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Feedforward control of both products requires two forward
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the operating range. But in order to achieve constant separation, a second forward loop is needed to adjust heat input proportionately to feed rate. A complete feedforward system for control of both products appears in Fig. 11.11. The scheme shown in Fig. 11.14 has an analyzer on the feed stream but none on either product. As the principal failing of feedforward control systems is insufficient accuracy, a feedback loop on product quality is usually of considerable worth. In fact, if feedback is available, the exactness of the feedforward model can often be relaxed, even to the extent of omitting a feed-stream analyzer. Figure 11.15 shows how feedback control of product composition might be added to the forward loops. Feedback from distillate composition is introduced through a divider in order to better compensate for the inverse relationship between distillate flow and composition, although exact correction is unnecessary. The principal factor is the maintenance of a D/F ratio, which in this case is adjusted as necessary by the output of the feedback controller.
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If maximum separation is desired at all feed rates, the heat input must be fixed at its maximum value, while feed rate is allowed to vary. It has been shown that this procedure can be used to control the quality of one
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F(t)
D(t)
*Set point to distillate flow l o o p (linear)
Bot;oms
Distillate
FIG 11.15. Incorporation of a feedback loop can reduce the complexity of the forward loop.
Distillation
FIG 11.16. The nonlinear relationship between D and F is characterized by the square-root converter and the summing amplifier.
product while maximizing t hat of the other. From Fig. 11.6, control of distillat e composition requires distillate to vary nonlinearly with feed rate, because bottoms composit ion varies. Although the relationship between D and F is somewhat complex, the curve can be modeled satisfactorily by a simple parabola:
D = x(aF - bFz) (11.15)
where a and h are const ants. The constants can be found by fitting the parabola to the curve of D vs. F. For t he example in Fig. 11.6
D = x(1.04F - 0.08F2)
A feedforward system designed to this model is shown in Fig. 11.16. If bot toms composition is the controlled variable, t he D vs. F curve bends upward, following the equation
D = z(aF + bP) (11.16)
In a situation where the flow of bottoms product is much less than the of distillate, bot)toms flow is manipulated from feed rat)e and composition. In this arrangement, the forward-loop calculation is complementary to that used when manipulating dist,illate flow. If no feedback loop is involved, accuracy of the forward loop is of paramount importance. Since the best accuracy will always be achieved by manipulating the smaller flow, the design of a control system cannot be intelligently undertaken without first writing a material balance across the tower.
flow
Multicomponent
Separations
general, feedforward control systems can be designed for multicomponent separations almost as easily as for binary separations. The first relevant question is how many product streams there are. If there are only two, distillate flow can be calculated as the sum of those components which pass overhead :
D = F(lclzl + lczzz + / csczs . . .) (11.17)
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where each k is the recovery factor of its respective component The recovery factor is identified as that fraction of a certain component in the feed stock which will be recovered in the distillate stream. If component 2 in Eq. (11.17) were the principal overhead product, its recovery factor would be somewhat less than 1.0. Each lighter component has a higher recovery factor. Recovery factors for components heavier than the principal overhead product (sometimes called the light key ) approach zero. From a material balance on a single component i, it will be observed t hat
In the special case of a binary system, where a material balance is made only on the light component,
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