Maximum mass flow for a polytropic expansion

When a gas at constant inlet conditions of pressure. Pi, and specific volume, vi, expands to a downstream pressure, p2, at downstream area, A2, flow will increase if the downstream pressure decreases until a maximum flow is reached. It follows from the form of equation (5.25) that maximum flow will occur at the maximum of the function, /, given by  

From calculus, this will occur when the differential of / with respect to the pressure ratio, (P2/P1). is zero, i.e. when 

Substituting from equation (5.40) into equation (5.25) gives the maximum flow as  

Equation (5.40) has specified the value of downstream pressure, p2, for maximum flow, but it has not specified the associated downstream area, A2. In fact, solving equation (5.25) for the area, A2, associated with 

It is instructive to examine how the maximum flow for any given inlet conditions varies with polytropic index, n. For fixed nozzle geometry and constant upstream conditions, the behaviour of will be determined by the behaviour of the function 

---