fluid_mechanics

226 Chapter 5 ■ Finite Control Volume Analysis
m • out
m • in
dA
V
Streamtube
F I G U R E 5.7
Streamtube flow.
Furthermore, if there is only one stream entering and leaving the control volume, then
aǔ p
cs
r V 2
gzb rV nˆ dA
2
aǔ p r V 2
2 gzb m # out aǔ p
out
r V 2
2 gzb m # in
in
(5.66)
Uniform flow as described above will occur in an infinitesimally small diameter streamtube as illustrated
in Fig. 5.7. This kind of streamtube flow is representative of the steady flow of a particle
of fluid along a pathline. We can also idealize actual conditions by disregarding nonuniformities
in a finite cross section of flow. We call this one-dimensional flow and although such uniform flow
rarely occurs in reality, the simplicity achieved with the one-dimensional approximation often justifies
its use. More details about the effects of nonuniform distributions of velocities and other fluid
flow variables are considered in Section 5.3.4 and in Chapters 8, 9, and 10.
If shaft work is involved, the flow must be unsteady, at least locally 1see Refs. 1 and 22. The
flow in any fluid machine that involves shaft work is unsteady within that machine. For example,
the velocity and pressure at a fixed location near the rotating blades of a fan are unsteady. However,
upstream and downstream of the machine, the flow may be steady. Most often shaft work is
associated with flow that is unsteady in a recurring or cyclical way. On a time-average basis for
flow that is one-dimensional, cyclical, and involves only one stream of fluid entering and leaving
the control volume, Eq. 5.64 can be simplified with the help of Eqs. 5.9 and 5.66 to form
m # c ǔ out ǔ in a p r b a p
out
r b in
V 2 out V 2 in
2
g1z out z in 2d Q # net
in
W # shaft
net in
(5.67)
We call Eq. 5.67 the one-dimensional energy equation for steady-in-the-mean flow. Note that Eq. 5.67
is valid for incompressible and compressible flows. Often, the fluid property called enthalpy, ȟ, where
(5.68)
ȟ ǔ p r
The energy equation
is sometimes
written in terms of
enthalpy.
is used in Eq. 5.67. With enthalpy, the one-dimensional energy equation for steady-in-the-mean
flow 1Eq. 5.672 is
m # c ȟ out ȟ in V 2 out V 2 in
2
g1z out z in 2d Q # net
in
W # shaft
net in
(5.69)
Equation 5.69 is often used for solving compressible flow problems. Examples 5.20 and 5.21
illustrate how Eqs. 5.67 and 5.69 can be used.
E XAMPLE 5.20
Energy—Pump Power
GIVEN A pump delivers water at a steady rate of 300 gal/min as
shown in Fig. E5.20. Just upstream of the pump [section (1)]
where the pipe diameter is 3.5 in., the pressure is 18 psi. Just
downstream of the pump [section (2)] where the pipe diameter is
1 in., the pressure is 60 psi. The change in water elevation across
the pump is zero. The rise in internal energy of water, ǔ 2 ǔ 1 , associated
with a temperature rise across the pump is 93 ft lb/lbm.
The pumping process is considered to be adiabatic.
FIND Determine the power (hp) required by the pump.