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MAE 320 - Chapter 7
Entropy
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
Objectives
- Define a new property called entropy to quantify the second-law
effects.
- Establish the increase of entropy principle.
- Calculate the entropy changes that take place during processes
for pure substances, incompressible substances, and ideal
gases.
- 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.
Definition of Entropy
300K 300k^ 300K
For the reversible engine:
0 300
K
kJ k
kJ T
Q
T
Q
L
L H
H
1000kJ (^) 1000kJ 1000kJ
300kJ 550kJ (^) 200kJ
For the irreversible engine:
For the impossible engine:
K
kJ k
kJ T
Q
T
Q
L
L H
H
K
kJ
k
kJ
T
Q
T
Q
L
L H
H
<0 irreversible engine
=0 reversible engine
>0 impossible engine
+ =∫ + = ∫ T
Q
T
Q
T
Q
T
Q
T
Q
L
L H
H L
L H
H (^ δ^ δ )^ δ
Clausius inequality: for any thermodynamic cycle, reversible or irreversible, the cyclic integral of δQ/T is always less than or equal to zero.
The equality in the Clausius inequality holds for totally or just internally reversible cycles and the inequality for the irreversible ones.
Definition of Entropy
If we extend this to a thermodynamic cycle, there is a new statement as follows:
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.
∫
can be viewed as the sum of all the differential amount of heat transfer divided by the temperature at the boundary.
For a device that undergoes an internally reversible cycle:
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. The net change in volume (a property) during a cycle is always zero.
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 ( δ Q/T)in, rev must
represent a property in the differential form.
Definition of Entropy
Clausius has realized this point and discovered a thermodynamic property named entropy
Note: C (^) v , Cp and R have the same unit kJ/kg.K
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.
In a special case (an internal reversible process), the entropy change between two specified states:
For a closed system that undergoes an irreversible process, the entropy change between two specified states:
A Special Case: Internally Reversible Isothermal Heat Transfer Processes
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.
Entropy
Recall that isothermal heat transfer processes are internally reversible:
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
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.
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 :
The term is equal to the entropy change S 1 -S 2 :
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.
The entropy generation S gen is always a positive quantity or zero. Its value depends on the process path, and thus it is not a property of a system
Rewrite the above equation
For an isolated system (an adiabatic closed system), the heat transfer is zero, thus
This equation indicates that the entropy of an isolated system during a process always increases. It never decreases. – Increase of entropy principle
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.
The 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:
(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 (S 2 -S 1 ) can be zero, positive, or negative
Some Remarks about Entropy
- 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, S gen ≥ 0. A process that violates this principle is impossible.
- 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.
- 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.
- Entropy can transferred by two ways: (i) heat transfer and (ii) mass transfer. Net work can not transfer entropy.
Entropy Change of Pure Substances Entropy Change of Pure Substances
Example 7-
Entropy Change of Pure Substances
Example 7- Table A- Superheated vapor
Table A-
Entropy Change of Pure Substances Example 7-
Isentropic Processes
During an internally reversible and adiabatic process, the entropy remains constant.
A process during which the entropy remains constant is called an isentropic process.
The isentropic process appears as a vertical line segment on a T-s diagram.
Isentropic Processes
Isentropic Processes
Example 7-
Isentropic Processes Example 7-
From Table A-5, the Tsat= 263.94 oC when P 1 = 5000 kPa.
T 1 =450 > Tsat= 263.94 oC , Therefore, it is superheated vapor under the State 1. So s 1 can be obtained from Table A-6, s 1 =6.8210 KJ/kg K and h 1 = 3317.2 kJ/kg k (page 922)
Isentropic Processes
Method (2), Figure A-9 in Page 926 to get the h 2
Example 7-
From Table A-5, the S (^) g= 6.4675 KJ/kg K when P 1 = 1400 kPa.
s 2 =s 1 = 6.8210> s (^) g@p=1400 kPa = 6.4675 kJ/kg K. Therefore, it is superheated vapor under the State 2. So h 2 can be obtained from Table A-6, h 2 =2967.4 kJ/kg K through interpolation (page 921)
Property Diagrams Involving Entropy
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
For an internal reversible process:
For an internal reversible isothermal process:
Temperature-entropy diagram
or
Property Diagrams Involving Entropy
P-v diagram of the Carnot Cycle The T-s diagram of the Carnot Cycle
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.
The T-s diagram serves as a valuable tool for visualizing the second law aspects of processes and cycles.
An isentropic process appears as a vertical line segment on a T- s diagram
The Entropy Change of Ideal Gases
From the first T ds relation From the second T ds relation
for ideal gas
Constant Specific Heats (Approximate Analysis)
Under the constant-specific-heat assumption, the specific heat is assumed to be constant at some average value.
Assuming constant specific heats for idea gases is a common approximation:
Variable Specific Heats (Exact Analysis)
We choose absolute zero as the reference temperature and define a function s ° as
The entropy of an ideal gas depends on both T and P. The function s represents only the temperature-dependent part of entropy.
On a unit–mole basis
On a unit–mass basis
Variable Specific Heats
Variable Specific Heats
Example 7-
Variable Specific Heats Example 7-
Isentropic Processes of Ideal Gases
Constant Specific Heats (Approximate Analysis)
Setting this eq. equal to zero, we get
The isentropic relations of ideal gases are valid for the isentropic processes of ideal gases only.
Isentropic Processes of Ideal Gases Variable Specific Heats (Exact Analysis)
Relative Pressure and Relative Specific Volume
T / Pr is the relative specific volume v (^) r
exp( s °/ R ) is the relative pressure Pr
The use of Pr data for calculating the final temperature during an isentropic process.
The use of vr data for calculating the final temperature during an isentropic process
S 2 - S 1 =
Isentropic Processes of Ideal Gases
Variable Specific Heats (Exact Analysis)
The use of Pr data for calculating the final temperature during an isentropic process.
The use of vr data for calculating the final temperature during an isentropic process
Table A-17 Table A-
Isentropic Processes of Ideal Gases
Isentropic Processes of Ideal Gases
Example 7-
Isentropic Processes of Ideal Gases
If we use k=1.400 at the initial temperature 295 K
T 2 = 667.7 K
Thus we can estimate the average temperature is 481 K At 481k, K=1.389 based on Table A-2b
T 2 = 662.4 K
T 2 =(295K) (8) 1.4-
Example 7-
Isentropic Efficiencies of Compressors
The h-s diagram of the actual and isentropic processes of an adiabatic compressor
When kinetic and potential energies are negligible
Isentropic Efficiencies of Compressors
Isentropic Efficiencies of Compressors
Compressors are sometimes intentionally cooled to minimize the work input.
Can you use isentropic efficiency for a non- adiabatic compressor? Can you use isothermal efficiency for an adiabatic compressor?
Isentropic Efficiencies of Compressors
For a reversible isothermal process, then we can define an isothermal efficiency:
W (^) t and W (^) a are the required work inputs to the compressor for the reversible isothermal and actual processes.
Isentropic Efficiencies of Pumps
Isentropic Efficiencies of Pumps:
When kinetic and potential energies of a liquid are negligible
Isentropic Efficiencies of Pumps
Isentropic Efficiencies of Pumps
Example 7-
Isentropic Efficiencies of Pumps Example 7- State 1:
Isentropic Efficiencies of Pumps
Example 7-
Isentropic Efficiency of Nozzles
The h-s diagram of the actual and isentropic processes of an adiabatic nozzle.
If the inlet velocity of the fluid is small relative to the exit velocity, the energy balance is
Isentropic Efficiency of Nozzles is defined as:
E in E out
2 2 2
2 1 1
a a
V
mh
V
m h + = +
Since Q W^ pe
Then
Isentropic process of Nozzles Isentropic process of Nozzles
Isentropic process of Nozzles
h 1 = 988 kJ/kg = 988,000 J/kg at 950K
h2s = 766 kJ/kg = 766,000 J/kg at 748K V 1 = 0 (^) Attention: unit conversion
(Table A-17)
Example 7-
Isentropic process of Nozzles
V 2 s = 2 ( h 1 − h 2 )= 2 ( 998000 − 766000 )= 666 ( m / s )
Alternatively, use a constant C (^) p
= a
h
h2a = 783.8 kJ/kg
From table A-17: h=800.03kJ/kg at 780K, and h=778.18 kJ/Kg at 760K
− T a
Interpolation: T (^) 2a = 765K
Example 7-
Entropy Balance of Closed Systems
A closed system involves no mass transfer across its boundary
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 Balance of Closed Systems Noting that any closed system and its surroundings can be treated as an adiabatic system, and then:
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
A closed system involves no mass transfer across its boundary
Energy balance:
( Q (^) in − Qout )+( Win − Wout )=Δ U +Δ KE +Δ PE
For a closed system without change in the KE and PE Q (^) net , in − Wnet , out =Δ U
Entropy balance:
Entropy Balance of Control Volumes As compared with a closed system, a control volume involves mass transfer across its boundary. Thus the entropy balance:
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
Entropy Balance of Control Volumes
For a general steady-flow process:
(kW/K)
For a single-stream, steady-flow device:
∑ ∑ ii ∑ ee gen k
k (^) ms ms S T
Q
∑ i e gen k
k (^) ms s S T
Q
( − )+ = 0
msi se S gen
For an adiabatic, single-stream, steady-flow device:
For an adiabatic, single-stream, steady-flow device that undergoes a reversible process: s (^) i = s (^) e
Balance Equations of Control Volumes
Mass balance :
For a general control volume:
Energy balance :
Entropy balance :
Balance Equations of Steady-flow Control Volumes
For a steady-flow control volume: Mass balance :
Energy balance :
Entropy balance :
∑ ∑ i i ∑ ee gen k
k ms ms S T
Q
Entropy Generation
Entropy Generation Example 7-
Entropy Generation
(Table A-6)
∑ ∑ ii ∑ ee gen k
k ms ms S T
Q
Example 7-
(Table A-6) Interpolation
Entropy Generation Entropy Generation Example 7-
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
