Da M L Vd

; thus the ratio becomes

Since the process (a-b) is reversible and adiabatic, then

This integrates to give for process (a-b) the

 mt} dT , Vb I --\ cv TL T ' Va i M Tf dT 1 Vd — I c = l^-i . K tH V ' T Vc

f c T = ln L. Therefore, since the limits of integration are J v T V

reversed, the equivalence gives J ln IL = ln Y± L or {- [ln Vc - ln Vb]}

 Vd] resulting in í ^ ln-c- V b y result of r -0L =TL 1 V 0h = —1. This upon substitution yields the expected d y This relation permits one to substitute temperatures for heat quantities in the determination of Carnot performance criteria which quickly give extremum values for the actual efficiencies for engines, q, as well as coefficient of performances for refrigerators, fi, and heat pumps, y. It must be remembered that one of the heat quantities is negative, out of a system, as it should be since the ratio of absolute temperatures is always positive. With reference to Ar-nas et al., (2003), we can examine the various details of this result particularly with respect to engine efficiency and refrigerator and heat pump coefficient of performances. For this demonstration, we will use the sign convention of Callen (1960) that all energy transfers into a system are positive and all out of the system are negative. Pi b adiabatic b adiabatic —Th, isothermal adiabatic d\ Tl, isothermal All processes are reversible —Th, isothermal adiabatic d\ Tl, isothermal Fig. 2.1 The Carnot cycle. 