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Unformatted text preview: The general relation for ds was expressed as (Eq. 12–41)
T2 s2 s1
T1 cp T P2 dT
P1 a 0v b dP 0T P where P1, T1 and P2, T2 are the pressures and temperatures of the gas at the initial and the final states, respectively. The thought that comes to mind at this point is to perform the integrations in the previous equation first along a constant line to zero pressure, then along the P 0 line to T2, and T1 cen84959_ch12.qxd 4/5/05 3:58 PM Page 672 672  Thermodynamics
finally along the T2 constant line to P2, as we did for the enthalpy. This approach is not suitable for entropychange calculations, however, since it involves the value of entropy at zero pressure, which is infinity. We can avoid this difficulty by choosing a different (but more complex) path between the two states, as shown in Fig. 12–17. Then the entropy change can be expressed as
s2 s1 1 s2 * sb 2 * 1 sb * s2 2 * 1 s2 * s1 2 * 1 s1 * sa 2 * 1 sa s1 2 (12–61) T T2 Actual process path 2 1* P2 P1 1 a* P0
b* 2* * * States 1 and 1* are identical (T1 T1 and P1 P1 ) and so are states 2 and 2*. The gas is assumed to behave as an ideal gas at the imaginary states 1* and 2* as well as at the states between the two. Therefore, the entropy change during process 1*2* can be determined from the entropychange relations for ideal gases. The calculation of entropy change between an actual state and the corresponding imaginary idealgas state is more involved, however, and requires the use of generalized entropy departure charts, as explained below. Consider a gas at a pressure P and temperature T. To determine how much different the entropy of this gas would be if it were an ideal gas at the same temperature and pressure, we consider an isothermal process from the actual state P, T to zero (or close to zero) pressure and back to the imaginary idealgas state P *, T * (denoted by superscript *), as shown in Fig. 12–17. The entropy change during this isothermal process can be expressed as
1 sP * sP 2 T 1 sP
P 0 T1 s* 2 T 0 a 1s* 0 * sP 2 T
0 P 0v b dP 0T P a 0 v* b dP 0T P Alternative process path s where v ZRT/P and v * and rearranging, we obtain
1 sP * sP 2 T videal
P RT/P. Performing the differentiations
Z2R RT 0 Zr a b d dP P 0T P FIGURE 12–17 An alternative process path to evaluate the entropy changes of real gases during process 12. 0 c 11 P PcrPR and rearranging, the entropy By substituting T TcrTR and P departure can be expressed in a nondimensionalized form as
Zs 1 s* s 2 T,P
0 PR Ru cZ 1 TR a 0Z b d d 1 ln PR 2 0 TR PR (12–62) – – The difference (s * s )T,P is called the entropy departure and Zs is called the entropy departure factor. The integral in the above equation can be performed by using data from the compressibility charts. The values of Zs are presented in graphical form as a function of PR and TR in Fig. A–30. This graph is called the generalized entropy departure chart, and it is used to determine the deviation of the entropy of a gas at a given P and T from the entropy of an ideal gas at the same P and T. Repl...
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 Spring '09
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