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constant dynamic gain, except for a process whose dominant element is a pipeline. Most analyzers are not so far from being linear that they materially affect the gain of the control loop. The notable exception is, of course, the pH measurement, whose general properties have already been presented. But analyzers are generally given a high order of sensitivity, because of the importance placed on quality control. As a result, the gain of a composition-control loop is invariably high. Objectively, composition is not as difficult to control as flow, for example, but the specifications placed on product quality are so stringent that ordinary performance is seldom acceptable. The impurity of a product stream leaving a fractionator, for example, may be specified at 1.0 f 0.2 percent. It is virtually impossible to regulate flow within + 1 percent in the unsteady state, yet the composition controller is asked to perform five times as well. This is perhaps the greatest single reason why composition control has the distinction of being a problem area. Because quality can be measured to 0.1 percent is apparently reason enough to expect it to be controlled to the same tolerance.
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Process Gain
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The dimensional gain of the process in Fig. 3.8 is the derivative of composition, Z, with respect to concentrate flow X. A material balance on the measured component is simply X = Fx Then,
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Since the nominal flow F has already been identified as a constant, process gain is also constant. (This is another illustration of the case where process steady-state gain varies with flow, but the time constant does too, so dynamic gain is invariant. Steady-state gain, as calculated above, is only meaningful at the rated flow F.) Dimensional gain of the composition process can always be found by writing a material balance across it. If composition of an effluent stream is controlled by manipulating an influent stream, as in this example, the process is linear. But if effluent composition is controlled by manipulating the efluent flow, the process is hyperbolic: x=1 F dx - XdF= F2
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(This was already encountered in the temperature-control example where coolant temperature was adjusted by manipulating its flow.) Examples of both linear and hyperbolic processes are common in both composition and temperature applications, because the controlled variable is always a function of the ratio of one varmble to another. If the manipulated variable happens to be in the numerator, the process is linear. example.3.6 The process in Fig. 3.8 is intended to deliver a solution at a nominal flow F, of controlled composition Z, by adding a manipulated flow X of concentrate to the diluent stream. Let the volume of the vessel be 100 gal and the nominal flow 20 gpm. If mixing is 95 percent complete, then O.O5V/F will be the effective dead time in the vessel: 7d = 0.05 100 = 0.25 min 20 The balance is a first-order lag: r1 = 0.95 g = 4.75 min Let the sampling time also be 0.25 min, with a 3.0-set analysis lag. total dead time in the loop is then 7,~ = 0.25 + 0.25 = 0.5 min Without the 3.0-set lag in the analyzer, the natural period would be 4rd = 2.0 min The phase shift of the 3.0-set lag at a period of 2.0 min is & = -tan-1 $!!&z = -9 A control valve with a 3.0-set lag will contribute another 9 . phase shift extends the natural period to approximately
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+ 9 + 9
= 2.2 min
The dynamic gain of the process is simply that of the principal time constant: 2.2 = 0.0737 G1 = Gl = 2a4.75 Dimensional process gain is the percent composition change brought about by a change in concentrate flow at the rated throughput: dx 1 -Z-C lOO%PO gpm = 5 %lgpm dX F
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