Page 303 - Aerodynamics for Engineering Students

Compressible flow 285

A slight rearrangement allows the mass flow, in a non-dimensional form, to be
expressed solely in terms of the pressure ratio, i.e.

(6.29)

Inspection of Eqn (6.29), or Eqn (6.22), reveals the obvious fact that riz = 0 when
p/po = 1, i.e. no flow takes place for zero pressure difference along the duct. Further
inspection shows that riz is also apparently zero when p/po = 0, i.e. under maximum
pressure drop conditions. This apparent paradox may be resolved by considering the
behaviour of the flow as p is gradually decreased from the value PO. As p is lowered
the mass flow increases in magnitude until a condition of maximum mass flow
occurs.
The maximum condition may be found by the usual differentiation process, Le.
from Eqn (6.29):
6 (Y+l)lY ] = 0 when ~ 2 is a maximum,
[
dE
(;)2’7-(;)

i.e.

which gives

(6.30)

It will be recalled that this is the value of the pressure ratio for the condition M = 1
and thus the maximum mass flow occurs when the pressure drop is sufficient to
produce sonic flow at the exit.
Decreasing the pressure further will not result in a further increase of mass
flow, which retains its maximum value. When these conditions occur the nozzle
is said to be choked. The pressure at the exit section remains that given by Eqn
(6.30) and as the pressure is further lowered the gas expands from the exit in
a supersonic jet.
From previous considerations the condition for sonic flow, which is the condition
for maximum mass flow, implies a throat, or section of minimum area, in the stream.
Further expansion to a lower pressure and acceleration to supersonic flow will be
accompanied by an increase in section area of the jet. It is impossible for the pressure
ratio in the exit section to fall below that given by Eqn (6.30), and solutions of
Eqn (6.29) have no physical meaning for values of