process dynamics and control

R1

R2

Tank
V, P

P1

P2

F1

F2
Fig. 1 A Pressure system with two resistances

Consider a pressure system with tank of volume V and varying pressure P at constant temperature. F1 is inlet flow through resistance R1 with source pressure P1. F2 is output flow through resistance R2 and flowing out at pressure P2. As the flows into and out of the tank are both influenced by the tank pressure, both flow resistances affect the time constant. A typical control problem would be to manipulate one flow rate (either in or out) to maintain a desired drum pressure.
Variables:
Controlled variable: P
Manipulated variable: F2
Disturbance variable: F1
Here we develop a model that describes how the tank pressure varies with the inlet and outlet flow rates.
Making a mass balance,
Accumulation in the tank = Input flow rate (F1) – Output flow rate
(F2)
Flows (Ohm's law is I =
F1=
F2=

1
?1
1
?2

(?1 − ?)
(? − ?2 )

?

?

=

Driving Force
??????????

)

?

V
V

??

=
??

??
??
??
??

?1 −?
?1

+P(

+P(

?1?2

V ( ?1+?2)

-

1
?1

?1 +?2
?1 ?2
??
??

?−?2
?2
1

?1

?2

?1

+

)=

)=

?1
?1
?1?2

+

+P= ( ?1+?2)

+

?2
?2

?2
?2
?1

?1?2

+ ( ?1+?2)
?1

?p ?? + P = K1 P1+ K2 P2
??
Where K1=

?2
?1 +?2

K2=

?2
?2

(Eq.1)

?1
?1 +?2

?p=

??1 ?2
?1 +?2

Taking Laplace transform of equation (Eq.1)

? Ps P’(s) + P’(s) = K1P1’(s) + K2P2’(s)

P’(s) (1+ ?Ps) = K1P1’(s) + K2 P2’(s)
?2
P’(s) = 1+ ? ? P1’(s) +
1+? ? ?
?
?1

P2’(s)

(Eq.2)

Equation 2 can be represented in a block diagram as below:

?1
???+1

P1’(s)

+
-

?2
?? ?+1

P2’(s)

A Block Diagram of a pressure system with two resistances

Assume:

?1 = 2 min/m3; ?2 =4min/m3; V=5m3

?p =

??1 ?2
?1 +?2

=

5(2)(4)
2+4

=

20
3

≈ 6.6676???

I. Processing Without Control

Where ? =

?’(?)
P1 ’(s)

?2
?1 +?2

=

=
?1

1+