fluid_mechanics

230 Chapter 5 ■ Finite Control Volume Analysis
where
is the heat transfer rate per mass flowrate, or heat transfer per unit mass. Note that Eq. 5.73 involves
energy per unit mass and is applicable to one-dimensional flow of a single stream of fluid
between two sections or flow along a streamline between two sections.
If the steady, incompressible flow we are considering also involves negligible viscous effects
1frictionless flow2, then the Bernoulli equation, Eq. 3.7, can be used to describe what happens between
two sections in the flow as
p out rV 2 out
2
q net
in
Q# net in
m #
gz out p in rV 2 in
2 gz in
(5.74)
where g rg is the specific weight of the fluid. To get Eq. 5.74 in terms of energy per unit mass, so
that it can be compared directly with Eq. 5.73, we divide Eq. 5.74 by density, r, and obtain
Minimizing loss is
the central goal of
fluid mechanical
design.
p out
r V 2 out
2 gz out p in
r V 2 in
2 gz in
A comparison of Eqs. 5.73 and 5.75 prompts us to conclude that
(5.75)
ǔ out ǔ in q net 0
(5.76)
in
when the steady incompressible flow is frictionless. For steady incompressible flow with friction,
we learn from experience (second law of thermodynamics) that
ǔ out ǔ in q net 7 0
in
In Eqs. 5.73 and 5.75, we can consider the combination of variables
p
r V 2
2 gz
(5.77)
as equal to useful or available energy. Thus, from inspection of Eqs. 5.73 and 5.75, we can conclude
that ǔ out ǔ in q net in represents the loss of useful or available energy that occurs in an incompressible
fluid flow because of friction. In equation form we have
ǔ out ǔ in q net
in
loss
For a frictionless flow, Eqs. 5.73 and 5.75 tell us that loss equals zero.
It is often convenient to express Eq. 5.73 in terms of loss as
(5.78)
p out
r V 2 out
2 gz out p in
r V 2 in
2 gz in loss
An example of the application of Eq. 5.79 follows.
(5.79)
E XAMPLE 5.23
Energy—Effect of Loss of Available Energy
GIVEN As shown in Fig. E5.23a, air flows from a room
through two different vent configurations: a cylindrical hole in
the wall having a diameter of 120 mm and the same diameter
cylindrical hole in the wall but with a well-rounded entrance.
The room pressure is held constant at 1.0 kPa above atmospheric
pressure. Both vents exhaust into the atmosphere. As discussed
in Section 8.4.2, the loss in available energy associated
with flow through the cylindrical vent from the room to the vent
exit is 0.5V 2 2/2 where V 2 is the uniformly distributed exit velocity
of air. The loss in available energy associated with flow
through the rounded entrance vent from the room to the vent exit
is 0.05V 2 2/2, where V 2 is the uniformly distributed exit velocity
of air.
FIND Compare the volume flowrates associated with the two
different vent configurations.