fluid_mechanics

598 Chapter 11 ■ Compressible Flow
T
p 0
T 0
s
T 0
T* =
( )
______ k
p* =
k – 1
p 0
______ 2
( k + 1 )
______ 2
( k + 1 )
F I G U R E 11.9 The relationship
between the stagnation and critical states.
For k 1.4, the nominal value of k for air, Eq. 11.61 yields
a p* 0.528
p 0
bk1.4
(11.62)
Because the stagnation pressure for our converging–diverging duct example is the atmospheric pressure,
p atm , the throat pressure for choked air flow is, from Eq. 11.62
We can get a relationship for the critical temperature ratio, T*T 0 , by substituting Ma 1
into Eq. 11.56. Thus,
or for k 1.4
For the duct of Fig. 11.6a, Eq. 11.64 yields
p* k1.4 0.528p atm
T*
2
T 0 k 1
a T* 0.833
T 0
bk1.4
(11.63)
(11.64)
The stagnation and
critical states are at
the same entropy
level.
T* k1.4
0.833T atm
The stagnation and critical pressures and temperatures are shown on the T–s diagram of
Fig. 11.9.
When we combine the ideal gas equation of state 1Eq. 11.12 with Eqs. 11.61 and 11.63, for
Ma 1 we get
r*
a p*
k1k12
r 0 T* b aT 0 2
b a
p 0 k 1 b a k 1
11k12
2
b a
2 k 1 b
For air 1k 1.42, Eq. 11.65 leads to
(11.65)
a r* 0.634
(11.66)
r 0
bk1.4
and we see that when the converging–diverging duct flow is choked, the density of the air at the
duct throat is 63.4% of the density of atmospheric air.
E XAMPLE 11.5
Isentropic Flow in a Converging Duct
GIVEN A converging duct passes air steadily from standard FIND Determine the mass flowrate through the duct and
atmospheric conditions to a receiver pipe as illustrated in Fig. sketch temperature–entropy diagrams for situations 1a2 and 1b2.
E11.5a. The throat 1minimum2 flow cross-sectional area of the converging
duct is 1 10 4 m 2 . The receiver pressure is 1a2 80 kPa
1abs2, 1b2 40 kPa 1abs2.