In the first equation above, B' again might more properly be written B". For a change in F to F' = F + AF:

But:

Since our programs are based on 1 mole feed per unit time, let us define:

Again, for simplicity of symbolism, we use B' instead of B". A Type B program, discussed under Case 1, should be used.

For the control system of Figure 19.3, we may wish to find the responses of xD and xB to changes in q. The following "prep" equations are required:

The Type B program (see Case 1) should then be employed.

For the control system of Figure 19.4, we may wish to determine overhead and base composition responses to changes in feed rate. The following "prep" equations are required:

Use the Type B program (see Case 1).

For the control system of Figure 19.4, we may wish to determine overhead and base composition responses to changes in feed enthalpy factor, q. The following "prep" equations are needed:

Use the Type B program (see Case 1).

For any of the control schemes, the effect of feed composition change from zF to z'F on xD and xB may be found simply by entering the new z'F into data storage for the Type B program (see Case 1).

For any of the control schemes, the effect of a column pressure change from P to P' on xD and xB may be found by recalculating a's; rerunning the program for the quadratic coefficients Aa, Ba, and Ca; and entering the new values of Aa, Ba, and Ca in the Type B program (see Case 1).

Column Gains for Feedforward Compensation

Here the objective is to find required feedforward compensator gain to hold terminal composition constant as various external factors vary.

For any of the control systems discussed—Figure 19.2, 19.3, or 19.4— assume that top composition and bottom composition are held constant. We wish to find the changes in LR, D, B, and F, required to hold compositions constant in the face of a feed composition change, AzF. This information could be used to design feedforward compensators to minimize transient changes in terminal compositions. The variables chosen for feedforward compensation will depend on which feedback control scheme is used—Figure 19.2,19.3, or 19.4.

The equations required to recalculate the other variables are: z'F = zF + A zF F = 1.00

Use the Type A program, described in Section 19.3.

Case 16

This is the same as Case 15 except that q is changed. The only equation needed is:

Use the Type A program (see Section 19.3).

This is the same as Case 15 except that column top pressure is changed to P' = P + A P. Relative volatilities should be recalculated and the quadratic coefficients Aa, Ba, and Ca reevaluated. Then run the Type A program (see Section 19.3).

Economic Penalty of Overrefluxing

If product purity at one end of the column, say xD, is set at a new, constant, higher value, to ensure that product purity is always at least as good as the original specification, we may wish to estimate the cost penalty of the increase boilup. We will need to know LR and Vs. If the column control scheme fixes either D or B, the other will also be fixed. Then if one new terminal composition is chosen, the other is readily calculated:

Use the Type A program (see Section 19.3).

If we have a situation similar to that of Case 18 except that both a new xô and a new x'B are chosen, we will get changes in D and B as well as LR and V,:

Use the Type A program (see Section 19.3).

Shinskey7 has pointed out that the differences between the design value of VJF and those required for higher values of xD are measures of the cost of overrefluxing.

If a column has automatic control of JR. = LR/D (it does not matter whether condensate receiver is level controlled via reflux or top product), the following "prep" equations apply for constant Vs:

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