Thermodynamics

Variable Specific Heats (Exact Analysis)When the constant-specific-heat assumption is not appropriate, the isentropicrelations developed previously yields results that are not quite accurate.For such cases, we should use an isentropic relation obtained from Eq.7–39 that accounts for the variation of specific heats with temperature. Settingthis equation equal to zero givesorChapter 7 | 359P 20 s° 2 s° 1 R lnP 1P 2s° 2 s° 1 R lnP 1where s° 2 is the s° value at the end of the isentropic process.(7–48)Relative Pressure and Relative Specific VolumeEquation 7–48 provides an accurate way of evaluating property changes ofideal gases during isentropic processes since it accounts for the variation ofspecific heats with temperature. However, it involves tedious iterationswhen the volume ratio is given instead of the pressure ratio. This is quite aninconvenience in optimization studies, which usually require numerousrepetitive calculations. To remedy this deficiency, we define two newdimensionless quantities associated with isentropic processes.The definition of the first is based on Eq. 7–48, which can berearranged asorP 2 s° 2 s° 1 expP 1 RP 2 exp 1s° 2 >R2P 1 exp 1s° 1 >R2The quantity exp(s°/R) is defined as the relative pressure P r . With this definition,the last relation becomesa P 2b P r2(7–49)P 1 sconst. P r1Note that the relative pressure P r is a dimensionless quantity that is a functionof temperature only since s° depends on temperature alone. Therefore,values of P r can be tabulated against temperature. This is done for air inTable A–17. The use of P r data is illustrated in Fig. 7–37.Sometimes specific volume ratios are given instead of pressure ratios.This is particularly the case when automotive engines are analyzed. In suchcases, one needs to work with volume ratios. Therefore, we define anotherquantity related to specific volume ratios for isentropic processes. This isdone by utilizing the ideal-gas relation and Eq. 7–49:P 1 v 1 P 2v 2S v 2 T 2 P 1T 1 T 2 v 1 T 1P 2 T 2P r1 T 2>P r2T 1 P r2 T 1 >P r1Process: isentropicGiven: P1, T1, and P2Find: T 2T P r. .. .PTread2 Pr 2 = 2P P r11. .Tread 1 Pr 1. .FIGURE 7–37The use of P r data for calculating thefinal temperature during an isentropicprocess.