EE 429
EXAM I
26 February 1996
Name:
ID#:
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO
Problem
Weight
Score
1
35
2
35
3
30
Total
100
This test consists of three problems.
Answer each problem on the exam itself; if you use
additional paper, repeat the identifying information above, and staple it to the rest of your exam
when you hand it in. The quality of your analysis and evaluation is as important as your answers.
Your reasoning must be precise and clear; your complete
English sentences should convey what you
are doing.
1
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Problem 1:
(35 points)
Fluid flows play an important role in many physical systems. As an example, hydraulic fluid flows are
used to move the control surfaces on airplanes.
In this problem you will analyze the water tank system
shown in Fig. 1. Water enters at the top of the tank with a mass flow rate of
ω
in
(
t
) [Kg/sec] and exits at
the bottom of the tank with flow rate
ω
out
(
t
) [Kg/sec]. The fluid level
h
(
t
) satisfies
˙
h
(
t
) =
1
A
T
ρ
(
ω
in
(
t
)

ω
out
(
t
)
)
where
A
T
[m
2
] is the crosssectional area of the tank and
ρ
[Kg/m
3
] is the density of water. This continuity
relation is simply a statement of the conservation of matter. If the input and output flow rates are equal,
then the fluid level remains constant. On the other hand, if
ω
in
> ω
out
, then the fluid level
h
(
t
) increases.
The fluid exits the tank through a discharge nozzle with crosssectional area
A
N
. The output flow rate
is given by Toricelli’s equation
ω
out
(
t
) =
ρA
N
2
g
1

A
2
N
A
2
T
h
(
t
)
,
where
g
= 9
.
8 m/sec
2
(use the symbol
g
in your analysis,
do not
substitute the numeric value for
g
).
.
.
h
(
t
)
A
N
A
T
ω
in
ω
out
Figure 1: Elementary system for studying fluid flow dynamics.
1. (5 points) Find a statespace representation using
x
(
t
) =
h
(
t
) as the state variable,
u
(
t
) =
ω
in
(
t
) as
the system input and
y
(
t
) =
h
(
t
) as the system output.
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 Spring '08
 SCHIANO
 Fluid Dynamics, National Income

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