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© 2005 by CRC Press LL
 
2
 
Engineering
 
Although various
formal study beg
 
the capacity of h
 
entropy
 
,
 
 
 
and with
 
thermodynamics
the manner that i
tool for engineer
 
Michael J. Mo
 
The Ohio State Un
 
0866_book.fm Page 1 Friday, August 6, 2004 2:25 PM
© 2005 by CRC Pre
2-11.50C
Thermodynamics
2.1 Fundamentals
Basic Concepts and Definitions • The First Law of 
Thermodynamics, Energy • The Second Law of 
Thermodynamics, Entropy • Entropy and Entropy Generation
2.2 Control Volume Applications
Conservation of Mass • Control Volume Energy Balance • 
Control Volume Entropy Balance • Control Volumes at Steady 
State
2.3 Property Relations and Data
Basic Relations for Pure Substances • P-v-T Relations • 
Evaluating ∆h, ∆u, and ∆s • Fundamental Thermodynamic 
Functions • Thermodynamic Data Retrieval • Ideal Gas Model • 
Generalized Charts for Enthalpy, Entropy, and Fugacity • 
Multicomponent Systems
2.4 Combustion
Reaction Equations • Property Data for Reactive Systems • 
Reaction Equilibrium
2.5 Exergy Analysis
Defining Exergy • Control Volume Exergy Rate Balance • 
Exergetic Efficiency • Exergy Costing
2.6 Vapor and Gas Power Cycles
Rankine and Brayton Cycles • Otto, Diesel, and Dual Cycles • 
Carnot, Ericsson, and Stirling Cycles
2.7 Guidelines for Improving Thermodynamic 
Effectiveness
 aspects of what is now known as thermodynamics have been of interest since antiquity,
an only in the early 19th century through consideration of the motive power of heat:
ot bodies to produce work. Today the scope is larger, dealing generally with energy and
 relationships among the properties of matter. Moreover, in the past 25 years engineering
 has undergone a revolution, both in terms of the presentation of fundamentals and in
t is applied. In particular, the second law of thermodynamics has emerged as an effective
ing analysis and design.
ran
iversity
ss LLC
 
2
 
-2
 
Chapter 2
 
2.1 Fundamentals
 
Classical thermo
gross characteris
The microstructu
thermodynamics
 
Basic Conce
 
Thermodynamic
interested in gain
escent quantities
 
Engineers are gen
 
facilitate this, en
which matter flo
 
System
 
In a thermodyna
 
specified quantit
surface. The defin
 
be movable or fi
 
referred to as a 
 
co
 
the system is call
 
interact in any w
 
State, Property
 
The condition of
 
described by the 
 
the state but not
type of physical 
 
Extensive 
 
prop
are examples of e
whole system equ
 
or extent of the s
 
A mole is 
 
a qua
the molecular we
 
kilogram mole, d
extensive proper
 
overbar is used to
per unit mass. Fo
 
two specific volu
 
Process, Cycle
 
Two states are id
of a system chan
 
a system in a give
it is said to have 
 
Phase and Pur
 
The term 
 
phase 
 
r
position and phy
 
all 
 
liquid
 
,
 
 
 
or all 
 
v
 
0866_book.fm Page 2 Friday, August 6, 2004 2:25 PM
© 2005 by CRC Pre
dynamics is concerned primarily with the macrostructure of matter. It addresses the
tics of large aggregations of molecules and not the behavior of individual molecules.
re of matter is studied in kinetic theory and statistical mechanics (including quantum
). In this chapter, the classical approach to thermodynamics is featured.
pts and Definitions
s is both a branch of physics and an engineering science. The scientist is normally
ing a fundamental understanding of the physical and chemical behavior of fixed, qui-
 of matter and uses the principles of thermodynamics to relate the properties of matter.
erally interested in studying systems and how they interact with their surroundings. To
gineers have extended the subject of thermodynamics to the study of systems through
ws.
mic analysis, the system is the subject of the investigation. Normally the system is a
y of matter and/or a region that can be separated from everything else by a well-defined
ing surface is known as the control surface or system boundary. The control surface may
xed. Everything external to the system is the surroundings. A system of fixed mass is
ntrol mass or as a closed system. When there is flow of mass through the control surface,
ed a control volume, or open, system. An isolated system is a closed system that does not
ay with its surroundings.
 a system at any instant of time is called its state. The state at a given instant of time is
properties of the system. A property is any quantity whose numerical value depends on
 the history of the system. The value of a property is determined in principle by some
operation or test.
erties depend on the size or extent of the system. Volume, mass, energy, and entropy
xtensive properties. An extensive property is additive in the sense that its value for the
als the sum of the values for its parts. Intensive properties are independent of the size
ystem. Pressure and temperature are examples of intensive properties.
ntity of substance having a mass numerically equal to its molecular weight. Designating
ight by M and the number of moles by n, the mass m of the substance is m = nM. One
esignated kmol, of oxygen is 32.0 kg and one pound mole (lbmol) is 32.0 lb. When an
ty is reported on a unit mass or a unit mole basis, it is called a specific property. An
 distinguish an extensive property written on a per-mole basis from its value expressed
r example, the volume per mole is , whereas the volume per unit mass is v, and the
mes are related by = Mv.
entical if, and only if, the properties of the two states are identical. When any property
ges in value there is a change in state, and the system is said to undergo a process. When
n initial state goes through a sequence of processes and finally returns to its initial state,
undergone a cycle.
e Substance
efers to a quantity of matter that is homogeneous throughout in both chemical com-
sical structure. Homogeneity in physical structure means that the matter is all solid, or
apor (or equivalently all gas). A system can contain one or more phases. For example,
v
v
ss LLC
 
Engineering Thermodynamics
 
2
 
-3
 
a system of liquid water and water vapor (steam) contains 
 
two 
 
phases. A 
 
pure substance 
 
is
 
 
 
one that is
uniform and invariable in chemical composition. 
 
A 
 
pure substance can exist in more than one phase,
but its chemical 
vapor form a syst
has the same com
 
rule 
 
(Section 2.3,
 
Equilibrium
 
Equilibrium mea
of forces, but als
thermodynamic 
 
mechanical 
 
equil
 
potentials (Sectio
 
chemical potenti
equilibrium mus
To determine i
the system from i
it may be conclud
said to be at an 
 
however, its state
properties, such a
the system is in e
is significant, a p
 
Temperature
 
A scale of temper
 
scale. The Kelvin
(Section 2.1, The
from the second 
the several 
 
empir
 
volume gas therm
 
gas thermometry
The empirical
 
all gases exhibit t
 
basis) if the press
level. On this bas
where 
 
T
 
 is tempe
 
of water 
 
(Section
 
temperature scale
 
The 
 
Celsius ter
 
the same magnit
 
the zero point on
the Celsius temp
On the Celsius sc
 
0866_book.fm Page 3 Friday, August 6, 2004 2:25 PM
© 2005 by CRC Pre
composition must be the same in each phase. For example, if liquid water and water
em with two phases, the system can be regardedas a pure substance because each phase
position. The nature of phases that coexist in equilibrium is addressed by the phase
 Multicomponent Systems).
ns a condition of balance. In thermodynamics the concept includes not only a balance
o a balance of other influences. Each kind of influence refers to a particular aspect of
(complete) equilibrium. Thermal equilibrium refers to an equality of temperature,
ibrium to an equality of pressure, and phase equilibrium to an equality of chemical
n 2.3, Multicomponent Systems). Chemical equilibrium is also established in terms of
als (Section 2.4, Reaction Equilibrium). For complete equilibrium the several types of
t exist individually.
f a system is in thermodynamic equilibrium, one may think of testing it as follows: isolate
ts surroundings and watch for changes in its observable properties. If there are no changes,
ed that the system was in equilibrium at the moment it was isolated. The system can be
equilibrium state. When a system is isolated, it cannot interact with its surroundings;
 can change as a consequence of spontaneous events occurring internally as its intensive
s temperature and pressure, tend toward uniform values. When all such changes cease,
quilibrium. At equilibrium. temperature and pressure are uniform throughout. If gravity
ressure variation with height can exist, as in a vertical column of liquid.
ature independent of the thermometric substance is called a thermodynamic temperature
 scale, a thermodynamic scale, can be elicited from the second law of thermodynamics
 Second Law of Thermodynamics, Entropy). The definition of temperature following
law is valid over all temperature ranges and provides an essential connection between
ical measures of temperature. In particular, temperatures evaluated using a constant-
ometer are identical to those of the Kelvin scale over the range of temperatures where
 can be used.
 gas scale is based on the experimental observations that (1) at a given temperature level
he same value of the product (p is pressure and the specific volume on a molar
ure is low enough, and (2) the value of the product increases with the temperature
is the gas temperature scale is defined by
rature and is the universal gas constant. The absolute temperature at the triple point
 2.3, P-v-T Relations) is fixed by international agreement to be 273.16 K on the Kelvin
. is then evaluated experimentally as = 8.314 kJ/kmol · K (1545 ft · lbf/lbmol · °R).
mperature scale (also called the centigrade scale) uses the degree Celsius (°C), which has
ude as the kelvin. Thus, temperature differences are identical on both scales. However,
 the Celsius scale is shifted to 273.15 K, as shown by the following relationship between
erature and the Kelvin temperature:
(2.1)
ale, the triple point of water is 0.01°C and 0 K corresponds to –273.15°C.
pv v
pv
T
R
pv
p
= ( )
→
1
0
lim
R
R R
T T°( ) = ( ) −C K 273 15.
ss LLC
 
2
 
-4
 
Chapter 2
 
Two other temperature scales are commonly used in engineering in the U.S. By definition, the 
 
Rankine
scale, 
 
the unit of which is the degree rankine (°R), is proportional to the Kelvin temperature according to
The Rankine scal
absolute zero of 
Kelvin or Rankin
A degree of th
 
point is shifted a
Substituting Equ
This equation sh
 
(100°C) is 212°F.
to 180 Fahrenhei
To provide a st
considerations, t
temperature mea
kelvin, to within 
is provided by Pr
 
The First La
 
Energy is a fund
neering analysis.
 
gravitational pot
 
another and 
 
tran
 
transfer. 
 
The tota
 
Work
 
In
 
 
 
thermodynam
 
system on anothe
if the 
 
sole effect 
 
o
 
whether a work i
that a force actua
 
a mass. The magn
raised. Since the 
mechanics is pre
displacement of 
Work done by
W < 0. The tim
convention.
Energy
A closed system 
riences an adiab
closed system is a
0866_book.fm Page 4 Friday, August 6, 2004 2:25 PM
© 2005 by CRC Pre
(2.2)
e is also an absolute thermodynamic scale with an absolute zero that coincides with the
the Kelvin scale. In thermodynamic relationships, temperature is always in terms of the
e scale unless specifically stated otherwise.
e same size as that on the Rankine scale is used in the Fahrenheit scale, but the zero
ccording to the relation
(2.3)
ation 2.1 and Equation 2.2 into Equation 2.3 gives
(2.4)
ows that the Fahrenheit temperature of the ice point (0°C) is 32°F and of the steam point
 The 100 Celsius or Kelvin degrees between the ice point and steam point corresponds
t or Rankine degrees.
andard for temperature measurement taking into account both theoretical and practical
he International Temperature Scale of 1990 (ITS-90) is defined in such a way that the
sured on it conforms with the thermodynamic temperature, the unit of which is the
the limits of accuracy of measurement obtainable in 1990. Further discussion of ITS-90
eston-Thomas (1990).
w of Thermodynamics, Energy
amental concept of thermodynamics and one of the most significant aspects of engi-
 Energy can be stored within systems in various macroscopic forms: kinetic energy,
ential energy, and internal energy. Energy can also be transformed from one form to
sferred between systems. For closed systems, energy can be transferred by work and heat
l amount of energy is conserved in all transformations and transfers.
ics, the term work denotes a means for transferring energy. Work is an effect of one
r that is identified and measured as follows: work is done by a system on its surroundings
n everything external to the system could have been the raising of a weight. The test of
nteraction has taken place is not that the elevation of a weight is actually changed, nor
lly acted through a distance, but that the sole effect could be the change in elevation of
itude of the work is measured by the number of standard weights that could have been
raising of a weight is in effect a force acting through a distance, the work concept of
served. This definition includes work effects such as is associated with rotating shafts,
the boundary, and the flow of electricity.
 a system is considered positive: W > 0. Work done on a system is considered negative:
e rate of doing work, or power, is symbolized by and adheres to the same sign
undergoing a process that involves only work interactions with its surroundings expe-
atic process. On the basis of experimental evidence, it can be postulated that when a
ltered adiabatically, the amount of work is fixed by the end states of the system and is
T T°( ) = ( )R K1 8.
T T°( ) = °( ) −F R 459 67.
T T°( ) = °( ) +F C1 8 32.
˙W
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Engineering Thermodynamics 2-5
independent of the details of the process. This postulate, which is one way the first law of thermodynamics
can be stated, can be made regardless of the type of work interaction involved, the type of process, or
the nature of the
As the work in
called energy can
adiabatic process
energy of a system
energy, KE, assoc
the change in gra
the Earth’s gravit
associated with t
extensive propert
In summary, t
an adiabatic proc
where 1 and 2 de
is in accordance 
assigned to the e
value of the energ
The specific e
kinetic energy, v2
where the velocit
and g is the accel
A property rel
or on an extensiv
Heat
Closed systems c
as, for example, a
a flame. This typ
A fundamenta
experiences preci
between the sam
these processes m
the amount of en
be summarized b
0866_book.fm Page 5 Friday, August 6, 2004 2:25 PM
© 2005 by CRC Pre
 system.
 an adiabatic process of a closed system is fixed by the end states, an extensive property
 be defined for the system such that its change between two states is the work in an
 that hasthese as the end states. In engineering thermodynamics the change in the
 is considered to be made up of three macroscopic contributions: the change in kinetic
iated with the motion of the system as a whole relative to an external coordinate frame,
vitational potential energy, PE, associated with the position of the system as a whole in
ational field, and the change in internal energy, U, which accounts for all other energy
he system. Like kinetic energy and gravitational potential energy, internal energy is an
y.
he change in energy between two states of a closed system in terms of the work Wad of
ess between these states is
(2.5)
note the initial and final states, respectively, and the minus sign before the work term
with the previously stated sign convention for work. Since any arbitrary value can be
nergy of a system at a given state 1, no particular significance can be attached to the
y at state 1 or at any other state. Only changes in the energy of a system have significance.
nergy (energy per unit mass) is the sum of the specific internal energy, u, the specific
/2, and the specific gravitational potential energy, gz, such that
(2.6)
y v and the elevation z are each relative to specified datums (often the Earth’s surface)
eration of gravity.
ated to internal energy u, pressure p, and specific volume v is enthalpy, defined by
(2.7a)
e basis
(2.7b)
an also interact with their surroundings in a way that cannot be categorized as work,
 gas (or liquid) contained in a closed vessel undergoing a process while in contact with
e of interaction is called a heat interaction, and the process is referred to as nonadiabatic.
l aspect of the energy concept is that energy is conserved. Thus, since a closed system
sely the same energy change during a nonadiabatic process as during an adiabatic process
e end states, it can be concluded that the net energy transfer to the system in each of
ust be the same. It follows that heat interactions also involve energy transfer. Denoting
ergy transferred to a closed system in heat interactions by Q, these considerations can
y the closed system energy balance:
(2.8)
KE KE PE PE U U W
ad2 1 2 1 2 1−( ) + −( ) + −( ) = −
specific energy v gz= + +u
2
2
h u pv= +
H U pV= +
U U KE KE PE PE Q W2 1 2 1 2 1−( ) + −( ) + −( ) = −
ss LLC
2-6 Chapter 2
The closed system energy balance expresses the conservation of energy principle for closed systems of
all kinds.
The quantity d
system during a 
an energy transf
surroundings and
is called an energ
The time rate of 
Methods base
recognize two ba
empirical relatio
convection. Furth
The quantities
different means 
properties, and i
economy of expr
transfer, respectiv
Power Cycles
Since energy is a 
any cycle
That is, for any 
energy transferre
of energy is trans
Power cycles a
of energy by hea
Combining the l
The thermal e
energy added by
The thermal effi
invariably rejecte
0866_book.fm Page 6 Friday, August 6, 2004 2:25 PM
© 2005 by CRC Pre
enoted by Q in Equation 2.8 accounts for the amount of energy transferred to a closed
process by means other than work. On the basis of experiments it is known that such
er is induced only as a result of a temperature difference between the system and its
 occurs only in the direction of decreasing temperature. This means of energy transfer
y transfer by heat. The following sign convention applies:
heat transfer, denoted by , adheres to the same sign convention.
d on experiment are available for evaluating energy transfer by heat. These methods
sic transfer mechanisms: conduction and thermal radiation. In addition, theoretical and
nships are available for evaluating energy transfer involving combined modes such as
er discussion of heat transfer fundamentals is provided in Chapter 4.
 symbolized by W and Q account for transfers of energy. The terms work and heat denote
whereby energy is transferred and not what is transferred. Work and heat are not
t is improper to speak of work or heat “contained” in a system. However, to achieve
ession in subsequent discussions, W and Q are often referred to simply as work and heat
ely. This less formal approach is commonly used in engineering practice.
property, over each cycle there is no net change in energy. Thus, Equation 2.8 reads for
cycle the net amount of energy received through heat interactions is equal to the net
d out in work interactions. A power cycle, or heat engine, is one for which a net amount
ferred out by work: Wcycle > 0. This equals the net amount of energy transferred in by heat.
re characterized both by addition of energy by heat transfer, QA, and inevitable rejections
t transfer, QR:
ast two equations,
fficiency of a heat engine is defined as the ratio of the net work developed to the total
 heat transfer:
(2.9)
ciency is strictly less than 100%. That is, some portion of the energy QA supplied is
d QR ≠ 0.
Q to
Q from
>
<
0
0
:
:
 heat transfer the system
 heat transfer the system
˙Q
Q W
cycle cycle=
Q Q Q
cycle A R= −
W Q Q
cycle A R= −
η = = −
W
Q
Q
Q
cycle
A
R
A
1
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Engineering Thermodynamics 2-7
The Second Law of Thermodynamics, Entropy
Many statements
a statement of th
every instance wh
it has been verifie
evidence.
Kelvin-Planck 
The Kelvin-Planc
reservoir is a syst
heat transfer. A re
of ways — by the
of thermal reserv
the reservoir tem
The Kelvin-Pla
to operate in a th
receiving energy b
of the second kind
where the words
reservoir as it exe
the system of int
present.
Irreversibilitie
A process is said
some way by whi
states. A process i
and its surround
an irreversible pr
the system restor
initial state.
There are man
not limited to, the
of a gas or liqui
compositions or 
flow through a re
term irreversibilit
Irreversibilitie
that occur within
normally the imm
is some arbitrarin
all irreversibilitie
irreversibilities is
be internally reve
all intensive prop
volume, and othe
actual and intern
0866_book.fm Page 7 Friday, August 6, 2004 2:25 PM
© 2005 by CRC Pre
 of the second law of thermodynamics have been proposed. Each of these can be called
e second law or a corollary of the second law since, if one is invalid, all are invalid. In
ere a consequence of the second law has been tested directly or indirectly by experiment
d. Accordingly, the basis of the second law, like every other physical law, is experimental
Statement
k statement of the second law of thermodynamics refers to a thermal reservoir. A thermal
em that remains at a constant temperature even though energy is added or removed by
servoir is an idealization, of course, but such a system can be approximated in a number
 Earth’s atmosphere, large bodies of water (lakes, oceans), and so on. Extensive properties
oirs, such as internal energy, can change in interactions with other systems even though
perature remains constant, however.
nck statement of the second law can be given as follows: It is impossible for any system
ermodynamic cycle and deliver a net amount of energy by work to its surroundings while
y heat transfer from a single thermal reservoir. In other words, a perpetual-motion machine
 is impossible. Expressed analytically, the Kelvin-Planck statement is
 single reservoir emphasize that the system communicates thermally only with a single
cutes the cycle. The “less than” sign applies when internal irreversibilities are present as
erest undergoes a cycle and the “equal to” sign applies only when no irreversibilities are
s
 to be reversible if it is possible for its effects to be eradicated in the sense that there is
ch both the system and its surroundings can be exactly restored to their respective initial
s irreversible if there is no way to undo it. That is, there is no means bywhich the system
ings can be exactly restored to their respective initial states. A system that has undergone
ocess is not necessarily precluded from being restored to its initial state. However, were
ed to its initial state, it would not also be possible to return the surroundings to their
y effects whose presence during a process renders it irreversible. These include, but are
 following: heat transfer through a finite temperature difference; unrestrained expansion
d to a lower pressure; spontaneous chemical reaction; mixing of matter at different
states; friction (sliding friction as well as friction in the flow of fluids); electric current
sistance; magnetization or polarization with hysteresis; and inelastic deformation. The
y is used to identify effects such as these.
s can be divided into two classes, internal and external. Internal irreversibilities are those
 the system, while external irreversibilities are those that occur within the surroundings,
ediate surroundings. As this division depends on the location of the boundary there
ess in the classification (by locating the boundary to take in the immediate surroundings,
s are internal). Nonetheless, valuable insights can result when this distinction between
 made. When internal irreversibilities are absent during a process, the process is said to
rsible. At every intermediate state of an internally reversible process of a closed system,
erties are uniform throughout each phase present: the temperature, pressure, specific
r intensive properties do not vary with position. The discussions to follow compare the
ally reversible process concepts for two cases of special interest.
W
cycle ≤ ( )0 single reservoir
ss LLC
2-8 Chapter 2
For a gas as the system, the work of expansion arises from the force exerted by the system to move
the boundary against the resistance offered by the surroundings:
where the force i
that Adx is the ch
This expression f
for an internally r
of the entire syst
specific volume v
an internally reve
When such a pro
specific volume, 
a-b-c′-d′ of Figu
Although imp
steps in this dire
For example, con
between them, a
source of irrevers
difference narrow
approaches ideal
amount of energ
amount of time, 
would require an
implications con
Carnot Corolla
The two corollar
irreversible powe
operates between
same two therma
are no irreversibi
and reservoirs oc
Kelvin Temper
Carnot corollary
thermal reservoi
substance makin
be concluded tha
of the substance 
0866_book.fm Page 8 Friday, August 6, 2004 2:25 PM
© 2005 by CRC Pre
s the product of the moving area and the pressure exerted by the system there. Noting
ange in total volume of the system,
or work applies to both actual and internally reversible expansion processes. However,
eversible process p is not only the pressure at the moving boundary but also the pressure
em. Furthermore, for an internally reversible process the volume equals mv, where the
 has a single value throughout the system at a given instant. Accordingly, the work of
rsible expansion (or compression) process is
(2.10)
cess of a closed system is represented by a continuous curve on a plot of pressure vs.
the area under the curve is the magnitude of the work per unit of system mass (area
re 2.3, for example).
roved thermodynamic performance can accompany the reduction of irreversibilities,
ction are normally constrained by a number of practical factors often related to costs.
sider two bodies able to communicate thermally. With a finite temperature difference
 spontaneous heat transfer would take place and, as noted previously, this would be a
ibility. The importance of the heat transfer irreversibility diminishes as the temperature
s; and as the temperature difference between the bodies vanishes, the heat transfer
ity. From the study of heat transfer it is known, however, that the transfer of a finite
y by heat between bodies whose temperatures differ only slightly requires a considerable
a large heat transfer surface area, or both. To approach ideality, therefore, a heat transfer
 exceptionally long time and/or an exceptionally large area, each of which has cost
straining what can be achieved practically.
ries
ies of the second law known as Carnot corollaries state: (1) the thermal efficiency of an
r cycle is always less than the thermal efficiency of a reversible power cycle when each
 the same two thermal reservoirs; (2) all reversible power cycles operating between the
l reservoirs have the same thermal efficiency. A cycle is considered reversible when there
lities within the system as it undergoes the cycle, and heat transfers between the system
cur ideally (that is, with a vanishingly small temperature difference).
ature Scale
 2 suggests that the thermal efficiency of a reversible power cycle operating between two
rs depends only on the temperatures of the reservoirs and not on the nature of the
g up the system executing the cycle or the series of processes. With Equation 2.9 it can
t the ratio of the heat transfers is also related only to the temperatures, and is independent
and processes:
W Fdx pAdx= =∫ ∫12 12
W pdV= ∫12
W m pdv= ∫12
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Engineering Thermodynamics 2-9
where QH is the e
and QC is the ene
emphasize that t
between the two
the function ψ in
The Kelvin tem
This equation de
assigning a nume
the gas scale: at t
is operated betwe
temperature T, th
where Q is the en
rejected to the re
valid over all ran
Carnot Efficien
For the special ca
and TC on the Ke
called the Carnot
reservoirs at TH a
ideal, could have
differ from Kelvi
scale of temperat
The Clausius I
The Clausius ine
uations of proce
Clausius inequal
where δQ repres
and T is the abso
Q
Q T T
C
H rev
cycle
C H



 = ( )ψ ,
0866_book.fm Page 9 Friday, August 6, 2004 2:25 PM
© 2005 by CRC Pre
nergy transferred to the system by heat transfer from a hot reservoir at temperature TH,
rgy rejected from the system to a cold reservoir at temperature TC. The words rev cycle
his expression applies only to systems undergoing reversible cycles while operating
 reservoirs. Alternative temperature scales correspond to alternative specifications for
 this relation.
perature scale is based on ψ(TC, TH) = TC /TH. Then
(2.11)
fines only a ratio of temperatures. The specification of the Kelvin scale is completed by
rical value to one standard reference state. The state selected is the same used to define
he triple point of water the temperature is specified to be 273.16 K. If a reversible cycle
en a reservoir at the reference-state temperature and another reservoir at an unknown
en the latter temperature is related to the value at the reference state by
ergy received by heat transfer from the reservoir at temperature T, and Q′ is the energy
servoir at the reference temperature. Accordingly, a temperature scale is defined that is
ges of temperature and that is independent of the thermometric substance.
cy
se of a reversible power cycle operating between thermal reservoirs at temperatures TH
lvin scale, combination of Equation 2.9 and Equation 2.11 results in
(2.12)
 efficiency. This is the efficiency of all reversible power cycles operating between thermal
nd TC. Moreover, it is the maximum theoretical efficiency that any power cycle, real or
 while operating between the same two reservoirs. As temperatures on the Rankine scale
n temperatures only by the factor 1.8, the above equation may be applied with either
ure.
nequality
quality provides the basis for introducing two ideas instrumental for quantitative eval-
sses of systems from a second law perspective: entropy and entropy generation. The
ity states that
(2.13a)
ents the heat transfer at a part of the system boundary during a portion of the cycle,
lute temperature at that part of theboundary. The symbol δ is used to distinguish the
Q
Q
T
T
C
H rev
cycle
C
H



 =
T QQ rev
cycle
=
′



273 16.
η
max
= −1
T
T
C
H
δQ
T b
  ≤∫ 0
ss LLC
2-10 Chapter 2
differentials of nonproperties, such as heat and work, from the differentials of properties, written with
the symbol d. The subscript b indicates that the integrand is evaluated at the boundary of the system
executing the cy
boundary and ov
statement of the 
there are no inte
internal irreversi
The Clausius i
where Sgen can be
internal irreversi
negative. Accord
cycle. In the next 
during the cycle.
Entropy and 
Entropy
Consider two cyc
from state 1 to s
cycle consists of 
C from state 2 to
where Sgen has bee
these equations l
Since A and B
reversible proces
be concluded, th
the symbol S to d
where the subscr
linking the two s
Since entropy 
the same for all p
words, once the 
entropy change f
0866_book.fm Page 10 Friday, August 6, 2004 2:25 PM
© 2005 by CRC Pre
cle. The symbol indicates that the integral is to be performed over all parts of the
er the entire cycle. The Clausius inequality can be demonstrated using the Kelvin-Planck
second law, and the significance of the inequality is the same: the equality applies when
rnal irreversibilities as the system executes the cycle, and the inequality applies when
bilities are present.
nequality can be expressed alternatively as
(2.13b)
 viewed as representing the strength of the inequality. The value of Sgen is positive when
bilities are present, zero when no internal irreversibilities are present, and can never be
ingly, Sgen is a measure of the irreversibilities present within the system executing the
section, Sgen is identified as the entropy generated (or produced) by internal irreversibilities
Entropy Generation
les executed by a closed system. One cycle consists of an internally reversible process A
tate 2, followed by an internally reversible process C from state 2 to state 1. The other
an internally reversible process B from state 1 to state 2, followed by the same process
 state 1 as in the first cycle. For these cycles, Equation 2.13b takes the form
n set to zero since the cycles are composed of internally reversible processes. Subtracting
eaves
 are arbitrary, it follows that the integral of δQ/T has the same value for any internally
s between the two states: the value of the integral depends on the end states only. It can
erefore, that the integral defines the change in some property of the system. Selecting
enote this property, its change is given by
(2.14a)
ipt int rev indicates that the integration is carried out for any internally reversible process
tates. This extensive property is called entropy.
is a property, the change in entropy of a system in going from one state to another is
rocesses, both internally reversible and irreversible, between these two states. In other
change in entropy between two states has been evaluated, this is the magnitude of the
or any process of the system between these end states.
∫
δQ
T
S
b
gen
  = −∫
δ δ
δ δ
Q
T
Q
T
S
Q
T
Q
T
S
A C
gen
B C
gen
1
2
2
1
1
2
2
1
0
0
∫ ∫
∫ ∫



 +



 = − =



 +



 = − =
δ δQ
T
Q
TA B1
2
1
2∫ ∫  =  
S S Q
T
rev
2 1
1
2
− =



∫ δ int
ss LLC
Engineering Thermodynamics 2-11
The definition of entropy change expressed on a differential basis is
Equation 2.14b i
energy by heat tr
from the system 
that an entropy t
transfer is the sa
system the entro
On rearrangem
Then, for an inte
When such a pro
the area under th
Entropy Balanc
For a cycle consi
are present, follo
form
where the first in
process. Since n
accounting for th
Applying the 
expressed as
Introducing this 
0866_book.fm Page 11 Friday, August 6, 2004 2:25 PM
© 2005 by CRC Pre
(2.14b)
ndicates that when a closed system undergoing an internally reversible process receives
ansfer, the system experiences an increase in entropy. Conversely, when energy is removed
by heat transfer, the entropy of the system decreases. This can be interpreted to mean
ransfer is associated with (or accompanies) heat transfer. The direction of the entropy
me as that of the heat transfer. In an adiabatic internally reversible process of a closed
py would remain constant. A constant entropy process is called an isentropic process.
ent, Equation 2.14b becomes
rnally reversible process of a closed system between state 1 and state 2,
(2.15)
cess is represented by a continuous curve on a plot of temperature vs. specific entropy,
e curve is the magnitude of the heat transfer per unit of system mass.
e
sting of an actual process from state 1 to state 2, during which internal irreversibilities
wed by an internally reversible process from state 2 to state 1, Equation 2.13b takes the
tegral is for the actual process and the second integral is for the internally reversible
o irreversibilities are associated with the internally reversible process, the term Sgen
e effect of irreversibilities during the cycle can be identified with the actual process only.
definition of entropy change, the second integral of the foregoing equation can be
and rearranging the equation, the closed system entropy balance results:
(2.16)
dS Q
T
rev
=
 
δ
int
δQ TdS
rev
( ) =int
Q m Tds
rev
int = ∫12
δ δQ
T
Q
T
S
b rev
gen
  +   = −∫ ∫12 21 int
S S Q
T
rev
1 2
2
1
− =
 ∫ δ int
S S Q
T
S
b
gen2 1
1
2
− =
  +∫ δ
______ ______ ______
entropy
change
entropy
transfer
entropy
generation
ss LLC
2-12 Chapter 2
When the end states are fixed, the entropy change on the left side of Equation 2.16 can be evaluated
independently of the details of the process from state 1 to state 2. However, the two terms on the right
side depend expl
the end states. T
during the proce
heat transfer. The
same sign conven
the system, and a
The entropy c
second term on 
irreversibilities ar
can be described
irreversibilities. T
generated by irre
Sgen measures the
the nature of the
When applyin
However, the valu
significance by it
entropy generati
the other compon
generation value
ordered. This allo
operation of the 
To evaluate th
heat transfer and
term is not alwa
unknown or und
In practical appli
immediate surrou
ambient tempera
present would no
generation term 
irreversibilities pr
A form of the
where dS/dt is th
of entropy trans
term accoun
For a system i
where Sgen is the t
in all actual proce
the entropy of th
˙Sgen
0866_book.fm Page 12 Friday, August 6, 2004 2:25 PM
© 2005 by CRC Pre
icitly on the nature of the process and cannot be determined solely from knowledge of
he first term on the right side is associated with heat transfer to or from the system
ss. This term can be interpreted as the entropy transfer associated with (or accompanying)
 direction of entropy transfer is the same as the direction of the heat transfer, and the
tion applies as for heat transfer: a positive value means that entropy is transferred into
 negative value means that entropy is transferred out.
hange of a system is not accounted for solely by entropy transfer, but is also due to the
the right side of Equation 2.16 denoted by Sgen. The term Sgen is positive when internal
e present during the process and vanishes when internal irreversibilities are absent. This
 by saying that entropy is generated (or produced) within the system by the action of
he second law of thermodynamics can be interpreted as specifying that entropy is
versibilities and conservedonly in the limit as irreversibilities are reduced to zero. Since
 effect of irreversibilities present within a system during a process, its value depends on
 process and not solely on the end states. Entropy generation is not a property.
g the entropy balance, the objective is often to evaluate the entropy generation term.
e of the entropy generation for a given process of a system usually does not have much
self. The significance is normally determined through comparison. For example, the
on within a given component might be compared to the entropy generation values of
ents included in an overall system formed by these components. By comparing entropy
s, the components where appreciable irreversibilities occur can be identified and rank
ws attention to be focused on the components that contribute most heavily to inefficient
overall system.
e entropy transfer term of the entropy balance requires information regarding both the
 the temperature on the boundary where the heat transfer occurs. The entropy transfer
ys subject to direct evaluation, however, because the required information is either
efined, such as when the system passes through states sufficiently far from equilibrium.
cations, it is often convenient, therefore, to enlarge the system to include enough of the
ndings that the temperature on the boundary of the enlarged system corresponds to the
ture, Tamb. The entropy transfer term is then simply Q/Tamb. However, as the irreversibilities
t be just those for the system of interest but those for the enlarged system, the entropy
would account for the effects of internal irreversibilities within the system and external
esent within that portion of the surroundings included within the enlarged system.
 entropy balance convenient for particular analyses is the rate form:
(2.17)
e time rate of change of entropy of the system. The term represents the time rate
fer through the portion of the boundary whose instantaneous temperature is Tj. The
ts for the time rate of entropy generation due to irreversibilities within the system.
solated from its surroundings, the entropy balance is
(2.18)
otal amount of entropy generated within the isolated system. Since entropy is generated
sses, the only processes of an isolated system that actually can occur are those for which
e isolated system increases. This is known as the increase of entropy principle.
dS
dt
Q
T
Sj
j
gen
j
= +∑ ˙ ˙
˙ /Q Tj j
S S S
isol gen2 1
−( ) =
ss LLC
Engineering Thermodynamics 2-13
2.2 Control Volume Applications
Since most appli
control volume 
especially import
energy, and entro
et al., 1960).
Conservation
When applied to 
of mass within th
the boundary. An
occur, each throu
The left side of t
volume, den
The volumetri
velocity compon
ρ(vn dA). The ma
For one-dimensio
equation become
where v denotes 
Control Volu
When applied to
lation of energy w
transfer and the t
and heat transfer
volume with one
where the underl
terms and
boundary (contr
m˙i
˙Q
cv
˙W
0866_book.fm Page 13 Friday, August 6, 2004 2:25 PM
© 2005 by CRC Pre
cations of engineering thermodynamics are conducted on a control volume basis, the
formulations of the mass, energy, and entropy balances presented in this section are
ant. These are given here in the form of overall balances. Equations of change for mass,
py in the form of differential equations are also available in the literature (see, e.g., Bird
 of Mass
a control volume, the principle of mass conservation states: The time rate of accumulation
e control volume equals the difference between the total rates of mass flow in and out across
 important case for engineering practice is one for which inward and outward flows
gh one or more ports. For this case the conservation of mass principle takes the form
(2.19)
his equation represents the time rate of change of mass contained within the control
otes the mass flow rate at an inlet, and is the mass flow rate at an outlet.
c flow rate through a portion of the control surface with area dA is the product of the
ent normal to the area, vn, times the area: vn dA. The mass flow rate through dA is
ss rate of flow through a port of area A is then found by integration over the area
nal flow the intensive properties are uniform with position over area A, and the last
s
(2.20)
the specific volume and the subscript n has been dropped from velocity for simplicity.
me Energy Balance
 a control volume, the principle of energy conservation states: The time rate of accumu-
ithin the control volume equals the difference between the total incoming rate of energy
otal outgoing rate of energy transfer. Energy can enter and exit a control volume by work
. Energy also enters and exits with flowing streams of matter. Accordingly, for a control
-dimensional flow at a single inlet and a single outlet,
(2.21)
ined terms account for the specific energy of the incoming and outgoing streams. The
account, respectively, for the net rates of energy transfer by heat and work over the
ol surface) of the control volume.
dm
dt
m mcv i
i
e
e
= −∑ ∑˙ ˙
m˙
e
m˙ dA
A
= ∫ ρvn
m˙ A A
v
= =ρv v
d U KE PE
dt
Q W m u m ucv
cv i
i
i e
e
e
+ +( )
= − + + +



 − + +



˙ ˙ ˙ ˙
___________ ___________
v
gz
v
gz
2 2
2 2
ss LLC
2-14 Chapter 2
Because work is always done on or by a control volume where matter flows across the boundary, the
quantity of Equation 2.21 can be expressed in terms of two contributions: one is the work associated
with the force of
denoted as , 
of the boundary
contributions to 
of application of
Equation 2.20) a
The terms (pv
respectively, and 
Substituting E
form of the cont
To allow for a
enters or exits, th
Equation 2.24 is 
rate of accumula
of energy transfe
as for closed syst
Control Volu
Like mass and e
transferred into o
the closed system
modifying Equat
where dScv /dt rep
account, re
with mass flow. O
˙W
˙W
cv
m˙i
d
d U(
m˙ s
e e
0866_book.fm Page 14 Friday, August 6, 2004 2:25 PM
© 2005 by CRC Pre
 the fluid pressure as mass is introduced at the inlet and removed at the exit. The other,
includes all other work effects, such as those associated with rotating shafts, displacement
, and electrical effects. The work rate concept of mechanics allows the first of these
be evaluated in terms of the product of the pressure force, pA, and velocity at the point
 the force. To summarize, the work term of Equation 2.21 can be expressed (with
s
(2.22)
i) and (peve) account for the work associated with the pressure at the inlet and outlet,
are commonly referred to as flow work.
quation 2.22 into Equation 2.21, and introducing the specific enthalpy h, the following
rol volume energy rate balance results:
(2.23)
pplications where there may be several locations on the boundary through which mass
e following expression is appropriate:
(2.24)
an accounting rate balance for the energy of the control volume. It states that the time
tion of energy within the control volume equals the difference between the total rates
r in and out across the boundary. The mechanisms of energy transfer are heat and work,
ems, and the energy accompanying the entering and exiting mass.
me Entropy Balance
nergy, entropy is an extensive property. And like mass and energy, entropy can be
r out of a control volume by streams of matter. As this is the principal difference between
 and control volume forms, the control volume entropy rate balance is obtained by
ion 2.17 to account for these entropy transfers. The result is
(2.25)
resents the time rate of change of entropy within the control volume. The terms and
spectively, for rates of entropy transfer into and out of the control volumeassociated
ne-dimensional flow is assumed at locations where mass enters and exits. represents
˙W
˙ ˙
˙
˙ ˙
W W p A p A
W m p v m p v
cv e e e i i i
cv e e e i i i
= + ( ) − ( )
= + ( ) − ( )
v v
m˙
e
U KE PE
dt
Q W m h m hcv
cv cv i i
i
i e e
e
e
+ +( )
= − + + +



 − + +



˙ ˙ ˙ ˙
v
gz
v
gz
2 2
2 2
KE PE
dt
Q W m h m hcv
cv cv i
i
i
i
i e
e
e
e
e
+ + )
= − + + +



 − + +



∑ ∑˙ ˙ ˙ ˙v gz v gz
2 2
2 2
dS
dt
Q
T
m s m s Scv j
jj
i
i
i e e
e
gen= + − +∑ ∑ ∑˙ ˙ ˙ ˙
_____ ______________________ _________
rate of
entropy
change
rate of
entropy
transfer
rate of
entropy
generation
m˙ si i
˙Qj
ss LLC
Engineering Thermodynamics 2-15
the time rate of heat transfer at the location on the boundary where the instantaneous temperature is
Tj; and accounts for the associated rate of entropy transfer. denotes the time rate of entropy
generation due to
of components, 
Control Volu
Engineering syste
in time. For a con
continuously, bu
The energy rate b
At steady state
Mass and ene
indicates that the
of the control vo
control volume e
2.26c shows that
the difference be
Applications fr
the control volum
average temperat
Equation 2.26a, r
Equation 2.26c r
When Equatio
plifications are u
other energy tran
the outer surface
be effective heat t
is small enough 
volume so quick
˙ /Q Tj j ˙Sgen
0866_book.fm Page 15 Friday, August 6, 2004 2:25 PM
© 2005 by CRC Pre
 irreversibilities within the control volume. When a control volume comprises a number
is the sum of the rates of entropy generation of the components.
mes at Steady State
ms are often idealized as being at steady state, meaning that all properties are unchanging
trol volume at steady state, the identity of the matter within the control volume change
t the total amount of mass remains constant. At steady state, Equation 2.19 reduces to
(2.26a)
alance of Equation 2.24 becomes, at steady state,
(2.26b)
, the entropy rate balance of Equation 2.25 reads
(2.26c)
rgy are conserved quantities, but entropy is not generally conserved. Equation 2.26a
 total rate of mass flow into the control volume equals the total rate of mass flow out
lume. Similarly, Equation 2.26b states that the total rate of energy transfer into the
quals the total rate of energy transfer out of the control volume. However, Equation
 the rate at which entropy is transferred out exceeds the rate at which entropy enters,
ing the rate of entropy generation within the control volume owing to irreversibilities.
equently involve control volumes having a single inlet and a single outlet, as, for example,
e of Figure 2.1 where heat transfer (if any) occurs at Tb: the temperature, or a suitable
ure, on the boundary where heat transfer occurs. For this case the mass rate balance,
educes to Denoting the common mass flow rate by Equation 2.26b and
ead, respectively,
(2.27a)
(2.28a)
n 2.27a and Equation 2.28a are applied to particular cases of interest, additional sim-
sually made. The heat transfer term is dropped when it is insignificant relative to
sfers across the boundary. This may be the result of one or more of the following: (1)
 of the control volume is insulated; (2) the outer surface area is too small for there to
ransfer; (3) the temperature difference between the control volume and its surroundings
that the heat transfer can be ignored; (4) the gas or liquid passes through the control
ly that there is not enough time for significant heat transfer to occur. The work term
˙Sgen
˙ ˙m mi
i
e
e
∑ ∑=
0
2 2
2 2
= − + + +



 − + +



∑ ∑˙ ˙ ˙ ˙Q W m h m hcv cv ii i i i ee e e e
v
gz
v
gz
0 = + − +∑ ∑ ∑˙ ˙ ˙ ˙QT m s m s Sjjj ii i e ee gen
˙ ˙ .m mi e= ˙ ,m
0
2
2 2
= − + −( ) + −

 + −( )






˙ ˙
˙Q W m h h
cv cv i e
i e
i e
v v
g z z
0 = + −( ) +˙ ˙ ˙QT m s s Scvb i e gen
˙Q
cv
ss LLC
2-16 Chapter 2
drops out of
electrical effects, 
changes in kineti
in the equation.
The special fo
when there is no
Accordingly, whe
as mass flows fro
passes through th
For no heat tr
A special form th
dropping the kin
In throttling devi
into a line throu
further to read
FIGURE 2.1 One
˙W
cv
0866_book.fm Page 16 Friday, August 6, 2004 2:25 PM
© 2005 by CRC Pre
 the energy rate balance when there are no rotating shafts, displacements of the boundary,
or other work mechanisms associated with the control volume being considered. The
c and potential energy of Equation 2.27a are frequently negligible relative to other terms
rms of Equation 2.27a and Equation 2.28a listed in Table 2.1 are obtained as follows:
 heat transfer, Equation 2.28a gives
(2.28b)
n irreversibilities are present within the control volume, the specific entropy increases
m inlet to outlet. In the ideal case in which no internal irreversibilities are present, mass
e control volume with no change in its entropy — that is, isentropically.
ansfer, Equation 2.27a gives
(2.27b)
at is applicable, at least approximately, to compressors, pumps, and turbines results from
etic and potential energy terms of Equation 2.27b, leaving
(2.27c)
ces a significant reduction in pressure is achieved simply by introducing a restriction
gh which a gas or liquid flows. For such devices = 0 and Equation 2.27c reduces
-inlet, one-outlet control volume at steady state.
s s
S
m
e i
gen
− = ≥
( )
˙
˙
0
no heat transfer
˙
˙W m h h
cv i e
i e
i e= −( ) + −

 + −( )






v v
g z z
2 2
2
˙
˙W m h h
compressors pumps turbines
cv i e= −( )
( ), , and 
˙W
cv
ss LLC
Engineering Thermodynamics 2-17
That is, upstream
A nozzle is a flo
in the direction 
devices, = 0
Equation 2.27b r
Solving for the o
TABLE 2.1 Energy and Entropy Balances for One-Inlet, One-Outlet 
Control Volumes at Steady State and No Heat Transfer
˙W
cv
0866_book.fm Page 17 Friday, August 6, 2004 2:25 PM
© 2005 by CRC Pre
(2.27d)
 and downstream of the throttling device, the specific enthalpies are equal.
w passage of varying cross-sectional area in which the velocity of a gas or liquid increases
of flow. In a diffuser, the gas or liquid decelerates in the direction of flow. For such
. The heat transfer and potential energy change are also generally negligible. Then
educes to
(2.27e)
utlet velocity
Energy balance
(2.27b)
Compressors, pumps, and turbinesa
(2.27c)
Throttling
(2.27d)
Nozzles, diffusersb
(2.27f)
Entropy balance
(2.28b)
a For an ideal gas with constant cp, Equation 1′ of Table 2.7 allows Equa-
tion 2.27c to be written as
‘ (2.27c′)
The power developed in an isentropic process is obtained with Equation
5′ of Table 2.7 as
(2.27c″)
where cp = kR/(k – 1).
b For an ideal gas with constant cp, Equation 1′ of Table 2.7 allows Equa-
tion 2.27f to be written as
(2.27f′)
The exit velocity for an isentropic process is obtained with Equation 5′
of Table 2.7 as
(2.27f″)
where cp = kR/(k – 1).
˙
˙W m h h
cv i e
i e
i e= −( ) + −

 + −( )






v v
g z z
2 2
2
˙
˙W m h h
cv i e= −( )
h h
e i≅
v v
e i i eh h= + −( )2 2
s s
S
m
e i
gen
− = ≥
˙
˙
0
˙
˙W mc T T
cv p i e= −( )
˙
˙W mc T p p s c
cv p i e i
k k
= − ( )  =( )
−( )1
1
v v
e i p i ec T T= + −( )2 2
v v
e i p i e i
k k
c T p p s c= + − ( )  =( )
−( )2 12 1
h h
throttling process
e i≅
( ) 
0
2
2 2
= − +
−
h hi e
i ev v
ss LLC
2-18 Chapter 2
(2.27f)
Further discussio
Themass, ene
multiple inlets an
heaters, and cou
ducted with Equ
provided by Mor
Example 1
A turbine receive
If heat transfer a
in kg/hr, for a tu
(b) the turbine e
Solution. With th
Steam table data
(a) At 0.2 MP
(b) For an int
2499.6 kJ/
Example 2
Air at 500°F, 150 
For a mass flow 
15 lbf/in.2. Mode
°R (k = 1.4).
Solution. The no
v = RT/p:
The exit velocity
Finally, with R =
v v
e i i eh h= + −( )2 2
v
e
=
=
=
0866_book.fm Page 18 Friday, August 6, 2004 2:25 PM
© 2005 by CRC Pre
n of the flow-through nozzles and diffusers is provided in Chapter 3.
rgy, and entropy rate balances, Equations 2.26, can be applied to control volumes with
d/or outlets, as, for example, cases involving heat-recovery steam generators, feedwater
nterflow and crossflow heat exchangers. Transient (or unsteady) analyses can be con-
ation 2.19, Equation 2.24, and Equation 2.25. Illustrations of all such applications are
an and Shapiro (1995).
s steam at 7 MPa, 440°C and exhausts at 0.2 MPa for subsequent process heating duty.
nd kinetic/potential energy effects are negligible, determine the steam mass flow rate,
rbine power output of 30 MW when (a) the steam quality at the turbine outlet is 95%,
xpansion is internally reversible.
e indicated idealizations, Equation 2.27c is appropriate. Solving, 
 (Table A.5) at the inlet condition are hi = 3261.7 kJ/kg, si = 6.6022 kJ/kg · K.
a and x = 0.95, he = 2596.5 kJ/kg. Then
ernally reversible expansion, Equation 2.28b reduces to give se = si. For this case, he =
kg (x = 0.906), and = 141,714 kg/hr.
lbf/in.2, and 10 ft/sec expands adiabatically through a nozzle and exits at 60°F, 15 lbf/in.2.
rate of 5 lb/sec determine the exit area, in in.2. Repeat for an isentropic expansion to
l the air as an ideal gas (Section 2.3, Ideal Gas Model) with specific heat cp = 0.24 Btu/lb ·
zle exit area can be evaluated using Equation 2.20, together with the ideal gas equation,
 required by this expression is obtained using Equation 2.27f′ of Table 2.1,
= 53.33 ft · lbf/lb · °R,
, nozzle diffuser( )
˙
˙ /( ).m W h h
cv i e= −
˙
. .
sec
,
m =
−( )








=
30
3261 7 2596 5
10
1
3600
1
162 357
3MW
kJ kg
kJ sec
MW hr
 kg hr
m˙
A
m m RT p
e
e
e
e e
e
= =
( )˙ ˙ν
v v
v
ft Btu
lb R
ft lbf
Btu
R lb ft sec
lbf
ft sec
2
i p i ec T T
s
+ −( )
  +
⋅
 
⋅


 °( )
⋅



2
2
2
10 2 0 24 778 17
1
440 32 174
1
2299 5
.
. .
.
 R /M
ss LLC
Engineering Thermodynamics 2-19
Using Equation 2
Then Ae = 3.92 i
Example 3
Figure 2.2 provid
kinetic/potential 
Solution. For this
Combining and 
Inserting steam t
Internally Rev
For one-inlet, on
rate and power i
FIGURE 2.2 Ope
0866_book.fm Page 19 Friday, August 6, 2004 2:25 PM
© 2005 by CRC Pre
.27f″ in Table 2.1 for the isentropic expansion,
n.2.
es steady-state operating data for an open feedwater heater. Ignoring heat transfer and
energy effects, determine the ratio of mass flow rates, 
 case Equation 2.26a and Equation 2.26b reduce to read, respectively,
solving for the ratio 
able data, in kJ/kg, from Table A.5,
ersible Heat Transfer and Work
e-outlet control volumes at steady state, the following expressions give the heat transfer
n the absence of internal irreversibilities:
n feedwater heater.
A
e
=
 
⋅
⋅ °
  °( )
   
=
5 53 3 520
2299 5 15
4 02
lb ft lbf
lb R
R
ft lbf
in.
in.
2
2sec
.
.
sec
.
v
ft
e
= ( ) + ( )( )( )( ) −  




=
10 2 0 24 778 17 960 32 174 1 15
150
2358 3
2
0 4 1 4
. . .
. sec
. .
˙ / ˙ .m m1 2
˙ ˙ ˙
˙ ˙ ˙
m m m
m h m h m h
1 2 3
1 1 2 2 3 30
+ =
= + −
˙ / ˙ ,m m1 2
˙
˙
m
m
h h
h h
1
2
2 3
3 1
=
−
−
˙
˙
. .
. .
.
m
m
1
2
2844 8 697 2
697 2 167 6
4 06= −
−
=
ss LLC
2-20 Chapter 2
(2.29)
(see, e.g., Moran
If there is no s
The specific volu
30b becomes
When the stat
described by a co
the area under th
ideal process is d
the magnitude o
a-b-c-d behind t
integral ∫pdv of E
FIGURE 2.3 Inter
˙Q
Tdscv



 =
2
0866_book.fm Page 20 Friday, August 6, 2004 2:25 PM
© 2005 by CRC Pre
(2.30a)
 and Shapiro, 1995).
ignificant change in kinetic or potential energy from inlet to outlet, Equation 2.30a reads
(2.30b)
me remains approximately constant in many applications with liquids. Then Equation
(2.30c)
es visited by a unit of mass flowing without irreversibilities from inlet to outlet are
ntinuous curve on a plot of temperature vs. specific entropy, Equation 2.29 implies that
e curve is the magnitude of the heat transfer per unit of mass flowing. When such an
escribed by a curve on a plot of pressure vs. specific volume, as shown in Figure 2.3,
f the integral ∫vdp of Equation 2.30a and Equation 2.30b is represented by the area
he curve. The area a-b-c′-d′ under the curve is identified with the magnitude of the
quation 2.10.
nally reversible process on p–v coordinates.
m˙
rev
  ∫int 1
˙
˙
W
m
dp g z zcv
rev



 = − +
−
+ −( )∫int ν v v12 2212 1 22
˙
˙
W
m
dp ke pecv
rev



 = − = =( )∫int ν ∆ ∆ 012
˙
˙
W
m
v p p vcv
rev



 = − −( ) =( )int 2 1 constant
ss LLC
Engineering Thermodynamics 2-21
2.3 Property Relations and Data
Pressure, temper
specific heats cv a
are certain other 
that are not so re
imental data of p
relations derived
sources are consi
tant substances.
Property data 
(formerly the U.S
Engineering (ASM
(ASHRAE), and 
Chemical. Handb
chapter are readi
online data bases
Basic Relatio
An energy balanc
the absence of ov
From Equation 
systems for whic
change, 
Introducing enth
H – TS, three ad
Equations 2.31
Similar expressio
δW
rev
( )int
0866_book.fm Page 21 Friday, August 6, 2004 2:25 PM
© 2005 by CRC Pre
ature, volume, and mass can be found experimentally. The relationships between the
nd cp and temperature at relatively low pressure are also accessible experimentally, as
property data. Specific internal energy, enthalpy, and entropy are among those properties
adily obtained in the laboratory. Values for such properties are calculated using exper-
roperties that are more amenable to measurement, together with appropriate property
 using the principles of thermodynamics. In this section property relations and data
dered for simple compressible systems, which include a wide range of industrially impor-
are provided in the publications of the National Institute of Standards and Technology
. Bureau of Standards), of professional groups such as the American Society of Mechanical
E), the American Society of Heating. Refrigerating, and Air Conditioning Engineers
the American Chemical Society, and of corporate entities such as Dupont and Dow
ooks and property reference volumes such as included in the list of references for this
ly accessed sources of data. Property data are also retrievable from various commercial
. Computer software is increasingly available for this purpose as well.
ns for Pure Substances
e in differential form for a closed system undergoing an internally reversible process in
erall system motion and the effect of gravity reads
2.14b, = TdS. When consideration is limited to simple compressible systems:
h the only significant work in an internally reversible process is associated with volume
= pdV, the following equation is obtained:
(2.31a)
alpy, H = U + pV, the Helmholtz function, Ψ = U – TS, and the Gibbs function, G =
ditional expressionsare obtained:
(2.31b)
(2.31c)
(2.31d)
 can be expressed on a per-unit-mass basis as
(2.32a)
(2.32b)
(2.32c)
(2.32d)
ns can be written on a per-mole basis.
dU Q W
rev rev
= ( ) − ( )δ δint int
δQ
rev
( )int
dU TdS pdV= −
dH TdS Vdp= +
d pdV SdTΨ = − −
dG Vdp SdT= −
du Tds pdv= −
dh Tds vdp= +
d pdv sdTψ = − −
dg vdp sdT= −
ss LLC
2-22 Chapter 2
Maxwell Relat
Since only prope
exact differential 
derivatives are eq
of the form u(s, 
in Table 2.2 can 
Example 4
Derive the Maxw
Solution. The dif
TABLE 2.2 Relations from Exact Differentials
0866_book.fm Page 22 Friday, August 6, 2004 2:25 PM
© 2005 by CRC Pre
ions
rties are involved, each of the four differential expressions given by Equations 2.32 is an
exhibiting the general form dz = M(x, y)dx + N(x, y)dy, where the second mixed partial
ual: (∂M/∂y) = (∂N/∂x). Underlying these exact differentials are, respectively, functions
v), h(s, p), ψ(v, T), and g(T, p). From such considerations the Maxwell relations given
be established.
ell relation following from Equation 2.32a.
ferential of the function u = u(s, v) is
ss LLC
Engineering Thermodynamics 2-23
By comparison w
In Equation 2.32
partial derivative
Since each of 
Table 2.2, four ad
These four relatio
from the fourth 
following from t
can be derived; s
Specific Heats 
Engineering ther
these properties. 
Among the en
required for ther
h(T, p), respectiv
Since u and h ca
heats can be sim
specific heat ratio
Values for cv a
can also be deter
du u ds u dv=   +  
∂ ∂
0866_book.fm Page 23 Friday, August 6, 2004 2:25 PM
© 2005 by CRC Pre
ith Equation 2.32a,
a, T plays the role of M and –p plays the role of N, so the equality of second mixed
s gives the Maxwell relation,
the properties T, p, v, and s appears on the right side of two of the eight coefficients of
ditional property relations can be obtained by equating such expressions:
ns are identified in Table 2.2 by brackets. As any three of Equations 2.32 can be obtained
simply by manipulation, the 16 property relations of Table 2.2 also can be regarded as
his single differential expression. Several additional first-derivative property relations
ee, e.g., Zemansky, 1972.
and Other Properties
modynamics uses a wide assortment of thermodynamic properties and relations among
Table 2.3 lists several commonly encountered properties.
tries of Table 2.3 are the specific heats cv and cp. These intensive properties are often
modynamic analysis, and are defined as partial derivations of the functions u(T, v) and
ely,
(2.33)
(2.34)
n be expressed either on a unit mass basis or a per-mole basis, values of the specific
ilarly expressed. Table 2.4 summarizes relations involving cv and cp. The property k, the
, is
(2.35)
nd cp can be obtained via statistical mechanics using spectroscopic measurements. They
mined macroscopically through exacting property measurements. Specific heat data for
s vv s∂ ∂
T u
s
p u
vv s
=
  − =  
∂
∂
∂
∂
,
∂
∂
∂
∂
T
v
p
ss v
  = − 
∂
∂
∂
∂
∂
∂
∂ψ
∂
∂
∂
∂
∂
∂ψ
∂
∂
∂
u
s
h
s
u
v v
h
p
g
p T
g
T
v p s T
s T v p
  =     =  



 =




  =  
,
,
c
u
Tv v
=
 
∂
∂
c
h
Tp p
=
 
∂
∂
k
c
c
p
v
=
ss LLC
2-24 Chapter 2
common gases, li
among the Chapt
of the incompress
function of temp
variation of cp wi
increases with in
The following
Their use is illus
Example 5
Obtain Equation
Solution. Identify
Applying Equatio
TABLE 2.3 Symbols and Definitions for Selected Properties
Property Symbol Definition Property Symbol Definition
Pressure
Temperature
Specific volume
Specific internal e
Specific entropy
Specific enthalpy
Specific Helmholt
Specific Gibbs fun
Compressibility fa
Specific heat ratio
0866_book.fm Page 24 Friday, August 6, 2004 2:25 PM
© 2005 by CRC Pre
quids, and solids are provided by the handbooks and property reference volumes listed
er 2 references. Specific heats are also considered in Section 2.3 as a part of the discussions
ible model and the ideal gas model. Figure 2.4 shows how cp for water vapor varies as a
erature and pressure. Other gases exhibit similar behavior. The figure also gives the
th temperature in the limit as pressure tends to zero (the ideal gas limit). In this limit cp
creasing temperature, which is a characteristic exhibited by other gases as well.
 two equations are often convenient for establishing relations among properties:
(2.36a)
(2.36b)
trated in Example 5.
 2 and Equation 11 of Table 2.4 from Equation 1.
ing x, y, z with s, T, and v, respectively, Equation 2.36b reads
n 2.36a to each of (∂T/∂v)s and (∂v/∂s)T ,
p Specific heat, constant volume cv
T Specific heat, constant pressure cp
v Volume expansivity β
nergy u Isothermal compressivity κ
s Isentropic compressibility α
h u + pv Isothermal bulk modulus B
z function ψ u – Ts Isentropic bulk modulus Bs
ction g h – Ts Joule-Thomson coefficient µJ
ctor Z pv/RT Joule coefficient η
k cp /cv Velocity of sound c
∂ ∂u T
v
( )
∂ ∂h T
p( )
1
v
v T
p
∂ ∂( )
− ( )1
v
v p
T
∂ ∂
− ( )1
v
v p
s
∂ ∂
− ( )v p v T∂ ∂
− ( )v p v
s
∂ ∂
∂ ∂T p h( )
∂ ∂T v
u
( )
− ( )v p v
s
2 ∂ ∂
∂
∂
∂
∂
x
y
y
x
z z




  = 1
∂
∂
∂
∂
∂
∂
y
z
z
x
x
yx y z
   



 = −1
∂
∂
∂
∂
∂
∂
T
v
v
s
s
Ts T v
      = −1
∂
∂ ∂ ∂ ∂ ∂
∂
∂
∂
∂
s
T T v v s
v
T
s
vv s T s T
  = − ( ) ( ) = −
   
1
ss LLC
Engineering Thermodynamics 2-25
Introducing the 
With this, Equati
Equation 11 of T
fixed temperatur
P-v-T Relatio
Considerable pre
important gases 
of state. Equation
TABLE 2.4 Specific Heat Relationsa
0866_book.fm Page 25 Friday, August 6, 2004 2:25 PM
© 2005 by CRC Pre
Maxwell relation from Table 2.2 corresponding to ψ(T, v),
on 2 of Table 2.4 is obtained from Equation 1, which in turn is obtained in Example 6.
able 2.4 can be obtained by differentiating Equation 1 with repect to specific volume at
e, and again using the Maxwell relation corresponding to ψ.
ns
ssure, specific volume, and temperature data have been accumulated for industrially
and liquids. These data can be represented in the form p = f (v, T ), called an equation
s of state can be expressed in tabular, graphical, and analytical forms.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
10)
(11)
(12)
a See, for example, Moran, M.J. and Sha-
piro, H.N. 1995. Fundamentals of Engineer-
ing Thermodynamics, 3rd ed. Wiley, New
York, chap. 11.
c
u
T
T s
Tv v v
=



 =




∂
∂
∂
∂
= −







T
p
T
v
Tv s
∂
∂
∂
∂
c
h
T
T s
Tp p p
=



 =




∂
∂
∂
∂
=







T
v
T
p
Tp s
∂
∂
∂
∂
c c T p
T
v
Tp v v p
− =








∂
∂
∂
∂
= −







T
v
T
p
vp T
∂
∂
∂
∂
2
=
Tvβ
κ
2
c T v
T
vp
J p
=



 −






1
µ
∂
∂
c T p
T
p
v
v
= −



 −




1
η
∂
∂
k
c
c
v
p
p
v
p
v T s
= =








∂
∂
∂
∂
∂
∂
∂
∂
c
v
T p
T
v
T v



 =




2
2∂
∂
∂
∂
c
p
T v
T
p
T p



 = −




2
2
∂
∂
∂
∂
∂
∂
s
T
v
T
p
Tv s v
  = −   
ss LLC
2-26
C
h
apter 2
FIGURE 2.4 cp of w oore, J.G. 1969 and 1978. Steam 
Tables — S.I. Units (E
0866_book.fm
 Page 26 Friday, A
ugust 6, 2004 2:25 PM
© 2005 by CRC Press L
ater vapor as a function of temperature and pressure. (Adapted from Keenan, J.H., Keyes, F.G., Hill, P.G., and M
nglish Units). John Wiley & Sons, New York.)
LC
Engineering Thermodynamics 2-27
P-v-T Surface
The graph of a fu
relationship for 
plane, called the 
Figure 2.5 has 
phase. Between t
The lines separat
represented by a 
the two-phase liq
state. The satura
state denoted by 
at the critical poin
FIGURE 2.5 Press
0866_book.fm Page 27 Friday, August 6, 2004 2:25 PM
© 2005 by CRC Pre
nction p = f (v, T) is a surface in three-dimensional space. Figure 2.5 shows the p-v-T
water. Figure 2.5b shows the projection of the surface onto the pressure-temperature
phase diagram. The projection onto the p–v plane is shown in Figure 2.5c.
three regions labeled solid, liquid, and vapor where the substance exists only in a single
he single phase regions lie two-phase regions, where two phases coexist in equilibrium.
ing the single-phase regions from the two-phase regions are saturation lines. Any state
point on a saturation line is a saturation state. The line separating the liquid phase and
uid-vapor region is the saturated liquid line. The state denoted by f is a saturated liquid
ted vapor line separates the vapor region and the two-phase liquid-vapor region. The
g is a saturated vapor state. The saturated liquid line and the saturated vapor line meet
t. At the critical point, the pressure is the critical pressure pc, and the temperature is the
ure-specific volume-temperature surface and projections for water (not to scale).
ss LLC
2-28 Chapter 2
critical temperature Tc. Three phases can coexist in equilibrium along the line labeled triple line. The
triple line projects onto a point on the phase diagram. The triple point of water is used in defining the
Kelvin temperatu
dynamics, Entrop
When a phase
constant as long 
constant pressur
temperature is ca
is called the satu
superheated vapo
for its pressure. T
because the liqui
When a mixtu
the vapor phase i
mv = mfvf + mgv
mass of the mixt
by x, called the q
where vfg = vg – v
For the case of
solid to vapor (su
phase change the
The Clapeyron eq
evaluated from p
where (dp/dT)sat 
temperature held
be written for su
0866_book.fm Page 28 Friday, August 6, 2004 2:25 PM
© 2005 by CRC Pre
re scale (Section 2.1, Basic Concepts and Definitions; The Second Law of Thermo-
y).
 change occurs during constant pressure heating or cooling, the temperature remains
as both phases are present. Accordingly, in the two-phase liquid-vapor region, a line of
e is also a line of constant temperature. For a specified pressure, the corresponding
lled the saturation temperature. For a specified temperature, the corresponding pressure
ration pressure. The region to the right of the saturated vapor line is known as the
r region because the vapor exists at a temperature greater than the saturation temperature
he region to the left of the saturated liquid line is known as the compressed liquid region
d is at a pressure higher than the saturation pressure for its temperature.
re of liquid and vapor coexists in equilibrium, the liquid phase is a saturated liquid and
s a saturated vapor. The total volume of any such mixture is V = Vf + Vg; or, alternatively,
g, where m and v denote mass and specific volume, respectively. Dividing by the total
ure m and letting the mass fraction of the vapor in the mixture, mg /m, be symbolized
uality, the apparent specific volume v of the mixture is
(2.37a)
f. Expressions similar in form can be written for internal energy, enthalpy, and entropy:
(2.37b)
(2.37c)
(2.37d)
 water, Figure 2.6 illustrates the phase change from solid to liquid (melting): a-b-c; from
blimation): a′-b′-c′; and from liquid to vapor (vaporization): a″-b″-c″. During any such
 temperature and pressure remain constant and thus are not independent properties.
uation allows the change in enthalpy during a phase change at fixed temperature to be
-v-T data pertaining to the phase change. For vaporization, the Clapeyron equation reads
(2.38)
is the slope of the saturation pressure-temperature curve at the point determined by the
 constant during the phase change. Expressions similar in form to Equation 2.38 can
blimation and melting.
v x v xv
v xv
= −( ) +
= +
1 f g
f fg
u x u xu
u xu
= −( ) +
= +
1 f g
f fg
h x h xh
h xh
= −( ) +
= +
1 f g
f fg
s x s xs
s xs
= −( ) +
= +
1 f g
f fg
dp
dT
h h
T v vsat
  =
−
−( )
g f
g f
ss LLC
Engineering Thermodynamics 2-29
The Clapeyron
signs of the speci
a phase change t
(dp/dT)sat is posi
volume decrease
negative, as illust
Graphical Rep
The intensive sta
independent inte
gravity. While an
These include th
diagram of Figur
the compressibili
Compressibilit
The p-v-T relatio
chart of Figure 2
reduced temperat
and
In these express
temperature, resp
FIGURE 2.6 Phas
0866_book.fm Page 29 Friday, August 6, 2004 2:25 PM
© 2005 by CRC Pre
 equation shows that the slope of a saturation line on a phase diagram depends on the
fic volume and enthalpy changes accompanying the phase change. In most cases, when
akes place with an increase in specific enthalpy, the specific volume also increases, and
tive. However, in the case of the melting of ice and a few other substances, the specific
s on melting. The slope of the saturated solid-liquid curve for these few substances is
rated for water in Figure 2.6.
resentations
tes of a pure, simple compressible system can be represented graphically with any two
nsive properties as the coordinates, excluding properties associated with motion and
y such pair may be used, there are several selections that are conventionally employed.
e p-T and p-v diagrams of Figure 2.5, the T-s diagram of Figure 2.7, the h-s (Mollier)
e 2.8, and the p-h diagram of Figure 2.9. The compressibility charts considered next use
ty factor as one of the coordinates.
y Charts
n for a wide range of common gases is illustrated by the generalized compressibility
.10. In this chart, the compressibility factor, Z, is plotted vs. the reduced pressure, pR,
ure, TR, and pseudoreduced specific volume, where
(2.39)
(2.40)
ions, is the universal gas constant and pc and Tc denote the critical pressure and
ectively. Values of pc and Tc are given for several substances in Table A.9. The reduced
e diagram for water (not to scale).
′vR ,
Z pv
RT
=
p p
p
T T
T
v
v
RT pR c
R
c
R
c c
= = ′ = ( ), ,
R
ss LLC
2-30 Chapter 2
isotherms of Figu
in developing the
of 5% and for m
Figure 2.10 giv
the critical comp
inaccurate in the
the correlation to
critical compress
variables other th
e.g., Reid and Sh
Generalized co
and in equation 
tabular, or equat
When accuracy i
substitute for p-v
of state.
Equations of S
Considering the 
might be express
that enjoy a the
pressure,
FIGURE 2.7 Tem
modynamics, Prent
Kell, G.S. 1984. NB
1 To determine 
be calculated using
0866_book.fm Page 30 Friday, August 6, 2004 2:25 PM
© 2005 by CRC Pre
re 2.10 represent the best curves fitted to the data of several gases. For the 30 gases used
 chart, the deviation of observed values from those of the chart is at most on the order
ost ranges is much less.1
es a common value of about 0.27 for the compressibility factor at the critical point. As
ressibility factor for different substances