Dependence of the polytropic exponent on nozzle efficiency

We may express the enthalpy differential for an isentropic expansion from pressure p to pressure p -I-dp as  

Now a gas undergoing an isentropic expansion will obey the relationship 

We may now use equation (14.12) along with equation (14.15) to yield the corresponding temperature differential for a frictionally resisted expansion from the same initial conditions (p, T)  

Integrating equation (14.16) between an upstream station 0 and any downstream station, with corresponding pressures po and p and temperatures Tq and T, gives  

Since the downstream station was chosen arbitrarily, it follows that the required relationship between specific volume for a frictionally resisted, adiabatic expansion is simply  

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