Batch Distillation: If a solution is distilled in a batch operation, then the amount of residue left after its concentration is reduced from x_F to x_W can be found from the following equation.

If the equilibrium relationship is given by constant relative volatility (α), then the above equation can be integrated and the result is given as

Where F and W are moles of feed and residue and x_F and x_W are mole fractions of more-volatile component in the feed and residue, respectively. This equation is non-linear in x_W. This equation can be rearranged into a simpler form and solved by using Newton's method.

Its derivative is given as

Where c₁ and c₂ are given as

Once x_W is known, x_D can be found as

Example 6.6: A liquid having a composition 50 mol % A, and 50 mol % B is subjected to a batch (differential) distillation at a pressure of 1 atm. About 60 mol % of the feed is distilled. Compute composition of the distillate and the residue. Use relative volatility, α, of 2.16 to generate the equilibrium data. Component A is more-volatile component.

Solution: Basis: 100 lbmol of feed.

x_F = 0.5, D = 60 lbmol.

Amount of residue, W, can be calculated as

W = F - D = 40 lbmol

The results of iteration are presented below:

I	xw	f(xw)	f'(xw)
1	0.5	-4.7	-36.3
2	0.369	-1.0	-24.2
3	0.328	-0.01	-23.9
4	0.328	1.4´10^-6	-23.9

Final residue composition, x_W, is found to be 0.328. The residue contains 32.8 mol % A and 61.2 % B. An overall mass balance yields the distillate composition.

Distillate composition, x_D, is found to be 0.615. The distillate contains 61.5 mol % A and 38.5 mol % B.