1. Entropy
Cengel & Boles, Chapter 6
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2. Entropy From the 1st Carnot principle:
this is valid for two thermal reservoirs
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3. The Clausius Inequality
For a system undergoing a cycle and communicating with N thermal reservoirs, it can be shown that
This can be further generalized into the Clausius Inequality:
where Q is the heat transfer at a particular location along the system boundary during a portion of the cycle and T is the absolute temperature at that location. This is called a cyclic integral.
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4. The Clausius Inequality, cont.
If the cycle is internally reversible, then
What does this mean? Consider:
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5. New Property: Entropy
Therefore, the quantity Q/T is a property of the system in differential form when the integration is performed along an internally reversible path. This property is known as entropy (S) :
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6. New Property: Entropy
This integral defines a new property of the system called entropy (S):
entropy is an extensive property with units of kJ/K; specific entropy is defined by s = S/m, with units of kJ/kg-K
the integration will only yield entropy change when carried out along an int. rev. path
like enthalpy, entropy is a convenient and useful property that has been introduced without physical motivation; its utility will be discovered as we learn more about its characteristics
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7. Entropy Change and Heat Transfer
Suppose we have a closed system undergoing an internally reversible process with heat transfer
if heat is added (Q>0), then S2>S1 or entropy increases
if heat is removed (Q<0), then S2<S1 or entropy decreases
if system is adiabatic (Q=0), then S2=S1 or entropy is constant (isentropic)
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8. Entropy Change and Heat Transfer, cont.
Entropy equation can be rearranged:
when temperature is plotted against entropy, the area under a process path is equal to the heat transfer when the process is internally reversible.
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9. Increase in Entropy Principle
Consider a cycle consisting of an irreversible process followed by a reversible one:
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10. Increase in Entropy Principle, cont.
The inequality can be turned into an equality by considering the “extra” contribution to the entropy change as entropy generated by the irreversibilities of the process:
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11. Increase in Entropy Principle, cont.
The increase in entropy principle states that an isolated system (or an adiabatic closed system) will always experience an increase in entropy since there can be no heat transfer, i.e.,
However, this principle does not preclude an entropy decrease, which may occur for a system that loses heat (Q < 0)
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12. Isentropic Processes
A process is isentropic if S (or s) is a constant, i.e.,
this corresponds to best performance (reversible) in the absence of heat transfer since irreversibilities are zero.
many engineering devices such as pumps, turbines, nozzles, and diffusers are essentially adiabatic, so they perform best when isentropic.
note that a reversible adiabatic process must be isentropic, but an isentropic process is not necessarily a reversible adiabatic process.
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13. The T-s Diagram
A T-s diagram displays the phases of a substance in much the same way as a P-v or T-v diagram:
Zero entropy is defined at sf(T = 0.01C) for H2O, sf(T = -40C) for R-134a, and s(T = 0K) for ideal gases
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14. The Tds Relations
Consider a closed, stationary system containing a simple compressible substance undergoing an int. rev. process:
substituting, we obtain:
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15. The Tds Equations, cont. Recalling H = U + PV, then substituting,
these are known as the Tds relations, which allow one to evaluate entropy in terms of more familiar quantities; the equations are also valid for irreversible processes because they involve only properties and so are path-independent.
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16. Calculating Entropy Change
From Tds relations,
these equations are used to evaluate entropy for H2O and R-134a in the text tables A-4 through A-13
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17. Entropy Change for Ideal Gases
similarly,
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18. Variable Specific Heats (Exact Analysis)
Define absolute entropy:
this is tabulated as a function of T in the ideal gas tables, A-17 through A-25
thus,
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19. Constant Specific Heats (Approximate Analysis)
If Cv, Cp  constant, then
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20. Entropy Change for Incompressible Substances
For incompressible substances (i.e., liquids and solids), v = constant and Cv = Cp = C
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21. Isentropic Processes for Ideal Gases
Recall for an ideal gas:
If process is isentropic,
Since adiabatic, isentropic processes are reversible and yield the best performance, solving this equation is important (e.g., finding T2 and h2)
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22. Variable Specific Heats (Exact Isentropic Analysis)
i) if T1, P1, and P2 are known, then T2 can be found from the ideal gas tables and
ii) if T1, P1, and T2 are known, then P2 can be found from
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23. Relative Pressure, Pr
The quantity exp(so/R) is tabulated in the ideal gas tables as a function of temperature - it is called the relative pressure, Pr
If the pressure ratio P2/P1 is known, then Pr is useful in finding the isentropic process by setting
The relative pressure values have no physical significance - they are only used as a “shortcut” in determining an isentropic process from the ideal gas tables
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24. Relative Volume, vr From the ideal gas law,
The quantity T/Pr is also tabulated in the ideal gas tables and is known as the relative volume, vr ; if the specific volume ratio is known, then vr is useful in finding the isentropic process by setting
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25. Constant Specific Heats (Approx. Isentropic Analysis)
Recall:
with k = Cp/Cv and Cp = Cv + R , the following relations result:
note that Pvk = constant for isentropic processes; thus, these processes are polytropic where n = k
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26. Reversible Steady-Flow Work
Recall relation between heat transfer and entropy for an int. rev. process:
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27. Reversible Steady-Flow Work, cont.
From CV energy balance,
for a reversible process,
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28. Reversible Steady-Flow Work, cont.
Note that if wrev = 0, we have the simple form of the Bernoulli equation
For turbines, compressors, and pumps with negligible KE, PE effects:
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29. Special Cases of Reversible Work
1) Pumps - fluid is incompressible (i.e., liquid) so v = constant:
2) Ideal gas (Pv = RT) compressor:
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30. Special Cases of Reversible Work, cont.
3) Polytropic compressor
substituting into reversible work equation and integrating yields:
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31. Special Cases of Reversible Work, cont.
4) Ideal gas and polytropic compressor:
4) Ideal gas and isothermal compressor:
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32. Isentropic Efficiencies of Steady-Flow Devices
Isentropic Efficiency is a perfor-mance measure of an adiabatic device that compares an actual process to an isentropic process, both having the same exit pressure
1) Turbines:
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33. Isentropic Efficiencies, cont.
2) Compressors:
3) Pumps:
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34. Isentropic Efficiencies, cont.
4) Nozzles:
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35. Entropy Balance for Closed Systems
From the increase in entropy principle,
If temperature is constant where heat transfer takes place, then
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36. Entropy Balance for Control Volumes
It can also be shown that the following entropy balance applies to steady-flow control volumes with single-inlet, single-exit:
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