This Demonstration analyzes a rectifying column for a binary mixture with a constant-relative volatility

. A binary mixture enters the bottom of the column with a mole fraction composition

. At the top of the column, the vapor is sent to a condenser. A portion of the condensed liquid is returned to the column. The mass balance for this operation by stages is described by the following nonlinear Riccati equation [1, 2]:

,

where

,

,

,

is the reflux ratio that measures the amount of liquid returned to the column, and

is the specified distillate composition leaving the condenser. This nonlinear Ricatti equation can be solved using

*Mathematica*'s

RSolve function. The solution then has two unknowns:

and an arbitrary constant

, which can be determined by specifying

at

and

at

. Hence the solutions of the Riccati equation for values of

in the range

are bounded by mole fractions

that lie in the range

.

Integer values of

in the range

define the liquid composition

that leaves an equilibrium stage such that the vapor leaving that stage has composition

. The largest integer value in the interval

determines the number of theoretical equilibrium stages (

) for a set value of

(i.e., mole fraction at the bottom of the rectifying section).

This Demonstration plots the McCabe and Thiele diagram and displays

(in red) found by solving the Riccati equation. You can change the values of the constant-relative volatility

, the reflux ratio

, the distillate mole fraction

, and the mole fraction at the bottom of the rectifying section

. The blue line in the plot defines the operating line for the column, the requirement that mass is conserved over an equilibrium stage.

Similar treatment can be performed for a stripping column.