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Unformatted text preview: s of the system. q i h 1 q 1 R 1 h 2 q 2 R 2 5 3
(a) What is the response q (t ) to a step input of magnitude 0.5 m /min in q (t ) if the system is 2 i init ially at steadystate corresponding to qi = q1 = q = 1 m 3 / min . 2
(b) The headflow relat ions for the tanks are 5 3 / min m 2 3 / min m q1 = h , q =
h 1
2
2
m m 3
What are the ult imate values o f each tank level aft er 1 m of liquid is suddenly added to the first tank? Why? Solution: The transfer funct ion between the input flowrate and the exit flowrate is given by, Q’ (s) i 1 5 + 1 s Q’ (s) 1 1 4 + 1 s Q’ (s) 2 Figure: A twotank surge process transfer funct ion model Q 2 ( ) ' s 1
=
Q i ( ) (5 + 1 ( s + 1) ' s s ) 4 If there is a step change in the input flowrate, the exit flowrate will be given as, Q 2 ( ) =
' s 1
0.5 (5 + 1 ( s + 1 s s ) 4 ) Either using the partial fract ion expansio ns, or the formulas for secondorder process responses, we can find the timedo main response o f the exit flowrate. I will use the latter, æ
t 1e  t / t1  t 2 e  t / t 2 ö
ç
÷
q ( t = KM 1  ')
t 1  t 2 è
ø
Where KM=0.5, and t 1 = 5 t 2 = 4 , as this is a slight ly overdamped process ( z = 1.006 ). The result is,
, 6 q ) = 1 + 0.5(1  5  t / 5 + 4  t / 4 ) ( t e e For part (b), we need to first find the transfer functio ns between the inlet flowrate and the levels in each tank. H 1 ( ) H ' ( ) Q 1 ( ) 1 1 ' s s ' s =1
=
Q i ( ) Q 1 ( ) Q i ( ) 5 5 + 1 ' s ' s ' s s
H 2 ( ) H 2 ( ) Q 2 ( ) 1 ' s ' s ' s 1
=
=
Q i ( ) Q 2 ( ) Q i ( ) 2 (5 + 1 ( s + 1) ' s ' s ' s s ) 4 The disturbance for this case is an impulse of magnitude 1. Hence the level responses will look like, 1 1 (1 )
5 5 + 1 s
1 1
H ' ( ) =
(1) 2 s 2 (5 + 1 ( s + 1 s ) 4 )
H ' ( ) =
1 s Using the Final Value Theorem, we can find the ultimate values of these levels, æ 1 1 ö
lim( H ' ( )) = lim s s 1 s ç
(1 = 0 )÷
è 5 5 + 1 ø
s 0 ®
s 0 ®
s
æ 1 1
ö
lim( H ' ( )) = lim s s 2 s (1 = 0 )÷
ç
s 0 ®
s 0 2 (5 + 1 ( s + 1 ®è
s ) 4 ) ø
This means that the levels go back to their original height after a transient deviation, as a result of the fact that the levels act like “selfregulating” process 7 8...
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 Fall '08
 Hjortso,M

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