fluid_mechanics

602 Chapter 11 ■ Compressible Flow
From Eq. 1 we obtain
V 10.4992 23386 1ft # lb21lbm # °R241488 °R211.662
279 1ft # lblbm2 1 2
or, using 1 lb 32.2 lbm # fts 2 ,
V 279 1ft # lblbm2 1 2 3132.2 lbm # fts 2 2lb4 1 2
1580 fts
(Ans)
isentropic flow along a pathline in a stagnation process. Even
though these equations and graph were developed for onedimensional
duct flows, they can be used for frictionless, adiabatic
pathline flows also.
Furthermore, while the Mach numbers calculated above are of
similar size for the air and helium flows, the flow speed is much
larger for helium than for air because the speed of sound in helium
is much larger than it is in air.
COMMENT Note that the isentropic flow equations and Fig.
D.1 for k 1.4 were used presently to describe fluid particle
The ratio of flow
area to the critical
area is a useful
concept for isentropic
duct flow.
Also included in Fig. D.1 is a graph of the ratio of local area, A, to critical area, A*, for different
values of local Mach number. The importance of this area ratio is clarified below.
For choked flow through the converging–diverging duct of Fig. 11.6a, the conservation of
mass equation 1Eq. 11.402 yields
or
From Eqs. 11.36 and 11.46, we obtain
and
rAV r*A*V*
A
A* ar* r b aV* V b
V* 1RT*k
V Ma 1RTk
By combining Eqs. 11.67, 11.68, and 11.69 we get
A
A* 1 Ma ar* b a r 0
r 0 r b 1T*T 0 2
B 1TT 0 2
The incorporation of Eqs. 11.56, 11.60, 11.63, 11.65, and 11.70 results in
A
A* 1 Ma e 1 31k 12 24Ma 2
1 31k 1224
1k12 321k124
f
(11.67)
(11.68)
(11.69)
(11.70)
(11.71)
Equation 11.71 was used to generate the values of AA* for air 1k 1.42 in Fig. D.1. These values
of AA* are graphed as a function of Mach number in Fig. 11.10. As is demonstrated in the
following examples, whether or not the critical area, A*, is physically present in the flow, the area
ratio, AA*, is still a useful concept for the isentropic flow of an ideal gas through a converging–
diverging duct.
2.0
A ___
A*
1.0
0 1.0
Ma
F I G U R E 11.10 The variation of area
ratio with Mach number for isentropic flow of an
ideal gas ( k 1.4, linear coordinate scales).