Page 83 - Mechanical Engineers' Handbook (Volume 4)

72 Fluid Mechanics

The stagnation pressure is reached reversibly and is thus the isentropic stagnation pres-
sure. It is also called the reservoir pressure, since for any flow a reservoir (stagnation)
pressure may be imagined from which the flow proceeds isentropically to a pressure p at a
Mach number M. The stagnation pressure p is a constant in isentropic flow; if nonisentropic,
0
but adiabatic, p decreases:
0
k /(k 1) 1 M k /(k 1)
p 0 T 0 k 1 2 2 3.5
p T 2 (1 0.2M ) for air
Expansion of this expression gives
p p 1 2 2 k 4 (2 k)(3 2k) M
2
V
6
0
2 1 4 M 24 M 192
where the term in brackets is the compressibility factor. It ranges from 1 at very low Mach
numbers to a maximum of 1.27 at M 1, and shows the effect of increasing gas density as
it is brought to a stagnation condition at increasingly higher initial Mach numbers. The
equations are valid to or from a stagnation state for subsonic flow, and from a stagnation
2
state for supersonic flow at M less than 2/(k 1), or M less than 5 for air.

10.2 Duct Flow

Adiabatic flow in short ducts may be considered reversible, and thus the relation between
2
velocity and area changes is dA/dV (A/V)(M 1). For subsonic flow, dA/dV is negative
and velocity changes relate to area changes in the same way as for incompressible flow. At
supersonic speed, dA/dV is positive and an expanding area is accompanied by an increasing
velocity; a contracting area is accompanied by a decreasing velocity, the opposite of incom-
pressible flow behavior. Sonic flow in a duct (at M 1) can exist only when the duct area
is constant (dA/dV 0), in the throat of a nozzle or in a pipe. It can also be shown that
velocity and Mach numbers always increase or decrease together, that temperature and Mach
numbers change in opposite directions, and that pressure and Mach numbers also change in
opposite directions.
Isentropic gas flow tables give pressure ratios p/p , temperature ratios T/T , density
0
0
ratios / , area ratios A/A*, and velocity ratios V/V* as functions of the upstream Mach
0
number M and the specific heat ratio k for gases.
x
The mass flow rate through a converging nozzle from a reservoir with the gas at a
pressure p and temperature T is calculated in terms of the pressure at the nozzle exit
0
0
˙
from the equation m (VA ) exit , where p /RT and the exit temperature is T
e
e
e
e
T (p /p ) (k 1)/k and the exit velocity is
e
0
0
2cT
(k 1) / k
p
e
V p 0 1 p 0
e
The mass flow rate is maximum when the exit velocity is sonic. This requires the exit
pressure to be critical, and the receiver pressure to be critical or below. Supersonic flow in
the nozzle is impossible. If the receiver pressure is below critical, flow is not affected in the
nozzle, and the exit flow remains sonic. For air at this condition, the maximum flow rate is
m 0.0404Ap / T 0 kg/sec.
˙
10
Flow through a converging–diverging nozzle (Fig. 25) is subsonic throughout if the
throat pressure is above critical (dashed lines in Fig. 25). When the receiver pressure is at