The quantitative treatment of batch distillation is difficult and complex because the distillate composition changes with time. One aspect of these distillations that can be treated more simply is that of differential distillation. In differential distillation, the liquid is vaporized and each segment of vapor is removed from contact with the liquid as it is formed. Although the vapor can be in equilibrium with the liquid as it is formed, the average vapor formed will not be in equilibrium with the liquid residue.

For any two components in a differential distillation,

(5-25)

where A and B are the moles in the still at time t, and A ₀ and B ₀ are the moles at the start of the distillation. The ? is the volatility of A relative to B.

Multi-stage batch colums can be designed by using a constant reflux operation. For such a case, at any given reflux ratio

(5-26)

and

(5-27)

where Mi, M _f are moles initially and finally present. Essentially the value of X _A,d is varied and corresponding values of X _A,b obtained.These are then used (50) with equations 5-26 and 5-27 to obtain X _{D ,AVG} for the overall batch distillation.

Figure 5-22 illustrates the procedure (operating lines of constant slope with varying X _A,d values).

Figure 5-22: Multistage batch column design (constant reflux) (40).

Another case is to use constant overhead composition (same X _A