fluid_mechanics

204 Chapter 5 ■ Finite Control Volume Analysis
z
F A
g
D 1 = 16 mm
x
Control volume
n
Control volume
w 1
p 1
A 1
Section (1)
w 1
h = 30 mm
z
w
w 2
F I G U R E E5.11a
D 2 = 5 mm
Section (2)
p 2 A 2
w 2
F I G U R E E5.11b
F A = anchoring force that holds
nozzle in place
n = weight of nozzle
w = weight of water contained in
the nozzle
p 1 = gage pressure at section (1)
A 1 = cross section area at
section (1)
p 2 = gage pressure at section (2)
A 2 = cross section area at
section (2)
w 1 = z direction velocity at
control volume entrance
w 2
= z direction velocity at
control volume exit
fluid density, r, is constant throughout. Proceeding further we obtain
for Eq. 1
To complete this example we use quantities given in the
problem statement to quantify the terms on the right-hand side
1m # 121w 1 2 m # of Eq. 6.
21w 2 2
From Eq. 5.6,
F A w n p 1 A 1 w p 2 A 2
(3)
where m # m # rw 1 A 1 rQ
rAV is the mass flowrate.
1999 kgm 3 210.6 liters2110 3 m 3 liter2
Note that w 1 and w are used because both of these velocities
are “down.” Also, m # 2
0.599 kgs
(7)
1 is used because it is associated with
flow into the control volume. Similarly, m # 2 is used because it is and
associated with flow out of the control volume. Solving Eq. 3 for
the anchoring force, F A , we obtain
w 1 Q Q
F A m # 1w 1 m # A 1 p1D142
2
2w 2 w n p 1 A 1 w p 2 A 2 (4)
From the conservation of mass equation, Eq. 5.12, we obtain
10.6 liter s2110 3 m 3 liter2
(8)
p116 mm2 2 411000 2 mm 2 m 2 2 2.98 m s
m # 1 m # 2 m #
(5) Also from Eq. 5.6,
which when combined with Eq. 4 gives
F A m # 1w 1 w 2 2 w n p 1 A 1 w w p 2 A 2
It is instructive to note how the anchoring force is affected
by the different actions involved. As expected, the nozzle
weight, w n , the water weight, w , and gage pressure force at
section 112, p 1 A 1 , all increase the anchoring force, while the
gage pressure force at section 122, p 2 A 2 , acts to decrease the
anchoring force. The change in the vertical momentum
flowrate, m # 1w 1 w 2 2, will, in this instance, decrease the anchoring
force because this change is negative 1w 2 7 w 1 2.
(6)
w 2 Q
Q
A 2 p1D 2 242
10.6 liter s2110 3 m 3 liter2
p15 mm2 2 411000 2 mm 2 m 2 2 30.6 m s
The weight of the nozzle, w n , can be obtained from the nozzle
mass, m n , with
w n m n g 10.1 kg219.81 ms 2 2 0.981 N
(9)
(10)
The weight of the water in the control volume, w , can be obtained
from the water density, r, and the volume of water, V w , in