Multistage batch distillation can obtain much higher product purity than simple batch distillation. The overall mass balance is:
,
where
is the initial molar quantity of the feed mixture,
is the molar quantity in the bottoms vessel (the "waste" product), and
is the total molar quantity of collected distillate. The component mass balance is:
,
where
is the initial mole fraction of component
in the feed,
is the average mole fraction of component
within the collected distillate, and
is the final mole fraction within the bottoms vessel.
Each tray in the column is assumed to be in vaporliquid equilibrium:
.
The function
represents an equilibrium curve, plotted on an

diagram. The composition of each stage lies somewhere upon this curve.
An operating line relates the composition of a stage to the adjacent stage composition. The operating line equation is:
,
where
is the vapor mole fraction of
at stage
,
is the reflux ratio, and
is the liquid mole fraction of
at stage
. The reflux ratio is:
,
where
is the flow rate of liquid returning from the condenser to the column, and
is the flow rate of distillate being collected. The Rayleigh equation can be used to solve for the final composition of distillate and bottoms:
,
but the relationship between distillate composition and bottoms composition
is complicated with multistage columns, and the mole fractions are obtained numerically.
[1] P. C. Wankat, "Batch Distillation,"
Separation Process Engineering: Includes Mass Transfer Analysis, 3rd ed., Upper Saddle River, NJ: Prentice Hall, 2012 pp. 329–347.