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From previous investigation of binary systems it was shown that
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D Z - X -=F Y--x
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Since the value of .C at constant y varies with the separation capability of the tower, k also varies with separation. It is not necessary to show how k varies, because this has already been demonstrated in Fig. 11.5: Ic is directly proportional to D/F. The most direct way of implementing a forward loop for a nmlticomponent system starts with analysis of the feed for the light key and lighter components, assuming the remainder to be heavy components, principally the heavy key. In a three component system,
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But, x3 = 1 - Z1 - x2 Therefore,
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D = F[(k, - k&1 + (kz - k&z + Ical
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The arrangement of the forward loop appears in Fig. 11.17. If feedback control is available, it should be made to adjust the coefficient of x2 through a divider (because kz varies as l/yt).
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A tower with a sidestream presents no particular problem as long as the mat erial balance is given foremost consideration. In this type of tower, sidestream composition is usually the controlled variable. Increasing dist illate flow mill reduce t.he concent,ration o f l i g h t components iI1 the sidestrcanl.
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FIG 11.17. The coefficients of ZI and z2 can be adjusted to obtain the lnoreaP- d e s i r e d p r o d u c t c o m p o s i t i o n .
ing bottoms flow will reduce t h e concentration of heavy components in the sidestream. Increasing the sidestream flov\- will increase the concentration of both light and heavy components in it.
Optimizing Programs
Opt imizing will be described here as that method of operating a tower which results in the least cost consistent, with specifications which have been set on the product. For every cont rolled variable at least one variable must be manipulated. If more t han one variable may be manipulat,ed, their relative values can be made to follow a least-cost, program while still maintaining the controlled variable at it,s desired value. III order to provide a least-cost program with only one manipulated variable, no variable can be controlled-~ all must float with the program. In short! the number of manipulated variables always exceeds the number of controlled variables in any optimizing situation. Suppose that a binary separation is taking place in a column whose heat input and cooling have no value: they are waste from other parts of the plant. In this situation, heat input must be maintained at its highest possible rate, wit hin t,hc limitations of the heat transfer surfaces and vapor and liquid carrying capability of t,hc t,ower. If t,he quality of one product is to be controlled, the quality of the other will be maximized at all rat es of feed, as in Fig. 11.6. Alnximizing t#he purity of the bottoms stream, for example, is the same thing as minimizing the amount of light component in that. stream. This is also S~CI~ to maximize the flow of distillate at all rates of feed, as l ig. 11.6 shows. Therefore this program results in cont rol of distillate quality with t,he maximum flow of distillate: an optimizing program. If cont,rol of both products is desired in t.his t,ower, minimum cost will only be achieved at maximum feed rate. This fixes all the variables at one value, and the tower is then operated at its constraints. If feed rate is variable, no optimizing is possible. If the qualit y of neither product needs to be controlled, as in a refinery
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where various stocks are blended anyway, minimum cost operation can be achieved by appropriate manipulation of D/F. Let vl be the value of the light component and v2 the value of the heavy. Then the cost of losses of light component with the bottom stream is Bxvl. Total losses are 1 = BXIJ~ + D(1 - y)vz Substituting for B, 1 = (F - D)xvl + D(1 - y)zf2 Dividing by F yields (11.18)
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