4. Switch to rectification operating line. 4.1 Calculate:
4.2 If the first calculated x shows that xk < xk-Y (Figure 19.1), assume a new and larger value of xB and resume at step 4. Continue until N =
5. Determine when the calculation is complete, that is, when the estimated x'D ^ y at the top tray. If [yT — is greater than the chosen tolerance, choose new x'B and repeat steps 1—5.
Note that changing B is equivalent for this case to changing D. We will term the procedure for steps 1-5 the Type B procedure in contrast to the Type A procedure.
The reader may have observed that although we need only the change in xD for a change in D to design the overhead composition control system, we have also calculated the change in xB in response to a change in D. This second gain is an interaction term. Note that we call the preceding Type B.
Since the original design calculations are unconverged, it is necessary, when the Type B program is to be used, first to obtain converged values of xR and xD for the specified reflux and boilup.
If we desire to control bottom composition by changing boilup as shown on Figure 19.2, we would start by assuming that both feed and top-product rates are fixed. This means that bottom product is also fixed. The "prep" equations for ultimately finding new base composition, xB, and new top composition, Xd, are therefore:
Then use the Type B routine.
To control base composition via boilup, we need the gain of xB with respect to Vs. The second gain, change in xD with respect to Vs, is another interaction term. The magnitudes of the interaction terms give clues to the need for decouplers.
Another commonly encountered control scheme is that of Figure 19.3. This is similar to that of Figure 19.2 except that overhead condensate receiver level is controlled by throttling distillate, while reflux is flow controlled, perhaps eventually cascaded from top composition control.
If feed and boilup are fixed at design conditions, top and bottom composition x'd, xb response to reflux changes may be found by starting with the following "prep" equations:
Lr = LR + A Lr B' = B + A Lr F = VJB' D' = D - ALr R' = Lr/D'
The Type B routine, described in the discussion of Case 1, would then be required.
Note that if the original design calculations were done on the assumption that reflux is not subcooled, the effect of subcooling may be considered as a change in LR:
If the base case assumes some subcooling (TR = TRi), and later the subcooling is different (TR = T^):
For the control system of Figure 19.3, if feed and reflux are fixed, and if it is desired to find the responses x'D,x'B to changes in boilup, the following "prep" equations are required:
B' = B - AF V; = VS + AF, ¡3' = V'JB' D' = D + AF, R' = LR/D'
The Type B routine, described in discussion of Case 1, is then required.
A third basic control scheme is that of Figure 19.4. Here overhead condensate receiver level is controlled by throttling distillate flow, while reflux is flow controlled, perhaps ultimately cascaded from overhead composition control. Base level is controlled by adjusting heating-medium flow control. Bottom-product flow is on flow control, perhaps ultimately cascaded from bottom-product composition control.
If feed and bottom rates are fixed, we wish to find the responses x'D, x'B to a change in reflux flow. The following "prep" equations are required:
If, for Figure 19.4, feed and reflux are fixed, we may wish to find the responses x'D,x'B to a change in bottom-product flow. The following "prep" equations are required:
B' = B + AS v; = Vs - AB j3' = v;/B' D' = D - AB R' - Lr/D'
The Type B routine, described under Case 1, is required.
Column Terminal Composition Sensitivity to Various Inputs
For the control system of Figure 19.2, we may wish to determine overhead and base composition responses to changes in feed rate. The following "prep" equations are needed:
In the first equation above, B' might better be labeled B". Since D is fixed:
But our program requires that all flows be relative to F = 1. Therefore let:
For simplicity of symbolism, we represent B" by B'.
For the control system of Figure 19.2, we may wish to find the responses x'D and x'B to changes in q. The following "prep" equations are needed:
Use the Type B program as described under Case 1.
For the control system of Figure 19.3, we may wish to determine overhead and base composition responses to changes in feed rate. The following "prep" equations are needed:
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