pressure cor
responding to the points B, J5',
and let A* denote the temperature difference indicated by the
segment B'B. If the steam were made to pass from the satura
tion state B' to the superheated state B at the constant pres
sure > 2 , the heat absorbed during the process would be c p Ai,
c p denoting the mean specific heat between B' and B. It follows
that
*J3 IB> ~ C P A,
that is, IA ia, = c A.
ART. 165] THE JOULETHOMSON EFFECT 275
In a similar manner the differences i A i cr) i A i Dl , etc. are ob
tained. The result is a relation between the heat content of
saturated steam at the original pressure p l (state A) and the
values of the saturation heat content for various lower pressures.
The temperatures corresponding
.to these pressures are now laid
off on an arbitrarily chosen line
MN, Fig. 98, and from the points
A, J5', (7', etc., the segments
etc. are laid off. A curve
through the points A, B n ', <7 ; , o [
D", etc. is an isolated segment of * 10 ' y8 '
the curve giving the relation between the heat content i and the
temperature t. Necessarily only relative values are thus obtained.
From the individual throttling curves Dr. Davis thus obtained
twentyfour overlapping segments of the itcurve, and by
properly coordinating these segments he obtained finally a
smooth curve covering the range 212 to 400 F. The curve
was found to be represented by the second degree equation
i=a + 0.3745(^212) 0.00055 (212) 2 ;
and from the experiments of Henning and Joly on the latent
heat of steam at 212 F., the value of the constant a was found
to be 1150.4.
165. The JouleThomson Effect. The classical porous plug
experiments of Joule and Lord Kelvin were undertaken for the
purpose of estimating the deviation of certain actual gases from
the ideal perfect gas. The gases tested were forced through a
porous plug and the temperatures on the two sides of the plug
were accurately determined. In the case of hydrogen the tem
perature after passing through the plug was slightly higher
than on the high pressure side ; air, nitrogen, oxygen, and car
bon dioxide showed a drop of temperature.
276 THROTTLING PROCESSES [CHAP, xiv
For an ideal perfect gas,
MJ = Jc^jf. U p and pv = BT' }
hence, (Jc v + ) ^ = (Jb, + JB) !F a
or ^1=^2
It follows that a perfect gas would show no change of tempera
ture in passing through the plug, and that the change of temper
ature observed in the actual gas is, in a way, a measure of the
degree of imperfection of the gas. The results of the experi
ments have been used to reduce the temperature scale of the
air thermometer to the Kelvin absolute scale.
The ratio of the observed drop in temperature to the drop in
pressure, that is, the ratio , is called the JouleThomson
coefficient and is denoted by /*. According to the experiments
of Joule and Kelvin //, varies inversely as the square of the
absolute temperature. That is,
It may be assumed that this relation holds good for air, nitro
gen, and other socalled permanent gases within the region of
ordinary observation and experiment. At very low tempera
tures it seems probable that p varies with the pressure as well
as with the temperature.
An expression for p in the case of superheated steam can
readily be derived from the formula for the heat content, namely:
Since i is constant in a throttling process, we may define the
JouleThomson coefficient more accurately as the derivative
fdT\
{ ] . From calculus, we have
ART. 166] EQUATION OF THE PERMANENT GASES
and from the definition of the heat content i,
277
Hence
dT *'
dp ji e f
or
The following table contains values of
Eq. (2).
(2)
calculated from
PUESSTTKE
Lit. PER
Sij. IN.
250 F.
300
850
400
450
500
550
000
15
0.668
0.492
0.369
0.282
0.220
0.176
0.143
0.119
100
0.327
0.261
0.208
0.169
0.140
0.118
300
0.191
0.162
0.138
0.118
It will be observed that the value of /j, varies with the pres
sure ; however, as the temperature rises, the influence of
pressure decreases ; hence for gases far removed from the satu
ration limit, such as were used in the porous plug experiments,
it seems probable that p is a function of the temperature only,
as found by Joule and Kelvin.
Dr. Davis has deduced from, the throttling experiments of
Grindley, Griessmann, Peake, and Dodge values of p for super
heated steam.* These were found by direct measurement of
the mean slopes of the throttling curves. The values thus
obtained agree very closely with those calculated from (2) and
shown in the preceding table.
166. Characteristic Equation of the Permanent Gases. From the cooling
effect shown in the JouleThomson's experiments for all gases except hydrogen,
it appears that those gases do not follow precisely the law expressed by the
equation pv = BT. By making use of the relation /*. = ~ it is possible to
derive a characteristic equation that represents more nearly the behavior of
bince the heat content i is constant during a throttling process, the gen
eral equation
di = c dT
takes the form
c <*T = A(T^V\ 0)
* dp V dT I
Differentiating both members of (1) with respect to T, we obtain
dT \ r ~dp 1 ~ y \dT dT 2 ~dT>
= AT. (2)
H HF$i
But we have
_ .
dp I 2
and from the general thermodynamic relations, \
p T
Substituting these expressions in (2), we obtain
3
whence
This is a partial differential equation, the general solution of which is the
equation
c p = T*<t>(T*3ap). (4)
Here ^> denotes an arbitrary function which must be determined from
physical considerations. Since at high temperatures c p for permanent gases
is given by the linear relation c p = a + bT, we have from (4),
whence (j>(T 3 ) = J+ j,,
and
Since the term 3 otp is small in comparison with the term T 3 , we have
approximately
ART. 166] EQUATION OF THE PERMANENT GASES 279
Introducing these expressions in (5) and substituting the resulting expres
sion for <f>(T 8 3 op) in (4), we obtain finally
(6)
It appears from (6) that the specific heat of the permanent gases varies
with the pressure and temperature. At very high temperatures the term
containing/) is small and the specific heat is given simply by a + bT; at
low temperatures, however, this term becomes appreciable and the specific
heat increases with the pressure. The specific heat curves have, therefore,
somewhat the form shown in Fig. 71.
From (6), we have by differentiation
_ AT 9 '_
~
_ /2a ,\
~**A~r ''
Integrating, we obtain
(7)
Vy
Introducing in (1) the expression for ^ given by (7), we obtain after
dJ
reduction
To determine the function /O), we assume that the perfect gas equation
pv = BT holds when T is very large. Hence f(p) = , and (8) becomes
Since the last term in the bracket is very small, it may be neglected, and (9)
may be written
The equation given by Joule and Thomson, namely ,
(ID
JouleThomson effect has been employed by Linde in a very
ingenious machine for the liquefaction of gases. A diagram
matic sketch of the
machine is shown in
Fig. 99. Air (or any
other gas that is to
be liquefied) is com
pressed to a pressure
of about 65 atmos
pheres and is dis
charged into a pipe
leading to the cham
ber c. A current of
cold water in the
vessel b cools the air
during its passage
from the compressor
to the receiving cham
ber. From c the air
passes through a valve
v into a vessel d, in which a pressure of about 22 atmospheres
is maintained. As a result of the throttling the temperature
of the air is lowered. Thus, if p 1 is the pressure in the chamber
c and j9 2 the pressure in the vessel cZ, the drop in temperature is
(1)
The air now passes from vessel d at temperature t 8 into the
space enclosing the chamber c and thence back to the compressor.
In passing back, the air absorbs heat from the air in c, and the
temperature rises from t B to the final temperature 4 , which is
nearly the same as the initial temperature t v Due to this
cooling, the air in c arrives at the valve v with a temperature t z ,
which becomes lower and lower as the process continues. As
the temperature 2 sinks the temperature 3 also sinks, but as
shown by (1), t% sinks more rapidly than t z . Ultimately, the
value of t 3 drops below the critical temperature of the gas,
FIG. 99.
PRINCIPLES
OF
THERMODYNAMICS
BY
G. A. GOODENOUGH, M.E.
PROFESSOR OF THERMODYNAMICS IN THE UNIVERSITY
OF ILLINOIS
SECOND EDITION, REVISED
Griessmann : Zeit. des Verein. deutsch. Ing. 47, 1852, 1880. 1903.
Grindley : Phil. Trans. 194 A, 1. 1900.
Peake : Proc. Royal Society 76 A, 185. 1905.
THE JOULETHOMSON EFFECT
Thomson and Joule : Phil. Trans. 143, 357 (1853) ; 144, 321 (1854) ; 152,
579 (1862).
Natanson : Wied. Annallen 31, 502. 1887.
Preston : Theory of Heat, 699.
Bryan : Thermodynamics, 128.
Lorenz : Technische Wiirmelehre, 273.
Davis : Proc. Amer. Acad. 45, 243.
CHARACTERISTIC EQUATION OF GASES
Zeuner: Technical Thermodynamics 2, 313.
Lorenz: Technische Warmelehre, 296.
Plank : Physikalische Zeit. 11, 633.
Bryan : Thermodynamics, 138.
CHAPTER XY
TECHNICAL APPLICATIONS, VAPOR MEDIA
THE STEAM ENGINE
168. The Carnot Cycle for Saturated Vapors. Since the
constant pressure line of a saturated vapor is also an iso
thermal, three of the processes of the Carnot cycle are ap
proximately attainable in a vapor motor, namely: isothermal
expansion, adiabatic expansion, and isothermal compression.
The adiabatic compression might also be accomplished by a
proper arrangement of the organs of the motor, though in
practice this is never attempted. Hence, the Carnot cycle is
D
\
s o
FIG. 100.
FIG. 101.
discussed in connection with vapor motors merely for the pur
pose of furnishing an ideal standard by which to compare the
cycles actually used.
The Carnot cycle on the T/Splane and p Fplanes, respec
tively, is shown in Fig. 100 and 101. The isothermal expan
sion AB occurs in the boiler, the adiabatic expansion BO in the
engine cylinder, the isothermal compression CD in the con
denser. To effect the adiabatic compression DA, the mixture
of liquid and vapor in the state D would have to be compressed
aaiabaticaily in a separate cylinder ana delivered to tne Doiier
in the state represented by point A.
The heat received from the boiler per unit weight of fluid is
2i = r i( x t> x <d ( area A^B^i) 00
that rejected to the condenser is
fe = r t (x a  x d } ; (area S 1 QDA^) (2)
and the heat transformed into work, represented by the cycle
area, is
AW=q, & = 5^^  xj. (3)
*!
The efficiency is
and the weight of fluid used per horsepowerhour is
2546 2546 g,
?i?. nc^^yjr, v  1
If point J. lies on the liquid line s' and point on the satu
ration curve s", then x a =0, # 6 = 1, and (3) and (5) become,
respectively,
2546
IV _
EXAMPLE. Let the upper and lower pressures "be respectively 125 Ib.
per square inch absolute, and 4 in. of mercury. Then from the steam table
TI = 804, T z = 585.1, n = 875.8 B.tu. From (4), the efficiency is
804  585.1
804
= 0.272.
The heat transformed into work is 875.8x0.272 = 238.2 B.t.u., and the
2546
238.2
steam consumption is ' = 10.7 Ib. per h. p.hour.
r 9:*R.9  1 r
169. The Rankine Cycle. In the actual engine the iso
thermal compression is continued until the vapor is entirely
condensed, thus locating the point D on the liquid curve s',
Fig. 102, The liquid is then forced into the boiler by a pump
and is there heated to the boiline 1 temnerature . This heat
JLU UO clSBUJUULCU. UHit U liUt!
! IJLttB LLU CM.BiUilLl.UO, liilt) V V 
diagram necessarily takes the form shown in Fig. 103.
D
O D l
B\
FIG. 102.
FIG. 103.
The heat supplied from the boiler per pound of steam is in
this case
(1)
(2)
(3)
and the heat rejected to the condenser is
Hence, the heat transformed into work is
and the efficiency of the cycle is
77 =
gi
(4)
It is evident that this efficiency is less than that of the Carnot
cycle.
The steam, consumption per h. p. hour is
,r 2546 2546 (5)
(6)
EXAMPLE. Using the data of the example of the last article, determine
the efficiency and steam consumption of an engine running in a Rankine
cycle with dry steam.
The quality at point C is determined from the relation
end of adiabatic expansion is
_ 0.4957  0.1739 + 1.0893 _
*' ~ L7497 ~
The available heat is
315.2  93.4 + 875.8  0.806 x 1023.7 = 272.4 B. t. u. ;
while the heat supplied in the boiler is
315.2  93.4 + 875.8 = 1097.6 B.t. u.
Hence the efficiency is
= i = 0.248,
7 1097.6
which may be compared with the efficiency 0.272 of the Carnot cycle under
similar conditions.
The steam consumption is
2o46
272.4
170. Rankine's Cycle with Superheated Steam. If super
heated steam is used, the Rankine cycle has the form shown in
Fig. 104. The heat q l supplied
from the source is increased by
the heat represented by the area
B^BEEy which comes from the
superheater; and the heat avail
able for transformation into work
is increased by the amount repre
sented by the area FBEO. Evi
dently the efficiency of the ideal
cycle is increased by the use of
superheated steam, but the in
~ s crease is small. The advantage of
FIG. 104. superheated steam lies in another
direction.
If T e denote the temperature of the superheated steam (i.e.
at point E\ the heat required for the superheating process BE
is i o p dT where c p is the specific heat of superheated steam.
for c p given by Eq. (9), Art. 133. Then the heat represented
by the area D^DA'BEE^ is given by the expression
2i = ft'fc / + ''i+Vz r . (1)
m
However, as has been shown, the sum 5'/+r 1 + ( c p dT is
practically equal to the heat content of the steam in the state E.
Hence we may write
n ,* i /n\
zl ( J2 \^J
and calculate i e from the general formula (5) Art. 135.
If the point O at the end of adiabatic expansion lies in the
saturated region, as is usually the case, we have, as in the first
case, g<2 = r z x c .
The heat transformed into work is, therefore,
and the efficiency is
'? = lT^ r  (4)
The value of x c is determined from the relation
where s is the entropy for the state J, and is calculated from
the general equation (3), Art. 137.
EXAMPLE. Find the efficiency of the Ranldne cycle, using the data of
the previous examples, but assuming the steam to be superheated to 1000
absolute.
From (6), Art. 135, the heat content of the superheated steam is
i = 1000(0.367 + 0.00005 x 1000)  125(1 + 0.0003 x 125)  0.0163 x 125
+ 886.7 = 1294.8 B. t. u. ;
and from (4), Art. 137, the entropy is
s = 0.8451 log 1000 + 0.0001 x 1000  0.2542 log 125
 125(1 + 0.0003 x 125) r~  396i = L7002 
Hence
=
1.7497
Heat supplied = i  q z > = 1294.8  93.4 = 1201.4 B. t. u.
Heat rejected = r z x c = 1023.7 x 0.872 = 892.7 B. t. u.
Available heat = 1201.4  892.7 = 308.7 B. t. u.
Efficiency = ^=0.257.
Steam consumption = = 8.25 Ib. per h. p.hour.
308.7
171. Incomplete Expansion. Because of the very large
specific volume of saturated steam at low pressures, it is usu
ally impracticable to continue the adiabatic expansion down
to the lower pressure p z . The exhaust valve opens and re
leases the steam at a pressure somewhat higher than p 2 . The
passage of the steam from the cylinder is an irreversible pro
cess in the nature of a free expansion and is indicated on the
pFdiagram by the drop in pressure EF (Fig. 106). The
O D, F, B i
FIG. 105.
F
FIG. 106.
actual process may be replaced by an assumed reversible pro
cess, cooling at constant volume. On the 5Wdiagrain the
cooling is represented by a constant volume line EF (Fig.
105) drawn as described in Art. 125.
Evidently this " cutting off the toe " of the diagram results
in a decrease in the ideal efficiency, but it is justified by the
smaller cylinder volume required (JDF instead of DC*) and by
other considerations.
Denoting by p 3 the pressure at E, the end of adiabatic
expansion, we have:
*' 1'
Heat rejected by medium
Heat transformed into work
Si  ft = ?i' + Vi ~ (&' + ^s)  s/fo ~ Pa)
The qualities x e and x f are found from the equations
and
(2)
(3)
(4)
If the steam is admitted throughout the stroke without cut
off, the adiabatic expansion is lacking, and the diagram takes
the form ABGrJ) (Figs. 105 and 106). The equations for this
case are readily derived from the preceding equations by
172. Effect of changing the Limiting Pressures. If the
upper pressure p 1 be raised to p^ while the lower pressure p z
is kept the same, the effect is to
increase both q v the heat absorbed,
and q q z , the available heat, by
an amount represented by the area
AAIB'B (Fig. 107). Evidently
the ideal efficiency is thus in
creased. If p z be lowered to p%,
keeping p l the same, q z is decreased
and q 1 q% increased without any
change in q r For the ideal
Rankine cycle the increase of avail
able heat would be that represented FIG. 107.
by the area D'DCC'. For the
modified cycle with incomplete expansion, however, the in
crease is represented by the relatively small area
We may draw the conclusion that in the actual steam engine
the limitation imposed by the cylinder volume prevents us
from realizing much improvement in efficiency by lowering the
back pressure p v Herein lies one important difference be
tween the steam, engine and steam turbine. With the turbine,
as will be shown, a lowering of the condenser pressure results
in a marked increase of efficiency.
173. Imperfections of the Actual Cycle. In the discussion of
the ideal Rankine cycle the following conditions are assumed:
1. That the wall of the cylinder and piston are nonconduct
ing, so that the expansion after cutoff is truly adiabatic.
2. Instantaneous action of valves and ample port area so
that free expansion or wiredrawing of the steam may not occur.
3. No clearance.
In the actual engine none of these conditions is fulfilled. The
metal of the cylinder and piston conducts heat and there is,
consequently, a more or less active interchange of heat, between
metal and working fluid, thus making adiabatic expansion im
possible. The cylinder must have clearance, and the effect of
the cushion steam has to be considered. The valves do not act
instantly and a certain amount of wiredrawing is inevitable.
It follows that the cycle of the actual engine deviates in many
ways from the ideal Rankine cycle, and that the actual efficiency
must be considerably less than the ideal efficiency. We must
regard the Rankine cycle as an ideal standard unattainable in
practice but approximated to more and more closely as the im
perfections here noted are gradually eliminated or reduced in
magnitude.
The effects of some of these imperfections may be shown
quite clearly by diagrams on the T$plane.
In Fig. 108 is shown the cycle of a noncondensing steam
engine. The feed water enters the boiler in the state represented
by point 6r and is changed into dry saturated steam at boiler
pressure, represented by point B. When this dry steam is
transferred to the engine cylinder, which has been cooled to
the temperature of the exhaust steam, it is partly condensed,
OYI^ 4Tia efafa rvF fVio mivHi r>a in fVio oirlivirl ov o4: mikriff ia Te>T>va_
sentea 07 point (J. me neat trnis aosorDeci by tne cylinder
walls is represented by the area 1 BOO 1 . CD represents the
adiabatic expansion, DE the assumed constantvolume cooling
of the steam, and JEF the condensation of the steam at the tem
perature corresponding to the back pressure, which is slightly
above atmospheric pressure. To close the cycle, the water at
the temperature represented by F
(somewhat above 212) must be
cooled to the original tempera
ture of the feed water ; this
process is represented by FG.
The heat supplied is repre
sented by the area G^GABB^
the heat transformed into work
by the area FAODE. It will
be observed that two segments
of the cycle, namely, GF and
CB, are traversed twice, and the
effect is a serious loss of effi
ciency. The loss due to starting the cycle at point G instead
of at point F may be obviated to a large extent by the use of a
feed water heater. The heat rejected in the exhaust is used to
heat the feed water to a temperature represented by point If,
which is only a little lower than the temperature of the ex
haust. The area Cr^GH^ represents the saving in the heat
that must be supplied. The loss due to cylinder condensation,
which is shown by the segment BO, cannot be wholly obviated ;
it may be reduced, however, by superheating and jacketing.
Losses due to wiredrawing and clearance are not shown on
the diagram. The drop of pressure in the steam main and in
the ports may be taken into account roughly by drawing a line
A'O' somewhat below the line AB, which represents full boiler
pressure.
174. Efficiency Standards. The ratio of the heat transformed
into useful work to the total heat supplied is usually termed the
thermal efficiency of the engine. The thermal efficiency, how
ever, does not give a useful criterion of the good or bad qualities
of an. engine for the reason that it does not take account of the
conditions under which the engine works. It has become cus
tomary, therefore, in estimating engine performance to make
use of certain other ratios.
Let q = heat supplied to the engine per pound of steam,
q R = heat 'transformed into work by an engine working
in an ideal Rankine cycle (Art. 169),
q a = heat transformed into work by actual engine under
the same conditions,
W a = work equivalent of heat q a , the indicated work,
W b = the work obtained at the brake.
We have then
r] R = = thermal efficiency of ideal Rankine engine,
?7 a = = thermal efficiency of actual engine,
3
77 Q
77. = = = efficiency ratio (based on indicated work),
VR ?.R
_ = brake efficiency ratio (based on work at
9.R , , ,
brake),
r) m  mechanical efficiency.
WA
The ratios ??, and % are sometimes called the potential efficiencies
of the engine, the first the indicated potential efficiency, the
second the brake potential efficiency. When the term efficiency
is used without qualification it usually means the efficiency ratio
or potential efficiency rather than the thermal efficiency.
It is clear that the useful criterion of the performance of an
engine is the ratio ?? 6 . We have
% = i?i X t] m .
Of the heat q supplied, only the heat q R could be trans
formed into work by the ideal engine using the Rankine cycle ;
hence the heat q R rather than the total heat q should be charged
to the engine. The ratio T?, = is a measure of the extent to
which the engine transforms into work the heat q R that may
possibly be thus transformed ; it may be called the cylinder
efficiency. The ratio r) m measures the mechanical perfection of
the engine. Hence, the product ^ x rj m measures the perform
ance of the engine both from the thermodynarnic and the
mechanical standpoints.
The efficiencies ??< and % may be given, other equivalent defi
nitions that are frequently useful.
Let N R = steam consumption of ideal Rankine engine per
h. p. hour.
N a steam consumption per h.p.hour of actual engine.
N b = steam consumption per b. h. p.hour of actual
engine.
&R N R
Ihen ^, % = ^
EXAMPLE. An actual engine operating under the conditions denned in
the example of Art. 169 shows a steam consumption of 14. 1 lb. per i. h.p.
hour and 18 lb. per b. h. p.hour. Since for the ideal engine the steam
consumption is 9.35 lb. per h. p.hour, we have
17, = f= 0.663, and ^ = ^ = 0.52.
EXERCISES
In Ex. 1 to 5 find the heat transformed into work, efficiency, and steam
consumption per h. p.hour.
1. Carnot cycle, p^ = 110 lb., p 2 = 15 lb. absolute, x b = 0.85.
2. Rankine cycle, same data as in Ex. 1.
3. Rankine cycle, p l = 110 lb., p 2 5 in. of mercury, steam superheated
to 450 F.
4. Rankine cycle p^ = 110 lb., p z = 15 lb., x b = 0.85 and adiabatic ex
pansion carried to 27 lb. per square inch.
5. Data the same as Ex. 4 except that steam is not cut off.
6. Let jt? 2 be fixed at 5 in. of mercury. Take x b = 1 and draw a curve
showing the relation between 17 and p\. Rankine's cycle.
7. Taking the data of Ex. 2, find the increase of available heat and effi
ciency when a condenser is attached and p^ is lowered to o in. of mercury.
8. Make the same calculation for the cycle with incomplete expansion,
y. JLne emciency rji or an engine is u.uo anu nue meciminc:<u muuiemj.y JN
0.85. If the heat transformed into work by the ideal Rankine engine is
190 B. t. u. per pound, what is the steam consumption of the actual engine
per b. h. p.hour?
10. The steam consumption of a Rankine engine is 9.2 11). per h. p.
hour, and the efficiency ratio 77, is 0.70. Find the heat transformed into
work by the actual engine per poxmd of steam.
THE STEAM TURBINE
175. Comparison of the Steam Turbine and Reciprocating En
gine. The essential distinction between the two types of'
vapor motors turbines and reciprocating engines lies in
the method of utilizing the available energy of the working
fluid. In the reciprocating engine this energy is at once util
ized in doing work on a moving piston ; in the turbine there is
an intermediate transformation, the available energy being first
transformed into the energy of a moving jet or stream, which is
then utilized in producing motion in the rotating element of the
motor.
While the turbine suffers from the disadvantage of an added
energy transformation with its accompanying loss of efficiency,
it has a compensating advantage mechanically. With any
motor the work must finally appear in the rotation of a shaft.
Hence, intermediate mechanism must be employed to transform
the reciprocating motion of the piston to the rotation required.
Evidently this is not the case with the turbine, which is thus
from the point of view of kinematics a much more simple ma
chine than the reciprocating engine. Many attempts have been
made to construct a motor (the socalled rotary engine) in
which both the intermediate mechanism of the reciprocating en
gine and the intermediate energy transformation of the turbine
should be obviated. These attempts have uniformly resulted
in failure.
With ideal conditions it is easily shown that the two methods
of working produce the same available work and, therefore,
give the same efficiency with the same initial and final con
ditions. Thus the Rankine ideal cycle, Fig. 102, gives the
maximum available work per pound of steam of a reciprocating
ART. 176] CLASSIFICATION OF STEAM TURBINES 295
engine with the pressures p and jP 2 . It likewise gives (Art.
152) the kinetic energy per pound of steam of a jet flowing
without friction from a region in which the pressure is p 1 into
o
a region in which it is p 2 . Hence if this kinetic energy 
g
is wholly transformed into work, the work of the turbine per
unit weight of fluid is precisely equal to that of the reciprocat
ing engine. Under ideal conditions, therefore, neither type of
motor has an advantage over the other in point of efficiency.
Under actual conditions, however, there may be a consider
able difference between the efficiencies of the two types. Each
type has imperfections and losses peculiar to itself. The re
ciprocating engine has large losses from cylinder condensation ;
the turbine, from friction between the moving fluid and the
passages through which it flows. It is a question which set of
losses may be most reduced by careful design.
Aside from the question of economy, the turbine has certain
advantages over the reciprocating engine in the matters of
weight, cost, and durability (associated with certain disadvan
tages) and these have been sufficient to cause the use of tur
bines rather than reciprocating engines in many new power
plants and also in some of the recently built steamships.
176. Classification of Steam Turbines. Steam turbines may
be divided broadly into two classes in some degree analogous
to the impulse water wheel and the water tur
bine, respectively. In the first class, of which
the de Laval turbine may be taken as typical,
steam expands in a nozzle until the pressure
reaches the pressure of the region in which the
turbine wheel rotates. The jet issuing from
the nozzle is then directed against the buckets
of the turbine wheel, Fig. 109, and the impulse
of the iet produces rotation. It will be noted
that with this type of turbine only a part of the
r*n rC\r}4c* r ri fill arl TITT fT~ o4an v** n 4 ortr i ri o4o r^f atran 1 T
The pressure of the steam is reduced during the
I passage through the blades both in the guide and
turbine wheels. In the turbine of the first type all
the available internal energy of the steam is trans
formed into kinetic energy of motion before the
steam enters the turbine wheel, while in the turbine
of the second type part of the internal energy is
transformed into work during the passage of the
fluid through the wheel.
The terms impulse and reaction have been used
FIG. no. to designate turbines of the first and second class,
respectively. Since, however, impulse and reaction
are both present in each type, these terms are somewhat mis
leading, and the more suitable terms velocity and pressure have
been proposed. Tims a de Laval turbine is a velocity turbine ;
a Parsons turbine is a pressure turbine.
177. Compounding. The high velocity of a steam jet result
ing from a considerable drop of pressure renders necessary
some method of compounding in order that the peripheral
speed of the turbine wheels may be kept within reasonable
limits without reducing the efficiency of the turbine. With
velocity turbines three methods of compounding are employed.
1. Pressure Compounding. The total drop of pressure jt^ p z
may be divided among several wheels, thus reducing the jet
velocity at each wheel. If, for example, the change of heat
content is % 2 and the expansion takes place in a single
nozzle, the ideal velocity of the jet is w = V2 g J^ ^' 2 ) ;
if, however, ^ i z is divided equally among n wheels, the jet
velocity is reduced to w = \' ^~ (^ i' 2 ) . The general arrange
ment of a turbine with several pressure stages is shown in
Fig. 111. Steam passes successively through orifices m v w 2 ,
etc. in partitions 5 r 5 2 , etc., which divide the interior of the
FIG. 111.
passing through the orifice m z the pres
sure drops from p 2 to p s ; as a result
the velocity is again increased and the
jet passes through the second wheel.
The pressure and velocity changes are
shown roughly in the diagram at the
bottom of the figure.
The method of compounding here
described is called pressure compound
ing. Each drop in pressure constitutes
a pressure stage.
2. Velocity Compounding. The steam
may be expanded in a single stage to
the back pressure p 2 , thus giving a rela
tively high velocity ; and the jet may
then be made to pass through a suc
cession of moving wheels alternating
with fixed guides. This system is shown diagrammatically in
Fig. 112. The jet passes into the first moving wheel, where
it loses part of its absolute velocity, as indicated by the
velocity curve w. It then passes through
the fixed guide g 1 with practically con
stant velocity and has its direction
changed so as to be effective on enter
ing the second moving wheel. Here
the velocity is< again reduced and the
decrease of kinetic energy appears as
work done on the wheel. This process
may be again repeated, if desired, by
adding a second guide and a third wheel.
However, the work obtainable from a
wheel is small after the second moving
wheel is passed, and a third wheel is
not usually employed.
3. Combination of Pressure and Velocity Compounding. Evi
dently the two methods of compounding may be combined in a
FIG. 112.
live sets oi nozzles delivering steam to a corresponding numoer
of wheels running in separate cliambers, and each wheel has
two sets of blades separated by guide vanes.
Pressure turbines are always of the multiple pressurestage
type, and the number of stages is large. The arrangement is
that shown in Fig. 113.
The steam flows through
alternate guides and moving
blades, its pressure falling
gradually as indicated by
the curve pp. The absolute
velocity of flow increases
through the fixed blades
and decreases in the moving
blades as indicated by the
velocity curve ww. This
curve, it will be observed,
rises as the pressure falls
much as if the turbine were
i a large diverging nozzle.
The . steam velocity with
this type of turbine is, however, relatively low even in the
last stages.
178. Work of a Jet. While the problems relating to the
impulse and reaction of fluid jets belong to hydraulics, it is
desirable to introduce here a brief discussion of the general case
of the impulse of a jet on a moving vane.
Let the curved blade have the velocity c in the direction in
dicated, Fig. 114, and let w denote the velocity of a jet directed
against the blade. The velocity w^ is resolved into two compo
nents, one equal to c, the velocity of the blade, the other, there
fore, the velocity a^ of the jet relative to the blade. The angle
of the blade and the velocity c should be so adjusted that the
direction of a is tanent to the ede of the blade at entrance.
The jet leaves the blade with a relative velocity a z equal in
magnitude to a^ neglecting friction, but of less magnitude if
friction is taken into account. This velocity a 2 combined with
the velocity c of the blade gives the absolute exit velocity w 2 .
It is convenient to draw all the velocities from one point as
shown in the velocity diagram.
The absolute entrance and exit velocities w l and w 2 may be
resolved into components w[ and w z ' in the direction of the
motion of the vane
and wJ 1 and w 9 " at
I M
right angles to this
direction, that is,
parallel to the axis of
the wheel that carries
the vane. These
latter may be termed
the axial components,
the former the pe
ripheral components.
The driving impulse
of the jet depends
upon the change in
the peripheral component only. To deduce an expression for
the impulse we proceed as follows :
Let Am denote the mass of fluid flowing past a given cross
.section in the time At ; then the stream of fluid in contact
with the blade may be considered as made up of a number of
mass elements Am, and in the time element At one mass ele
ment enters the vane with a peripheral velocity w^ and another
leaves it with a peripheral velocity w z '. The effect is the same
as if a single element Am by contact with the blade had its
velocity decreased from 10^ to w z ' in the time At. From the
fundamental principle of mechanics, the force required to pro
duce the acceleration
FIG. 114.
At
Am
is
W,
At
(1)
; an equal ana opposite xorce is, tnereiore, wie impulse or
Am on the vane.
If M denotes the weight of steam flowing per second, then
Aw = At, and we have for the force exerted by the jet on
9
the vane in the direction of the velocity 0,
= C '>'") (2*)
Evidently this equation holds equally well when the weight M
flowing from the nozzle is divided among several moving vanes.
The product pc of the peripheral force and peripheral veloc
ity of vane gives the work per second ; therefore,
work per second =  (w^ w 2 '), (3)
y
and
work per pound of fluid =  (w/ w 2 ') 00
y
When, as is usually the case, the direction of w 2 r is opposite
to that of w^, the sign of w 2 ' must be considered negative and
the algebraic difference tv^ w 2 in (2), (3), and (4) becomes
the arithmetic sum w/ + w 2 '.
179. Singlestage Velocity Turbine. In analyzing the action
of the singlestage velocity turbine, it is convenient to start
with an ideal frictionless tur
bine and then take up the
case of the actual turbine.
Let the jet emerge from
the nozzle with the velocity
w v Fig. 115, at an angle a
with the plane of the wheel. Combining w 1 with <?, the periph
eral velocity of the blade, the velocity ^ of the jet relative
to the blade is obtained. The angle /3 between the direction
of &J and the plane of the wheel determines the angle of the
blade at entrance. If the blade is symmetrical, the exit relative
velocity a 2 makes the same angle /3 with the plane of the wheel,
and since the frictionless case is assumed, a z = a r Combining
2 and <?, the result is the absolute exit velocity w z .
an 2
The energy of the jet with the velocity w^ is i per pound
rt/j 2
oi medium flowing; and the jet at exit has the energy ^.
^9
The work absorbed by the wheel per unit weight of steam in
this ideal frictionless case is, therefore,
W = W ? ~ W *\ (1)
2 ff
and the ideal efficiency is
(2)
w
2
i
From the triangle OAE, Fig. 115, we have
W 2 2 =w 1 2 + (2c) 2 2w 1 (2c) cos a; (3)
whence wf m 2 2 = ^(w^ c cos a c 2 ). (4)
Combining (2) and (4), we get,
\ ' (5)
l iv \J
Equation (5) shows that the efficiency is greater the smaller
the angle oc ; and that with a given constant angle , the effi
ciency depends upon the ratio . It is readily found that vj
W^
rt
takes its maximum value ?? max = cos 2 when the ratio takes
w l
the value  cos a.
As an example, let K = 20, whence cos a =0.9397 and cos 2 = 0.883. If
w = 3600 ft. per second, then to get the nmximnm efficiency 0.883, the ratio
must be  cos a = 0.47, whence c = 0.47 x 3600 = 1692 ft. per second, a
wi 2
value too high for safety. If c be given the permissible value 1200 ft. per
second, we have = J, and 77 = 4 x  (0.9397  0.3333) = 0.809.
zi>j o 3 .
In the actual turbine, friction in the nozzle and blades reduces
the efficiency considerably below the value given by (5). The
velocity diagram with friction is shown in Fig. 116. The ideal
>*
actual jeu veiOCliy W^ uumumeu. wiuu veiuui^y u^ivoo imc iciauivc
velocity a v as before. The exit relative velocity a z is smaller
than G&! because of friction
in the blades, and as a
result the absolute exit ve
locity w% is smaller than in
the ideal case.
The work per pound of
4 ^ f 1
steam may be found from
the velocity diagram either by calculation or by direct measure
ment. Having the components wj and wj, the work per
pound of steam is given by the expression
\s
(6)
This work may be compared with the work obtained from the
ideal Motionless turbine given by (1) or with the energy of
Alt 2
the jet per pound of steam, namely, ~ .
* 9
180. Multiplestage Velocity Turbine. In the Rateau turbine
and in others of similar construction, the principle of pressure
compounding is employed. The turbine consists essentially
of several de Laval turbines in series, running in separate cham
bers. See Fig. 111. The action of this type of turbine is con
veniently studied in connection with a Mollier diagram, Fig. 117.
Let the initial state of the steam entering the turbine
at the pressure p 1 be that indicated by the point A. If p 2
is the pressure in the first chamber, a Motionless adiabatic
expansion from p^ to p z is represented by AB, and the
decrease in the heat content ^ 2 is represented by the
length of the segment AB. Under ideal conditions, this
drop in the heat content would all be transformed into
kinetic energy of the jet of steam flowing into the chamber,
and this in turn would be given up to the wheel. Actu
ally, however, friction losses are encountered and the jet
has an exit velocity w z , thereby carrying away the kinetic
an A
energy . The velocity diagram for the single wheel under
consideration is similar to that shown in Fig. 116. The work
lost in overcoming friction in the nozzles and blades and the
nn
exit energy ^ are transformed into heat, and this heat, except
^9
a small fraction that is radiated, is expended in further super
heating (or raising the quality of) the steam. Hence, instead
of the final state B, we have a final state on the same con
stantpressure curve. Referring to Fig. 117, AC' represents
FIG. 117.
the part of the heat drop that is utilized by the wheel, while
O'B represents the part that is rendered unavailable by internal
losses of various kinds.
The steam in the state flows into the second chamber where
the pressure is p y Frictionless adiabatic expansion would give
the second state D, but the actual state is represented by the
point E. Again CE' represents the effective drop of heat con
tent in this stage, while E'D represents the part of the drop
going back into the steam.
The same process is repeated in succeeding stages until
finally the steam drops to condenser pressure in the last stage.
The final state is represented by the point K, and the curve
AEK represents the change of state of the steam during its
passage through the turbine. The final state under ideal fric
tionless conditions is represented by point M, The segment
AM represents the ideal heat drop, which, as has been shown, is
AN represents the heat drop utilized. The ratio
AM
de
pends upon the magnitude of the internal losses, such as friction
in nozzles and blades, leakage from stage to stage, windage,
exit velocity, etc. Roughly, this ratio may lie between 0.50
and 0.80.
181. Turbine with both Pressure and Velocity Stages. In
certain turbines, notably the Curtis turbine, velocity compound
ing is employed. There are relatively few (three to seven)
pressure stages,
but in each cham
ber there are two
or three rows of
moving blades at
tached to the
F IG . us. wheel rim and
these are sepa
rated by alternate rows of guide blades, as shown in Fig. 112.
The velocity diagram for a single pressure stage with two
velocity stages is shown in Fig. 118. The velocities in relation
to the successive sets of blades are shown in Fig. 119. The jet
emerges from the nozzle with an absolute velocity w v which is
smaller than the ideal W Q
because of friction in the
nozzle. Combining w 1 with
the peripheral velocity c of
the first moving blade m p the
result is the velocity a^ of
jet relative to blade m v The
angle a between a^ and the
plane of rotation is the proper
entrance angle of the blade
m v The exit relative velocity 2 , which is smaller than a v
due to friction in the blade, is combined with the velocity c,
giving the absolute exit velocity w 2 which makes the angle /3
with the plane of rotation. The jet enters the stationary
guide blade s with the velocity w z and emerges with a smaller
FIG. 119.
blade m v Combination of w 3 with c gives the relative
velocity a z and the entrance angle y for the blade m z . The
exit velocity 4 is determined from a s and the friction in the
blade, and by combining a 4 and c, the absolute exit velocity w 4
is obtained.
In the diagram, Fig. 119, the blades have been taken as
symmetrical. Sometimes, however, the exit angles of the last
sets of blades are made smaller than the entrance angles. The
diagram can easily be modified to suit this condition.
The work per pound of steam for this wheel is readily deter
mined from the velocity diagram. From the first set of blades
m 1 the work (w^ w z '~) and from the second set of blades m 2
i/
the work  (wJ w /) is obtained. Hence the total work per
9
pound of steam is
W= c  (w /  < + wj  O . (1)
9
Care must be taken that w z ' and w/ be given their proper alge
braic signs.
The state of the fluid as it passes through the turbine may
be shown by the Mollier diagram precisely similar to that
shown in Fig. 117. Starting with an initial state indicated by
point .A, the available drop from the initial pressure p to the
pressure p z in the first chamber is represented by AB. The
heat utilized in useful work Tfas given by (1) is represented
by AC'. Hence projecting C' horizontally to on the line of
constant pressure p v we get the state of the steam as it enters
the second stage nozzles.
182. Pressure Turbine. In the pressure type of turbine
there is always a large number of stages, the guide blades and
moving blades alternating in close succession. The fact that
the pressure falls continuously, both through the guide blades
and the moving blades, makes the velocity diagram essentially
different from that of the velocity turbine, lief erring to Fig. 120,
let w l denote the absolute velocity of the steam entering the
stationary Diaae S 1? ana w z tne aosoiute exit veiocicy. JLI unere
were no, change of pressure, w z would be smaller than w l be
cause of friction ; but the drop in pressure Ajt? causes a decrease
in heat content Ai, and as a result, there
is an increase of velocity given by the
relation
FIG. 120.
Thus the exit velocity w 2 is greater than
the entrance velocity w^. Combining w z
with c, the velocity of the moving blade,
we obtain 2 , the velocity of entrance
relative to the moving blade. Now the
pressure drops through the moving
blades also ; hence as a result the velocity of exit a z is greater
than a v just as e# 2 , is greater than w. Combining a z with (?,
the result is zv^, the absolute velocity of entrance into the
next row of fixed blades.
The work done in any single stage, consisting of one set of
stationary blades and one set of moving blades, is obtained from
the velocity diagram for that stage in the usual way. Thus,
if we have the diagram shown in
Fig. 120 for a particular stage,
the work per pound of steam
for that stage is given by the
product
G (w' z w'J.
u
If the fixed and moving blades
have the same entrance angles
and exit angles, it may be as
sumed that the velocity diagram
has the symmetrical form shown in Fig. 121 ; that is, w^ = j
and w 2 a z . In this case, the work may be obtained by a simple
graphical construction. Using point B as a center and with a
radius BA let a circular arc ADC be described and from ^let
a perpendicular be dropped cutting this arc in _D. Denoting the
length JED by h, we have
G
FIG. 121.
It follows that the work per pound of steain is given by the
h 2
expression provided h is measured to the same scale as the
y
velocity vectors w v w z .
183. Influence of High Vacuum. In Art. 172 it was pointed
out that the reciprocating engine is unable to take advantage
of a very low back pressure for the reason that the cylinder
volume cannot be made sufficiently large to permit the expan
sion of the steam to the condenser pressure. No such restriction
applies to the steam turbine. The blades in the final stages
may be made long enough to pass the required volume of steain
at the lowest pressures obtainable. The advantage of the tur
bine in this respect is shown graphically in Fig. 107. Since
the cylinder volume of the reciprocating engine is limited to
the volume indicated by the point _Z?, the effect of lowering the
back pressure from p 2 to p z ' is the addition of the area D'DFF'
to the area of the original cycle. The turbine, however, can
accomodate volumes indicated by points and 0' ; hence if the
pressure is lowered from p 2 to p 2 ', the area of the ideal cycle is
increased by the area D'DCG 1 . It is evident, therefore, that
high vacuum is much more effective in the case of the steam
turbine than in the case of the reciprocating engine.
The superior efficiency of the steain turbine at low pressures
and the ability of the turbine to make effective use of high
vacuum has led to the introduction of the lowpressure turbine
in combination with the reciprocating engine. The engine
takes steam at boiler pressure and exhausts into the turbine at
about atmospheric pressure. In general, the combination is
more efficient than either the engine alone or the turbine alone
using the entire range of pressure.
EXERCISES
1. In a singlestage velocity turbine the jet emerges from the nozzle with
a velocity of 3150 ft. per second and the direction of the jet makes an angle
that will give maximum efficiency, (b) Find the efficiency if the circum
ferential velocity is 1100 ft. per second.
2. Find the work per pound of steam in case (b) of Ex. 1.
3. Using the data of Ex. 1 and 2 assume that the exit relative velocity is
reduced 10 per cent by friction in the blades. Draw a velocity diagram and
by measurement or calculation find the work done per pound of steam,
Compare this result with that found for the ideal frictionless case.
4. A reciprocating engine receives steam at a pressure of 160 Ib. per
square inch, superheated 120. The steam expands adiabatically to a pres
sure of 16 in. of mercury and is then discharged into a low pressure turbine
where it expands adiabatically to a pressure of 2 in. of mercury. Find the
percentage by which the efficiency is increased by the addition of the tur
bine. Assume ideal conditions.
5. A turbine of the Curtis type has three pressure stages. The initial
pressure is 140 Ib. with the steam superheated 120 F., and the condenser
pressure is 3 in. of mercury, The loss of energy due to friction, etc., is 30
per cent of the total available energy, (a) Find the condition of the steam
entering the condenser. (&) Find the consumption per h. p.hour. (c)
Determine the intermediate pressures in the cells on the assumption that the
work developed in each stage shall be approximately the same.
REFIGERATION WITH VAPOE MEDIA
184. Compression Refrigerating Machines. The essential
organs of a compression machine using vapor as a medium
are shown in Fig.
122. The action of
the machine may be
studied to advan
tage in connection
with the ^dia
gram, Fig. 123.
The medium is
drawn into the
compressor cylinder
through the suction
pipe from the coils
in the brine tank.
It may be assumed that the medium entering is in the saturated
state at the temperature T^ which may be taken equal to the
i f P 41 "K >* Tl "44" 411 "j. 7~>
Expansion
Vulvo
FIG. 122.
M
Jbig. 126. Ihe vapor is compressed adiabatically to a final
pressure p z , which is determined by the upper temperature T z
that may be obtained with the cooling water available. The
adiabatic compression is represented by B Q, The superheated
vapor in the state (7 is discharged into the coils of the cooler
or condenser, where heat is abstracted from it. The coils are
surrounded by cold water which
flows continuously. First the
gas is cooled to the state of
saturation ; this process is rep T E{ ^ YD .i
resented by the curve CD, and
the heat abstracted by the area r /j\A_T l
CiCDDv Then heat is further
removed at the constant tern II\
perature T z (and pressure p%)
and the vapor condenses. At
the end of the process, the
medium is liquid and its state 
is represented by the point E Fl{} 123>
on the liquid curve.
It should be noted that there are two parts of the fluid circuit :
one including the discharge pipe and coils at the higher pres
sure p z , and one including the brine coils and the suction pipe
at the lower pressure p r These are separated by a valve called
the expansion valve. The liquid in the state represented by
point H is allowed to trickle through the valve into the region
of lower pressure. The result of this irreversible free expan
sion is to bring the medium to a new state represented by point
A. In this state the medium, which is chiefly liquid with a small
percentage of vapor, passes into the coils in the brine tank or
in the room to be cooled. The temperature of the brine being
higher than that of the medium, heat is absorbed by the medium,
and the liquid vaporizes at constant pressure. This process is
represented by the line AB and the heat absorbed from the
surrounding brine by the area A l ABQ r
The position of the point A is determined as follows : The
passage of the liquid through the expansion valve is a case of
throttline or wiredrawing of the character discussed in Art. 162.
Hence, the heat content at A must be equal to the heat content
at E, that is,
^ 2 = ^i + ^ r i
Graphically, the area OHGAA l is equal to the area OHEE^, or
taking away the common area OHGFE^ the rectangle E l FAA l
is equal to the triangle GEI. (See Art. 162).
Since the throttling process represented by EA is assumed to
be adiabatic, the work that must be done 011 the medium is the
difference between Q^ the heat absorbed, and Q z , the heat rejected
to the condenser. We have then
Q 1 = area
W = area O 1 ODEE 1  area A 1 AB0 1
If the expansion valve be replaced by an expansion cylinder,
permitting a reversible adiabatic expansion from p z to p v as in
dicated by the line JEF, we have
= area
W= area BCDEFB.
The effect of using the expansion valve rather than the expansion
cylinder is thus to decrease the heat removed by the area E^FAA^
and to increase the work done by an equivalent amount.
185. Vapors used in Refrigeration. The three vapors that are
used to any extent as refrigerating media are ammonia, sulphur
dioxide, and carbon dioxide. Of these, ammonia is used almost
exclusively in America and largely in Europe. The other two
are used to a small extent chiefly in Europe.
The choice of vapor to be used depends chiefly upon two things :
(1) The suction and discharge pressures that must be employed
to give proper lower and upper temperatures T^ and T z . The
lower temperature must be such as to keep the proper temperature
in the brine or the space to be kept cool, while the upper
temnerature is fixed bv the temoerature of the ooolino
ailable. (2) The volume of the medium required for a given
louiit of refrigeration. This determines the bulk of the
iichine.
If the upper temperature be taken as 68 F. (T z = 528) and
13 lower temperature as 14 F., the pressures and the volume
lion for tlio three vapors mentioned are about as follows:
Nils SO. COj
ol.ion proHHuni, 11). pur sq. in. 41.5 14.75 385
s<:]i:ir^(! pressure, II). per sq. in. 124 47.61 826
>lumi!, taking Uuiti of (.X) 3 an 1 4.4 12 1
It appears that carbon dioxide requires for proper working
!iy high pressures, so high, in fact, as to be practically prohib
ve except in maehines of small size. With sulphur dioxide
e pressures are low, but the necessary volume of medium is
gh, being nearly three times that required by ammonia and
,'elvo times that required by carbon dioxide. With ammonia,
c pressures are reasonable and the volume of medium is not
:<:essive; hence from these considerations, ammonia is seen to
! most advantageous.
From the point of view of economy, ammonia and sulphur
oxide are about equal. Carbon dioxide shows a somewhat
Killer el'l'ieiem:/ than the others under similar conditions be
,use, on account of the small latent heat of carbon dioxide, the
sses due to superheating and the passage through the expan
jn valve are a larger per cent of the total effect.
186. Calculation of a Vapor Machine. The following analysis
>plies to the ideal, cycle shown in Fig. 123. Denoting by T
ie temperature at the end of compression indicated by the
>int (7, the heat that must be removed per minute from the
iperheated vapor to bring it to the saturation state (the heat
presented by the area O^DD^ is
v .
i Avlm.li * p denotes the specific heat of superheated vapor, and
E the weight of the medium required per minute^ i he heat
ejected by the vapor during condensation (area 1)^^) is
fr a . Hence the heat rejected by the medium per minute is
^ *rr , .. rrn _ TV1. (1)
Denoting by x l the quality of the mixture of liquid and vapor
in the state represented by point A, we have for the heat ab
sorbed by the medium from the brine or cold room (repre
sented by the area A^ABO^)
Q^Mr^lxJ. (2)
But area OHGfAA^ area OHEU V that is,
2i' + r i x i = &' ; ( 3 )
whence combining (3) and (2),
Q, = M^  qj + ft ') = M( qi  2a '). 00
The work required per minute is, therefore,
TT= JX<? a 0i) = ^[2 a "<?i" + 'p(^ 2*)], (5)
and the net horsepower required to drive the machine is,
Combining (6) and (4), we have
778^[^^+ gp (r e2
83000( ?1 " 2a ') ' . ^ }
To the horsepower thus calculated should be added perhaps
10 to 20 per cent to allow for imperfections of the cycle, and to
the gross horsepower must be added 10 to 20 per cent to allow
for the friction of the mechanism.
Assuming the vapor entering the compressor to be dry and
saturated, as indicated by point 33, Fig. 123, the volume of
vapor entering the compressor per stroke is
Mv "
V _1_ /^
v c ft (0)
where v^' is the specific volume of vapor at the pressure p l
and N the number of working strokes per minute. If the
medium enters the compressor as a mixture of quality x m as in
dicated by point M, then approximately
The net cylinder volume as determined by (8^ or (9) must
JL
The weight of cooling water required per minute is readily
found from (1) when the initial and final temperatures of the
water are fixed. Denoting this weight by Cr and the initial and
final temperatures by t" and t\ respectively, we have
<?(*"  O = M fa + c p (T  5i)]. (10)
To determine the value of Q z from (1) the temperature T c at
the end of compression must be obtained. For adiabatic .com
pression T c may be found by the following method. Eeferring
to Fig. 123, the decrease of entropy in passing from to D is
the same as passing from B to D. If e p , the specific heat along
curve CD, is assumed to be constant, we have
T
UUt Sjj Sd S[ \ fif i "2 ' rn
LI \ J 2 .
hence c p log e  = s^ f ~
Since <? p , 2j, jP 2 , s/, s 2 , r 15 and r 2 are known quantities, ^ is
easily calculated.
EXAMPLE. Required the dimensions and the horsepower of an ammonia
refrigerating machine that is to abstract 15,000 B. t. u. per minute from a
cold chamber which is to be kept at a temperature of 30 3?. The tempera
ture of the ammonia in the condenser may be taken as 85 F. and that of
the ammonia in the brine coils 20 F. Assume one doubleacting com
pressor making 75 r. p. m.
From the table of the properties of saturated ammonia, we have the fol
lowing values corresponding to ti = 20 and fe = 85 :
pi = 47.46 Ib. per square inch, ri = 500 B. t. u., q\ = 13 B. t. u.,
ji" = 547 B. t. u., si' =  0.027, g = 1.168, v{' = 6.01 cu. ft.,
M.
p z = 166.8 Ib. per square inch, r 2 = 496 B.t. u., q z ' = 01 B. fc.u.,
0," = 557 B.t.u., s 2 ' = 0.118, ^.= 0.910, t? 2 " = 1.78.
end of compression, we have, from (11),
0.51 log e p^4 =  0.027 + 1.168  (0.118 + 0.910)= 0.113,
whence log T c = log 544.6 + 0.4343 x 9^ = 2.83231,
T = 679.7,
and 4 = 679.7  459.6 = 220.1 F.
The weight of ammonia that must be circulated per minxtte is, from (4),
isooo
o47  61
The net horsepower is, from (6),
778 x 80.86 r
33000
_ 547 0<51(22 o.l _ 85)] = 57.4.
V ^
Adding 15 per cent for cycle imperfections, the compressor will requii
about 66 horsepower. The steam engine required to drive the compressc
should develop, say, 80 horsepower.
The volume of the compressor cylinder is, from (S),
30.86 x 6.01 = mcu . ft .
2 x 75
Adding 15 per cent for clearance, etc., the required volume is 1.43 cu. f
This is given by a stroke of 20 in. and a cylinder diameter of 12 in.
TABLE I
PROPERTIES OF SATURATED STEAM
PRESSURE
t3 INCHES OP
HG.
TEMP.
FAHU.
t
HEAT CONTENT
LATENT HEAT
ENTROPY
VOLUME
OF ONE
POUND
V
of Liquid
i'
of Vapor
i"
Total
r
Internal
P
of Liquid
s'
of Vapor
ization
r
T
of Vapor
s"
0.5
58.81
26.9
1087.1
1060.2
1002.9
.0532
2.0431
2.0963
1259.3
1.0
79.12
47.2
1096.7
1049.5
989.8
.0916
1.9482
2.0398
656.7
1.6
91.90
59.9
1102.5
1042.6
982.2
.1150
1.8905
2.0055
443.0
2.0
101.27
69.2
1106.6
1037.4
975.9
.1317
1.8497
1.9814
338.3
2.5
108.81
76.7
1109.9
1033.2
970.8
.1451
1.8178
1.9629
274.3
3.0
115.15
83.1
1112.7
1029.6
966.5
.1561
1.7915
1.9476
231.2
3.5
120.63
88.5
1115.0
1026.5
962.8
.1656
1.7692
1.9348
200.1
4.0
125.48
93.4
1117.1
1023.7
959.5
.1739
1.7497
1.9236
176.6
4.5
129.85
97.7
1118.9
1021.2
956.5
.1813
1.7325
1.9138
158.1
5.0
133.81
101.7
1120.6
1018.9
953.7
.1880
1.7170
1.9050
143.2
6
140.83
108.7
1123.4
1014.7
94S.8
.1997
1.6901
1.8898
120.7
7
146.90
114.8
1125.9
1011.1
944.7
.2097
1.6672
1.8769
104.4
8
152.28
120.2
1128.0
1007.9
940.9
.2186
1.6473
1.8659
92.2
9
157.12
125.0
1130.0
1005.0
937.5
.2265
1.6296
1.85G1
82.6
10
161.52
129.4
1131.7
1002.3
934.3
.2336
1.6138
1.8474
74.8
11
165.57
133.4
1133.3
999.9
931.5
.2401
1.5994
1.8395
68.38
12
169.31
137.2
1134.7
997.6
928.8
.2460
1.5862
1.8322
63.03
13
172.80
140.6
1136.0
995.4
926.3
.2516
1.5739
1.8254
58.48
14
176.07
143.9
1137.3
993.4
924.0
.2568
1.5627
1.8195
54.55
15
179.16
147.0
1138.5
991.5
921.8
.2616
1.5522
1.8138
51.13
16
182.08
150.0
1139.6
989.6
919.6
.2662
1.5423
1.8085
48.11
17
184.84
152.7
1140.6
987.9
917.6
.2705
1.5330
1.8035
45.46
18
187.47
155.4
1141.6
986.2
915.7
.2746
1.5242
1.7988
43.09
19
189.99
157.9
1142.5
984.6
913.8
.2785
1.5158
1.7943
40.96
20
192.38
160.3
1143.4
983.1
912.1
.2822
1.5079
1.7901
39.04
21
194.69
162.6
1144.2
981.6
910.4
.2857
1.5003
1.7860
37.29
22
196.91
164.8
1145.0
980.2
908.8
.2891
1.4931
1.7822
35.68
23
199.04
167.0
1145.8
978.8
907.2
.2923
1.4862
1.77S5
34.22
24
201.10
169.0
1146.5
977.5
905.7
.2955
1.4796
1.7751
32.88
25
203.09
171.0
1147.2
976.2
904.2
.2985
1.4732
1.7717
31.65
26
205.01
173.0
1147.9
974.9
902.7
.3014
1.4670
1.7674
30.52
27
206.87
174.9
1148.6
973.7
901.4
.3042
1.4611
1.7653
29.46
28
208.68
176.7
1149.2
972.5
900.0
.3069
1.4554
1.7623
28.47
29
210.43
178.4
1149.8
971.4
898.8
.3095
1.4499
1.7594
27.55
PHEHIUIKK
TUMP.
JL1KAT V .
Ln. J'Klt
S. In.
FA.UU.
)(' Liquid
of VHIHT
Toliil
Internal
,1 i,,,,,.i.l
Vupon
/ution
<>! Viijior
P
t
''
7."
r
'
K 1
7'
K"
14.7
212.0
180.0
1150.4
970.4
85)7.7
.3121
1.4415)
1.7570
15
213.0
181.1
1.150,8
5)09.7
85)0.8
.3137
1.4417
1.7551
16
21(5.3
184,4
1152.0
5)07.0
85)4,5
.318(5
1.4315
1.7501
17
219.4
187.5
1153.0
5X55.5
892.1
.323 1
1.4220
1. 745 1
4 O
,),)>) A
1 1 54 1
5)03.0
8X9 9
3275
1 1 125)
1 7404
lo
19
225/2
1SKU
iibr>!o
887.7
.3317
1.4043
1.7300
20
227.9
15)0.1
1 155.9
5)59.8
885.7
.3357
1.3901
1.73 IX
21
230.5
198.7
1 150.8
5)58.1
883.8
.335)5
1 .3X83
1.727X
22
233.0
201.2
1 157.7
5)5(5.5
882.0
.3431
1.3X09
1.7210
23
235.4
203.0
1 158.5
5)54,8
880.0
.3407
1 .3738
1.7205
24
237,8
20(i'.()
1 159.2
5)53.2
878.0
.3501
1.30(55)
1.7170
25
240.0
208.3
1100.0
5)51.7
87(5.2
.3533
1.31504
1.7137
26
242 2
210.5
1 1(50.7
5)50.2
874.5
.35(55
1.3540
1.7105
27
244,3
212.7
11(51.5
5)48.8
Uto
.355)5
1.347!)
1.7071
28
24(5.4
214,8
11(52.1
/fl.{7.3
\7l.2 ,
.3(525
1.341!)
1.7041
29
30
248.4
250.3
21(5,8
218,8
1JW2.N
HS3.4
kjibfcO '
SOi).S
8(58/2
.3(553
.3(581
1.330'J
1.3307
1.7015
1.0988
31
252/2
220.7^
11(54,0
5)13.3
800.H
.3708
1 .3253
1.05)01
32
254.0
222.5 \
V 104*0
5)42. 1
805.1
.373.1
1.3'JOl
l.( 55)35
33
255,8
224,3
N&.1
5)40,8
8(54.0
.37(50
1.3151
1.155) 11
34
257.0
22(5.1
V 105.7
5)39.0
8(52.7
.3784
1.3102
1.15X8(5
35
259.3
227.8
\1(5(5.2
5)38. 1
,8(51.3
.3808
1.3054
1 .0X03
36
2(51.0
229.5
1.1(5(5,8
5)37.3
8(50.1
.3832
1.300X
1. (5810
37
2(52.0
231.2
11(57.3
93(5.1
858.8
.3855
1 .251(53
1. (5,8 1 8
38
2(54/2
232.8
11(57.8
5)35.0
857.0
.387S
1.25)18
1.075 Hi
39
2(55.7
234.4
1108.3
5)33.5)
85(5.4
.35)00
1.287(5
1 .(577(5
40
2(57.3
23(5.0
1 1 (58.8
5)32.8
855. 1
.35)21
1 .2834
1.0755
41
268.8
237.5
11(59.3
931.8
854.0
.3912
1 .275)3
1.0735
42
270.2
239.0
1109.7
5)30.7
8513,8
.35)02
1 .2753
1. 07 1 5
43
271.7
240.5
1170.1
92!).7
851.7
.35)82
1.2714
1.005X5
44
273.1
241.9
1170.0
5)28.7
S50.0
.1002
1.2(57(5
1 .(5078
45
274.5
243.3
1171.0
927.7
849.5
.4021
1 .2(538
1. (5(559
46
275.8
244.0
11.71.4
5)2(5.8
818.5
..1010
1.2(502
1. (5(542
' 47
277.2
24(5.0
1.171.8
5)25,8
847.4
.4059
1.25(5(5
1. (5(525
48
278.5
247.3
1172.2
5)24,9
84(5.4
.1077
1 .253 1
1.0008
49
279.8
248.7
1172.0
923.!)
815.3
.1095
1.219(5
1.055)1
50
281.1
250.0
1173.0
923.0
814,4
.4112
1/2103
1.0575
51
282.3
251.3
1173.4
5)22.1
843.4
.4130
1.2425)
1.0555)
52
283.5
252.0
1.173,8
5)21.2
842.4
.4147
1.235)7
1.0514
53
284.8
253.8
1174,2
5)20.4
811.5
.41(51
1.23(55
1.0525)
54
280.0
255.0
1174.5
919.5
840.5
.1180
1 .2333
1.0511?
55
287.1
250.2
1174,5)
918.7
835U5
.4151(5
1/230: 5
1. (515)5)
56
288.3
257.4
1175/2
917,8
838.7
.4212
1 .2272
1. (5481
57
289.4
258.0
1175.0
917.0
837.8
.4228
1.2213
1.0171
58
290.6
259.7
1175.5)
910.2
83(5.5)
.4243
1.2213
1. (54 5(5
59
291.7
2(50.8
1170/2
5)15.4
83(5.0
.1258
1.2181
1.0442
JIi.vi f
'n,\r.sr
LATKN
r HKAT
ENTKOPV
Tl.MI'.
i
_
VOLUME
OF ONE
I''AUH.
if Liuuiit
u( Vnn.r
Tot ul
Intonml
of Liquid
Vapori
zation
of Vapor
POUND
t
T'
i"
r
P
'
r
T
s"
v"
21)2.8
2(52.0
117(i.(i
014.0
S85.2
.4273
1.215(5
1.6429
7.168
1208.0
2(58. 1
1 170.0
013.S
834,3
.4288
1.2128
1.0416
7.057
201.0
12(5 4.2
1177.2
1)13.0
833.4
.4302
1.2101
1.0403
6.949
12015.0
120,"). 3
1 177."'
012 2
S3'' '"5
481(5
1 2074
1 (i'WO
K VAK
207.0
1177.8
S31.S
.4330
j. *\j t T:
1.2047
JL .IJOt/V/
1.6378
U.OTrU
6.744
2118.0
2(57.4
117S.1
010.7
S80.0
.4344
1.2021
1.6365
6.646
120!>.l
17S.I
5)10.0
830.2
.4358
1.105)5
1.6353
6.551
800. 1
17K.7
000.2
820.8
.4372
1.1069
1.6341
6.459
801.1
27<Y.f>
170.0
U08.5
S2s!r>
.4385
1.1044
1.6329
6.370
802.0
271.5
170.8
007.S
S27.7
.4308
1.1020
1.6318
6.283
308.0
1272.4
170.5
007.1
S27.0
.441 1
1.1895
1.6306
6.198
308J>
273.4
170.S
000..4
82(5.2
.4424
1.1871
1.6295
6.115
301.!)
1274.4
ISO.]
!)05?7
S25.5
.4437
1.1847
1.6284
6.035
305.8
127.">. 4
1S0.4
005.01
S24.7
.4450
1.1823
1.6273
5.957
8015.S
27(5.3
ISO. (5
i)o4 !n
823.9
.44(52
1.1800
1.0262
5.882
307.7
277.8
ISO.!)
008.15
823:2
.4474
1.1777
1.6251
5.809
308.15
27S.2
181. 1
002.0
S22'.4
. .448(5
1.1755
1.6241
5.737
305). 5
27! I.I
LSI 4
<)()'> 'i
S'M S
.445)8
1.1732
1.6230
5.666
310.4
12SIU)
IS 1.0
001.15
821.0*
.4510'
1.1710
1.6220
5.597
311.2
12SO.O
IS 1.0
001
S20.4
.4522
1.1688
1.6210
5.530
812.1
281.8
1 1S2.1
000.3
SI 0.0
.4533
1.1667
1.6200
5.464
313.S
1 1S2.7
SO!)
81 8.2
.455(5
1.1625
1.6181
5.338
315.5
2sr>'3
1 183.1
S07!s
81(5.1)
.4578
1.1584
1.0162
5.219
317.2
12S7
.4(500
1.1543
1.6143
5.104
31S.S
288.7
IS'LO
SOfvl
814.3
.4022
1.1503
1.0125
4.995
320.4
><)() 3
1 1 S4 r >
S04.2
818.0
.4042
1.1465
1.6107
4.890
321.0
323.4
25) L5)
1S4!<)
S03.0
811.7
810.4
.4(503
.4(583
1.1427
1.1390
1.6090
1.6073
4.789
4.692
824.0
320.5
LJOOJi
ish!7
1815.1
siwhV
S00.3"
SOS.O
.4703
.4723
1.1353
1.1317
1.6056
1.6040
4.599
4.511
327.0
320.8
330.7
882.1
833.5
2! )!>.(!
801.0
802.5
18(5.5
1S7.7
188.1
888.4.
SSf>!2
SS4.2
80(5.8
S05.0
803!4
S02.3
.4742
.4701
.4770
.4707
.4815
1.1282
1.1248
1.1214
1.1181
1.1149
1.6024
1.6009
1.5993
1.5978
1.5964
4.425
4.343
4.264
4.188
4.114
334.8
83(5.2
887.5
338.S
840.1
80S.O
310/7
ISS.4
1SS.S
1 8' ) 5
1 S! ) S
SS3.1
SS2. 1
881.1
SSO.l
801.1
800.0
7!)7!l)
70(5.8
.4833
.4850
.4807
.4884
.4901
1.1117
1.1085
1.1054
1.1024
1.0994
1.5950
1.5935
1.5921
1,5908
1.5895
4.043
3.975
3.909
3.845
3.784
341.3
342.(5
343.S
345.0
il
1 00.1
100.5
inn .1
S7S.2
S77.2
875!3
705.S
704.8
703.S
71)2.7
701.S
.4917
.4033
.4040
.40(55
.4980
1.0965
1.0936
1.0907
1.0879
1.0851
1.5882
1,5869
1.5856
1.5844
1.5831
3.724
3.666
3.610
3,555
3,502
Ll). I'BU
SQ. IN.
TEMP.
FAUK.
HKAT GONTKNT
LATKNT HKAT
KNTHOl'Y
>f Liquid
of Vupor
Total
Internal
of Liiiuid
Vfipori
zutioii
of Vupor
P
t
i'
i"
r
P
'
T
130
347.4
318.2
1191.7
873.5
700.S
.4995
1.0824
1.581!)
132
348.0
319.4
1192.0
872.15
78!).!)
.5010
1.0797
1.5807
134
349.7
320.0
1102.3
871.7
788.9
.5025
.0770
1 .5795
136
350.8
321.8
1192.0
870.8
788.0
.5039
.0744
1 .5783
138
352.0
323.0
1192.9
800.0
787.0
.5054
.071!)
1 .5773
140
353.1
324.2
1193.2
8(59.0
78(5.1
.50(58
.0(593 ! 1.57(51
142
354.2
325.3
1193.5
808.2
785.2
.5082
.0(5(58 1.5750
144
355.3
32(5.5
1103.X
8(57.3
784,3
.5090
.0(544
1.5740
146
350.4
327.0
1104.0
8(5(5.5
783.4
.5110
.0(519
1.572!)
148
357.4
328.7
1194.3
805.0
782.4
.5123
1.0595
1.5718
150
358.5
329.8
1104.0
8(54,8
781. (i
.5137
1.0571
1.5708
160
303.0
335.0
1195.8
8(50.8
777.4
.5202
1.045(5
1.5(558
170
308.5
340.2
1107.1
85(5.9
773.2
.5203
1.034!)
1.5(512
180
373.1
345.0
1108.2
853.2
7(59.3
.5321
1.024(5
1.55(57
190
377.0
349.0
1199.3
849.0
7(55.5
.5377
1.014!)
1.552(5
200
381.8
354,1
1200.3
84(5.2
7(52.0
.5430
1.0057
1.5487
210
385.9
358.4
1201.3
842.!)
758.5
.5481
.9908
1 .544!)
220
389.5)
3(52.5
1202.2
839.0
755. 1
.5530
.9884
1.5414
230
393.7
3(5(5.5
1203.0
83(5.5
751.8
.5577
.9803
1.5380
240
397.4
370.4
1203.9
833.5
748.7
.5(522
.972(5
1.5348
250
401.0
374,1
1204,7
830.0
745.7
.5(500
.9(551
1.5317
260
404.5
377.8
1205.5
827.7
742.7
.5708
.9579
1 .5287
270
407.8
381.3
120(5.2
824.!)
730.S
.574!)
.9510
1.5259
280
411.1
384,7
120(5.9
822.2
737.0
.5788
.9443
1.5231
290
414.3
388.1
1207.0
819.5
734.2
.582(5
.9378
1.5204
300
417.4
391.3
1208.3
817.0
731.5
.58(53
.9315
1.5178
TABLE II
tt OF SATURATED STEAM BELOW 212 F.
mini:
'(IMIMK OF
)NK 1'OIINl)
((Ju. FT.)
VBIOHT OF ONIJ CUBIC
FOOT
TOTAL
HEAT
1"
LATENT
HEAT
r
'BMP.
t
Pounds
7
Grains
0.1 S02
32XX
0.000304
2.129
1073.7
1073.7
32
0. 1955
3047
0.000328
2.297
1074.7
1072.7
34
().2\'\l)
282(5
0.000354
2.477
1075.8
1071.7
36
0.22!) 1
2(523
0.000381
2.669
1076.8
1070.8
38
0.217X
2137
0.000410
2.872
1077.8
1069.8
40
0.2(577
22(55
0.000442
3.091
1078.8
1068.8
42
0.2X91
2107
0.000475
3.322
1079.8
1067.7
44
0.311!)
Mil
0.000510
3.570
1080.8
1066.7
46
0.33(5(5
1 827
0.000547
3.831
1081.8
1065.7
48
0.3(527
1703
0.000587
4.110
1082.8
1064.7
50
0.3905
15S7
0.000(530
4.411
1083.8
1063.7
52
0.1202
14X3
0.000(574
4.720
1084.7
1062.7
54
1 3X5
0.000722
5.054
1085.7
1061.6
56
o''lX r (i
1 204
0.000773
5.410
1086.7
1060.6
58
(K5217
1210
0.000827
5.785
1087.6
1059.6
60
0.55i)X
OJ5I4
1132
10(50
993
0.000883
0.000943
0.001007
6.043
6.604
7.048
1088.6
1089.6
1090.5
1058.5
1057.5
1056.4
62
64
66
()'.(5X!)
0.73X
931
X73
0.001074
0.001145
7.52
8.02
1091.5
1092.4
1055.4
1054.3
68
70
0.7X9
O.X.M
O.!)03
0.9()l
1.02!)
X20
770
723
(5X0
(53!)
0.001220
0.001300
0.001383
0.001471
0.0015(54
8.54
9.10
9.68
10.30
10.95
1093.3
1094.3
1095.2
1096.1
1097.1
1053.3
1052.2
1051.2
1050.1
1049.0
72
74
76
78
80
1.0! )X
1.171
I. IMS
1.32!)
(501.4
5(55.7
532.2
500.X
471.4
0.001603
0.001708
0.001879
0.001997
0.002121
11.64
12.37
13.15
13.98
14.85
1098.0
1098.9
1099.8
1100.7
1101.6
1048.0
1046.9
1045.8
1044.7
1043.6
82
84
86
88
90
1.50(5
1.1502
1 .704
LSI 2
1.925
443.9
41X.2
394.2
371.X
350.9
0.002253
0.002391
0.002537
0.00265)0
0.002850
15.77
16.74
17.76
18.79
19.95
1102.5
1103.4
1104.3
1105.2
1106.1
1042.5
1041.4
1040.3
1039.2
1038.1
.
92
94
96
98
100
TEMP.
FAHH.
t
PHESSUUE
VOLUME OF
ONE POUND
(Cu. FT.)
v"
WEIGHT OF UNB UUBIC
FOOT
TOTAL
HEAT
q"
LATENT
HEAT
r
Lb. per
Sq. In.
P
Inches of
Eg.
Pounds
7
Grains
102
1.004
2.044
331.4
0.003017
21.12
1107.0
1037.0
104
1.066
2.171
313.2
0.003193
22.35
1107.9
1035.9
106
1.131
2.303
296.2
0.003376
23.63
1108.7
1034.8
108
1.199
2.441
280.4
0.003566
24.96
1109.6
1033.7
110
1.271
2.588
265.6
0.003765
26.36
1110.5
1032.5
120
1.689
3.439
203.4
0.004916
34.42
1114.8
1026.9
130
2.219
4.518
157.5
0.00635
44.45
1119.0
1021.1
140
2.885
5.874
123.1
0.00812
56.86
1123.1
1015.2
150
3.714
7.56
97.2
0.01029
72.0
1127.1
1009.3
160
 4.737
9.64
77.4
0.01293
90.5
1131.1
1003.2
170
5.988
12.19
62.09
0.01611
112.7
1135.0
997.1
180
7.506
15.28
50.23
0.01991
139.4
1138.8
990.9
190
9.335
90.01
40.94
0.02443
171.0
1142.5
984.6
200
11.523
23.46
33.60
0.02976
208.3
1146.2
978.2
210
14.122
28.75
27.77
0.03601
252.1
1149.7
971.7
212
14.697
29.92
26.75
0.03738
261.7
1150.4
970.4
FQ
W .
PH H
s
a o
fc O
a I
J H
^ *
>o
eO<N(M THrH
I 1 I
rH Tl (N CJ CO CO ^ * U3 1O O ? l> t 00 00 <T
ThiCOCOCOCOC^C<l<MC<li ITHI it li iO
OOO>C^O3O3OG
ooooooooooooooooooooooooc
I I I I I I I I I I I I I

CD CD lO O * >* CO CO <N i
1 II 1 I 1 1 II
rH (N (M CO CO * ^ O C3
TH * (M CO CO iM 00 <M O IH CD 00 O <N
INDEX
[The numbers refer to pages]
Absolute scale, Kelvin's, 55.
temperature, 18.
zero, 18.
Acoustic velocity, 257.
Adiabatic change, defined, 40.
expansion of gas, 103.
of vapor mixture, 185, 189.
of superheated steam, 218.
irreversible, 75.
of air and steam mixture, 233.
of superheated steam, approximation
to, 220.
of vapor mixture, approximation to,
190.
on TSplane, 70.
with variable specific heat, 126.
Air and steam, mixture of, 232, 236.
compression, 152.
engine cycles, analysis of, 140.
engines, classification of, 137.
moist, constants for, 230.
moisture in, 228.
refrigeration, 149.
required for combustion, 119.
Allen denseair refrigerating machine,
150.
Ammonia, saturated, 180.
superheated, 223.
Andrews' experiments, 198.
Atomic weights, 111.
Availability of energy, 46.
Available energy of a system, 56.
Bertrand's formulas, 168.
Biot's formula, 167.
Boltzmann's interpretation of the second
law, 65.
Boyle's law, 89.
Brayton cycle, 145.
Callendar's equation for superheated
steam, 204.
Calorimeter, throttling, 271.
Caloric theory, 3.
Carbon dioxide, saturated, 182.
Carnot cycle, 50, 134.
for saturated vapors, 283.
on TSplane, 73.
engine, efficiency of, 54.
Carnot's principle, 52.
Characteristic equation, 16.
of gases, 93, 277.
surface, 20.
Charles' law, 90.
Chemical energy, 5.
ClapeyronClausius formula, 178.
Clausius' equation, 200.
inequality of, 63.
statement of the second law, 50.
Combustion, 117.
air required for, 119.
products of, 119.
temperature of, 127.
Compound compression of air, 156.
Compounding of steam turbines, 296.
Compressed air, 152.
engines, 158.
Compression, compound, 156.
refrigerating machine, 308.
Conduction of heat, waste in, 57.
Conservation of energy, 6.
Constant energy curve of mixture, 187.
Constant volume curve, 186.
Continuity, equation of, 244.
Coordinates defining state of system, 15.
Critical states, 197.
temperature, volume, and pressure, 199.
Cycle, Carnot, 50, 134.
Diesel, 146.
Joule, 145.
Lenoir, 162.
Otto, 142.
processes, 72, 133.
Rankine, 284.
rectangular, 73.
Cycles, isoadiabatic, 136.
of actual steam engine, 290.
of air engines, analysis of, 140.
of gas engines, comparison of, 148.
with irreversible adiabatics, 75.
Cylinder efficiency, 293.
Curtis type of steam turbine, 304.
Curve, constant volume, of steam, 186.
of heating and cooling, 70.
polytropic, 71.
saturation, 166, 182.
Curves, specific heat, superheated steam,
209, 211.
324
INDEX
Dalton's law, 114, 228.
Davis formula for heat content, 177, 274.
Degradation of energy, 7.
Degree of superheat, 165, 196.
De Laval nozzle, 258.
Derivative ^ 170.
Design of nozzles, 264.
Diesel cycle, 146.
Differential equations of thermodynam
ics, 82, 84.
expressions, interpretation of, 28.
inexact, 30.
Differentials of u, i, F and $, 79 .
Dissociation, 197.
DupreHertz formula, 168.
Efficiency, conditions of maximum, 135.
cylinder, 293.
of Carnot engine, 54.
potential, 292.
ratio, 292.
thermal, 291.
standards, 291.
Electrical energy, 5.
Energy, availability of, 46.
chemical, 5.
conservation of, 6.
degradation of, 7.
dissipation of, 8.
electrical, 5.
Energy equation, 36.
applied to cycle process, 39.
applied to vaporization, 170.
integration of, 38.
Energy, heat, 3.
high grade, and low grade, 7.
mechanical, 2.
of gases, 97.
of saturated vapor, 172.
of superheated steam, 214.
relativity of, 2.
transformations of, 5.
units of, 8.
units, relations between, 10.
Engine, compressed air, 158.
Ericsson's, 139.
Stirling's, 138.
Engines, gas, 142.
hotair, 138.
steam, 283.
Entropy, as a coordinate, 68.
first definition of, 59.
of gases, 100.
of liquid, 179.
of superheated steam, 215.
Equation of Clausius, 200.
of perfect gas, 17.
of van der Waals, 20, 200.
of vapor mixture, 184.
Equations for gases, 94.
for discharge of air and steam, 255.
for superheated steam, 203.
general, of thermodynamics, 79.
Equilibrium of thermodynamics systems,
87.
Ericsson's air engine, 139.
Exact differentials, 30.
Expansion of gases, adiabatic, 103.
at constant pressure, 101.
isothermal, 102.
Expansion valve, 272, 309.
Exponent n, determination of, 108.
External work of a system, 37.
First law of thermodynamics, 35.
Fliegner's equations for flow of air, 255.
Flow of air, equations for, 255.
Flow of fluids, assumptions, 244.
experiments on, 243, 254.
formulas for discharge, 255
fundamental equations, 244.
graphical representation, 247.
through orifices, 252.
Flow of steam, Grashof's equation, 256.
Rateau's equation, 256.
Napier's equation, 257.
Free expansion of gases, 58.
Friction in nozzles, 262.
Frictional processes, 74.
Fuels, 118.
Gas, characteristic equation of, 93, 277.
constant B, value of, 92.
constant, universal, 113.
constants, relations between, 112.
free expansion of, 58.
permanent, 89.
Gasengine cycles, comparison of, 148.
Gases, entropy of, 100.
general equations for, 94.
heat content of, 99.
intrinsic energy of, 97.
laws of, 89.
mixtures of, 114.
specific heat of, 96, 124.
Graphical representation of energy equa
tion, 43.
of flow of fluids, 247.
Grashof's equation, flow of steam, 256.
iat content 01 gases, yy.
of saturated vapor, 173, 177.
of superheated steam, 210.
iat, effects of, 35.
Intent, 20.
mechanical equivalent of, 11.
mechanical theory of, 3.
af liquid, 171, 174.
sf vaporization, 171, 175.
specific, 24.
total, 172, 177, 213.
units of, 9.
jilting of air by internal combustion,
141.
sating value of fuels, 118.
inning's formula for latent heat, 176.
>lborn and Hcnning's experiments,
205.
>tair engines, 138.
imidity, 229.
equality of Clausius, 63.
(xsrnul combustion, heating by, 141.
trinsic energy, 30.
of gases, 97.
of superheated steam, 214.
of vapors, 172.
evorsiblo adiabatics, 75.
processes, 47.
processes, waste in, 57.
Kidiabatic cycles, 130.
nlynamic change of vapor, 190.
processes, 42.
>rnetric lines, 22.
>piestio lines, 22.
>thermal, definition of, 21.
sxpansion of gases, 102.
of superheated steam, 217.
of vapor mixture, 188.
311 7Y>plane, 70.
sf steam and air mixture, 232. '
i, work of, 298.
ulc's cycle, 145.
experiments, 11.
law, 90.
tileThomson coefficient, 276.
effect, 275.
ilvin's absolute scale, 55.
statement of the second law, 50.
loblauch's experiments, 201.
loblauch and Jakob's experiments, 205.
ngcn's equations for specific heat, 124,
205.
tent heat, 26.
external, 172.
Hemline's formula for, 176.
Latent heat, internal, 172.
of expansion, 27.
of pressure variation, 27.
of vaporization, 171, 175.
Lenoir cycle, 162.
Linde's process for liquefaction, 280.
Liquefaction of gases, 280.
Liquid curve, 166.
Mallard and Le Chatelier's experiments,
205.
Marks' formula, 170.
Maxwell's thermodynamic relations, SO
Mean specific heat, 210.
Mechanical energy, units of, 9.
Mechanical equivalent of heat, 11.
theory of heat, 3.
Mixture of gases and vapors, 228.
of gases, specific heat of, 125.
of steam and air, 232, 236.
Moist air, constants for, 230.
Moisture in atmosphere, 228.
Molecular specific heat, 123.
weights, 111.
Mollier's chart, 223.
use in flow of fluids, 251.
use in steam turbines, 302.
Munich experiments, 201.
Napier's equations, flow of steam, 257.
Nozzle, De Laval, 258.
Nozzles, design of, 264.
friction in, 262.
Otto cycle, 142, 148.
Peake's throttling curves, 273.
Perfect gas, definition of, 18.
equation of, 17.
Permanent gas, explanation of term,
89.
Perpetual motion of first class, 6.
of second class, 8.
Polytropic change of state, 104.
changes, specific heat in, 106.
curve, 71.
Potential efficiency, 292.
thermodynamic, 77, 87.
Pressure and temperature, relation be
tween, 167.
Pressure compounding, 296.
critical, 199.
turbines, action of, 298, 305.
Products of combustion, 119.
Quality of mixture, 165.
variation of, 185.
's cycle, 284.
effect of changing pressure,
289.
incomplete expansion, 288.
with superheated steam,
286.
a, 168.
i formula, flow of steam, 286.
alar cycle, 73.
iting machine, analysis of, 311.
ition, air, 149.
i used in, 310.
apor media, 308.
I heat engine, 74.
le processes, 47.
3 and Moorby's experiments, 12.
.'a experiments, 11.
ingine, 294.
:nant's hypothesis, 254.
d vapor, 165.
energy of, 172.
entropy of, 179.
heat content of, 173, 177.
latent heat of, 171, 175.
specific heat of, 182.
surface representing, 166.
total heat of, 172, 177.
>n curve, 166, 182.
nature, 165.
iw of thermodynamics, 50.
tiann's interpretation of, 65.
ticat, 24.
curves, 209, 211.
in polytropic changes, 106.
Langen's formulas for, 124.
mean, 210.
heat, molecular, 123.
of gaseous mixture, 125.
of gaseous products, 123.
of gases, 96.
of saturated vapor, 182.
of superheated steam, 204, 273.
volume of vapors, 177.
id air, mixture of, 232, 236.
I temperature of, 199.
3 volume of, 177.
180.
il properties of, 173.
teat of, 172, 177.
irbine, 294.
classification of, 295.
compared with reciprocating
engine 294,
compounding, 296.
Curtis type, 304.
impulse and reaction, 296.
influence of high vacuum, 307.
low pressure, 307.
Steam turbine multiple stage, 302.
pressure type, 298, 305.
single stage, 300.
velocity and pressure, 296,
Stirling's engine, 138.
Sulphur dioxide, saturated, 182.
superheated, 223.
Superheat, degree of, 165, 196.
Superheated ammonia, 223.
Superheated steam, 165, 196.
changes of state, 216.
energy of, 214.
entropy of, 215.
equations for, 203.
heat content of, 210.
specific heat of, 204, 273.
tables and diagrams, 221.
total heat of, 213.
Superheated sulphur dioxide, 223.
vapor, characteristics of, 196.
Surface, characteristic, 20.
representing saturated vapor, 166
System, defined, 15.
state of, 15.
Temperature, absolute, 18.
and pressure, relation between, 167.
critical, 199.
Kelvin scale of, 55.
Temperature of combustion, 127.
saturation, 165.
scales, comparison of, 91.
Temperature entropy representation, 68.
Thermal capacities, relation between, 27.
capacity defined, 24.
efficiency, 291.
energy, 4.
lines, 21.
properties of steam, 173.
Thcrmodynamic degeneration, 8.
potentials, 77, 87.
relations, 80.
Thermodynamics, first law of, 35.
general equations of, 84.
scope of, 1.
second law of, 50.
Throttling calorimeter, 271.
curves, 273.
loss due to, 269.
processes, 268.
Total heat of saturated vapor, 172, 177.
of superheated steam, 213.
Transformations of energy, 5.
Tumlirtz equation for superheated steam,
204.
Turbine, steam, see Steam turbine.
Units of energy, 8.
of heat, 9.
Universal gas constant, 113.
uum, influence of, on steam turbine,
307.
i der Waals' equation, 20, 200.
ior, energy of, 172.
itropy of, 179.
[tit content of, 173, 177.
itent heat of, 171, 175.
ior mixture, acliabatic expansion of,
189.
instant volume change, 189.
jrvos on TiSplanc, 186.
cmnil equation of, 184.
lodynamic of, 190.
lothurniiil expansion of, 188.
>or refrigerating machine, 311.
jporhotitcd, 196.
:>tal heat of, 172, 177.
)orization, heat of, 171, 175.
Vaporization, process of, 164.
Vapors used in refrigeration, 310.
Velocity compounding, 297.
Volume, critical, 199.
specific, of vapor, 177.
Waste in irreversible processes, 57.
Water, critical temperature of, 199.
jacketing, 155.
vapor, thermal properties of, 173.
Wiredrawing, 268.
Work, conversion of, into heat, 57.
external, of expansion, 37.
of a jet, 298.
Zero curve, 186.
Zeuner's equation for superheated steam,
204.