Question

57. A common application of control systems is in regulating the temperature of a chemical process (Figure P6.13). The flow of a chemical reactant to a process is controlled by an actuator and valve. The reactant causes the temperature in the vat to change. This temperature is sensed and com- pared to a desired set-point temperature in a closed loop, where the flow of reactant is adjusted to yield the desired temperature. In Chapter 9, we will learn how a PID controller is used to improve the performance of such process control systems. Figure P6.13 shows the control system prior to the addition of the PID controller. The PID controller is replaced by the shaded box with a gain of unity. For this system, prior to the design of the PID controller, find the range of amplifier gain, K, to keep the system stable. Chemical Future PID and heat controller Amplifier valve process Actual set point + temperature 52 + 1.7s +0.3 Actuator Desired temperature 0.7 K S + 0.5 0.1 S + 0.1 Temperature sensor FIGURE P6.13 Block diagram of a chemical process control system

Answer 1

Given control system future PID desired chemical heat Process controllo amplifer actuator and valve temperature + OY 1 st 1.45 +0.3 actual temperatur S+0.5 sensor closed - 0.1 S+01 temperature loop transfer function can be determined by using mason's Gain formula (we can apply this formula directly to the block diagrams also (1) CK) (540) (2 1.45 +0.3) 1480.3 (LTF = Hashes) a C.LTF = + 0.7K (S+0.5) ('71-75 +0.3) (s +0.19 0.7k (5+0.5) (511-45 +0.3) (5+0.5) (5 +1:45+0.3) (sto.1) + (0.1K) (0.1) (Stoo!) (s+0,5) ($71.45 +0.3) 0.YK (s+ 0.1) (6+015) (871.75 +0. 3) (540.1) + (0.7R(0.1) characteristic equation is Hq (s)H(s) = 0 [sto.5) (5 +1.75 +0.3) (5+0-1) + 0.07% = =>

+ 0,55 +0.855 +0.15) (+0) + 0.02 K = 0 » 63+2.25 +1.155 +0.15) (s+0.1 +0.07K = 0 > $4+0.153 +2.253 + 0.22 + 155+ 0.1155 +0.155+ 0.015 +0.01K = 0 ► sy + 213 33 + 1.375 +1.265 + 0.015 +0.07ko out the from RH criteria range we can find of k. RH table table 0.015 +0.07R 1.37 1-265 2-3 0.015+0.04K 0.82 1 1-0343 - 2-3 (0.015+0.04(e) 0.82 So 0.015+0.07% for stability - first coloumn of RH table should be greater than zero. UUTUU 0.015 +0.07K >O 0.0tk > -0.015 K > . 0.2 as gain should be greater than o kol

1.0373 - 2-3 (0.015+0.07kJ 0.82 > 10373 - 2-3 (0.015+0.07) >O 2-3 (0.015 +0.07k) < 1.0373 0.015+0.07% < 0.45/ o.ork < 0.436 K< 6.22 to keep system stable is in range of amplifier gain Tocks 6.22 |