WOLFRAM|DEMONSTRATIONS PROJECT

Analyzing a Rectifying Column Using the Calculus of Finite Differences

​
relative volatility α
2.5
distillate mole fraction
x
0
0.9
reflux ratio R
1.5
bottom composition
x
f
0.5
This Demonstration analyzes a rectifying column for a binary mixture with a constant-relative volatility
α
. A binary mixture enters the bottom of the column with a mole fraction composition
x
f
. 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]:
x
n
x
n+1
+A
x
n+1
+B
x
n
+C=0
,
where
A=
x
0
(α-1)-α(R+1)
(α-1)R
,
B=
1
α-1
,
C=
x
0
R(α-1)
,
R
is the reflux ratio that measures the amount of liquid returned to the column, and
x
0
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:
n
and an arbitrary constant
C
, which can be determined by specifying
x
n
=
x
0
at
n=0
and
x
n
=
x
f
at
n=
n
max
. Hence the solutions of the Riccati equation for values of
n
in the range
0≤n≤
n
max
are bounded by mole fractions
x
that lie in the range
x
0
≤x≤
x
f
.
Integer values of
n
in the range
0≤n≤
n
max
define the liquid composition
x
n
that leaves an equilibrium stage such that the vapor leaving that stage has composition
y
n
=α
x
n
/1+
x
n
(α-1)
. The largest integer value in the interval
0≤n≤
n
max
determines the number of theoretical equilibrium stages (
n
theoretical
) for a set value of
x
f
(i.e., mole fraction at the bottom of the rectifying section).
This Demonstration plots the McCabe and Thiele diagram and displays
n
max
(in red) found by solving the Riccati equation. You can change the values of the constant-relative volatility
α
, the reflux ratio
R
, the distillate mole fraction
x
0
, and the mole fraction at the bottom of the rectifying section
x
f
. 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.