Unformatted text preview: e component balance on the nonreact ing species in the first tank looks like: d ( rVc1 ) = rqc  rqc 0
1
d t The densit y, vo lume and the vo lumetric flowrate are constants, thus, the equat ion reduces to: V dc 1
= q 0  c ) (c 1
d t This is a linear equation, so one can rewrite it in deviat ion variable form trivially, V dc 1
= q 0  c ) (c 1
d t We are interested in the transfer funct ion that relates the exit concentration to the inlet concentration. Taking the Laplace transform yields, V
sc ( = c (  c ( 1 s)
0 s)
1 s)
q c1 ( s )
1 = c ( s (V / q + 1 )s
0)
Recognizing that all tanks are the same, we can generalize to the fo llowing relat ionships: c ( s 1 i)
=
c 1 ( s (V / q + 1 )s
i )
c ( s 5 c ( s æ
)
ö
1 5)
=Õ i
=ç
ç (V / q + 1 ÷
÷
c ( s i =1 c 1 ( s è )s
ø
0)
i ) 5 If the input is changed by 0.15, then, the so lut ion in t ime do main can be written using partial fract ion expansio ns, leading to the standard result (with V / q = 10 ),
2 3 4 é
æ
t 1 æ t ö
1 æ...
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 Fall '08
 Hjortso,M
 funct ion, FPTD Linear, transfer funct ion, Alternat ive Linear, ive Linear model, inco ming stream.

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