Problem 4.32
[Difficulty: 2]
Given:
Data on flow through device
Find:
Velocity
V
3
; plot
V
3
against time; find when
V
3
is zero; total mean flow
Solution:
Governing equation:
For incompressible flow (Eq. 4.13) and uniform flow
A
→
V
→
⌠
⎮
⎮
⌡
dV
→
∑
A
→
⋅
=
0
=
Applying to the device (assuming
V
3
is out)
V
1
−
A
1
⋅
V
2
A
2
⋅
−
V
3
A
3
⋅
+
0
=
V
3
V
1
A
1
⋅
V
2
A
2
⋅
+
A
3
=
10 e
t
2
−
⋅
m
s
⋅
0.1
×
m
2
⋅
2 cos 2
π
⋅
t
⋅
()
⋅
m
s
⋅
0.2
×
m
2
⋅
+
0.15 m
2
⋅
=
The velocity at
A
3
is
V
3
6.67 e
t
2
−
⋅
2.67 cos 2
π
⋅
t
⋅
⋅
+
=
The total mean volumetric flow at
A
3
Q
0
∞
t
V
3
A
3
⋅
⌠
⎮
⌡
d
=
0
∞
t
6.67 e
t
2
−
⋅
2.67 cos 2
π
⋅
t
⋅
⋅
+
⎛
⎜
⎝
⎞
⎠
0.15
⋅
⌠
⎮
⎮
⎮
⌡
d
m
s
m
2
⋅
⎛
⎝
⎞
⎠
⋅
=
Q
∞
t
2
−
e
t
2
−
⋅
1
5
π
⋅
sin 2
π
⋅
t
⋅
⋅
+
⎛
⎜
⎝
⎞
⎠
lim
→
2
−
−
=
2m
3
⋅
=
Q2
m
3
⋅
=
The time at which
V
3
first is zero, and the plot of
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 Fall '07
 Lear
 Fluid Dynamics, Fluid Mechanics, Zagreb, 2m, Incompressible Flow, 2 m

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