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the computer in this domain is to manipulate those variables which can best offset the influence of the above uncontrolled variables on the plant economy. Complicated as all this seems to be, often a fairly simple relation can be derived in an effort to optimize part of the plant, or to partially optimize the whole plant. As an example, consider a simplified absorption tower where a gas stream F, containing z percent of a valuable material, is absorbed by a liquid stream L, which is to be manipulated to maintain minimum-cost operation. Assume that gas-exit composition y varies as follows:
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where k = absorption rate coetlicient. The principal debits 1 associated with such an operation might be losses of valuable material in the exit gas and costs of processing the absorbent: 1 = vlFy i- VZL where v1 = product value v2 = processing cost The debit equation may be rewritten on the basis of independent variabIes alone : 1 = vlkx; + VZL The minimum point on a curve of debits vs. L would occur where the slope db/dL is zero. dl -= -I&~ + 212 = 0 dL L2 This defines the locus of optimum L. L, ,, = VlkZ F2 v,
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Having solved the minimum-cost equation, it is only necessary to build a computer which will program L2 to follow variations in F2 and x in accordance with current figures of v1 and v2. (The manipulated variable has been left in the form L2 because flow rate is most commonly measured by a differential meter.) Adjustable coefficients should be available for perfecting the model in actual operation. Note that if no solution exists to the derivative, the process exhibit s no minimum. This would be the case if v2 were zero.
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FIG 8.18. The locus of minimum cost can be drawn across a contour plot such as this.
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Often such rigorous and simple expressions cannot be established. Shotgun patterns of data sometimes must be analyzed, involving all combinations of wild and manipulated variables until some measure of relationship can be envisioned. This usually culminates in contour plots such as Fig. 8.18. Contours of debit are shown as a function of a wild and a manipulated variable. If such a plot can be made, a line may be drawn across it representing the locus of minimum debit. A model of this line can then be made to program the manipulated variable as a function of the wild variable. Again, adjustable coefficients may be incorporated to perfect the model inasmuch as some doubt always accompanies.relationships derived from real plant data. Normally the operating conditions of any plant are surrounded by constraints and limitations. It is not surprising to learn that the optimum conditions for many plants lie outside equipment limitations. Many applications would not result in enough remuneration to pay for the computer or the engineering involved. These two facts severely restrict the number of processes that could benefit by optimizing control. The total debits which can be expected to be recovered from a given operation with a given computer, divided by the installed cost of the computer, is the payout in percent per year. A simple analog like the one just discussed cannot be expected to recover all the existing debits in a process, though 50 to 75 percent recovery should not be difficult to realize. A more complex analog designed to a more exact model might be able to recoup an additional 10 to 20 percent, but at perhaps twice the installed cost. It is easy to see that the simplest optimizing computer will nearly always result in the greatest rate of payout.
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There is absolutely no question that feedforward is the most powerful technique that has ever been brought to bear in the regulation of difficulty processes. Where certain unusual feedback modes like complementary
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