The batch distillation of binary mixtures will be considered fpr the cases of (1) no rectification, (2) rectification without liquid holdup in the column, and (3) rectification with holdup.

No Rectification, Batch distillation without rectification corresponds to the simple distillations of Chap. 6, and the calculations of the concentrations as a function of the amount distilled can be made by Eqs. (6-3) and (6-7).

Rectification without Liquid Holdup in Column. Finite Reflux Ratio. In this case, it is assumed that the distillation is carried out with a fractionating column, that the hoI^juQof liquid in the column is negligi-

ble in comparison to the liquid in thej^^m, and that the rate of £han^ a plate is negligible in com parison to the rate of flow of that component through the plate. Thus the change, in the quantity of each component in the column can be neglected in the differential material balances.

Consider the system shown in Fig. 14-1. On the basis of the assumptions made, an over-all differential material balance gives dD = -dL where L represents the mols of liquid in still, and a component balance gives xpdD = ~d(LxL)

= — L dxL — Xh dL = —L dxh + xL dD dD _ dxh L

Fig. 14-1. Schematic diagram of batch distillation system.

Fig. 14-1. Schematic diagram of batch distillation system.

This equation is equivalent to the Ray-leigh equation but differs in that the denominator is xl — Xd instead of xl — Vl. The integration of Eq. (14-1) involves determining a relation between xL and xD. Neglecting the rate of change of holdup of a component on the plates and in the condenser, a balance between the n and n + 1 plates gives

At any time these equations are identical to those for the continuous distillation and can be applied to determine the relation between xL and xD for that instant. By applying the equations repeatedly, the value of Xl — Xd can be obtained as a function of xL and the integration of Eq. (14-1) performed. There are a number of ways such a distillation can be made, but the two most common cases involve (1) operating at constant reflux ratio and taking cuts that average the desired composition and (2) operating at variable reflux ratio to give a constant product composition while making the desired product. In the first case the value of (0n/7n+i) will remain constant during the distillation, and a series of lines of this slope can be drawn on the usual y,x diagram for various assumed values of xD} and the value of xL can be determined by stepping down each line the number of theoretical

Fig. 14-2 for a column and still equivalent to four theoretical plates. In the second case the value of xD is fixed, a series of operating lines of different slopes is drawn through it, and the plates are stepped off on each line to determine the value of xL. This procedure is illustrated in Fig. 14-3. By these procedures, Eq. (14-1) can be integrated, giving the relation between the amount distilled and the composition of the

Fig. 14-2 for a column and still equivalent to four theoretical plates. In the second case the value of xD is fixed, a series of operating lines of different slopes is drawn through it, and the plates are stepped off on each line to determine the value of xL. This procedure is illustrated in Fig. 14-3. By these procedures, Eq. (14-1) can be integrated, giving the relation between the amount distilled and the composition of the liquid remaining in the still. In the second case with xD constant, the integration of Eq. (14-1) amounts to only simple material balance, and the data from Fig. 14-3 are not needed for this purpose. In addition to the information obtained by integration of Eg. (14-1), it is frequently necessary to have data on the vapor required or the average composition of a fraction produced.

For the constant reflux case, the calculation of the vapor requirement can be made as follows:

and the average composition of any fraction is

For the variable reflux case, dV = dO + dD

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