fluid_mechanics

which for dTds 0 1point b2 gives
(11.118)
The flow at point b is subsonic 1Ma b 6 1.02. Recall that k 7 1 for any gas.
To learn more about Rayleigh flow, we need to consider the energy equation in addition to
the equations already used. Application of the energy equation 1Eq. 5.692 to the Rayleigh flow
through the finite control volume of Fig. 11.21 yields
m # c ȟ 2 ȟ 1 V 2 2 V 1
2
2
11.5 Nonisentropic Flow of an Ideal Gas 623
Ma b B
1
k
01negligibly small
for gas flow2
g1z 2 z 1 2d Q # net
in
01flow is steady
throughout2
or in differential form for Rayleigh flow through the semi-infinitesimal control volume of Fig. 11.21
dȟ V dV dq
W # shaft
net in
where dq is the heat transfer per unit mass of fluid in the semi-infinitesimal control volume.
By using dȟ c p dT Rk dT1k 12 in Eq. 11.119, we obtain
(11.119)
dV
V dq
c p T c V dT
T
dV V 2 1k 12
kRT
1
d
(11.120)
Fluid temperature
reduction can accompany
heating a
subsonic Rayleigh
flow.
Thus, by combining Eqs. 11.36 1ideal gas speed of sound2, 11.46 1Mach number2, 11.1 and 11.77
1ideal gas equation of state2, 11.79 1continuity2, and 11.112 1linear momentum2 with Eq. 11.120 1energy2
we get
dV
(11.121)
V dq 1
c p T 11 Ma 2 2
With the help of Eq. 11.121, we see clearly that when the Rayleigh flow is subsonic 1Ma 6 12,
fluid heating 1dq 7 02 increases fluid velocity while fluid cooling 1dq 6 02 decreases fluid velocity.
When Rayleigh flow is supersonic 1Ma 7 12, fluid heating decreases fluid velocity and fluid
cooling increases fluid velocity.
The second law of thermodynamics states that, based on experience, entropy increases with
heating and decreases with cooling. With this additional insight provided by the conservation of
energy principle and the second law of thermodynamics, we can say more about the Rayleigh
line in Fig. 11.22. A summary of the qualitative aspects of Rayleigh flow is outlined in Table
11.2 and Fig. 11.23. Along the upper portion of the line, which includes point b, the flow is subsonic.
Heating the fluid results in flow acceleration to a maximum Mach number of 1 at point
a. Note that between points b and a along the Rayleigh line, heating the fluid results in a temperature
decrease and cooling the fluid leads to a temperature increase. This trend is not surprising
if we consider the stagnation temperature and fluid velocity changes that occur between
points a and b when the fluid is heated or cooled. Along the lower portion of the Rayleigh curve
the flow is supersonic. Rayleigh flows may or may not be choked. The amount of heating or
cooling involved determines what will happen in a specific instance. As with Fanno flows, an
abrupt deceleration from supersonic flow to subsonic flow across a normal shock wave can also
occur in Rayleigh flows.
TABLE 11.2
Summary of Rayleigh Flow Characteristics
Ma 6 1
Ma 7 1
Heating
Acceleration
Deceleration
Cooling
Deceleration
Acceleration