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Problem 4.99
[Difficulty: 4]
Given:
Data on flow in wind tunnel
Find:
Mass flow rate in tunnel; Maximum velocity at section 2; Drag on object
Solution:
Basic equations: Continuity, and momentum flux in x direction; ideal gas equation
p
ρ
R
⋅
T
⋅
=
Assumptions:
1) Steady flow
2) Uniform density at each section
From continuity
m
flow
ρ
1
V
1
⋅
A
1
⋅
=
ρ
1
V
1
⋅
π
D
1
2
⋅
4
⋅
=
where m
flow
is the mass flow rate
We take ambient conditions for the air density
ρ
air
p
atm
R
air
T
atm
⋅
=
ρ
air
101000
N
m
2
⋅
kg K
⋅
286.9 N
⋅
m
⋅
×
1
293 K
⋅
×
=
ρ
air
1.2
kg
m
3
=
m
flow
1.2
kg
m
3
⋅
12.5
×
m
s
⋅
π
0.75 m
⋅
()
2
⋅
4
×
=
m
flow
6.63
kg
s
=
Also
m
flow
A
2
ρ
2
u
2
⋅
⌠
⎮
⎮
⌡
d
=
ρ
air
0
R
r
V
max
r
R
⋅
2
⋅
π
⋅
r
⋅
⌠
⎮
⎮
⌡
d
⋅
=
2
π
⋅
ρ
air
⋅
V
max
⋅
R
0
R
r
r
2
⌠
⎮
⌡
d
⋅
=
2
π
⋅
ρ
air
⋅
V
max
⋅
R
2
⋅
3
=
V
max
3m
flow
⋅
2
π
⋅
ρ
air
⋅
R
2
⋅
=
V
max
3
2
π
⋅
6.63
×
kg
s
⋅
m
3
1.2 kg
⋅
×
1
0.375 m
⋅
⎛
⎝
⎞
⎠
2
×
=
V
max
18.8
m
s
=
For x momentum
R
x
p
1
A
⋅
+
p
2
A
⋅
−
V
1
ρ
1
−
V
1
⋅
A
⋅
⋅
A
2
ρ
2
u
2
⋅
u
2
⋅
⌠
⎮
⎮
⌡
d
+
=
R
x
p
2
p
1
−
A
⋅
V
1
m
flow
⋅
−
0
R
r
ρ
air
V
max
r
R
⋅
⎛
⎝
⎞
⎠
2
⋅
2
⋅
π
⋅
r
⋅
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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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