fluid_mechanics

5.4 Second Law of Thermodynamics—Irreversible Flow 239
shaft work. This application of both Eqs. 5.54 and 5.82 allows us to ascertain the amount of loss that
occurs in incompressible turbomachine flows as is demonstrated in Example 5.28.
E XAMPLE 5.28
Energy—Fan Performance
GIVEN Consider the fan of Example 5.19.
FIND Show that only some of the shaft power into the air
is converted into useful effects. Develop a meaningful efficiency
equation and a practical means for estimating lost shaft
energy.
SOLUTION
We use the same control volume used in Example 5.19. Application
of Eq. 5.82 to the contents of this control volume yields
p 2
r V 2 2
2 gz 2 p 1
r V 2 1
2 gz 1 w shaft loss
net in
As in Example 5.26, we can see with Eq. 1 that a “useful effect”
in this fan can be defined as
useful effect w shaft loss
net in
a p 2
r V 2 2
2 gz 2b a p 1
r V 1 2
2 gz 1b
(1)
(2) (Ans)
In other words, only a portion of the shaft work delivered to the
air by the fan blades is used to increase the available energy of the
air; the rest is lost because of fluid friction.
A meaningful efficiency equation involves the ratio of shaft
work converted into a useful effect 1Eq. 22 to shaft work into the
air, w shaft net in . Thus, we can express efficiency, h, as
h
w shaft
net in loss
w shaft
net in
(3)
However, when Eq. 5.54, which was developed from the momentof-momentum
equation 1Eq. 5.422, is applied to the contents of
the control volume of Fig. E5.19, we obtain
w shaft U 2 V u2
net in
Combining Eqs. 2, 3, and 4, we obtain
h 531p 2r2 1V 2 222 gz 2 4
31p 1r2 1V 2 122 gz 1 46U 2 V u2
(4)
(5) (Ans)
Equation 5 provides us with a practical means to evaluate the efficiency
of the fan of Example 5.19.
Combining Eqs. 2 and 4, we obtain
loss U 2 V u2 ca p 2
r V 2 2
2 gz 2b
a p 1
r V 2 1
2 gz 1bd
(6) (Ans)
COMMENT Equation 6 provides us with a useful method of
evaluating the loss due to fluid friction in the fan of Example
5.19 in terms of fluid mechanical variables that can be measured.
5.4 Second Law of Thermodynamics—Irreversible Flow 5
The second law of
thermodynamics
formalizes the notion
of loss.
The second law of thermodynamics affords us with a means to formalize the inequality
ǔ 2 ǔ 1 q net 0
(5.90)
in
for steady, incompressible, one-dimensional flow with friction 1see Eq. 5.732. In this section we
continue to develop the notion of loss of useful or available energy for flow with friction. Minimization
of loss of available energy in any flow situation is of obvious engineering importance.
5.4.1 Semi-infinitesimal Control Volume Statement
of the Energy Equation
If we apply the one-dimensional, steady flow energy equation, Eq. 5.70, to the contents of a control
volume that is infinitesimally thin as illustrated in Fig 5.8, the result is
m # c dǔ d a p r b d aV2 2 b g 1dz2d dQ# net
in
(5.91)
5 This entire section may be omitted without loss of continuity in the text material.