1.

Show that shock compression of a gas causes its entropy to rise.

Answer»


Solution :Consider the cycle depicted in Fig. Since the gas returns to the original state, the change in the entropy during the complete cycle is zero. But during the quasi-static adiabatic expansion of gas (section 2-3) its entropy does not change, during the isochoric cooling of the gas (section 3 - 1) its entropy DECREASES. Therefore its entropy MUST have increased during the shock compression (section 1- 2). PERFORM hte calculation for the compression ratio of the gas `x = rho_2//rho_1`. Putting as in Problem 19.7. `alpha = (gamma + 1)/(gamma -1)` , we obtain with the aid of hte Hugoniot and the Poission equations
`(p_2)/(p_1) = (alpha x - 1)/(alpha -x) , (p_2)/(p_1) = x^(gamma) ,` therefore `(p_3)/(p_1) = (alphax - 1)/((alpha - x) x^(gamma))`
The change in entropy as a resultof hte shock compression is equal to the change in entropy as a result of the isochoric cooling, only with the opposite sign. Making use of hte result of Problem. 18.14. we obtain
`DeltaS_(sh) = - DeltaS_V = -m/M C_(MV) ln (T_1)/(T_2) = m/M C_(mV) ln (P_3)/(P_1)`
`= m/M C_(mV) ln [(alphax - 1)/((alpha - x) x^(gamma))]`


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