fluid_mechanics

622 Chapter 11 ■ Compressible Flow
s s p T
3000
(psia) ( R) 1
[( ft lb) ( lbm R)]
2500
13.5 969 9.32
12.5 1459 202
11.5 1859 251
10.5 2168 285
9.0 2464 317
8.0 2549 330
7.6 2558 333
7.5 2558 334
7.0 2544 336
6.3 2488 338
6.0 2450 338
5.5 2369 336
5.0 2266 333
4.5 2140 328
4.0 1992 321
2.0 1175 259
1.0 633 181
T, °R
2000
1500
1000
500
100 200 300
(ft
s – s 1 , ________
• lb)
(lbm•°R)
F I G U R E E11.15
COMMENT Depending on whether the flow is being heated or
cooled, it can proceed in either direction along the curve.
At point a on the Rayleigh line of Fig. 11.22, dsdT 0. To determine the physical importance
of point a, we analyze further some of the governing equations. By differentiating the linear
momentum equation for Rayleigh flow 1Eq. 11.1102 we obtain
or
dp rV dV
dp
r
V dV
(11.112)
The maximum
entropy state on the
Rayleigh line corresponds
to sonic
conditions.
Combining Eq. 11.112 with the second T ds equation 1Eq. 11.182 leads to
T ds dȟ V dV
For an ideal gas 1Eq. 11.72 dȟ c p dT. Thus, substituting Eq. 11.7 into Eq. 11.113 gives
(11.113)
T ds c p dT V dV
or
ds
(11.114)
dT c p
T V dV
T dT
Consolidation of Eqs. 11.114, 11.112 1linear momentum2, 11.1, 11.77 1differentiated equation of
state2, and 11.79 1continuity2 leads to
ds
dT c p
T V T
1
31TV2 1VR24
Hence, at state a where dsdT 0, Eq. 11.115 reveals that
V a 1RT a k
Comparison of Eqs. 11.116 and 11.36 tells us that the Mach number at state a is equal to 1,
(11.115)
(11.116)
Ma a 1
(11.117)
At point b on the Rayleigh line of Fig. 11.22, dTds 0. From Eq. 11.115 we get
dT
ds 1
dsdT 1
1c pT2 1VT231TV2 1VR24 1