# lec6-1 - Chapter 6 Using Entropy Photo courtesy of U.S...

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1 Chapter 6: Using Entropy Combining the 1 st and the 2 nd Laws of Thermodynamics Photo courtesy of U.S. Military Academy photo archives. Introducing Entropy • The Clausius and Kelvin-Plank Statements constitute the 2 nd law of thermodynamics • Carnot corollaries apply to systems communicating with a hot and cold reservoir only • They are used to establish the max efficiency and COP for various thermodynamic cycles • Another corollary, known as Clausius Inequality , will be introduced that can be applied to any cycle independent of the bodies it interacts with. • This corollary will lead to the definition of entropy .

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2 Clausius Inequality 0 no irreversibilities present within the system 0 irreversibilities present within the system 0 impossible cycle cycle cycle σ = > < cycle b T Q σ = ⎛ δ • Inequality: : cyclic integral over the boundary and cycle • Unlike energy, entropy in the presence of irreversibilities, is always produced . T b δ Q Sketch of Proof b r b r T T Q Q = δ δ 0 δ r Q Temperature Scale: By the 2 nd Law: 0 ⎛ δ b T Q δ Q r δ Q b Reversible cycle δ W Reservoir, T r system system T b
3 Defining Entropy • The Clausius inequality states that for an internally reversible cycle • This implies that is independent of the path from state 1 to 2. • Therefore can be considered a property change • This integral is defined to be the entropy at state 2 relative to 1: 0 = ⎛ δ b T Q δ 2 1 T Q ⎛ δ 2 1 rev int T Q ⎛ δ = 2 1 rev int 1 2 T Q S S Finding Entropy Data For Water and Refrigerants: Motivation: Application of the 2 nd Law of Thermodynamics frequently requires finding specific entropy data. Using Tables A-2 – A-18 For example: P 1 = 3 MPa T 1 = 500 o C Use Table A-4; s 1 = 7.2338 kJ/kg K (1 ) ( ) f gf s xs x s s s = −+ = +− Saturation Data: Liquid Data: In the absence of tables, use the Compressed Liquid Rule

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4 Graphical Entropy Data Temperature-Entropy Diagram Enthalpy-Entropy (Mollier) Diagram T dS Equations TdS dU PdV = + TdS dH VdP = Tds du Pdv = + Per Unit Mass: Per Unit Mass: Tds dh vdP =
5 Finding Entropy Data For Ideal Gases: Motivation: Application of the 2 nd Law of Thermodynamics frequently requires finding specific entropy data. For use with Table A-22 () v p du C T dh C T P vR T = = = For an Ideal Gas: Using these relations with the T ds equations yields: 2 22 11 2 1 1 ,, ( ) ( ) l n P sTP sTP sT sT R P ⎛⎞ −= ⎜⎟ ⎝⎠ oo 2 2 1 1 ( ) ( ) l n P P For use with Table A-23 Finding Entropy Data Internally Reversible Processes Internally reversible processes are idealizations, but are found in all Carnot cycles.

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lec6-1 - Chapter 6 Using Entropy Photo courtesy of U.S...

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