Multicomponent Mixtures
The state of the art for the batch distillation of multicomponent mixtures is even less satisfactory than for binary mixtures, and except for total reflux, no accurate and practical method is available even without liquid holdup in the column. This difficulty arises from the fact that, starting with a given liquid composition in the still, it is possible to calculate the equilibrium vapor leaving the still, but the composition of the overflow to the still from the first plate cannot be calculated without knowing the composition of the distillate leaving the system. In a binary system, it is possible to choose the composi tion of the distillate arbitrarily and calculate back to the liquid in the still, and the liquid composition in the actual distillation must pass through this condition. However, in a multicomponent mixture, the still composition calculated in that manner for an assumed overhead product will probably not occur in an actual distillation. For example, consider the distillation of a mixture of components A, B, and C and assume an overhead composition of Xda., xdb} and xDc. For these assumed values, the calculations can be carried down the column for a given reflux ratio assuming no holdup, and values of xla, Xlb, and Xlg will be obtained. However, in the actual distillation when component A has the value it is very improbable that the ratio of B to C will be the same as calculated on the basis of the assumed overhead composition. By a laborious trialanderror procedure, a consistent calculation could be made, but it is doubtful that it would justify the effort. A laboratory distillation would probably give a better and cheaper evaluation.
As a guide to the characteristics of multicomponent batch distillation, the case of (1) total reflux with no liquid holdup in the column and (2) finite reflux ratio with no liquid holdup by an approximate method, will be considered.
Rectification without Liquid Holdup in Column. Total Reflux. For the case of no liquid holdup, Eq. (141) applies to each of the components, but the difficulty involves the evaluation of xD as a function of xL.
In this case the stepwise calculations can be carried out starting at the still, because the composition of the distillate does not affect the operating line. Thus for a given xL the value of xD can be calculated, but the calculation is still difficult for the general case because the concentration pattern followed by the liquid in the still is unknown. The pattern can be approximated by a stepwise integration, but the calculations are tedious. The general principles will be developed for the case in which all the relative volatilities remain constant, because in this case direct integration is possible. It is simpler to apply the relative volatility form of the simple distillation of Eq. (66) than to use Eq. (141). Thus, for any two components of a multicomponent mixture at total reflux,
and, for a column with a total condenser,
Let a differential amount, dV, be vaporized and set
where A, B are mols of A and B in still, respectively.
integrating from A0 to A, B0 to B
Ao \Bo) Likewise,
and the same for the other components.
For a given fraction of A vaporized, it is possible to calculate (1) the fraction of all components vaporized, (2) the composition of the liquid remaining in the still, (3) the instantaneous composition of the distillate, and (4) the average composition of all of the distillate.
Batch Rectification of Multicomponent Mixture at Total Reflux. As an example, consider the fractionation of an equimolal mixture of A, B, C, and D at total reflux with relative volatility <*ab = 2.0, «ao = 4.0, and «ad = 8.0. The column is equivalent to three theoretical plates and holdup of liquid will be neglected. Solution. Basis: 100 mols originally charged to still, a. First distillate composition:
4,096 n oqq yA ~ 4,096 + 256 + 16 + 1 " °'938 Vb «■ 0.0586 yo  0.00366 yr> « 0.00023
b. 50 per cent of A distilled:
Do c
 0.5*4" « 0.9973 ~  0.51/4 0®« « 0.99983
The compositions of (1) liquid remaining in still, (2) average distillate, and (3) instantaneous distillate are given in Table 143.
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