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FIG 11.13. The response of distillate composition to a step increase in distillate flow is the sum of an incident and a reflected wave.
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standing waves. A change in distillate flow first changes the material distribution in the top of the tower, which starts affect ing composition. But the new distribution is propagated downward, too, by the resulting change in reflux rate, and then reflectBed from the bot,tom, returning sometime later. The st ep response is bhe sum of tn-o dead-time plus capacity combinations, the second dead time being much longer than the first. Nonetheless, the first dead t,ime dominates the control loop. It may be 5 t o 30 min in duration, depending principally on the distance of the loop between the distillate valve and t he analyzer. (The volume of the accumulator contributes significantly to the response.) The closed loop can then be expected to oscillate at some period between 20 min and 2 hr. The ratio of effective dead time to effective time constant, will generally be found in the region of 0.15 to 0.30, as Fig. 2.4 indicated. The dynamic: gain of the tower can be determined from this ratio, which, combined with known values of process and transmitter gain, can provide an estimate of the required proportional band. In the numerical example which has been used thus far, dy ___ = - 0.9 W/F) at the normal operating point. dD= Then the process gain is
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It is customary to manipulate distillate flow through a flow controller, to prevent load changes in the stream from affecting composition. The gain G, of this flow loop to a set-point change takes the place of valve gain in this composition loop. In the case of a linear flowmeter, the gain is the maximum flow per 100 percent. Since D/F is 0.5, a maximum flow of 0.5F is reasonable, making the gain O.SF/lOO%. Since the normal value of y has been taken to be 0.95, an analyzer range of 0.90 to 1.00 seems reasonable. This is a span of 0.10. Trans-
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mitter gain is then
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~ = l,OOO% 0.1
The required proportional band is
p = 4OOrd dy T&T dD GfGT
If a value of 0.25 is assumed for 7d/71,
p = 400(0.25) 0.9 OAF
~ Fi@jq l,OOO% = 143% 7r 0
In this example, the specificat,ion on distillate purity was not severe. In many towers, though, extreme purit,y is demanded of t he products, and narrow-span analyzers are used. If the span of the distillate analyzer in this case were 0.01, the proportional band would need to be 1,430 percent. To improve speed of response, the analyzer sometimes samples t he material on a tray closer to the cent,er of the tower. But as with a temperature element, this does not ensure that the product itself mill always be of the desired quality. Improved response speed can be combined with accurate composition It requires a kmperature controller control by means of a cascade 10op.~ whose sensing element is located somewhere between the end of the t,ower and the feed tray, manipulating distillat,e flow. The set point of the temperature controller is then positioned in cascade by a composition controller sensing product qualit,y.
Control of Two Products
Feedback control over the quality of two products leaving a tower encounters severe coupling. It is not often tried and has, under certain circumstances, failed altogether. Derivation of t he relative-gain matrix will reveal the reasons behind the difficulty. Selecting D and V as the manipulated variables, Eqs. (11.3) and (11.6) can be solved for y and IC in terms of D/F and X. Differentiating,
x - x dY (Y - x) a(D,'F) z = - (D/F)2 X - X (1 - Y12X dy _ l - x 1 + .; - 1) - [l + $- l)] = as z
(y - x)2 -Y - 1 -D,F+ ( ;:;;$2= - Y--z W/F) 2/ 8X Y(l - Y) x2(1 - Y) ds,= - [Y + w - Y)12 = Y
Although the four gains derived above represent closed-loop conditions, their inverse can be used to find relative gain in the same way as openloop terms, following the procedure of Eq. (7.16). Accordingly,