Page 370 - Bird R.B. Transport phenomena

352 Chapter 11 The Equations of Change for Nonisothermal Systems

Next we substitute the integrated continuity equation into the equation of motion and in-
tegrate once to obtain
-.- иъ .
р + + с (1L4 66)
Evaluation of the constant C ni from upstream conditions, where dv /dx = 0, gives C m =
x
v
P\ \ + Pi = P\\v\ + (RT-i/M)]. We now multiply both sides by v x and divide by p v . Then,
} }
with the help of the ideal gas law, p = pRT/M, and Eqs. 11.4-61 and 65, we may eliminate p
from Eq. 11.4-60 to obtain a relation containing only v x and x as variables:
(11.4-67)
Vx Vx
dx
This equation can, after considerable rearrangement, be rewritten in terms of dimensionless
variables:
ф — = /ЗМа^ф — 1)(ф ~ <У) (11.4-68)

The relevant dimensionless quantities are

ф = -^y = dimensionless velocity (11.4-69)
x
£ = — = dimensionless coordinate (11.4-70)
Л
Ma, = = Mach number at the upstream condition (11.4-71)

1
(11.4-72)
(11.4-73)

2
The reference length Л is the mean free path defined in Eq. 1.4-3 (with d eliminated by use of
Eq. 1.4-9):
1 irM
Л = (11.4-74)
Pi
We may integrate Eq. 11.4-68 to obtain

(11.4-75)
(Ф ~ a) a
This equation describes the dimensionless velocity distribution ф(£) containing an integration
f
constant £ 0 = XQ/A, which specifies the position of the shock wave in the nozzle; here 0 is con-
sidered to be known. It can be seen from the plot of Eq. 11.4-85 in Fig. 11.4-5 that shock waves

1.0
0.9 -
L — A,—H -
0.8
Г
0.7 -
\
Ma, =2 -
T, = 530 R
0.4
0.3 - -
0.2
0.1 - -
1 Fig. 11.4=5. Velocity distri-
0
-1.5 -1.0 -0.5 0 + 0.5 +1.0 bution in a stationary shock
(x-x 0 ), cm x 10 5 wave.