Problem 4.37
[Difficulty: 3]
Given:
Velocity distribution in annulus
Find:
Volume flow rate; average velocity; maximum velocity; plot velocity distribution
Solution:
Governing equation
For the flow rate (Eq. 4.14a) and average velocity (Eq.
4.14b)
Q
A
→
V
→
⌠
⎮
⎮
⌡
d
=
V
av
Q
A
=
The given data is
R
o
5 mm
⋅
=
R
i
1 mm
⋅
=
∆
p
L
10
−
kPa
m
⋅
=
μ
0.1
N s
⋅
m
2
⋅
=
(From Fig. A.2)
u r
( )
∆
p
−
4
μ
⋅
L
⋅
R
o
2
r
2
−
R
o
2
R
i
2
−
ln
R
i
R
o
⎛
⎜
⎝
⎞
⎠
ln
R
o
r
⎛
⎜
⎝
⎞
⎠
⋅
+
⎛
⎜
⎜
⎜
⎝
⎞
⎟
⎠
⋅
=
The flow rate is
Q
R
i
R
o
r
u r
( ) 2
⋅
π
⋅
r
⋅
⌠
⎮
⌡
d
=
Considerable mathematical manipulation leads to
Q
∆
p
π
⋅
8
μ
⋅
L
⋅
R
o
2
R
i
2
−
⎛
⎝
⎞
⎠
⋅
R
o
2
R
i
2
−
⎛
⎝
⎞
⎠
ln
R
o
R
i
⎛
⎜
⎝
⎞
⎠
R
i
2
R
o
2
+
⎛
⎝
⎞
⎠
−
⎡
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎦
⋅
=
Substituting values
Q
π
8
10
−
10
3
⋅
(
)
⋅
N
m
2
m
⋅
⋅
m
2
0.1 N
⋅
s
⋅
⋅
5
2
1
2
−
(
)
⋅
m
1000
⎛
⎜
⎝
⎞
⎠
2
⋅
5
2
1
2
−
ln
5
1
⎛
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '07
 Lear
 Fluid Mechanics, Velocity, RO, Ri, Annulus

Click to edit the document details