fluid_mechanics

614 Chapter 11 ■ Compressible Flow
If the linear momentum equation 1Eq. 5.222 is applied to the Fanno flow through the control
volume sketched in Fig. 11.19a, the result is
p 1 A 1 p 2 A 2 R x m # 1V 2 V 1 2
where is the frictional force exerted by the inner pipe wall on the fluid. Since and
m # R x A 1 A 2 A
rAV constant, we obtain
p 1 p 2 R x
A rV1V 2 V 1 2
(11.84)
The differential form of Eq. 11.84, which is valid for Fanno flow through the semi-infinitesimal
control volume shown in Fig. 11.19b, is
dp t wpD dx
A
rV dV
(11.85)
The wall shear stress, t w , is related to the wall friction factor, f, by Eq. 8.20 as
Friction forces in
Fanno flow are
given in terms of
the friction factor.
By substituting Eq. 11.86 and A pD 2 4 into Eq. 11.85, we obtain
or
(11.86)
(11.87)
(11.88)
Combining the ideal gas equation of state 1Eq. 11.12, the ideal gas speed-of-sound equation 1Eq.
11.362, and the Mach number definition 1Eq. 11.462 with Eq. 11.88 leads to
Since V Ma c Ma 1RTk, then
or
dp
p f rV 2 dx
p 2 D r d1V 2 2
0
p 2
dp
p fk 2
dp fr V 2
f 8t w
rV 2
2
dx
D
rV dV
dx
Ma2
D k Ma2 d1V 2 2
0
2 V 2
V 2 Ma 2 RTk
d1V 2 2
d1Ma2 2
dT
V 2 Ma 2 T
(11.89)
(11.90)
Section (1) Section (2)
Flow
Control volume
p 1 A 1 p 2 A 2
R x
(a)
Semi-infinitesimal control volume
Flow
D
(b)
pA
τw
π
δx
(p + δ p)A
D
δ x
F I G U R E 11.19 (a) Finite
control volume. (b) Semi-infinitesimal control
volume.