Problem 6.17
[Difficulty: 3]
Given:
Flow in a pipe with variable area
Find:
Expression for pressure gradient and pressure; Plot them
Solution:
Assumptions: 1) Incompressible flow 2) Flow profile remains unchanged so centerline velocity can represent average velocity
Basic equations
QV
A
⋅
=
Given data
ρ
1250
kg
m
3
⋅
=
A
0
0.25 m
2
⋅
=
a
1.5 m
⋅
=
L5
m
⋅
=
u
0
10
m
s
⋅
=
p
0
300 kPa
⋅
=
For this 1D flow
Qu
0
A
0
⋅
=
uA
⋅
=
Ax
()
A
0
1e
x
a
−
+
e
x
2a
⋅
−
−
⎛
⎜
⎝
⎞
⎠
⋅
=
so
ux
() u
0
A
0
A
⋅
=
u
0
x
a
−
+
e
x
⋅
−
−
⎛
⎜
⎝
⎞
⎠
=
a
x
u
x
u
∂
∂
⋅
v
y
u
∂
∂
⋅
+
=
u
0
x
a
−
+
e
x
⋅
−
−
⎛
⎜
⎝
⎞
⎠
x
u
0
x
a
−
+
e
x
⋅
−
−
⎛
⎜
⎝
⎞
⎠
⎡
⎢
⎢
⎣
⎤
⎥
⎥
⎦
∂
∂
⋅
=
u
0
2
e
x
⋅
−
⋅
2e
x
⋅
−
⋅
1
−
⎛
⎜
⎝
⎞
⎠
⋅
⋅
e
x
a
−
e
x
⋅
−
−
1
+
⎛
⎜
⎝
⎞
⎠
3
⋅
=
For the pressure
x
p
∂
∂
ρ
−
a
x
⋅
ρ
g
x
⋅
−
=
ρ
u
0
2
⋅
e
x
⋅
−
⋅
x
⋅
−
⋅
1
−
⎛
⎜
⎝
⎞
⎠
⋅
⋅
e
x
a
−
e
x
⋅
−
−
1
+
⎛
⎜
⎝
⎞
⎠
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 Fall '07
 Lear
 Fluid Dynamics, Fluid Mechanics, Following, Incompressible Flow, ∂x

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