Entropy

1. Entropy Cengel & Boles, Chapter 6 ME 152

2. Entropy • From the 1st Carnot principle: • this is valid for two thermal reservoirs ME 152

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. ME 152

4. The Clausius Inequality, cont. • If the cycle is internally reversible, then • What does this mean? Consider: ME 152

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): ME 152

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 ME 152

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) ME 152

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. ME 152

9. Increase in Entropy Principle • Consider a cycle consisting of an irreversible process followed by a reversible one: ME 152

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: ME 152

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) ME 152

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. ME 152

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 ME 152

14. The Tds Relations • Consider a closed, stationary system containing a simple compressible substance undergoing an int. rev. process: • substituting, we obtain: ME 152

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. ME 152

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 ME 152

17. Entropy Change for Ideal Gases • For ideal gases, • similarly, ME 152

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, ME 152

19. Constant Specific Heats (Approximate Analysis) • If Cv, Cp constant, then ME 152

20. Entropy Change for Incompressible Substances • For incompressible substances (i.e., liquids and solids), v = constant and Cv= Cp= C ME 152

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) ME 152

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 ME 152

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 ME 152

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 ME 152

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 ME 152

26. Reversible Steady-Flow Work • Recall relation between heat transfer and entropy for an int. rev. process: ME 152

27. Reversible Steady-Flow Work, cont. • From CV energy balance, • for a reversible process, ME 152

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: ME 152

29. Special Cases of Reversible Work 1) Pumps - fluid is incompressible (i.e., liquid) so v = constant: 2) Ideal gas (Pv = RT) compressor: ME 152

30. Special Cases of Reversible Work, cont. 3) Polytropic compressor • substituting into reversible work equation and integrating yields: ME 152

31. Special Cases of Reversible Work, cont. 4) Ideal gas and polytropic compressor: 4) Ideal gas and isothermal compressor: ME 152

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: ME 152

33. Isentropic Efficiencies, cont. 2) Compressors: 3) Pumps: ME 152

34. Isentropic Efficiencies, cont. 4) Nozzles: ME 152

35. Entropy Balance for Closed Systems • From the increase in entropy principle, • If temperature is constant where heat transfer takes place, then ME 152

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: ME 152