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BBaassiicc aanndd AApppplliieedd TThheerrmmooddyynnaammiiccss
CChhaapptteerr--55 EEnnttrrooppyy
PPrreeppaarreedd BByy
BBrriijj BBhhoooosshhaann
AAsssstt.. PPrrooffeessssoorr
BB.. SS.. AA.. CCoolllleeggee ooff EEnngggg.. AAnndd TTeecchhnnoollooggyy
MMaatthhuurraa,, UUttttaarr PPrraaddeesshh,, ((IInnddiiaa))
SSuuppppoorrtteedd BByy::
PPuurrvvii BBhhoooosshhaann
In This Chapter We Cover the Following Topics
Art. Content Page
5.1 Two Reversible Adiabatic Paths Cannot Intersect Each Other 3
5.2 Reversible Cycles and Clausius Theorem
Clausius Theorem
Clausius Inequality
4
4
5
5.3 Entropy 6
5.4 Principle of Entropy Increase 8
5.5 Temperature Entropy Diagram & 2nd Law of a Control Volume 11
5.6 Applications of Entropy Principle
Mixing of Two Fluids
Maximum Work Obtainable from Two Finite Bodies at Temperatures T1
and T2
Maximum Work Obtainable from a Finite Body and a TER
12
12
13
15
5.7 Entropy Generation in a Closed System 18
5.8 Entropy Generation in an Open System 20
5.9 First and Second Laws Combined 22
5.10 Entropy Change for Some Elementary Processes
Isothermal Process
Reversible Adiabatic Work in a Steady Flow System
Isentropic Process
For Ideal/Perfect Gases
Polytropic Process
23
23
24
25
26
27
5.11 Third Law of Thermodynamics 28
References:
1. M. J. Moran and H. N. Shapiro, Fundamentals of Engineering Thermodynamics, 6e,
John Wiley & Sons, Inc., New York, 2008.
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2 Chapter 5: Entropy
2. G. J. Van Wylen, R. E. Sonntag, C. Borgnakke, Fundamentals of Thermodynamics,
John Wiley & Sons, Inc., New York, 1994.
3. J. P. Holman, Thermodynamics, 4e, McGraw-Hill, New York, 1988.
4. F. W. Sears, G. L. Salinger, Thermodynamics, Kinetic theory, and Statistical
Thermodynamics, 3e, Narosa Publishing House, New Delhi, 1998.
5. Y. A. Cengel and M. A. Boles, Thermodynamics: An Engineering Approach, 2e,
McGraw-Hill, New York, 1994.
6. E. Rathakrishnan, Fundamentals of Engineering Thermodynamics, 2e, PHI Learning
Private Limited, New Delhi, 2008.
7. P. K. Nag, Basic and Applied Thermodynamics, 1e, McGraw-Hill, New Delhi, 2010.
8. V Ganesan, Gas Turbine, 2e, Tata McGraw-Hill, New Delhi, 2003.
9. Y. V. C. Rao, An Introduction to Thermodynamics, 1e, New Age International (P)
Limited, Publishers, New Delhi, 1998.
10. Onkar Singh, Applied Thermodynamics, 2e, New Age International (P) Limited,
Publishers, New Delhi, 2006.
Please welcome for any correction or misprint in the entire manuscript and your
valuable suggestions kindly mail us [email protected] .
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3 Basic and Applied Thermodynamics By Brij Bhooshan
Till now the detailed explanation of Zeroth law, first law and second law of
thermodynamics have been made. Also we have seen that the first law of
thermodynamics defined a very useful property called internal energy. For overcoming
the limitations of first law, the second law of thermodynamics had been stated. Now we
need some mathematical parameter for being used as decision maker in respect of
feasibility of process, irreversibility, nature of process etc. The first law of
thermodynamics was stated in terms of cycles first and it was shown that the cyclic
integral of heat is equal to the cyclic integral of work. When the first law was applied for
thermodynamic processes, the existence of a property, the internal energy, was found.
Similarly, the second law was also first stated in terms of cycles executed by systems.
When applied to processes, the second law also leads to the definition of a new property,
known as entropy. ‘Entropy’ is the outcome of second law and is a thermodynamic
property. Entropy is defined in the form of calculus operation, hence no exact physical
description of it can be given. However, it has immense significance in thermodynamic
process analysis. In fact, thermodynamics is the study of three E's, namely, energy,
equilibrium and entropy.
5.1 TWO REVERSIBLE ADIABATIC PATHS CAN’T INTERSECT EACH OTHER
Let it be assumed that two reversible adiabatics AC and BC intersect each other at point
C (Diagram. 5.1). Let a reversible isotherm AB be drawn in such a way that it intersects
the reversible adiabatics at A and B. The three reversible processes AB, BC, and CA
together constitute a reversible cycle, and the area included represents the net work
output in a cycle. But such a cycle is impossible, since net work is being produced in a
cycle by a heat engine by exchanging heat with a single reservoir in the process AB,
which violates the Kelvin-Planck statement of the second law. Therefore, the
assumption of the intersection of the reversible adiabatics is wrong. Through one point,
there can pass only one reversible adiabatic.
Diagram 5.1
Since two constant property lines can never intersect each other, it is inferred that a
reversible adiabatic path must represent some property, which is yet to be identified.
Diagram 5.2
P
Rev. isotherm
Rev. adiabatic
Rev. adiabatic
P
Rev. isotherm
Rev. adiabatics
C
B
A
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4 Chapter 5: Entropy
5.2 REVERSIBLE CYCLES AND CLAUSIUS THEOREM
First of all we Substitution of a reversible process by the reversible isothermal and
reversible adiabatic processes. Let a system be taken from an equilibrium state i to
another equilibrium state f by following the reversible path i-f refer to Diagram 5.2. Let
a reversible adiabatic i-a and another reversible b-f be drawn in such a manner that a
and b can be joined by a reversible isotherm. Also, these reversible adiabatic and
reversible isothermal lines are such that the area under iabf is equal to the area under
i-f.
Applying the first law for i-f,
Applying the first law for iabf,
We know Wif = Wiabf (area under the curves are same).
From Eqn. (5.1) and (5.2)
Since Qia and Qbf are zero, then
Thus, any reversible path may be substituted by a reversible adiabatic, a reversible
isotherm and a reversible adiabatic between the same end states such that the heat
transferred during the isothermal process is the same as that transferred during the
original process.
Clausius Theorem
Let a smooth closed curve representing a reversible cycle (Diagram 5.3) be considered.
Let the closed cycle be divided into a large number of strips by means of reversible
adiabatics. Each strip may be closed at the top and bottom by reversible isotherms. The
original closed cycle is thus replaced by a zigzag closed path consisting of alternate
adiabatic and isothermal processes, such that the heat transferred during all the
isothermal processes is equal to the heat transferred in the original cycle. Thus the
original cycle is replaced by a large number of Carnot cycles. If the adiabatics are close
to one another and the number of Carnot cycles is large, the saw-toothed zigzag line will
coincide with the original cycle.
Diagram 5.3
P
Rev. isotherms
Rev. adiabatics
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5 Basic and Applied Thermodynamics By Brij Bhooshan
Consider a system undergoing a reversible cycle. The given cycle may be sub-divided by
drawing a family of reversible adiabatic lines. Every two adjacent adiabatic lines may be
joined by two reversible isotherms (refers to Diagram 5.3).
Now,
Also, a1-b1-c1-d1 is a Carnot cycle which receives heat dQ1 during the a1b1 process and
rejects heat dQ2 during the c1d1 process. Let the heat addition be at temperature T1 and
the heat rejection to be at temperature T2.
Then it is possible to write,
and
or,
Since dQ2 is negative, it reduces to
Similarly for the cycle e1, f1, g1, h1,
If similar equations are written for all the elementary cycles, then
or,
The cyclic integral of dQ/T for a reversible cycle is equal to zero. This is known as
Clausius' theorem. The letter R emphasizes the fact that the equation is valid only for a
reversible cycle.
Clausius Inequality
Let us go back to the cycle a1, b1, c1, d1, from Diagram 5.3,
Now, the efficiency of a general cycle will be equal to or less than the efficiency of a
reversible cycle. Now ηIR ≤ ηR where, ηIR = 1 ‒ dQ2/dQ1 and this is not equal to 1 ‒ T2/T1.
Or
Since
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6 Chapter 5: Entropy
or
for any process a1, b1, c1, d1, reversible or irreversible.
For the reversible cycle,
Hence, for any process a1, b1, c1, d1,
Then for any cycle
Similarly, for the irreversible cycle e1, f1, g1, h1,
Summing up all elementary cycles
Since entropy is a property and the cyclic integral of any property is zero. The above two
conclusions about reversible and irreversible cycles can be generalized as
The equality holds good for a reversible cycle and the inequality holds good for an
irreversible cycles. The complete expression is known as Clausius Inequality.
If ∮dQ/T = 0, the cycle is reversible,
If ∮dQ/T < 0, the cycle is irreversible and possible
If ∮dQ/T > 0, the cycle is impossible, since it violates the second law.
5.3 ENTROPY
Clausius inequality can be used to analyze the cyclic process in a quantitative manner.
The second law became a law of wider applicability when Clausius introduced the
property called entropy. By evaluating the entropy change, one can explain as to why
spontaneous processes occur only in one direction.
Diagram 5.4
Let a system be taken from an initial equilibrium state i to a final equilibrium state f by
following the reversible path R1, (Diagram 5.4). The system is brought back from f to i
by following another reversible path R2. Then the two paths R1 and R2 together
constitute a reversible cycle. From Clausius' theorem
P
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The above integral may be replaced as the sum of two integrals, one for path R1 and the
other for path R2,
or
Since R2 is a reversible path
Since R1 and R2 represent any two reversible
Is independent of the reversible path connecting i and f. Therefore, there exists a
property of a system whose value at the final state f minus its value at the initial state i
is equal to
This property is called entropy, and is denoted by S. lf Si is the entropy at the initial
state i, and Sf is the entropy at the final state f then
When the two equilibrium states are infinitesimally near
where dS is an exact differential because S is a point function and a property. The
subscript R in dQ indicates that heat dQ is transferred reversibly.
The word 'entropy' was first used by Clausius, taken from the Greek word 'tropee'
meaning 'transformation'. It is an extensive property, and has the unit J/K. The specific
entropy
If the system is taken from an initial equilibrium state i to a final equilibrium state f by
an irreversible path, since entropy is a point or state function, and the entropy change is
independent of the path followed, the non-reversible path is to be replaced by a
reversible path to integrate for the evaluation of entropy change in the irreversible
process (Diagram 5.5).
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8 Chapter 5: Entropy
Integration can be performed only on a reversible path.
Diagram 5.5
Calculation of Entropy Change
The following facts should be kept in mind while calculating the change in entropy for a
process
1. ds = (dQ/T)R for a reversible process
2. Entropy is a state function. The entropy change of a system is determined by its
initial and final states only, irrespective of how the system has changed its state.
3. In analyzing irreversible processes, it is not necessary to make a direct analysis
of the actual process. One can substitute the actual process by a reversible
process connecting the final state to the initial state, and the entropy change for
the imaginary reversible process can be evaluated.
Diagram 5.6
Mathematical formulation for entropy (dQrev = T·dS) can be used for getting property
diagrams between “temperature and entropy” (T – S), “enthalpy and entropy” (h – S).
Area under process curve on T–S diagram (Diagram 5.6) gives heat transferred, for
internally reversible process
5.4 PRINCIPLE OF ENTROPY INCREASE
Refer to Diagram 5.7. Let a system change from state 1 to state 2 by a reversible process
A and return to state 1 by another reversible process B. Then 1A2B1 is a reversible
cycle.
Diagram 5.7
C
1
P 2
B
A
2
1
Rev. path which replaces
the irrev. path Actual irrev. path
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9 Basic and Applied Thermodynamics By Brij Bhooshan
Therefore, the Clausius inequality gives
If the system is restored to the initial state 1 from state 2 by an irreversible process C,
then 1A2C1 is an irreversible cycle. Then the Clausius inequality gives
Subtracting (5.15) from (5.16)
Since the process 2B1 is reversible
or,
Thus, for any irreversible process,
whereas for a reversible process
Therefore, for the general case, we can write
Where the equality sign holds good for a reversible process and the inequality sign holds
good for an irreversible process.
Diagram 5.8
Now let us apply the above results to evaluate the entropy of the universe when a
system interacts with the surroundings. Refer to Diagram 5.8.
Let Temperature of the surroundings = Tsur
Temperature of the system = Tsys and,
If the energy exchange takes place, dQ will be the energy transfer from the
surroundings to the system.
Surrounding
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10 Chapter 5: Entropy
Since Tsur > Tsys, then
For any infinitesimal process undergone by a system, we have from Eq. (5.17) for the
total mass
If the system is isolated, that is, when there is no energy interaction between the system
and the surroundings, dQ = 0, then
For a reversible process,
For a irreversible process,
Through generalization we can write for an isolated system
Examine the statement critically. For a reversible process it is equal to zero. For an
irreversible process it is greater than zero.
It is thus proved that the entropy of an isolated system can never decrease. It always
increases and remains constant only when the process is reversible. This is known as
the principle of increase of entropy, or simply the entropy principle. It is the quantitative
general statement of second law from the macroscopic viewpoint.
An isolated system can always be formed by including any system and its surroundings
within a single boundary (Diagram 5.9). Sometimes the original system which is then
only a part of the isolated system is called a 'subsystem'.
Diagram 5.9 Isolated system
The system and the surroundings together (the universe or the isolated system) include
everything which is affected by the process. For all possible processes that a system in
the given surroundings can undergo
The equality sign holds good if the process undergone is reversible; the inequality sign
holds good if the process undergone is irreversible.
Entropy may decrease locally at some region within the isolated system, but it must be
compensated by a greater increase of entropy somewhere within the system so that the
net effect of an irreversible process is an entropy increase of the whole system. The
entropy increase of an isolated system is a measure of the extent of irreversibility of the
process undergone by the system.
Surrounding isolated
(composite) system System
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Clausius summarized these as:
Die Energie der Welt ist konstant.
Die Entropie der Welt strebt einem Maximum zu.
This means:
The energy of the universe (world) in constant (first law).
The entropy of the universe (world) tends towards a maximum (second law).
The entropy of an isolated system always increases and becomes a maximum at the
state of equilibrium. If the entropy of an isolated system varies with some parameter x,
then there is a certain value of xe which maximizes the entropy (when dS/dx = 0) and
represents the equilibrium state (Diagram 5.10). The system is then said to exist at the
peak of the entropy hill, and dS = 0. When the system is at equilibrium, any conceivable
change in entropy would be zero.
Diagram 5.10
5.5 TEMPERATURE ENTROPY DIAGRAM AND 2ND LAW OF A CONTROL
VOLUME
Entropy change of a system is given by dS = (dQ/T)R. During the reversible process, the
energy transfer as heat to the system from the surroundings is given by
Refer to Diagram 5.11. Here T and S are chosen as independent variables. The ∮TdS is
the area under the curve.
Diagram 5.11
The first law of thermodynamics gives dU = dQ ‒ dW. Also for a reversible process, we
can write,
Therefore,
For a cyclic process, the above equation reduces to
For a cyclic process ∮T.dS represents the net heat interaction which is equal to the net
work done by the system. Hence the area enclosed by a cycle on a T−S diagram
1
2
3 4
Equilibrium state
Possible
Impossible
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12 Chapter 5: Entropy
represents the net work done by a system. For a reversible adiabatic process, we know
that
or,
or,
Hence a reversible adiabatic process is also called an isentropic process. On a T−S
diagram, the Carnot cycle can be represented as shown in Diagram 5.11. The area under
the curve 1-2 represents the energy absorbed as heat Q1 by the system during the
isothermal process. The area under the curve 3-4 is the energy rejected as heat Q2 by the
system. The shaded area represents the net work done by the system.
We have already seen that the efficiency of a Carnot cycle operating between two
thermal reservoirs at temperatures T1 and T2 (T2 < T1) is given by
This was derived assuming the working fluid to be an ideal gas. The advantage of T−S
diagram can be realized by a presentation of the Carnot cycle on the T−S diagram. Let
the system change its entropy from SA to SB during the isothermal expansion process 1-
2. Then,
and,
and,
or,
This demonstrates the utility of T-S diagram.
5.6 APPLICATIONS OF ENTROPY PRINCIPLE
The principle of increase of entropy is one of the most important laws of physical
science. It is the quantitative statement of the second law of thermodynamics. Every
irreversible process is accompanied by entropy increase of the universe, and this entropy
increase quantifies the extent of irreversibility of the process. The higher the entropy
increase of the universe, the higher will be the irreversibility of the process. A few
applications of the entropy principle are illustrated in the following.
Application 5.1: (Mixing of Two Fluids): A mass m of water at T1 is isobarically and
adiabatically mixed with an equal mass of water T2.
Show that and prove that this is non-negative.
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Diagram 5.12 Mixing of two fluids
Solution: Consider a general case, Subsystem 1 having a fluid of mass m1, specific heat
c1, and temperature T1, and subsystem 2 consisting of a fluid of mass m2, specific heat c2,
and temperature T2, comprise a composite system in an adiabatic enclosure (Diagram
5.12). When the partition is removed, the two fluids mix together, and at equilibrium, let
Tf be the final temperature, and T2 < Tf < T1. Since energy interaction is exclusively
confined to the two fluids, the system being isolated
Entropy change for the fluid in subsystem 1
This will be negative, since Tf < T1.
Entropy change for the fluid in subsystem 2
This will be positive, since T2 < Tf.
ΔSuni will be positive definite, and the mixing process is irreversible.
Although the mixing process is irreversible, to evaluate the entropy change for the
subsystems, the irreversible path was replaced by a reversible path on which the
integration was performed.
If m1 = m2 = m and c1 = c2 = c.
This is always positive, since the arithmetic mean of any two numbers is always greater
than their geometric mean.
Application 5.2: (Maximum Work Obtainable from Two Finite Bodies at Temperatures
T1 and T2): A heat engine is working between the starting temperature limits of T1 and
T2 of two bodies. Working fluid flows at rate ‘m’ kg/s and has specific heat at constant
pressure as Cp. Determine the maximum obtainable work from engine.
Solution: Let us consider two identical finite bodies of constant heat capacity at
temperatures T1 and T2 respectively, T1, being higher than T2. If the two bodies are
Adiabatic enclosure
Partition
Subsystem 1
Subsystem 2
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14 Chapter 5: Entropy
merely brought together into thermal contact, delivering no work, the final temperature
Tf reached would be the maximum
Diagram 5.13
If a heat engine is operated between the two bodies acting as thermal energy reservoirs
(Diagram 5.13), part of the heat withdrawn from body 1 is converted to work W by the
heat engine, and the remainder is rejected to body 2. The lowest attainable final
temperature Tf corresponds to the delivery of the largest possible amount of work, and is
associated with a reversible process.
As work is delivered by the heat engine, the temperature of body 1 will be decreasing
and that of body 2 will be increasing. When both the bodies attain the final temperature
Tf, the heat engine will stop operating. Let the bodies remain at constant pressure and
undergo no change of phase.
Total heat withdrawn from body 1
where Cp is the heat capacity of the two bodies at constant pressure.
Total heat rejected to body 2
Amount of total work delivered by the heat engine
For given values of Cp, T1, and T2, the magnitude of work W depends on Tf. Work
obtainable will be maximum when Tf is minimum.
Now, for body 1, entropy change ΔS1 is given by
For body 2, entropy change ΔS2 would be
Since the working fluid operating in the heat engine cycle does not undergo any entropy
change, ΔS of the working fluid in heat engine = ∮dS = 0.
Applying the entropy principle
HE
Body 1
Body 2
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15 Basic and Applied Thermodynamics By Brij Bhooshan
From equation (5.25), for Tf to be a minimum
Since Cp ≠ 0,
For W to be a maximum, Tf will be . From equation (5.24)
The final temperatures of the two bodies, initially at T1, and T2, can range from (T1 +
T2)/2 with no delivery of work to with maximum delivery of work.
Application 5.3: (Maximum Work Obtainable from a Finite Body and a TER)
Suppose that one of the bodies considered in the previous application be a thermal
energy reservoir. The finite body has a thermal capacity Cp and is at temperature T and
the TER is at temperature T0, such that T > T0. Let a heat engine operate between the
two (Diagram 5.14).
Diagram 5.14
As heat is withdrawn from the body, its temperature decreases. The temperature of the
TER would, however, remain unchanged at T0. The engine would stop working, when
the temperature of the body reaches T0. During that period, the amount of work
delivered is W, and the heat rejected to the TER is (Q ‒ W). Then
Applying the entropy principle
or
TER T0
HE
Body T
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16 Chapter 5: Entropy
or
Application 5.4: Two identical bodies of constant heat capacity are at the same initial
temperature T1. A refrigerator operates between these two bodies until one body is
cooled to the temperature T2. If the bodies remain at constant pressure and undergo no
change of phase, obtain an expression for the minimum amount of work required to
achieve this.
[Engg. Services 1987]
Diagram 5.15
Solution: Suppose Cp be Specific heat of the body at constant pressure.
For minimum work, the refrigerator should operate on a reversible cycle. If it is so, then
entropy change for the total cycle is equal to zero.
If Tf is the final temperature of the high level reservoir, then the above equation can be
written as
Cyclic work done = Cyclic heat added
work done = heat rejected at the higher temperature − heat picked up at the lower
temperature.
HP
Body B
Body A
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17 Basic and Applied Thermodynamics By Brij Bhooshan
Utilizing Eqn. (i) and (ii), then we have
Application 5.5: An ideal gas is heated from temperature T1 to T2 by keeping its
volume constant. The gas is expanded back to it's initial temperature according to the
law pvn = constant. If the entropy changes in the two processes are equal, find the value
of ‘n’ in terms of adiabatic index γ.
[Engg. Services-1997]
Diagram 5.16
Solution: During constant volume process change in entropy
Change in entropy during polytropic process,
Since the entropy change is same, so
Application 5.6: A cool body at temperature T1 is brought in contact with high
temperature reservoir at temperature T2. Body comes in equilibrium with reservoir at
constant pressure. Considering heat capacity of body as C, show that entropy change of
universe can be given as;
Solution: Since body is brought in contact with reservoir at temperature T2, the body
shall come in equilibrium when it attains temperature equal to that of reservoir, but
there shall be no change in temperature of the reservoir.
Entropy change of universe ΔSuni = ΔSbody + ΔSreservoir
1 2 3
T
3
2
1
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18 Chapter 5: Entropy
as, heat gained by body = Heat lost by reservoir
Thus, ΔSuni is
or, rearranging the terms,
5.7 ENTROPY GENERATION IN A CLOSED SYSTEM
The entropy of any closed system can increase in two ways:
(a) by heat interaction in which there is entropy transfer
(b) internal irreversibilities or dissipative effects in which work (or K.E.) is
dissipated into internal energy increase.
If dQ is the infinitesimal amount of heat transferred to the system through its boundary
at temperature T, the same as that of the surroundings, the entropy increase dS of the
system can be expressed as
where deS is the entropy increase due to external heat interaction and diS is the entropy
increase due to internal irreversibility. From Eq. (5.28),
The entropy increase due to internal irreversibility is also called entropy production or
entropy generation, Sgen.
In other words, the entropy change of a system during a process is greater than the
entropy transfer (dQ/T) by an amount equal to the entropy generated during the
process within the system (diS), so that the entropy balance gives:
Entropy change = Entropy transfer + Entropy generation
which is a verbal statement of Eq. (5.28).
It may so happen that in a process (e.g., the expansion of a hot fluid in a turbine) the
entropy decrease of the system due to heat loss to the surroundings (‒∫dQ/T) is equal to
the entropy increase of the system due to internal irreversibilities such as friction, etc.
(‒∫diS), in which case the entropy of the system before and after the process will remain
the same (∫dS = 0). Therefore, an isentropic process need not be adiabatic or reversible.
But if the isentropic process is reversible, it must be adiabatic. Also, if the isentropic
process is adiabatic. it cannot but be reversible. An adiabatic process need not be
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19 Basic and Applied Thermodynamics By Brij Bhooshan
isentropic, since entropy can also increase due to friction etc. But if the process is
adiabatic and reversible, it must be isentropic.
For an infinitesimal reversible process by a closed system,
If the process is irreversible,
Since U is a property,
The difference (pdV ‒ dW) indicates the work that is lost due to irreversibility, and is
called the lost work d(LW), which approaches zero as the process approaches
reversibility as a limit. Equation (5.30) can be expressed in the form
Thus the entropy of a closed system increases due to heat addition (deS) and internal
dissipation (diS ).
Diagram 5.17
In any process executed by a system, energy is always conserved, but entropy is
produced internally. For any process between equilibrium states 1 and 2 (Diagram 5.17),
the first law can be written as
Energy transfer Energy change
By second law,
It is only the transfer of energy as heat which is accompanied by entropy transfer, both
of which occur at the boundary where the temperature is T. Work interaction is not
accompanied by any entropy transfer. The entropy change of the system (S2 ‒ S1)
exceeds the entropy transfer
The difference is produced internally due to irreveersibility. The amount of entropy
generation Sgen is given by
Surrounding
(Work Transfer)
(Entropy Transfer)
(Heat Transfer)
T
(Boundary
temperature)
Boundary System 1→2
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20 Chapter 5: Entropy
The second law states that, in general, any thermodynamic process is accompanied by
entropy generation.
Diagram 5.18
Process 1-2, which does not generate any entropy (Sgen = 0), is a reversible process
(Diagram 5.18). Paths for which Sgen > 0 are considered irreversible. Like heat transfer
and work transfer during the process 1-2, the entropy generation also depends on the
path the system follows. Sgen is, therefore, not a thermodynamic property and dSgen is an
inexact differential, although (S2 ‒ S1) depends only on the end states. In the differential
form, Eq. (5.31) can be written as
The amount of entropy generation quantifies the intrinsic irreversibility of the process. If
the path A causes more entropy generation than path B (Diagram 5.18), i.e.
the path A is more irreversible than path B and involves more 'lost work'.
If heat transfer occurs at several locations on the boundary of a system, the entropy
transfer term can be expressed as a sum, so Eq. (5.31) takes the form
where Qj/Tj is the amount of entropy transferred through the portion of the boundary at
temperature Tj.
On a time rate basis, the entropy balance can be written as
where dS/dτ is the rate of change of entropy of the system, is the rate of entropy
transfer through the portion of the boundary whose instantaneous temperature is Tj,
and is the rate of entropy generation due to irreversibilities within the system.
5.8 ENTROPY GENERATION IN AN OPEN SYSTEM
In an open system, there is transfer of three quantities: mass, energy and entropy. The
control surface can have one or more openings for mass transfer (Diagram 5.19). It is
rigid, and there is shaft work transfer across it.
The continuity equation gives
Rev
1
2
B A
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21 Basic and Applied Thermodynamics By Brij Bhooshan
Diagram 5.19
Net mass transfer rate = rate of mass accumulation in the CV
The energy equation gives
Net rate of energy transfer = rate of energy accumulation in the CV
The second law inequality or the entropy principle gives
Net rate of entropy transfer = rate of increase of entropy of the CV
Here represents the rate of heat transfer at the location of the boundary where the
instantaneous temperature is T. The ratio /T accounts for the entropy transfer along
with heat. The terms ṁisi and ṁese account, respectively, for rates of entropy transfer
into and out of the CV accompanying mass flow. The rate of entropy increase of the
control volume exceeds, or is equal to, the net rate of entropy transfer into it. The
difference is the entropy generated within the control volume due to irreversibility.
Hence, the rate of entropy generation is given by
By the second law,
If the process is reversible, = 0. For an irreversible process, > 0.
The magnitude of quantifies the irreversibility of the process. If systems A and B
operate so that
it can be said that the system A operates more irreversibly than system B.
At steady slate, the continuity equation gives
the energy equation becomes
CS
(Entropy Transfer Rate)
Surrounding
Shaft Work Transfer Rate
(Heat Transfer Rate)
Surface temperature, T
S
M E
Control volume
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22 Chapter 5: Entropy
and the entropy equation reduces to
These equations often must be solved simultaneously, together with appropriate
property relations.
Mass and energy are conserved quantities, but entropy is not generally conserved. The
rate at which entropy is transferred out must exceed the rate at which entropy enters
the CV, the difference being the rate of entropy generated within the CV owing to
irreversibilities.
For one-inlet and one-exit control volumes, the entropy equation becomes
5.9 FIRST AND SECOND LAWS COMBINED
By the second law
dQ = T.dS
and by the first law, for a closed non-flow system,
dQ = dU + p.dV
T.dS = dU + p.dV [5.43]
Again, the enthalpy
H = U + pV
dH = dU + p.dV + V.dp = T.dS + V.dp
T.dS = dH – V.dp [5.44]
Equations (5.43) and (5.44) are the thermodynamic equations relating the properties of
the system.
Let us now examine the following equations as obtained from the first and second laws:
(a) dQ = dE + dW→ This equation holds good for any process, reversible or
irreversible, and for any system.
(b) dQ = dU + pdW→ This equation holds good for any process undergone by a closed
stationary system.
(c) dQ = dU + pdV→ This equation holds good for a closed system when only pdV-
work is present. This is true only for a reversible (quasi-static) process.
(d) dQ = TdS→ This equation is true only for a reversible process.
(e) TdS = dU + pdV→ This equation holds good for any process reversible or
irreversible, undergone by a closed system, since it is a relation among properties
which are independent of the path,
(f) TdS = dH ‒ Vdp→ This equation also relates only the properties of a system.
There is no path function term in the equation. Hence the equation holds good for
any process.
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23 Basic and Applied Thermodynamics By Brij Bhooshan
The use of the term 'irreversible process' is doubtful, since no irreversible path or
process can be plotted on thermodynamic coordinates. It is more logical to state that 'the
change of state is irreversible, rather than say 'it is an irreversible process'. A natural
process which is inherently irreversible is indicated by a dotted line connecting the
initial and final states, both of which are in equilibrium. The dotted line has no other
meaning, since it can be drawn in any way. To determine the entropy change for a real
process, a known reversible path is made to connect the two end states, and integration
is performed on this path using either equation (e) or equation (f'), as given above.
Therefore, the entropy change of a system between two identifiable equilibrium states is
the same whether the intervening process is reversible or the change of state is
irreversible.
For constant pressure process, dh = T·ds
or
which means slope of constant pressure line on enthalpy – entropy diagram (h – s) is
given by temperature.
Also from above two relations
For a constant pressure process above yields
It gives the slope of constant pressure line on T – s diagram.
Similarly, for a constant volume process,
It gives the slope of constant volume line on T – s diagram.
It can be concluded from the above mathematical explanations for slope that slope of
constant volume line is more than the slope of constant pressure line as cp > cv.
Diagram 5.20
5.10 ENTROPY CHANGE FOR SOME ELEMENTARY PROCESSES
Isothermal Process
Let us find out entropy change for isothermal heat addition process. As isothermal
process can be considered internally reversible, therefore entropy change shall be;
Constant pressure line
Constant volume line
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24 Chapter 5: Entropy
where Qa‒b is total heat interaction during state change a – b at temperature T.
Reversible Adiabatic Work in a Steady Flow System
In the differential form, the steady flow energy equation per unit mass is given by,
For a reversible process, dQ = Tds
Using the property relation, Eq. (5.45), per unit mass,
in Eq. (5.45), we have
On integration
If the changes in KE and P.E. are neglected, Eq. (5.47) reduces to
If dQ = 0, implying ds = 0, the property relation gives
or
From Eqs (5.48) and (5.49),
The integral
represents an area on the p-v plane (Diagram 5.21a). To make the
integration, one must have a relation between p and v such as pvn = constant
Diagram 5.21 Reversible work interaction
(a) Steady flow
2
1
(b) A closed system
2
1
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25 Basic and Applied Thermodynamics By Brij Bhooshan
Equation (5.50) holds good for a steady flow work-producing machine like an engine or
turbine as well as for a work-absorbing machine like a pump or a compressor, when the
fluid undergoes reversible adiabatic expansion or compression.
For closed system (Diagram 5.21b), the reversible work done would be
Isentropic Process
It is the process during which change in entropy is zero and entropy remains constant
during process.
It indicates that when ΔSa‒b = 0, then Qa‒b = 0
which means there is no heat interaction during such process and this is adiabatic
process.
Hence, it can be said that "a reversible isentropic process shall be adiabatic, where as if
isentropic process is adiabatic then it may or may not be reversible’’.
Thus, adiabatic process may or may not be reversible. It means in reversible adiabatic
process all states shall be in equilibrium and no dissipative effects are present along
with no heat interaction whereas in adiabatic process there is no heat interaction but
process may be irreversible.
Finally, it can be concluded that an adiabatic process may or may not be isentropic
whereas a reversible adiabatic process is always isentropic.
An adiabatic process of non isentropic type is shown in Diagram 5.22 where
irreversibility prevails, say due to internal friction.
Diagram 5.22
Here a – b is reversible adiabatic expansion of isentropic type.
Non-isentropic or adiabatic expansion is shown by a – b'.
Isentropic expansion efficiency may be defined as ratio of actual work to ideal work
available during expansion.
Similarly, isentropic and non-isentropic compression process are shown as c–d and c–d'
respectively.
Isentropic compression efficiency can be defined on same lines as,
Absorption of energy by a constant temperature reservoir: A certain amount of heat is
added to a constant temperature reservoir. The actual process can be replaced by a
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26 Chapter 5: Entropy
reversible path in which an equivalent amount of energy is added to the reservoir. Then,
the entropy change of the reservoir is given by
Heating or Cooling of matter: The heating can be carried out either at constant pressure
or at constant volume. From the first law of thermodynamics
Q = ΔU for constant volume heating/cooling process
Q = ΔH for constant pressure heating/cooling process
or,
Similarly,
or,
Phase change at constant temperature and pressure
Melting:
Evaporation:
For Ideal/Perfect Gases
Suppose a certain quantity of a perfect gas being heated by any thermodynamics
process. Suppose
m = mass of gas,
p1 = initial pressure of a gas,
v1 = initial volume of a gas,
T1 = initial temperature of a gas,
p2, v2, T2 are corresponding final values of the gas,
dT = small change in temperature, and
dv = small change in volume.
Now from first law of thermodynamics
Dividing by T
On integration
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27 Basic and Applied Thermodynamics By Brij Bhooshan
Now from general equation of gas
Using equation (5.51a), then we have
Now we know that Cp ‒ Cv = R, then
Again from general equation of gas
Using equation (5.51a), then we have
Polytropic Process
Entropy change in a polytropic process having governing equation as pvn = constant, can
be obtained as below,
For polytropic process between 1 and 2,
Also, from gas laws,
Above two pressure ratios give,
Now using equation (5.61a), then we have
For perfect gas R = cp – cv, then R = cv (γ ‒1)
Substituting R in entropy change
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28 Chapter 5: Entropy
5.11 THIRD LAW OF THERMODYNAMICS
‘Third law of thermodynamics’, an independent principle uncovered by ‘Nernst’ and
formulated by ‘Planck’, states that the “Entropy of a pure substance approaches zero at
absolute zero temperature.” This fact can also be corroborated by the definition of
entropy which says it is a measure of molecular disorderness. At absolute zero
temperature substance molecules get frozen and do not have any activity, therefore it
may be assigned zero entropy value at crystalline state. Although the attainment of
absolute zero temperature is impossible practically, however theoretically it can be used
for defining absolute entropy value with respect to zero entropy at absolute zero
temperature. Second law of thermodynamics also shows that absolute zero temperature
can’t be achieved. Third law of thermodynamics is of high theoretical significance for the
sake of absolute property definitions and has found great utility in thermodynamics.