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FIG 2.22. to influent effluent.
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Changing the ratio of acid adjusts the pH of the
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FIG 2.13. An uncompensated pH loop tends to exhibit a distorted oscillation.
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load change of any magnit ude can then drive the pH far enough up the curve to reduce the loop gain to the point where recovery is extremely slow. Compensation for such a severe nonlinearity is essential if a satisfactory degree of performance is to be obtained. The nonlinearity is so severe that even poor compensation is noticeably effective. Reduction in loop-gain variation from 100: 1 to 5 : 1, for example, is bound to improve the situation. The shape of the neutralization curve is a function of the reacting subst ances. If the ingredients of the influent are subject to change, the slope of the curve at a given pH may also change. This amounts to a second gain variation superimposed on the first. This is not a predictable characteristic. The response of a typical uncompensated pH loop to a load change is pictured in Fig. 2.13. Oscillations are flattened by the change in gain. Recovery is considerably retarded by t he reduced loop gain away from the vicinity of the set point. Alore space is devoted to pH control in Chap. 10.
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Figure 2.14 is a record of temperature and flow of a product stream leaving a heat exchanger. Temperature was being controlled by manipulating the flow of st eam to the exchanger. Notice that the temperature record at 80 percent flow is overdamped, whereas at 40 percent flow,
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2 180 2
E 140 r r- 100
FIG 2.14. Both magnitude and damping of the transient are functions of flow.
; 4 0 0 Time
n 1 Ud erstanding Feedback Control
damping is slightly heavier than s/4-amplitude. (It appears that the loop will be only marginally stable at 20 percent flow.) It is as if the proportional band had been changed. But the more lightly damped curve departs farther from the set point-contrary to the effects of changing proportional band as in Fig. 1.7. Therefore it is the process gain which has changed: the process is more lightly damped and more sensitive to disturbance at low rates of flow. The problem has been identified as variable dynamic gain. It is a common problem, not often recognized, still less often anticipated. I t occurs in processes where the values of the secondary dynamic elements, principally dead time, vary with flow. These variations cause proportionate changes in the period of the loop, which affects the dynamic gain of the principal capacity. Consider the heat exchanger as a single-ca.pacity plus dead-time process where the dynamic gain of the capacity is expressed as
GI =-27&F
Let dead time vary with flow through tubing of volume v:
where f = fractional flow F = maximum flow of product The period of oscillation varies with Td:
Dynamic gain is then G1 = 4vlff 2aV/F _ 2v rV.f
Dynamic gain is inversely proportional to product flow. As flow approaches zero, gain approaches infinity. If uncompensated, this gain variation will cause serious problems, particularly during startup, when flow is low. If adjusted for low flow, the controller will perform poorly at higher rates, as Fig. 2.14 substantiates. Notice that the response is in no way similar to that of Fig. 2.13, the nonlinear process. To distinguish between the two, this characteristic will be referred to as variable gain. It is more like the response that would be encountered with a nonlinear valve, or valve-plus-piping characteristic-a function of flow, not of the measurement.
Characteristics of Real Processes
Fortunately, the correction is so readily applied that in most cases the correction is inadvertent and there is little evidence of the problem s ever having existed. The gain of the process varies inversely with product flow. The gain of an equal-percentage valve varies directly with the manipulated flow. As long as the ratio of the two is constant, the gain product will be constant. Let js be the fraction of maximum steam flow Fs. Then faFs is proportional to product flow by the temperature rise AT of t he product as it passes through the exchanger: j-SF, = fFC AT where C is a constant. The gain of the equal-percentage steam valve is G, = 4f,F, = 4fFC AT Gain product, is then 8CATv GG 1 =&fFCATv =z1 TV/F TVf (2.19)
The loop gain is no longer a function of product flow, because of the compensating nonlinearity of the equal-percentage valve. But a t rade has been made: loop gain is now a function of AT, where it was not before. In most heat exchangers, however, AT never varies as much as 2: 1, whereas flow commonly does. So the trade is distinctly in the best interests of the loop. Equal-percentage valves are so widely used to combat line drop, that in many cases they are also compensating for variable dynamic gain without the user being aware of it. When the period of oscillation varies, derivative time ought to be changed accordingly. But stable control can be achieved with an incorrect value of derivative if the gain is appropriately adjusted. Consequently gain compensation for the variable dynamic element is mandatory, whereas derivative compensation can only be classified as desirable.
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