fluid_mechanics

11.5 Nonisentropic Flow of an Ideal Gas 609
Constant area duct
Fluid flow
F I G U R E 11.14
flow.
Constant area duct
texts on compressible flows and gas dynamics 1for example, Refs. 4, 5, and 62 for additional
material on this subject.
F l u i d s i n t h e N e w s
Rocket nozzles To develop the massive thrust needed for space
shuttle liftoff, the gas leaving the rocket nozzles must be moving
supersonically. For this to happen, the nozzle flow path must first
converge, then diverge. Entering the nozzle at very high pressure
and temperature, the gas accelerates in the converging portion of
the nozzle until the flow chokes at the nozzle throat. Downstream
of the throat, the gas further accelerates in the diverging portion of
the nozzle (area ratio of 77.5 to 1), finally exiting into the atmosphere
supersonically. At launch, the static pressure of the gas
flowing from the nozzle exit is less than atmospheric and so the
flow is overexpanded. At higher elevations where the atmospheric
pressure is much less than at launch level, the static pressure of
the gas flowing from the nozzle exit is greater than atmospheric
and so now the flow is underexpanded, the result being expansion
or divergence of the exhaust gas as it exits into the atmosphere.
(See Problem 11.49.)
11.4.3 Constant Area Duct Flow
For steady, one-dimensional, isentropic flow of an ideal gas through a constant area duct 1see
Fig. 11.142, Eq. 11.50 suggests that dV 0 or that flow velocity remains constant. With the energy
equation 1Eq. 5.692 we can conclude that since flow velocity is constant, the fluid enthalpy
and thus temperature are also constant for this flow. This information and Eqs. 11.36 and 11.46
indicate that the Mach number is constant for this flow also. This being the case, Eqs. 11.59
and 11.60 tell us that fluid pressure and density also remain unchanged. Thus, we see that a
steady, one-dimensional, isentropic flow of an ideal gas does not involve varying velocity or
fluid properties unless the flow cross-sectional area changes.
In Section 11.5 we discuss nonisentropic, steady, one-dimensional flows of an ideal gas
through a constant area duct and also a normal shock wave. We learn that friction andor heat transfer
can also accelerate or decelerate a fluid.
11.5 Nonisentropic Flow of an Ideal Gas
Fanno flow involves
wall friction with
no heat transfer
and constant crosssectional
area.
Actual fluid flows are generally nonisentropic. An important example of nonisentropic flow involves
adiabatic 1no heat transfer2 flow with friction. Flows with heat transfer 1diabatic flows2 are generally
nonisentropic also. In this section we consider the adiabatic flow of an ideal gas through a constant
area duct with friction. This kind of flow is often referred to as Fanno flow. We also analyze the
diabatic flow of an ideal gas through a constant area duct without friction 1Rayleigh flow2. The concepts
associated with Fanno and Rayleigh flows lead to further discussion of normal shock waves.
11.5.1 Adiabatic Constant Area Duct Flow
with Friction (Fanno Flow)
Consider the steady, one-dimensional, and adiabatic flow of an ideal gas through the constant area
duct shown in Fig. 11.15. This is Fanno flow. For the control volume indicated, the energy equation
1Eq. 5.692 leads to
01negligibly 01flow is adiabatic2
small for 01flow is steady
gas flow2
throughout2
m # c ȟ 2 ȟ 1 V 2 2 V 2 1
2
g1z 2 z 1 2d Q # net W # shaft
in. net in