ME3514 - HWK - Burdisso - 68
ME 3514 – System Dynamics
HOMEWORK #7-Solution
DUE: 10/31/07
1.
(10pts) In the liquid level system shown in the following figure, the head is kept at
1m
for
0
t
<
.
The inflow valve opening is changed at
0
=
t
, and the inflow rate is
sec
05
.
0
3
m
for
0
≥
t
.
Determine the time needed to fill the tank to 2.5-m level. Assume that the outflow rate
Q
s
m
3
and head
H
m are related by
H
Q
02
.
0
=
The capacitance of the tank is
2
2
m
.
Remark: To solve this problem, please linearize the equation of motion around
1
H
m
=
.
Figure:
Liquid-level system
Solution:
The equation of motion in terms of the absolute level
( )
H t
is (conservation of mass)
( )
( )
i
dH
C
Q t
Q t
dt
=
−
The valve equation is then
( )
0.02
( )
Q t
H t
=
There are two options:
a) Replace
( )
0.02
( )
Q t
H t
=
into EOM:
( )
0.02
( )
( )
i
dH t
C
H t
Q t
dt
+
=
(1a)
b) Differentiate
( )
0.02
( )
Q t
H t
=
w.r.t. and then replace:
( )
0.02
( )
2
( )
dQ t
dH t
dt
dt
H t
=
2
( )
( )
( )
( )
0.02
i
H t
dQ t
C
Q t
Q t
dt
+
=
(1b)
NAME:
( )
( )
Q t
Q
q t
=
+
( )
( )
i
i
Q t
Q
q t
=
+
( )
( )
H t
H
h t
=
+

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