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Being a 2 by 2 matrix, all other elements are determinable from &,D. Notice that x,D varies between 0 and 1; for the example of the binary separation used throughout this chapter, &,D = 0.5. Because separation is a function of V/F and nothing else, X,v = 1,s; dimensional gain and hence nonlinearity disappear in the normalization procedure. To see whether bottoms flow is, under certain conditions, more favorable to manipulate than distillate flow, a matrix of gains of x and y with respect to B/F and S may be prepared. But relative gain &,I] turns out to be identical to XUD. Thus the values of II: and y determine which composition controller should manipulate heat input and which should manipulate the material balance. In general, the purer product should be controlled by manipulating heat input. If reflux and heat input are chosen t o be the two manipulated variables, coupling is considerably different. Starting with a simplified description of the column parameters, observe what happens when V - L is substituted for D:
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y = X,DD + (1 - X,D)V = --X,DL + I 2 = (1 - x,D)D + x,DV = (&o - l)L + V
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The coefficients of L and V are then normalized, following Eq. (7.16), to produce XUL:
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Observe that XV0 and x,~ are mutually exclusive:
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Coupling for manipulation of L and V is worse for all cases except 0 and 1 . For the cited example, control of both products is impossible with this choice of variables-another point in favor of material-balance control.
Distillation
APPLYING FEEDFORWARD CONTROL
Feedback control of product quality from a column is not always satisfactory even when the control loops are properly arranged. Proper arrangement only protects the process from upsets in heat input, feed enthalpy, and rcflux flow and enthalpy. The most significant disturbances to quality control are generally variations in feed rate and composition. From the previous example, it was pointed out that a composition controller may need a proportional band as high as 1,000 percent. A n d because the period of the closed loop may be from 20 min to 2 hr, reset time of 10 min to 1 hr is commonly encountered. The integrated error caused by a load change was shown earlier to equal the product of the proportional band times reset time. Distillation is characterized by a large proportional-reset, product , compared to other processes. And because integrated error in product quality can be costly, distillation is a prime candidate for feedforward control. An on-line analysis of product composition is not always available. In these instances, there is no measurement to feedback from, so a forward loop can be a great help in maintaining control in the face of disturbances. Furthermore, if the real cont,rolled variable is profit or loss, an optimum control program can be based on a feedforward model. Consequently the feedforward approach to control is of utmost importance in distillation processes, whatever the nature of the separation.
The Material Balance
The basis for feedforward. control of any mass transfer operation is the material balance. Earlier in this chapter the distillate to feed ratio was shown to be the principal factor affecting composition of either product stream. The feedforward control model is nothing more than an on-line solution to the material balance:
(11.14)
Distillate rate is the manipulated variable; feed rate F is one component of load and feed composition x is the other. Either distillate composition y, or bottoms composition x is the set point, while the other depends on separation. The most significant factor is that distillate flow is proportional to feed rate. Even if the forward loop is reduced to the simplicity of this ratio, it will bc of help, because feed-rate changes can occur instantaneously. If separation is constant, we know that x and y can both be controlled. In this event distillat,e is directly proportional to feed rate throughout
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