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Problem 4.190
[Difficulty: 3]
NOTE ERROR: Retarding torque is 0.05 N.m!
Given:
Data on rotating spray system
Find:
Differential equation for motion; steady speed; troque to stop
Solution:
Basic equation: Rotating CV
Assumptions: 1) No surface force; 2) Body torques cancel; 3) 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
⋅
=
T
0.05 N
⋅
m
⋅
=
For each branch
V
1
2
Q
π
4
d
2
⋅
⋅
=
V
6.37
m
s
=
A
π
4
d
2
⋅
=
A
19.6 mm
2
⋅
=
The basic equation reduces to a single scalar equation (FOR EACH BRANCH)
T
2
V
r
→
2
ω
→
⋅
V
→
×
r
→
×
α
→
r
×
+
()
×
ρ
⋅
⌠
⎮
⎮
⌡
d
−
A
→
r
V
xyz
→
⎯⎯
×
ρ
⋅
V
xyz
→
⎯⎯
⋅
⌠
⎮
⎮
⌡
d
=
where T is the retarding torque
α
is the angular
acceleration
But
r
2
ω
→
⋅
V
→
×
r
×
α
→
r
×
+
×
2
ω
⋅
r
⋅
V
⋅
α
r
2
⋅
+
=
(r and
α
perpendicular)
The volume integral is then
V
r
→
2
ω
→
⋅
V
→
×
r
→
×
α
→
r
→
×
+
×
ρ
⋅
⌠
⎮
⎮
⌡
d
−
ω
R
2
⋅
V
⋅
α
R
3
3
⋅
+
⎛
⎜
⎝
⎞
⎠
−
ρ
⋅
A
⋅
=
For the surface integral (FOR EACH BRANCH)
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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

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