fluid_mechanics

228 Chapter 5 ■ Finite Control Volume Analysis
SOLUTION
We use a control volume that includes the steam in the turbine
from the entrance to the exit as shown in Fig. E5.21b. Applying
Eq. 5.69 to the steam in this control volume we get
0 (elevation change is negligible)
0 (adiabatic flow)
m # c ȟ 2 ȟ 1 V 2 2 V 2 1
2
g1z 2 z 1 2d Q # net
in
The work output per unit mass of steam through-flow, w shaft net in , can
be obtained by dividing Eq. 1 by the mass flow rate, m # , to obtain
W # shaft
net in
w shaft
net in m # ȟ 2 ȟ 1 V 2 2 V 2 1
2
Since w shaft net out w shaft net in , we obtain
W # shaft
net in
(1)
(2)
Section (1)
V 1 = 30
^
m/s
h 1 = 3348 kJ/kg
Thus,
Control volume
Steam turbine
w shaft = ?
F I G U R E E5.21b
Section (2)
V 2 = 60
^
m/s
h 2 = 2550 kJ/kg
w shaft 3348 kJkg 2550 kJkg 1.35 kJkg
net out
797 kJ/kg
(Ans)
or
w shaft ȟ 1 ȟ 2 V 2 1 V 2 2
net out
2
w shaft
3348 kJkg 2550 kJkg
net out
3130 m s2 2 160 ms2 2 431 J1Nm24
231 1kgm21Ns 2 2411000 JkJ2
COMMENT Note that in this particular example, the change
in kinetic energy is small in comparison to the difference in enthalpy
involved. This is often true in applications involving steam
turbines. To determine the power output, , we must know
the mass flowrate, m #
W # shaft
.
If the flow is steady throughout, one-dimensional, and only one fluid stream is involved, then the
shaft work is zero and the energy equation is
V5.12 Pelton wheel
turbine
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
(5.70)
We call Eq. 5.70 the one-dimensional, steady flow energy equation. This equation is valid for incompressible
and compressible flows. For compressible flows, enthalpy is most often used in the
one-dimensional, steady flow energy equation and, thus, we have
m # c ȟ out ȟ in V 2 out V 2 in
2
An example of the application of Eq. 5.70 follows.
g1z out z in 2d Q # net
in
(5.71)
E XAMPLE 5.22
Energy—Temperature Change
GIVEN The 420-ft waterfall shown in Fig. E5.22a involves
steady flow from one large body of water to another.
FIND
flow.
Determine the temperature change associated with this
SOLUTION
To solve this problem we consider a control volume consisting of
a small cross-sectional streamtube from the nearly motionless
surface of the upper body of water to the nearly motionless surface
of the lower body of water as is sketched in Fig. E5.22b. We
need to determine T 2 T 1 . This temperature change is related to
the change of internal energy of the water, ǔ 2 ǔ 1 , by the relationship
T 2 T 1 ǔ 2 ǔ 1
č
(1)