Air at a stagnation pressure of 1 MPa and a stagnation temperature of 300 K flow

Question

Air at a stagnation pressure of 1 MPa and a stagnation temperature of 300 K flows isentropically through a convergent-divergent nozzle which feeds an adiabatic frictional constant-area duct. sonic at the nozzle throat . If the area ratio of the nozzle is 3.0, and the average friction factor is 0.005, determine the length to diameter ratio of the duct if the flow is sonic at the exit and no shocks or expansions are present. What would happen (qualitative explanation) if the back pressure were increased? What would happen if the back pressure were decreased? In your explanations, discuss the mass flow rate through the system and the development of possible shocks or expansion waves.

Explanation / Answer

From air isentropic flow table corresponding to A/A* = 3, the two possible Mach numbers at exit are

MeC= 0.198

MeD=2.64

We use M = 2.64 for shortest L*/D

From Fanno Line Table, for M=2.64

fL*/D = 0.4604

f=0.005

Hence L*/D = 0.4604/0.005 = 92.08

Hence after L/D = 92.08 the flow will become sonic without shock.

If the back pressure pb at duct exit is

a) If the back pressure is increased above the choking pressure there will be oblique shock outside the duct if pbp< pe, sh where pe,sh is shock pressure. If pb>pe,sh there will be shock inside the duct.

b) If the pressure is decreased, i.e the flow is underexpanded, there will be expansion outside the duct.

The mass flow rate is dictated by the duct exit choking flow conditions for the case a) above. For case b) however, there is no choking at duct exit and flow is dictated by choking at nozzle throat.

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