Variables: Controlled variable: P Manipulated variable: F2 Disturbance variable: Fl 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 (Fl) - Output flow rate (F2) Flows (Ohm''s law is I = Driving Force Resistance dp dt RIR2 V (Rl+R2) RI +R2 RI R2 ( RI +R2) + ( Rl+R2) rp dP+p = KI Pl+K2P2 Where Kl= (Eq. l) VRI R2 Taking Laplace transform of equation (Eq. l) ps P(S) + P(S) = KI PI''(S) + K2P2''(S)
P(S) (1+ IPS) = KI PI''(S) + K2 P2''(S) 1+1 s + KI P2''(S) (Eq. 2) Equation 2 can be represented in a block diagram as below: rps+l rp S+l A Block Diagram of a pressure system with two resistances Assume: RI = 2 rntn,''t-n3; R2 v=5rn3 2+4 20 3 6. 6676mtn l. Processing Without Control Where K — 4 = 0. 667; Assuming step change from 2 m3/min to 3 m3/min in the disturbance variable: S(3S+1) Disturbance response without Control 0. 8 0. 4 0. 2 10 50 time (min) II. Proportional Integral Control for the Disturbance Changes Process:
Process Transfer function: Gp(s) = Disturbance Transfer function: Gd(s) = I-I(S) Pressure Sensor-Transmitter (Analyzer): Gm(s) = Km m s+l but negligible dynamics > > > Gm(s) = Km Pm(s) = Gm(s)Pa(s) Block diagram for the pressure sensor-tramsmitter pa(S) m S+l Pm(s) Controller (PI controller): GC(S) = ) pa(S) = Control Valve: GV(S) = KV Pt''(S) o. '' S+l ; but negligible dynamics r > > > Comparator: E(s) = P'' sp (s) - P'' m (s) P'' sp (s) = K m P'' sp (s) Block diagram for the entire process: Rd(S
Psp(s) P'' sp (s) KC(I + ) pa(S) KIP PLI(S) Where: Y=controlled variable U=manipulated variable D=disturbance variable (also referred to as load variable) P=controller output E=error signal Ym=measured value of Y Ysp=set point sp=internal set point (used by the controller) Yu=change in Y due to U Yd=change in Y due to D Gc=controller transfer function GFtransfer function for final control element (including K , if required) IP Gp=process transfer function Gd=disturbance transfer function
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