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b. But the concentration of benzene obtained is only 58 mole percent, and c. The concentration of benzene in the still only drops to 10 mole percent (not a very pure toluene).

2. Constant reflux ratio a. Requires about 50 percent more heat input than simple distillation to achieve approximately the same distribution of benzene, but b. The benzene concentration in the product is much better than that obtained by simple distillation (84 percent vs 58 mole percent).

3. Varying reflux ratio a. Permits specification of the purity of the overhead product, but b. Requires the most heat input of the three methods, and c. Leaves the highest benzene concentration in the still.

Although these last two points seem significant, the heat requirements for the separation actually achieved in the variable reflux ratio case is only about 20 percent greater than the heat requirement of the constant reflux case. The benzene concentration in the still residue can be reduced by decreasing reflux ratio to some intermediate point (say 2:1 or 3:1) and continuing the distillation until the residual concentration is reduced to some acceptable value. Of course, the concentration of the benzene recovered in the overhead would be less than 95 mole percent, but this material would not be combined with the product. Instead, this so-called "slop" cut or "tailings" cut would be collected in a different receiver and be recycled to the still during recharging for the next distillation.

Thus, as so frequently is the case, the chemical engineer must choose between alternatives: should high product recoveries with low energy expenditures and low product concentrations, or modest product recoveries with high energy inputs to achieve high product purities be chosen, or is some intermediate case best? Such questions can only be answered on an ad hoc basis.

Several final points remain to be considered. Fir.;t, all of the discussions above ignored the effect of liquid holdup in the column and condenser, but because the combined volumes of these two are quite small as compared to the volumes in the still and receiver (particularly because so many batch stills include packed sections rather than trays), this assumption does not seem unwarranted, particularly for binary mixtures.

When column holdup is significant, and especially when multi-component mixtures must be considered, the analysis of the effect of holdup is extremely complex. This problem has been addressed by a number of investigators (11,14, 16, 17), however, and it has been concluded the column holdups on the order of 10 to 15 percent actually are beneficial in obtaining better separation. The interested reader should consult the references cited.

Second, analysis of the effect of holdup is so complex that large computers are invariably involved in the process (7, 12, 13). Perhaps the most versatile program is the one developed by Boston et al. (5). The basic problem involved in the computer analysis is that the differential equations describing holdup in the column have small time constants, while the variation in the still content varies over a period of hours. Thus, the combination of equations is "stiff," i.e., difficult to integrate numerically. Because the equations are stiff, small time increments must be used, and the software must contain built-in stability controls. Therefore, in the simulation of batch distillation, "...computer time hundreds of times longer than the time for steady state distillations are obtained and design costs are inappropriate to the size of the equipment being designed" (19).

Third, the examples developed above were presented on the basis of a small number of theoretical stages, primarily for clarity of the figures illustrating the calculation methods, and no mention was made of the effect of some other number of theoretical stages. However, it should be obvious that more stages will give sharper separations—just as one would expect in a continuous still.

Last, the point was made early on that the separation could be calculated on the basis of a light "key," but all of the examples were based on a binary mixture. Simple distillation of a multicomponent mixture is analyzed on the basis of the light key as follows.

By material balance, the moles of liquid leaving the still at any time are equal to the moles of liquid entering the receiver (6).

For component A (using the symbol I for liquid)

For component B

Dividing 5.26 into 5.23

Recalling the definition of relative volatility, Eq. (5.6)

= yA(l - xA) 01 *a(1 " y a) Substituting 1 - xB and 1 - yB for xA and yA

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