= 1 - ; XVI + ; (1 - y)v2 F ( > in .NET framework

Make QR Code JIS X 0510 in .NET framework = 1 - ; XVI + ; (1 - y)v2 F ( >

1 = 1 - ; XVI + ; (1 - y)v2 F ( >
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(11.19)
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The curves of J: and y vs. D/F from Fig. 11.3 were modified by mulCplying x by 1 - D/F and 1 - y by D/F, and then assigning values VI and v2. The results appear in Pig. 11.18. In this example, v1 was chosen to be four times ~2. Because the slopes of the intersecting curves change monotonically, the minimum value of their sum occurs at their intersection. Having found the optimum value of D/F, the corresponding ratio of 1 - y to x can be calculated: xv1 = p (1 - y)vz ( )opb = : [ (o:t )opt - ] (11.20)
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The optimum value of D/F is, of course, directly proportional to feed composition x, which was omitted in the previous explanation. Should
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FIG 11.18. Optimum D/F occurs at the intersection of the two curves.
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Distillation
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FIG 11.19. Cost of heating plays a major role in determining the optimum V/F ratio.
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separation change, D/F remains constant only if (1 - 1;l)/x = 1.0. If it is greater than 1.0, D/F must decrease with decreasing separation (increasing feed rate). Unfortunately, too many possibilities present themselves to touch on all of them. The intent of this section is to suggest how solutions to particular optimizing problems might be approached. A second class of optimizing applications is characterized by a heat input whose value compares to t hatt of t,he products. In this situation, the loss equation becomes ;= (++(l -;)Z~l+$(l -y)vz (11.21)
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where v0 is the cost of generating one unit of vapor. If y is to be controlled, D and V can be manipulated together to minimize total loss as calculated above. Figure 11.5 shows t,hat there is a value of D/F which can maintain control of 7~ for each value of V/F. Bottoms composition z is seen to be dependent on V/F. Data taken from Fig. 11.5 were used to generate the three component)s of the 10s~ equation, which are plotted and summed in Fig. 11.19. As before, this particular set of curves is based on a feed composition z of 0.50. Should z change, D/F must change, which will shift the location of optimum V/F. A control program can readily be written manipulating D on the basis of F and x, and setting V/F as a fur&ion of the calculated value of D/F. Since there is always an upper limit placed 011 V, it is entirely possible that the optimum V,/F may be unobt,ainable at high rates of feed. The loss plots of Figs. 11.18 and 11.19 are two-dimensional. Threedimensional contour plots of T7/F vs. D/F, with l/F as a parameter, can
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FIG 11.20. The locus of optimum V/F represents the minimum loss for each value of D/F.
readily be prepared, from which a locus of minimum loss can be found. Figure 11.20 is a combination of Figs. 11.18 and 11.19 in three dimensions.
Dynamic Compensation
As is typical of feedforward cont,rol loops, dynamic compensation is necessary to ensure that the effect of a distillate-rate change be manifest at the same time as the feed-rate change which promoted it,. Because feed enters the tower at a location considerably removed from where distillat,e is withdrawn, their dynamic effects upon composition diff cr by a corresponding amount. The response of a tower due to a change in feed rate appears as the sum of an incident and a reflected wave, just as is the case with distillate rate, but the incident path is longer and the reflected path is shorter. Figure 11.21 illustrates the difference in the length of the paths. The response of dist,illate composition to equivalent step changes in feed and distillate flow rates is shown in Fig. 11.22. Because the
FIG 11.21. The incident path is longer but the reflected path is shorter for a feed-rate change.
Distillation
FIG 11.22. The response to a step increase in feed rate exhibits a longer incident dead time but shorter reflected dead time.
ay(D,t)
Time
incident path front the feed t ruy is longer, the dead time of feed-rate response is longer. But, the corresponding reflect ed wave travels a shortel path, resulting in a total elapsed time that is less t han for an equivalent change in distillate rat e. A feedforward loop without dynamic compensation (i.e., constant D/F) would produce a transient step response t,hat is the sum of these two curves: dy dY (t> = dF @) dF (t) + & dD (0 The sum is plotted in Fig. 11.23. Although the response of an uncompensat,ed loop represents t,he difference in dynamic response between t he t,wo variables, the proper compensation corresponds to a ratio of the two. A ratio of the change in y that is due to F to the reverse of the change in 1/ t hxt is due to D (I ig. 11.22) is presented in Fig. 11.24. The step response of a compensator consisting of two lags and a lead is also included in Fig. 11.21. Alt hough the model is not an exact representation of the process, it is the best approximation available within a three-component structure. A higherorder model would not only cost more, but would also be more difficult, to adjust. The foregoing compcnsution applies specifically to the case of withdrawal of distillate flow from a flooded condenser. It assumes constant liquid inventory. If reflux is manipulated from accumulator level, programming it with respect to distillate-flow changes according to Fig. 11.12 provides the required lead compensation. This source of lead action is recommended because it acts on rcflux rather than distillate.
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