fluid_mechanics

620 Chapter 11 ■ Compressible Flow
For this example,
or
so that
By using the value from Eq. 1 and Fig. D.2, we get
and
f 1/ 2 / 1 2
D
f 1/* / 12
D
10.02211 m2
0.1 m 0.4 f 1/* / 22
D
f 1/* / 2 2
0.2
D
Ma 2 0.70
p 2
p* 1.5
f 1/* / 22
D
(1)
(Ans)
(2)
We obtain
p 2
from
p 2 a p 2
p* b ap* p 1
b a p 1
p 0,1
b 1p 0,1 2
where p 2p* is given in Eq. 2 and p*p 1 , p 1p 0,1 , and p 0,1 are
the same as they were in Example 11.12. Thus,
p 2 11.52 a 1
1.7 b 10.7623101 kPa1abs24
68.0 kPa1abs2
(Ans)
COMMENT A larger back pressure [68.0 kPa1abs2] than the
one associated with choked flow through a Fanno duct [45 kPa1abs2]
will maintain the same flowrate through a shorter Fanno duct with
the same friction coefficient. The flow through the shorter duct is not
choked. It would not be possible to maintain the same flowrate
through a Fanno duct longer than the choked one with the same friction
coefficient, regardless of what back pressure is used.
Rayleigh flow involves
heat transfer
with no wall friction
and constant
cross-sectional area.
11.5.2 Frictionless Constant Area Duct Flow with Heat Transfer
(Rayleigh Flow)
Consider the steady, one-dimensional, and frictionless flow of an ideal gas through the constant
area duct with heat transfer illustrated in Fig. 11.21. This is Rayleigh flow. Application of the
linear momentum equation 1Eq. 5.222 to the Rayleigh flow through the finite control volume
sketched in Fig. 11.21 results in
01frictionless flow2
or
p 1 A 1 m # V 1 p 2 A 2 m # V 2 R x
p 1rV22 constant
(11.110)
r
Use of the ideal gas equation of state 1Eq. 11.12 in Eq. 11.110 leads to
p 1rV22 RT
constant
(11.111)
p
Since the flow cross-sectional area remains constant for Rayleigh flow, from the continuity equation
1Eq. 11.402 we conclude that
rV constant
For a given Rayleigh flow, the constant in Eq. 11.111, the density–velocity product, rV, and the
ideal gas constant are all fixed. Thus, Eq. 11.111 can be used to determine values of fluid temperature
corresponding to the local pressure in a Rayleigh flow.
To construct a temperature–entropy diagram for a given Rayleigh flow, we can use Eq. 11.76,
which was developed earlier from the second T ds relationship. Equations 11.111 and 11.76 can
be solved simultaneously to obtain the curve sketched in Fig. 11.22. Curves like the one in Fig.
11.22 are called Rayleigh lines.
Frictionless and adiabatic
converging–diverging duct
Semi-infinitesimal
control volume
Frictionless duct with
heat transfer
Flow
D = constant
Section (1)
Finite
control volume
Section (2)
F I G U R E 11.21
Rayleigh flow.