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Problem 4.172
[Difficulty: 4]
CS
(moves
at speed U)
c
d
F
f
R
y
y
x
Given:
Water jet striking moving vane
Find:
Plot of terminal speed versus turning angle; angle to overcome static friction
Solution:
Basic equations: Momentum flux in x and y directions
Assumptions:
1) Incompressible flow 2) Atmospheric pressure in jet 3) Uniform flow 4) Jet relative velocity is constant
Then
F
f
−
Ma
rfx
⋅
−
u
1
ρ
−
V
1
⋅
A
1
⋅
()
⋅
u
2
ρ
V
2
⋅
A
2
⋅
⋅
+
=
VU
−
−
ρ
−
⋅
A
⋅
[]
⋅
−
(
) cos
θ
⋅
ρ
−
⋅
A
⋅
⋅
+
=
a
rfx
ρ
−
2
A
⋅
1
cos
θ
−
⋅
F
f
−
M
=
(1)
Also
R
y
Mg
⋅
−
v
1
ρ
−
V
1
⋅
A
1
⋅
⋅
v
2
ρ
⋅
V
2
⋅
A
2
⋅
+
=
0V
U
−
(
) sin
θ
⋅
ρ
−
⋅
A
⋅
⋅
+
=
R
y
⋅
ρ
−
2
A
⋅
sin
θ
⋅
+
=
At terminal speed a
rfx
= 0 and F
f
= μ
k
R
y
.
Hence in Eq 1
0
ρ
t
−
2
⋅
A
⋅
1
cos
θ
−
⋅
μ
k
⋅
ρ
t
−
2
⋅
A
⋅
sin
θ
⋅
+
⎡
⎣
⎤
⎦
⋅
−
M
=
ρ
t
−
2
⋅
A
⋅
1
cos
θ
−
μ
k
sin
θ
⋅
−
⋅
M
μ
k
g
⋅
−
=
or
t
−
μ
k
M
⋅
g
⋅
ρ
A
⋅
1
cos
θ
−
μ
k
sin
θ
⋅
−
⋅
=
U
t
V
μ
k
M
⋅
g
⋅
ρ
A
⋅
1
cos
θ
−
μ
k
sin
θ
⋅
−
⋅
−
=
The terminal speed as a function of angle is plotted below; it can be generated in
Excel
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This note was uploaded on 10/19/2011 for the course EGN 3353C taught by Professor Lear during the Fall '07 term at University of Florida.
 Fall '07
 Lear
 Fluid Mechanics, Flux

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