Thermodynamics filled in class notes_Part_5

Thermodynamics filled in class notes_Part_5 - P v v v v v T...

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Chapter 2 Cycle analysis Read BS, Chapters 7, 8, 10, 11, 12. 2.1 Carnot The Carnot cycle is the most well-known thermodynamic cycle. It is a useful idealization, but is very diFcult to realize in practice. Its real value lies in serving as a standard to which other cycles can be compared. These are usually fully covered in introductory courses. The cycle can be considered as follows 1 2: isentropic compression, 2 3: isothermal expansion, 3 4: isentropic expansion, and 4 1: isothermal compression. This forms a rectangle in the T s plane The P v plane is more complicated. Both are shown in ±igure 2.1. ±or this discussion, consider a calorically perfect ideal gas. The isotherms are then straightforward and are hyperbolas described by P = RT 1 v . (2.1) The slope of the isotherms in the P v plane is found by di²erentiation:
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Unformatted text preview: P v v v v v T = RT 1 v 2 , (2.2) = P v . (2.3) The slope of the isentrope is found in the following way. Consider rst the Gibbs equation, Eq. (1.1): Tds = du + Pdv. (2.4) 17 18 CHAPTER 2. CYCLE ANALYSIS v P T s 1 2 3 4 1 2 3 4 w cycle = P dv = q cycle = T ds isentrope isotherm isotherm i s e n t r o p h m is Figure 2.1: Sketch of P v and T s planes for a CPIG in a Carnot cycle. Because the gas is calorically perfect, one has du = c v dT, (2.5) so the Gibbs equation, Eq. (2.4), becomes Tds = c v dT + Pdv. (2.6) Now for the ideal gas, one has Pv = RT, (2.7) Pdv + vdP = RdT, (2.8) Pdv + vdP R = dT. (2.9) CC BY-NC-ND. 18 November 2011, J. M. Powers....
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Thermodynamics filled in class notes_Part_5 - P v v v v v T...

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