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Problem 4.56
[Difficulty: 4]
Given:
Data on draining of a funnel
Find:
Diameter that will drain in 1 min.; plot diamter versus depth y
0
Solution:
0
=
⋅
+
∂
∂
∫
∫
CS
CV
A
d
V
V
d
t
r
r
ρ
Basic equation
Assumptions:
1) Uniform flow
2) Incompressible flow
3) Neglect air density
Treating the funnel as the CV the basic equation becomes
t
0
y
y
ρ
A
funnel
⋅
⌠
⎮
⌡
d
∂
∂
ρ
V
⋅
A
opening
⋅
+
0
=
For the funnel
A
funnel
π
r
2
⋅
=
π
y tan
θ
()
⋅
2
⋅
=
Hence
ρπ
⋅
tan
θ
2
⋅
t
0
y
y
y
2
⌠
⎮
⌡
d
∂
∂
⋅
ρ
V
⋅
π
4
⋅
d
2
⋅
+
0
=
or
tan
θ
2
t
y
3
3
⎛
⎜
⎝
⎞
⎠
d
d
⋅
2g
⋅
y
⋅
−
d
2
4
⋅
=
Then
tan
θ
2
y
2
⋅
dy
dt
⋅
⋅
y
⋅
−
d
2
4
⋅
=
Separating variables
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 Fall '07
 Lear
 Fluid Mechanics

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