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

kr 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

kr 3 Entropy (2/3) 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, an d 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, an d Similarly, the same procedure can be applied for both reversible and irreversible refrigeration cycles. Proof of the Inequality of Clausius

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

kr 5 Principle of the Increase of Entropy (1/2) From the Clausius Inequality or Here, you can find that Entropy generation Increase of Entropy Principle 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 : 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. Lost Work → Exergy (Chapter 8)

kr 6 Principle of the Increase of Entropy (2/2) 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 Discussions on Entropy Generation

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

kr 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

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

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

kr 11 Thermodynamic Property Relations Gibbs Equations (T–ds Relations) 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 and For a unit mass, and

kr 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.

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

kr 14 Reversible Polytropic Process for an Ideal Gas If n is a constant, constant Polytropic Process 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 or and : Polytropic Relation

kr 15 Heat Transfer and Entropy Generation

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

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

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

kr 19 Homework #7 Solve the Examples 7–1 ~ 7–23 Jellabukdo