Objectives
• Define a new property called entropy to quantify the second-law
effects.
• Establish the increase of entropy principle.
MAE 320 - Chapter 7
• Calculate the entropy changes that take place during processes
for pure substances, incompressible substances, and ideal
gases.
Entropy
• Examine a special class of idealized processes, called isentropic
processes, and develop the property relations for these
processes.
• Derive the reversible steady-flow work relations.
• Develop the isentropic efficiencies for various steady-flow
devices.
• Introduce and apply the entropy balance to various systems.
The content and the pictures are from the text book: Çengel, Y. A. and Boles, M. A., “Thermodynamics:
An Engineering Approach,” McGraw-Hill, New York, 6th Ed., 2008
Definition of Entropy
Definition of Entropy
For the reversible engine:
If we extend this to a thermodynamic cycle, there is a new statement as
follows:
QH QL 1000kJ − 300kJ
+
=
+
=0
TH TL
1000k
300 K
1000kJ
1000kJ
1000kJ
Clausius inequality: for any thermodynamic cycle, reversible or irreversible,
the cyclic integral of δQ/T is always less than or equal to zero.
For the irreversible engine:
QH QL 1000kJ − 550kJ
+
=
+
TH TL
1000k
300 K
The symbol ∫ (integral symbol with a circle in the middle) is used to indicate
that the integration is to be performed over the entire cycle.
= −0.83 < 0
300kJ
For the impossible engine:
Q H QL 1000kJ − 200kJ
+
=
+
TH TL
1000k
300 K
550kJ
300k
300K
200kJ
can be viewed as the sum of all the differential amount of heat transfer
divided by the temperature at the boundary.
300K
For a device that undergoes an internally reversible cycle:
= 0.33 > 0
δQ
δQ
QH QL
δQ
+
= ( H + L)=∫
TH TL ∫ TH
TL
T
<0 irreversible engine
=0 reversible engine
>0 impossible engine
Definition of Entropy
The equality in the Clausius inequality holds for totally or just internally
reversible cycles and the inequality for the irreversible ones.
Definition of Entropy
Entropy is a property, like all other properties, it has a fixed value at a fixed
sate. The entropy change between two specified states is the same whether
the process is reversible or irreversible.
A quantity whose cyclic integral is zero (i.e., a
property like volume) typically depends on the
state and not the process path. Thus such a
quantity is a property. Therefore (δQ/T)in, rev must
represent a property in the differential form.
In a special case (an internal reversible
process), the entropy change between two
specified states:
Clausius has realized this point and discovered
a thermodynamic property named entropy
Entropy (S) is an extensive property of a system.
Entropy per unit mass, designated s, is an
intensive property with the unit kJ/kg.K.
Note: Cv, Cp and R have the same unit kJ/kg.K
For a closed system that undergoes an
irreversible process, the entropy change
between two specified states:
The net change in volume
(a property) during a cycle
is always zero.
+Δ
1
Entropy
The Increase of Entropy Principle
A Special Case: Internally Reversible Isothermal Heat Transfer Processes
Considering a cycle that is made of two
processes: Process 1-2, which is
arbitrary (reversible or irreversible), and
Process 2-1, which is internally reversible.
From the Clausius inequality:
Recall that isothermal heat transfer processes are internally reversible:
This equation is particularly useful for determining the entropy changes of
thermal energy reservoirs that can adsorb or supply heat indefinitely at a
constant temperature.
Heat transfer to a system increases the entropy of a system, whereas heat
transfer from a system decreases it. In fact, losing heat is the only way the
entropy of a system can be decreased.
The Increase of Entropy Principle
The term is equal to the entropy change S1-S2:
A cycle composed of
a reversible and an
irreversible process.
(1).The equality holds for an internally
reversible process. The entropy change
becomes equal to
, which represent
entropy transfer with heat.
(2). The inequality for an irreversible process.
The Increase of Entropy Principle
The inequality indicate that the entropy change of a closed system during an
irreversible process is greater than entropy transfer. In other words, some
entropy is generated or created during an irreversible process. The entropy
generated is called entropy generation, Sgen.
Rewrite the above equation
The entropy generation Sgen is always a positive quantity or zero. Its value
depends on the process path, and thus it is not a property of a system
For an isolated system (an adiabatic closed system), the heat transfer is
zero, thus
The entropy change of an
isolated system is the sum of the
entropy changes of its
components, and is never less
than zero.
A system and its surroundings
form an isolated system.
This equation indicates that the entropy of an isolated system during a process
always increases. It never decreases. – Increase of entropy principle
The Increase of Entropy Principle
Some entropy is generated or created during an irreversible process, and
this generation is due entirely to the presence of irreversibilities.
The Increase of entropy principle can summarized as follows:
Some Remarks about Entropy
1. Processes can occur in a certain direction only, not in any direction. A
process must proceed in the direction that complies with the increase of
entropy principle, that is, Sgen ≥ 0. A process that violates this principle is
impossible.
2. Entropy is a nonconserved property, and there is no such thing as the
conservation of entropy principle. Entropy is conserved during the
idealized reversible processes only and increases during all actual
processes.
(1). The entropy generation, Sgen, is not
a property of a system. Its value
depends on the process path. It is
always a positive quantity or zero.
(2). The entropy, S, is a property. It is
independent on the path. The entropy
change (S2-S1) can be zero, positive, or
negative
3. The performance of engineering systems is degraded by the presence of
irreversibilities, and entropy generation is a measure of the magnitudes
of the irreversibilities during that process. It is also used to establish
criteria for the performance of engineering devices.
4. Entropy can transferred by two ways: (i) heat transfer and (ii) mass
transfer. Net work can not transfer entropy.
2
Energy balance & Entropy balance of closed system
The Increase of Entropy Principle
Conservation of energy is applied to a closed system that undergoes any processes
regardless of reversible of irreversible, hence the energy balance equation:
E in − E out = ΔE sys
Q in − Q out + W in − W out = Δ E system = Δ U + Δ PE + Δ KE
There is no conservation of entropy for a closed system that undergoes any
processes regardless of reversible of irreversible, hence the entropy balance equation:
Qin Qout
−
+ S gen = ΔS sys = S 2 − S1
Tin Tout
In a special case, a closed system undergoes an internal reversible processes, then
the entropy balance equation becomes:
Qin Qout
−
= ΔS sys = S 2 − S1
Tin Tout
The Increase of Entropy Principle
The Increase of Entropy Principle
Example 7-2
Example 7-2
Entropy Change of Pure Substances
Entropy Change of Pure Substances
Entropy is a property, and thus the value of entropy of a system is fixed once
the state of the system is fixed.
The value of entropy at a specific state is determined just like any other
proper.
In the saturated region:
Especially, the T-s
diagram of water is shown
in Figure A-9 in Pg. 926
The entropy of a compressed liquid:
P=
co
ns
ta n
t
The entropy of a pure
substance is determined
from the tables (like other
properties).
T-s diagram of water.
3
Entropy Change of Pure Substances
Entropy Change of Pure Substances
Example 7-3
Entropy Change of Pure Substances
Entropy Change of Pure Substances
Example 7-3
Example 7-3
Table A-13
Superheated
vapor
Table A-12
Isentropic Processes
Isentropic Processes
A process during which the entropy remains constant is called an isentropic
process.
During an internally reversible
and adiabatic process, the
entropy remains constant.
The isentropic process appears as a
vertical line segment on a T-s
diagram.
4
Isentropic Processes
Isentropic Processes
Example 7-5
Example 7-5
From Table A-5, the Tsat= 263.94 oC when P1= 5000 kPa.
T1=450 > Tsat= 263.94 oC, Therefore, it is superheated vapor under the State 1. So
s1 can be obtained from Table A-6, s1=6.8210 KJ/kg K and h1= 3317.2 kJ/kg k
(page 922)
Property Diagrams Involving Entropy
Isentropic Processes
For an internal reversible process:
Example 7-5
From Table A-5, the Sg= 6.4675 KJ/kg K when P1= 1400 kPa.
On a T-S diagram, the area under
the process curve represents the
heat transfer for internally reversible
processes. But the area has no
meaning for an irreversible process
s2=s1= 6.8210> sg@p=1400 kPa = 6.4675 kJ/kg K. Therefore, it is superheated vapor
under the State 2. So h2 can be obtained from Table A-6, h2=2967.4 kJ/kg K
through interpolation (page 921)
Temperature-entropy diagram
Method (2), Figure A-9 in Page 926 to get the h2
For an internal reversible isothermal process:
or
Property Diagrams Involving Entropy
Property Diagrams Involving Entropy
An isentropic process appears
as a vertical line segment on a
T-s diagram. This expected
since an isentropic process
involves no heat transfer, and
therefore the area under the
process path must be zero.
P-v diagram of the Carnot Cycle
The T-s diagram serves as a
valuable tool for visualizing the
second law aspects of
processes and cycles.
The T-s diagram of the Carnot Cycle
An isentropic process appears as a
vertical line segment on a T-s diagram
5
Property Diagrams Involving Entropy
What is Entropy?
Entropy is a measure of molecular disorder or molecular randomness
The h-s diagram is useful
analysis of adiabatic steadyflow devices, such as
turbines, compressors and
nozzles:
(i). The vertical distance ∆h
(between the inlet and the exit
states) on an h-s diagram is a
measure of work.
(ii). The horizontal distance ∆s
is a measure of irreversibilities
associated with the process.
Mollier diagram: The h-s diagram
Fig. A-10 in Appendix in Page 927
The T ds Relations
The level of molecular disorder
(entropy) of a substance increases
as it melts or evaporates.
A pure crystalline substance at
absolute zero temperature is in
perfect order, and its entropy is zero
(the third law of thermodynamics).
The T ds Relations
For a closed system that undergoing an internal reversible process:
For a closed system that undergoing an internal reversible process:
Tds – PdV = du
the first T ds, or Gibbs equation
Tds – PdV = du
the first T ds, or Gibbs equation
the second T ds equation
The T ds Relations
The T ds relations are derived from an internal reversible process. However,
it is valid for both reversible and irreversible processes and for both closed
and open systems, since entropy is a property and the change in a property
between two states is independent on the type of process the system
undergoes.
The T ds relations are valid
for both reversible and
irreversible processes and
for both closed and open
systems.
the second T ds equation
Entropy Change of Liquids and Solids
Since
for liquids and solids
Liquids and solids can be
approximated as
incompressible substances
since their specific volumes
remain nearly constant during
a process.
An isentropic process of an incompressible substance is also an isothermal
process:
6
The Entropy Change of Ideal Gases
Constant Specific Heats (Approximate Analysis)
From the first T ds relation
Assuming constant specific heats for idea gases is a common approximation:
From the second T ds relation
for ideal gas
Under the constant-specific-heat
assumption, the specific heat is
assumed to be constant at some
average value.
Variable Specific Heats (Exact Analysis)
Variable Specific Heats
We choose absolute zero as the reference
temperature and define a function s° as
On a unit–mass basis
On a unit–mole basis
Variable Specific Heats
Example 7-9
The entropy of an ideal
gas depends on both T
and P. The function s
represents only the
temperature-dependent
part of entropy.
Variable Specific Heats
Example 7-9
7
Isentropic Processes of Ideal Gases
Isentropic Processes of Ideal Gases
Constant Specific Heats (Approximate Analysis)
Variable Specific Heats (Exact Analysis)
S2- S1 =
Setting this eq. equal to zero, we get
Relative Pressure and Relative Specific Volume
exp(s°/R) is the relative pressure Pr
The use of Pr data for calculating the final
temperature during an isentropic process.
The isentropic relations of ideal gases
are valid for the isentropic processes
of ideal gases only.
T/Pr is the relative specific volume vr
The use of vr data for calculating the final
temperature during an isentropic process
Isentropic Processes of Ideal Gases
Isentropic Processes of Ideal Gases
Variable Specific Heats (Exact Analysis)
Table A-17
The use of Pr data for calculating
the final temperature during an
isentropic process.
Table A-17
The use of vr data for calculating
the final temperature during an
isentropic process
Isentropic Processes of Ideal Gases
Example 7-10
Isentropic Processes of Ideal Gases
Example 7-10
If we use k=1.400 at the initial temperature 295 K
T2=(295K) (8)1.4-1
T2 = 667.7 K
Thus we can estimate the average
temperature is 481 K
At 481k, K=1.389 based on Table A-2b
T2= 662.4 K
8
Isentropic Efficiencies of Steady-flow Devices
Isentropic Efficiencies of Steady-flow Devices
Isentropic process: internal reversible, adiabatic
The isentropic process involves no irreversibilities and serves as the ideal
process for adiabatic devices.
Isentropic Efficiency of Turbines:
Isentropic Efficiency of Turbines:
Isentropic efficiency is also called the second law efficiency, which is a
measure of the deviation of actual processes from the corresponding
idealized ones.
Thermal efficiency is
called the first law
efficiency.
Typically a turbine under
the steady operation, the
inlet state of the working
fluid and the exhaust
pressure is fixed
Isentropic Efficiencies of Steady-flow Devices
The h-s diagram for the actual
and isentropic processes of an
adiabatic turbine.
Isentropic Efficiencies of Steady-flow Devices
3. For a turbine under the steady operation,
the inlet state of the working fluid and the exhaust pressure is fixed.
Isentropic Efficiencies of Steady-flow Devices
Example 7-14
Isentropic Efficiencies of Steady-flow Devices
Example 7-14
, State 2s is in the liquid-vapor region
9
Isentropic Efficiencies of Compressors
Isentropic Efficiencies of Compressors
Isentropic Efficiencies of Compressors
Isentropic Efficiencies of Compressors
When kinetic and potential energies are
negligible
Can you use isentropic efficiency for a nonadiabatic compressor?
Can you use isothermal efficiency for an
adiabatic compressor?
For a reversible isothermal process, then we
can define an isothermal efficiency:
Wt and Wa are the required work inputs to the
compressor for the reversible isothermal and
actual processes.
The h-s diagram of the actual and
isentropic processes of an
adiabatic compressor
Isentropic Efficiencies of Pumps
Compressors are
sometimes intentionally
cooled to minimize the
work input.
Isentropic Efficiencies of Pumps
Isentropic Efficiencies of Pumps:
When kinetic and potential energies
of a liquid are negligible
Isentropic Efficiencies of Pumps
Example 7-15
Isentropic Efficiencies of Pumps
Example 7-15
State 1:
10
Isentropic Efficiencies of Pumps
Example 7-15
Isentropic Efficiency of Nozzles
Isentropic Efficiency of Nozzles is defined as:
•
•
E in = E out
•
Since Q = 0,
•
W = 0,
Δpe ≅ 0
V2
V
m( h1 +
) = m( h2 a + 2 a )
2
2
•
2
1
•
If the inlet velocity of the fluid is small
relative to the exit velocity, the energy
balance is
The h-s diagram of the actual and
isentropic processes of an
adiabatic nozzle.
Then
Isentropic process of Nozzles
Isentropic process of Nozzles
Isentropic process of Nozzles
Isentropic process of Nozzles
Example 7-16
Example 7-16
V2 s = 2(h1 − h2 ) = 2(998000 − 766000) = 666
(m / s )
Alternatively, use a constant Cp
0.92 =
h1 = 988 kJ/kg = 988,000 J/kg at 950K (Table A-17)
h2s = 766 kJ/kg = 766,000 J/kg at 748K
V1 = 0
Attention: unit conversion
988 − h2 a
988 − 766
h2a = 783.8 kJ/kg
From table A-17: h=800.03kJ/kg at 780K, and h=778.18 kJ/Kg at 760K
Interpolation:
780 − T2 a 800.03 − 783.8
=
780 − 760 800 − 778.18
T2a = 765K
11
Isentropic process of Nozzles
Entropy Balance
Example 7-16
Alternatively, use a constant Cp
Entropy Change of a System, ∆Ssystem
When the properties of the system
are not uniform
Energy and entropy balances for a system
Entropy Generation, Sgen
Mechanisms of Entropy Transfer
Attention: the unit for following three forms:
Entropy can be transferred to or from a system by two mechanisms: heat
transfer and mass flow.
Entropy transfer by heat:
No entropy accompanies work as it
crosses the system boundary. But
entropy may be generated within the
system as work is dissipated into a
less useful form of energy.
(1). For an irreversible process, the entropy generation (Sgen) is greater
than zero
(2). For a reversible process, the entropy generation (Sgen) is zero and thus
the entropy change of a system is equal to the entropy transfer
Mechanisms of Entropy Transfer
Entropy transfer by work:
Heat transfer is always accompanied by
entropy transfer in the amount of Q/T,
where T is the boundary temperature.
Entropy Change of a System
Entropy transfer by mass:
When the properties of the mass change during the process
Mass contains entropy as well as
energy, and thus mass flow into or out
of system is always accompanied by
energy and entropy transfer.
The entropy change between the final state and the initial state of
system that undergoes a process:
1) The entropy transfer by heat
2) The entropy transfer by mass flow
3) The irreversibility extent of the process
Mechanisms of entropy transfer for a general system.
12
Entropy Balance of Closed Systems
Entropy Balance of Closed Systems
A closed system involves no mass transfer across its boundary
Noting that any closed system and its surroundings can be treated as an
adiabatic system, and then:
For an adiabatic process, no heat transfer across the boundary of a closed
system, the entropy change of the system only depends on the irreversibility
of the process:
Entropy generation outside system
boundaries can be accounted for by
writing an entropy balance on an
extended system that includes the
system and its immediate
surroundings.
surr
Balance Equations of Closed Systems
Entropy Balance of Control Volumes
A closed system involves no mass transfer across its boundary
As compared with a closed system, a control volume involves mass transfer
across its boundary. Thus the entropy balance:
Energy balance:
(Qin − Qout ) + (Win − Wout ) = ΔU + ΔKE + ΔPE
For a closed system without change in the KE and PE
Qnet ,in − Wnet ,out = Δ U
Entropy balance:
Entropy Balance of Control Volumes
Qk
•
•
∑ T + ∑m s − ∑m
i
i
Mass balance :
•
e
se + S gen = 0
Balance Equations of Control Volumes
For a general control volume:
For a general steady-flow process:
•
The entropy change within a control volume
during a process is equal to the sum:
(1) Entropy transfer by heat
(2) the net entropy transfer by mass flow
(3) the entropy generation as result from
irreversibility
(kW/K)
k
For a single-stream, steady-flow device:
•
Energy balance :
•
•
Q
∑ T k + m( si − se ) + S gen = 0
k
For an adiabatic, single-stream, steady-flow device:
•
•
m( si − se ) + S gen = 0
Entropy balance :
For an adiabatic, single-stream, steady-flow device that undergoes a
reversible process:
si = se
13
Balance Equations of Steady-flow Control Volumes
Entropy Generation
For a steady-flow control volume:
Mass balance :
Energy balance :
•
Entropy balance :
Qk
•
•
∑ T + ∑ m s − ∑m
i
i
e
•
se + S gen = 0
k
Entropy Generation
Example 7-18
Entropy Generation
Example 7-18
•
Qk
•
•
∑ T + ∑m s − ∑m
i
i
e
•
se + S gen = 0
k
(Table A-6)
(Table A-6)
Interpolation
Entropy Generation
Entropy Generation
Example 7-19
14
Entropy Generation
Entropy Generation
Example 7-19
Now we take the lake as the closed system.
Cavg=
The heat transfer form the iron block to the lake while the lake keeps a
constant temperature (285K). Therefore, we can consider this isothermal
process is an internal reversible. Thus Sgen =0
For the iron block:
Qnet ,in − Wnet ,out = ΔU
Entropy Generation
Entropy Generation
Example 7-19
Entropy Generation
Entropy Generation
Example 7-20
Energy Balance:
•
Qk
•
•
∑ T + ∑ m s − ∑m
i
i
e
•
se + S gen = 0
k
h1 = ?
s1 = ?
15
Summary
• Entropy
• The Increase of entropy principle
• Entropy change of pure substances
• Isentropic processes
• Property diagrams involving entropy
• The T ds relations
• Entropy change of liquids and solids
• The entropy change of ideal gases
• Isentropic efficiencies of steady-flow devices
• Entropy balance
16