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Problem 4.188
[Difficulty: 3]
Given:
Data on rotating spray system
Find:
Torque required to hold stationary; steadystate speed
Solution:
Basic equation: Rotating CV
Assumptions: 1) No surface force; 2) Body torques cancel; 3) Sprinkler stationary; 4) Steady flow; 5) Uniform flow; 6) L<<r
The given data is
Q1
5
L
min
⋅
=
R
225 mm
⋅
=
d5
m
m
⋅
=
ρ
999
kg
m
3
⋅
=
For each branch
V
1
2
Q
π
4
d
2
⋅
⋅
=
V
6.37
m
s
=
The basic equation reduces to a single scalar equation (FOR EACH BRANCH)
T
shaft
V
r
α
→
r
×
()
×
ρ
⋅
⌠
⎮
⎮
⌡
d
−
A
→
r
V
xyz
→
⎯⎯
×
ρ
⋅
V
xyz
→
⎯⎯
⋅
⌠
⎮
⎮
⌡
d
=
where
α
is the angular acceleration
But
r
α
→
r
×
×
r
2
α
⋅
=
(r and
α
perpendicular); the volume integral is
V
r
α
→
r
×
×
ρ
⋅
⌠
⎮
⎮
⌡
dV
r
2
α
⋅
ρ
⋅
⌠
⎮
⎮
⌡
d
=
R
3
3
α
⋅
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 Fall '07
 Lear
 Fluid Mechanics

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