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1 1 = 1 + klVIF Co + F(l + klV/F)m V = F+k,V
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(11.3)
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BLOCK DIAGRAM OF A CHEMICAL-REACTOR CONTROL SYSTEM
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At steady state, dclldt = 0, and Eq. (11.3) becomes Cls = 1
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where s refers to steady state. Subtracting Eq. (11.4) from (11.3) and introducing the deviation variables Cl = Cl - Cl$ co = co - cos
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M=m-m,
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-++1= * d t
1 + ;J,F Co + F(l + :,VIF)M
Taking the transform of Eq. (11.5) yields the transfer function of the first reactor:
Cl(S) =
l/(l + klV/F) 71s + 1
c (s) + W U + W WIMM(sj
71s + 1
(11.6)
A material balance on A around tank 2 gives
V% = F(cl - c2) - k2Vc2
As with tank 1, this last equation can be written in terms of deviation variables and arranged to give 1 dC2 -+c2= 1 + k2VIF Cl r2 d t where
V *2 = F+k2V c2 = c:! - C&
Taking the transform of Eq. (11.8) gives the transfer function for the second reactor: l/(l + k2V/F) (11.9) C2(s) = Cl(S) 72.7 + 1 To obtain some numerical results, we shall assume the following data to apply to the system: Molecular weight of A = 100 lb/lb mole PA = 0.8 lb mole/ft3 QS = 0.1 lb mole A/ft3 F = 100 cfm ms = 1.0 lb mole/min
LINEAR CLOSED-LOOP SYSTEMS
k2 = 3 min-
v = 300 ft3 Substituting these constants into the parameters of the problem yields the following values: 71 = 2 min 72 = 1 min CL = 0.0733 lb mole Alft3 c2s = 0.0244 lb mole Alft3 m,/pA = 1.25 cfm
Control Valve
Assume that the control valve selected for the process has the following characteristics: The flow of A through the valve varies linearly from zero to 2 cfm as the valve-top pressure varies from 3 to 15 psig. The time constant 7, of the valve is so small compared with the other time constants in the system that its dynamics can be neglected. From the data given, the valve sensitivity is computed as
K, zz -
2 - 0 15 - 3
i cfmpsi
Since m,/pA = 1.25 cfm, the normal operating pressure on the valve is
ps=3+ ,(15 - 3) = 10.5 psi
(11.10)
The equation for the valve is therefore m = [1.25 + K,(p - 10.5)]p.4 In terms of deviation variables, this can be written
M = K,pAP
(11.11)
(11.12)
where A4 = m - 1.25~~ P = p - 10.5 Taking the transform of Eq. (11.12) gives
M(s) = K,,~A P(s)
(11.13)
as the valve transfer function.
BLOCK
DIAGRAM
CHEMICAL-REACTOR
CONTROL
SYSTEM
Measuring Element
For illustration, assume that the measuring element converts concentration of A to a pneumatic signal. Specifically, the output of the measuring element varies from 3 to 15 psig as the concentration of A varies from 0.01 to 0.05 lb mole A/ft3. We shall assume that the concentration measuring device is linear and has negligible lag. The sensitivity (or gain) of the measuring device is therefore 15 - 3 K, 0.05 - 0.01 = 300 psi/(lb mole/ft3) Since czS is 0.0244 lb mole/ft3, the normal signal from the measuring device is 0.0244 - 0.01 0.05 - 0.01 (15 - 3 ) + 3.0 = 4.32 + 3.0 = 7.32 psig The equation for the measuring device is therefore b = 7.32 + Km(c2 - 0.0244) (11.14) where b is the output pressure (psig) from the measuring device. In terms of deviation variables, Eq. (11.14) becomes B = K,C2 (11.15)
whereB = b-7.32andC2
= c2-Q. The transfer function for the measuring device is therefore (11.16)
A measuring device that changes the units between input and output signals is called a transducer; in the present case, the concentration signal is transduced to a pneumatic signal.
Controller
For convenience, we shall assume the controller to have proportional action, in which case the relation between controller output pressure and error is (11.17) p = ps + Kc(c~ - b) = ps + K,E where CR = desired pneumatic signal (or set point), psig K, = controller sensitivity, psig/psig E =error = CR-b,PSifJ
In terms of deviation variables, Eq. (11.17) becomes
P = K,E
(11.18)
The transform of this equation gives the transfer function of the controller (11.19)
LINEAR CIJXED-LOOP
SYSTEMS
Assuming the set point and the signal from the measuring device to be the same when the system is at ,steady state under normal conditions, we have for the reference value of the set point CRS = b = 7.32 psig The corresponding deviation variable for the set point is CR = CR - cRs Tb-ansportation Lag
A portion of the liquid leaving tank 2 is continuously withdrawn through a sample line, containing a concentration-measuring element, at a rate of 0.1 cfm. The measuring element must be remotely located from the process, because rigid ambient conditions must be maintained for accurate concentration measurements. The sample line has a length of 50 ft, and the cross-sectional area of the line is 0.001 ft2. The sample line can be represented by a transportation lag with parameter volume rd=flowrate= (m0.001) z o 5 min 0.1 *
The transfer function for the sample line is, therefore, e-Tds = e-o.5s Block Diagram We have now completed the analysis of each component of the control system and have obtained a transfer function for each. These transfer functions can now be combined so that the overall system is represented by the block diagram in Fig. 11.2. In Fig. 11.2, a block containing the transfer function K, is placed at the positive inlet of the comparator in order to relate the set point in concentration units to a pneumatic signal, which matches the units of the feedback signal B. If the pneumatic controller in Fig. 11.2 were replaced by an electronic or computer-
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