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pletely mixed : 7~7~ V F rd=,Fm There are several observations to be made from these two derivations. First, it may be noted that V Tl+Td = F (3.25)
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(3.24)
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Placing 7d in the same terms helps to compare the two components:
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This is reasonable, because it confirms that the average particle cannot be retained in the vessel longer than its residence time V/F, whether mixed with the rest of the contents or not. Furthermore it concurs with Eq. (2.4). Second, the difficulty of control, T~/T~, varies only with F and F,: F -=71 F,
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Notice that difficulty is not a function of volume. This contradicts the commonly accepted rule that relates controllability to volume. In fact, increasing the volume of a system while retaining the same flow and agitation serves only to reduce its speed of. response, because 7d would be increased proportionately. The effect can be more readily visualized if carried to extremes: it would be no easier to control composition in a lake than in a small tank, using the same agitator, and response would certainly be slower. The actual mechanism by which mixing takes place is obviously not discrete, as the dead-time plus lag model would suggest. Flow from the agitator is not in a single direction, as it would be in a pipe, and even if it were, the velocity profile could not be perfectly flat. Furthermore, turbulence is what produces the actual mixing, and turbulence seems to be an omnidirectional effect. Even without an agitator, some mixing always takes place through diffusion and a token amount of turbulence resulting from flow through the vessel. Tests4 conducted on stirred tanks show that the response of the effluent to a step change in concentrate flow resembles that of a system comprised of multiple interacting capacities. Figure 3.9 shows the response a typical vessel might produce both with and without agitation. These response curves are typical of diffusive and distributed processes, as was mentioned in Chap. 2. It was also pointed out how capably this
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FIG 3.9. Agitation reduces the effective dead time while increasing the effective time constant.
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Time
sort of response could be represented by dead time plus a single capacity. Thus the simple model just postulated is quite valid, if imperfect.
The Analyzer
Dynamics associated with the analysis play an important role in the performance of the loop. The foremost limitation in the speed of analysis is generally t hat of transport ing the sample to the detector. Fortunately some composition measurements can be made without withdrawing a sample: electrolytic conduckivity, density, and pH are notable examples. But any analysis requiring the withdrawal of a sample, particularly if that sample must undergo a certain amount of preparation, results in a significant accumulat,ion of dead time (see the example cited at the close of Chap. 2). Naturally any effort spent in minimizing the sampling time will be rewarded by bot h tighter control and faster response. Some analyzers are discont,inuous. They produce only one analysis in a given time interval. This characteristic is worthy of much more attention, because it periodically interrupts t,he control loop. Process chromatographs are the principal, but not sole, constituents of this group. The response of this kind of control loop will be given extensive roverage in Chap. 4, and methods for coping with it will be presented. A few analyzers exhibit a time lag in addition to the dead t ime associated with sample t,ransport. Sormally this property is of little consequence, except when the process itself consists of nothing but the volume of a pipeline, whose time const,ant may be less than that, of the analyzer. Those measurements which are fast, are by the same t,oken subject to noise. Conductivity and pH are usually in this category, because they are fast enough to react to an incompletely mixed solution, or particles of an immiscible phase. Dead time in sample lines is understandably constant. Dead t ime in a pipe carrying the main st,ream varies with flow. Dead t,ime within a stirred tank is slight,ly affected by flow, to the extent of F/F,; in most systems t,his variation would not be significant. The natural period of the composition loop would therefore be virtually constant, producing
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