fluid_mechanics

5.3 First Law of Thermodynamics—The Energy Equation 237
SOLUTION
Application of Eq. 5.87 to the contents of the control volume
shown in Fig. E5.26 leads to
or solving Eq. 1 for loss we get
0 1change in gz is negligible2
To proceed further, we need values of w shaft net in , V 1 , and V 2 . These
quantities can be obtained as follows. For shaft work
power to fan motor
w shaft
net in m #
or
p 2
r a 2V 2 2
gz
2 2 p 1
r a 1V 2 1
gz
2 1
loss w shaft
net in
For the average velocity at section 112, V 1 , from Eq. 5.11 we obtain
V 1 m#
rA 1
m #
(4)
r1pD 2 142
0.479 ms
For the average velocity at section 122, V 2 ,
(a) For the assumed uniform velocity profiles 1a 1 a 2 1.02,
Eq. 2 yields
loss w shaft
net in
loss w shaft
net in
a p 2 p 1
b a 1V 2 1
a 2V 2 2
r 2 2
w shaft 10.14 W2311 N # ms2W4
160 smin2
net in 0.1 kgmin
84.0 N # mkg
10.1 kg min2 11 min60 s2 11000 mmm2 2
11.23 kgm 3 23p160 mm2 2 44
V 2 10.1 kg min2 11 min60 s2 11000 mmm2 2
11.23 kgm 3 23p130 mm2 2 44
1.92 ms
a p 2 p 1
b V 2 1
r 2 V 2 2
2
Using Eqs. 3, 4, and 5 and the pressure rise given in the problem
statement, Eq. 6 gives
loss 84.0 N # m
kg
10.1 kPa211000 Pa kPa211 Nm 2 Pa2
1.23 kgm 3
10.479 ms2 2
11.92 ms2 2
231 1kg # m21N # s 2 24 231 1kg # m21N # s 2 24
(1)
(2)
(3)
(5)
(6)
Turbulent
flow
or
(Ans)
(b) For the actual velocity profiles 1a 1 2, a 2 1.082, Eq. 1
gives
If we use Eqs. 3, 4, and 5 and the given pressure rise, Eq. 7
yields
or
Section (2)
α 2 = 1.08
D 2 = 30 mm
Control volume
loss 84.0 N # mkg 81.3 N # mkg
0.115 N # mkg 1.84 N # mkg
0.975 N # mkg
loss w shaft
net in
loss 84 N # mkg 10.1 kPa211000 Pa kPa211 Nm 2 Pa2
1.23 kgm 3
210.479 ms2 2
1.0811.92 m s2 2
231 1kg # m21N # s 2 24 231 1kg # m21 # s 2 24
loss 84.0 N # mkg 81.3 N # mkg
0.230 N # mkg 1.99 N # mkg
0.940 N # mkg
(7)
(Ans)
COMMENT The difference in loss calculated assuming uniform
velocity profiles and actual velocity profiles is not large
compared to for this fluid flow situation.
w shaft net in
D 1 = 60 mm
Section (1)
α 1 = 2.0
F I G U R E E5.26
Laminar flow
m • = 0.1 kg/min
a p 2 p 1
b a V 1 2
r
1
2 a V 2 2
2
2