fluid_mechanics

11.5 Nonisentropic Flow of an Ideal Gas 625
This relationship is graphed in the margin on the previous page for air.
From Eqs. 11.125, 11.126, and 11.128 we see that
2.0
r a
r V 11 k2Ma
Ma c
V a 1 kMa d 2
(11.129)
V___
Va
r
___ a ,
ρ
1.0
0.0
0.1
1.0
Ma
10
This relationship is graphed in the margin for air.
The energy equation 1Eq. 5.692 tells us that because of the heat transfer involved in Rayleigh
flows, the stagnation temperature varies. We note that
T 0
T 0,a
a T 0
T b a T T a
b a T a
T 0,a
b
(11.130)
Unlike Fanno flow,
the stagnation temperature
in Rayleigh
flow varies.
We can use Eq. 11.56 1developed earlier for steady, isentropic, ideal gas flow2 to evaluate T 0T and
T aT 0a because these two temperature ratios, by definition of the stagnation state, involve isentropic
processes. Equation 11.128 can be used for TT a . Thus, consolidating Eqs. 11.130, 11.56, and
11.128 we obtain
2.0
21k 12Ma 2 a1 k 1 Ma 2 b
T 0
2
T 0,a 11 kMa 2 2 2
(11.131)
___ T 0
T 0,a
1.0
0.0
0.1
1.0
Ma
10
This relationship is graphed in the margin for air.
Finally, we observe that
p 0
p 0,a
a p 0
p b a p p a
b a p a
p 0,a
b
(11.132)
We can use Eq. 11.59 developed earlier for steady, isentropic, ideal gas flow to evaluate p 0p and
p ap 0,a because these two pressure ratios, by definition, involve isentropic processes. Equation
11.123 can be used for pp a . Together, Eqs. 11.59, 11.123, and 11.132 give
p 0 ___
2.0
1.0
p 0
p 0,a
11 k2
11 kMa 2 2
2
ca
k 1 b a1 k 1
2
k1k12
Ma 2 bd
(11.133)
p 0,a
GIVEN
0.0
0.1
1.0
Ma
10
This relationship is graphed in the margin for air.
Values of pp a , TT a , rr a or VV a , T 0T 0,a , and p 0p 0,a are graphed in Fig. D.3 of Appendix D
as a function of Mach number for Rayleigh flow of air 1k 1.42. The values in Fig. D.3 were calculated
from Eqs. 11.123, 11.128, 11.129, 11.131, and 11.133. The usefulness of Fig. D.3 is illustrated
in Example 11.16.
See Ref. 7 for a more advanced treatment of internal flows with heat transfer.
E XAMPLE 11.16
Effect of Mach Number and Heating/Cooling
for Rayleigh Flow
The information in Table 11.2 shows us that subsonic
Rayleigh flow accelerates when heated and decelerates when
cooled. Supersonic Rayleigh flow behaves just opposite to subsonic
Rayleigh flow; it decelerates when heated and accelerates
when cooled.
FIND Using Fig. D.3 for air 1k 1.42, state whether velocity,
Mach number, static temperature, stagnation temperature, static
pressure, and stagnation pressure increase or decrease as subsonic
and supersonic Rayleigh flow is 1a2 heated, 1b2 cooled.