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Problem 6.19
[Difficulty: 3]
Given:
Diffuser geometry
Find:
Acceleration of a fluid particle; plot it; plot pressure gradient; find
L
such that pressure gradient is less than
25 kPa/m
Solution:
The given data is
D
i
0.25 m
⋅
=
D
o
0.75 m
⋅
=
L1
m
⋅
=
V
i
5
m
s
⋅
=
ρ
1000
kg
m
3
⋅
=
For a linear increase in diameter
Dx
()
D
i
D
o
D
i
−
L
x
⋅
+
=
From continuity
QV
A
⋅
=
V
π
4
⋅
D
2
⋅
=
V
i
π
4
⋅
D
i
2
⋅
=
Q
0.245
m
3
s
=
Hence
Vx
π
4
⋅
2
⋅
Q
=
4Q
⋅
π
D
i
D
o
D
i
−
L
x
⋅
+
⎛
⎜
⎝
⎞
⎠
2
⋅
=
or
V
i
1
D
o
D
i
−
LD
i
⋅
x
⋅
+
⎛
⎜
⎝
⎞
⎠
2
=
The governing equation for this flow is
or, for steady 1D flow, in the notation of the problem
a
x
V
x
V
d
d
⋅
=
V
i
1
D
o
D
i
−
i
⋅
x
⋅
+
⎛
⎜
⎝
⎞
⎠
2
x
V
i
1
D
o
D
i
−
i
⋅
x
⋅
+
⎛
⎜
⎝
⎞
⎠
2
d
d
⋅
=
Hence
a
x
x
2V
i
2
⋅
D
o
D
i
−
⋅
D
i
L
⋅
1
D
o
D
i
−
D
i
L
⋅
x
⋅
+
⎡
⎢
⎣
⎤
⎥
⎦
5
⋅
−
=
This can be plotted in
Excel
(see below)
From Eq. 6.2a, pressure gradient is
x
p
∂
∂
ρ
−
a
x
⋅
=
x
p
∂
∂
2
ρ
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 Fall '07
 Lear
 Fluid Mechanics

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