Lecture Notes on Thermodynamics 2008

1. Lecture Notes on Thermodynamics 2008 Chapter 7 Entropy Prof. Man Y. Kim, Autumn 2008, ⓒmanykim@chonbuk.ac.kr, Aerospace Engineering, Chonbuk National University, Korea

2. Entropy (1/3) • The Inequality of Clausius • The inequality of Clausius is a corollary or a consequence of the 2nd law of thermodynamics. • It is valid for all possible cycles, including both reversible and irreversible ones • The entropy is defined from this formulation, i.e., and

3. Entropy (2/3) • Proof of the Inequality of Clausius Consider first a reversible (Carnot) heat engine cycle : From the definition of absolute temperature scale ( ) If , and Finally, we conclude that for all reversible heat engines, and Now consider an irreversible cycle heat engine : Consequently, for the irreversible cycle engine, and If we make the engine become more and more irreversible, but keep , , and fixed, and Finally, we conclude that for all irreversible heat engine cycles, and Similarly, the same procedure can be applied for both reversible and irreversible refrigeration cycles.

4. Entropy (3/3) • Entropy – A Property of a System Reversible process along path A-B Reversible process along path C-B Subtracting the second equation from the first, we have is independent of the path → point function → property This property is called entropy and

5. Principle of the Increase of Entropy (1/2) • Increase of Entropy Principle From the Clausius Inequality or Here, you can find that Entropy generation • Entropy Generation where, :entropy generation due to irreversibility occurring inside the system ( because of friction, unrestricted expansion, internal energy transfer over a finite temp. difference, etc.) Reversible process : and Irreversible process : 1st law : Thermodynamic property relation : Lost Work → Exergy (Chapter 8) Thus we have an expression for the change of entropy for an irreversible process as an equality, whereas in the last slide we had an inequality.

6. Principle of the Increase of Entropy (2/2) • Discussions on Entropy Generation Discussion 1 : There are 2 ways in which the entropy of a system can be increased by (1) transferring heat to the system (2) having an irreversible process Note : There is only one way in which entropy can be decreased by transferring heat from the system Discussion 2 : For an adiabatic system, the increase of entropy is always associated with the irreversibility Discussion 3 : The presence of irreversibility will cause the work to be smaller than the reversible work

7. Entropy Change of a Pure Substance • see Examples 7–3 (p.326) and 7–4 (p.327) • Isentropic Process or

8. Isentropic Relations Consider the case of an ideal gas undergoing an isentropic process, However, , where : specific heat ratio Finally we can obtain , and : Isentropic Relation Note : constant is a special case of a polytropic process in which the polytropic exponent n is equal to the specific heat ratio k

9. T–s Diagram of the Carnot Cycle Consider the Carnot cycle, i.e., ① → ② : reversible isothermal heat addition process Area 1-2-b-a-1 : heat transferred to the working fluid during the process ② → ③ : reversible adiabatic process → isentropic process ③ → ④ : reversible isothermal heat rejection process Area 3-4-a-b-3 : heat transferred from the working fluid to the low-temperature reservoir. ④ →① : reversible adiabatic process → isentropic process Area 1-2-3-4-1 : net work of the cycle Efficiency Comments on efficiency : • see Example 7–6

10. What is Entropy ? Figure 7–23 Figure 7–22 Figure 7–21 Figure 7–20 Figure 7–27 Figure 7–24 Figure 7–25 Figure 7–26

11. Thermodynamic Property Relations • Gibbs Equations (T–ds Relations) and For the simple compressible substance with no motion or gravitational effects, the 1st law becomes For a reversible process of a simple compressible substance, and Since enthalpy is defined as For a unit mass, and

12. Entropy Change during Irreversible Process Reversible cycle : reversible process along path A-B Irreversible cycle : irreversible path C and reversible path B Subtracting the second equation from the first, we have As path C was arbitrary, the general result is (both reversible and irreversible cases) and This is one of the most important equations of thermodynamics ! and Therefore, we can find that the entropy change for an irreversible process is larger than the change in a reversible process for the same and T.

13. Entropy Change for a Solid(Liquid) and Ideal Gas • For a Solid or Liquid specific volume is very small, and • For an Ideal Gas We know that , and and Similarly, , and and If we assume that the specific heat is constant, and

14. Reversible Polytropic Process for an Ideal Gas • Polytropic Process constant If n is a constant, and : Polytropic Relation • Work done during a reversible polytropic process and constant constant • Isobaric process (P=constant) : n=0 • Isothermal process (T=constant) : n=1 • Isentropic process (s=constant) : n=k • Isochoric process (v=constant) : n=∞ for any value of n except n=1 • The reversible isothermal process constant constant or

15. Heat Transfer and Entropy Generation

16. Examples (1/3) • Turbine : Example 7–14

17. Examples (2/3) • Compressor : Example 7–14

18. Examples (3/3) • Nozzle : Example 7–16

19. Saemangum @ Jellabukdo Homework #7 Solve the Examples 7–1 ~ 7–23