I am a grader for a fluid mechanics class, and have come across a problem that I don't understand the answer to, and I was hoping someone could explain it to me. I've quoted the question below.
Consider compressible flow through the converging nozzle below, that has a low velocity at the inlet and a sonic velocity (Ma=1) at the exit. the nozzle exit diameter is reduced by hlaf while the inlet temperature and pressure are maintained the same. the nozzle exit velocity will:
- a) remain the same
- b) double
- c) quadruple
- d) go down by 1/4th
- e) go down by half
(The included picture is rather useless, it just shows two nozzles one with a smaller opening at one end. I'd take the effort of posting it if it were relevant)
According to the solutions I have, the answer is supposedly a) remain the same but I don't understand why. Based on the continuity equation, shouldn't it be c) quadruple? Or does that assume it is an incompressible fluid?
From the area-Mach number relation it follows that when you reduce the area the flow stays choked at M =1. Then, in both cases, the stagnation temperature at exit is the same as the stagnation tmperature at the inlet because it is an adiabatic flow. It follows that the exit static temperature is also the same because in both cases M=1 at exit. Hence a the speed of sound is also the same and since u = Ma, the velocity remains the same.
The velocity would quadruple if the fluid were incompressible - but here the density at exit would increase to keep (rho u A) constant.