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Problem 1
From table 7.1 in the textbook, the Laplace transform of y(t)=te
t
is:
t
2
1
Lt
e
y
(
s
)
(s 1)
−
⎡⎤
==
⎣⎦
+
(S1.1)
Moreover, the Laplace transform of the unitimpulse
δ
(t) is:
[ ]
L(
t
) u
(
s
)1
δ=
=
(S1.2)
Since u(t)=0 all the time but at t=t
o
, assuming that y(t) is in deviation form (i.e. y
s
=0), the
transfer function of the system in exam can be written as follows:
2
y(s)
1
G(s)
u(s)
(s 1)
+
(S1.3)
Problem 2
Assuming constant density and reactor volume, the overall material balance can be
written as follows:
12
ii
dh(t)
A
F(t) F
F(t) 8h(t)
dt
=−
(S2.1)
As initial condition for equation (S2.1), we chose the steady state value of the hydrostatic
pressure h
s
given by:
2
si
s
h(0)
h
F 64
(S2.2)
Linearizing the right hand side of equation (S2.1) about the steady state value (S2.2), one
obtains:
[]
[ ]
s
s
s
4h
h(t) h
−
−≅
−
−
−
(S2.3)
Defining the deviation variables H(t)=h(t)h
s
and Q(t)=F
i
(t)F
s,
and considering (S2.3),
equation (S2.2) becomes
ss
Ah
h
dH(t)
H(t)
Q(t)
4d
t
4
+=
(S2.4)
subject to the following initial condition
H(0)
0
=
(S2.5)
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This note was uploaded on 10/04/2009 for the course CHE 470 taught by Professor Smith during the Spring '09 term at Rice.
 Spring '09
 Smith

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