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where the individual transfer functions are determined by one of the methods indicated above. Forcing functions in the original Rippin and Lamb equations were limited to LR, Vs, zF, and F; the authors have added terms for p and q. Symbolism such as means the open-loop transfer function of yT with respect to zF. Equations (18.43) and (18.44) may now be represented in signal flow diagram form as shown in Figure 18.7.

Rippin and Lamb developed a stepping procedure, later extended by Luyben and others,9-14 for computing these transfer functions in the frequency domain with a digital computer. Applying asymptote techniques to the Bode plots, they determined approximate Laplace transforms. Since these transfer functions were mosdy of relatively low order—second to fourth order—they could be simulated on an analog computer with much less hardware than would be necessary with more conventional distillation-column simulations. Feedback control for top and bottom composition could then be designed with ease. The advent, however, of more powerful and less expensive digital computers has shifted the emphasis to digital simulation. More commonly today the individual tray differential equations are combined and solved in the time domain.

A comparison of the computational efficiency of stepping and matrix-inversion techniques has been carried out by Shunta and Luyben. For large columns in particular, it is shown that the former is much faster.

It should be noted that some of the work of modeling column composition dynamics has been concerned with multicomponent rather than binary separations.12-14

It should also be noted that the open-loop transfer functions given in equations (18.43) and (18.44) are valid, strictly speaking, for only one combination of holdup volumes in the condensate receiver and the column base. Studies to date, however, suggest that if these holdups are no larger than the total holdup on the trays, then the open-loop transfer functions are relatively insensitive to variations in terminal volumes. This topic definitely needs more study. In the meantime we should determine terminal volumes by material-balance and protective-control calculations before calculating column composition dynamics.

Other Models

Two other models for binary columns are those proposed by Wahl and Harriott16 and by Waller.17 These are similar in that they are both based on circulation rates and liquid holdup in the column. Low-order transfer functions are calculated from steady-state data. The two models give essentially the same results except that the Waller method leads to higher order—and intuitively more accurate—transfer functions. For columns with moderate relative volatilities—say 2 to 5—and for terminal purities not greater than 98 to 99 percent, predicted behavior checks with that of more rigorous models. But for high-purity, nonideal separations, accuracy falls off. More work needs to be done to check the range of validity of these models.

A comprehensive review of the literature of distillation dynamics and control through about 1974 is given by Rademaker, Rijnsdorp, and Maarleveld.18 Tolliver and Waggoner19 have published an exhaustive review of more recent additions to the literature.

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