Problem 4.153
[Difficulty: 4]
Given:
Data on vane/slider
Find:
Formula for acceleration, speed, and position; plot
Solution:
Apply x momentum
Assumptions:
1) All changes wrt CV
2) Incompressible flow 3) Atmospheric pressure in jet 4) Uniform flow 5) Constant jet area
The given data is
ρ
999
kg
m
3
⋅
=
M3
0
k
g
⋅
=
A
0.005 m
2
⋅
=
V2
0
m
s
⋅
=
k
7.5
Ns
⋅
m
⋅
=
Then
k
−
UM
a
rf
⋅
−
u
1
ρ
−
VU
−
()
⋅
A
⋅
[]
⋅
u
2
m
2
⋅
+
u
3
m
3
⋅
+
=
where
a
rf
dU
dt
=
u
1
−
=
u
2
0
=
u
3
0
=
Hence
k
−
U
⋅
M
dU
dt
⋅
−
ρ
−
−
2
⋅
A
⋅
=
or
dU
dt
ρ
−
2
⋅
A
⋅
M
kU
⋅
M
−
=
The acceleration is thus
a
ρ
−
2
⋅
A
⋅
M
⋅
M
−
=
The differential equation for
U
can be solved analytically, but is quite messy.
Instead we use a simple numerical method  Euler's
method
Un
1
+
U
n
ρ
n
−
2
⋅
A
⋅
M
kUn
⋅
M
−
⎡
⎢
⎣
⎤
⎥
⎦
∆
t
⋅
+
=
where
∆
t is the time step
For the position x
dx
dt
U
=
so
xn 1
+
x
n
() Un
∆
t
⋅
+
=
The final set of equations is
1
+
U
n
ρ
n
−
2
⋅
A
⋅
M
⋅
M
−
⎡
⎢
⎣
⎤
⎥
⎦
∆
t
⋅
+
=
an
ρ
n
−
2
⋅
A
⋅
M
⋅
M
−
=
+
x
n
∆
t
⋅
+
=
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View Full DocumentThe results can be plotted in
Excel
t
(s )
x
(m)
U
(m/s)
a
(m/s
2
)
0.0
66.6
0.1
6.7
28.0
0.2
0.7
9.5
16.1
0.3
1.6
11.1
10.5
0.4
2.7
12.1
7.30
0.5
3.9
12.9
5.29
0.6
5.2
13.4
3.95
6.6
13.8
3.01
0.8
7.9
14.1
2.32
0.9
9.3
14.3
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 Fall '07
 Lear
 Fluid Mechanics, UCI race classifications, Tour de Georgia, simple numerical method, Constant jet area

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