fluid_mechanics

The ideal gas equation of state 1Eq. 11.12 leads to
11.5 Nonisentropic Flow of an Ideal Gas 617
p
p* r r*
T
T*
(11.106)
5.0
and merging Eqs. 11.106, 11.105, and 11.101 gives
p ___
p*
p
p* 1 Ma e 1k 122
12
1 31k 1224Ma f 2
(11.107)
0.0
0.1
1.0
Ma
10
This relationship is graphed in the margin for air.
Finally, the stagnation pressure ratio can be written as
p 0
p* 0
a p 0
p b a p p* b ap* p* 0
b
(11.108)
For Fanno flow,
thermodynamic and
flow properties can
be calculated as a
function of Mach
number.
which by use of Eqs. 11.59 and 11.107 yields
p 0
1 p* 0 Ma ca 2
k 1 b a1 k 1 31k1221k124
Ma 2 bd
2
(11.109)
Values of f 1/* /2D, TT*, VV*, pp*, and p 0p* 0 for Fanno flow of air 1k 1.42 are
graphed as a function of Mach number 1using Eqs. 11.99, 11.101, 11.103, 11.107, and 11.1092 in Fig.
D.2 of Appendix D. The usefulness of Fig. D.2 is illustrated in Examples 11.12, 11.13, and 11.14.
See Ref. 7 for additional compressible internal flow material.
E XAMPLE 11.12
Choked Fanno Flow
GIVEN Standard atmospheric air 3T 0 288 K, p 0 101
kPa1abs24 is drawn steadily through a frictionless, adiabatic converging
nozzle into an adiabatic, constant area duct as shown in
Fig. E11.12a. The duct is 2 m long and has an inside diameter of
0.1 m. The average friction factor for the duct is estimated as being
equal to 0.02.
FIND What is the maximum mass flowrate through the duct?
For this maximum flowrate, determine the values of static temperature,
static pressure, stagnation temperature, stagnation pressure,
and velocity at the inlet [section 112] and exit [section 122] of
the constant area duct. Sketch a temperature–entropy diagram for
this flow.
SOLUTION
We consider the flow through the converging nozzle to be isentropic
and the flow through the constant area duct to be Fanno
flow. A decrease in the pressure at the exit of the constant area
duct 1back pressure2 causes the mass flowrate through the nozzle
and the duct to increase. The flow throughout is subsonic. The
maximum flowrate will occur when the back pressure is lowered
to the extent that the constant area duct chokes and the Mach
number at the duct exit is equal to 1. Any further decrease of back
pressure will not affect the flowrate through the nozzle–duct
combination.
p 0 = 101 kPa (abs)
Frictionless and
adiabatic nozzle
T, K
Standard atmospheric air
T 0 = 288K
300
290
280
270
260
250
240
230
1
Fanno line
p 0.1 =
101 kPa (abs)
p 1 =
77 kPa (abs)
T 1 = 268 K
0 10 20 30 40 50
J
s – s 1 , _____
(kg•K)
(b)
F I G U R E E11.12
Adiabatic duct with friction
factor f = 0.02
2
Control volume
Section (1) D = 0.1 m Section (2)
= 2 m
(a)
p 0.2 =
84 kPa (abs)
T 0 = 288 K
p 2 =
45 kPa (abs)
T 2 = 240 K