WOLFRAM|DEMONSTRATIONS PROJECT

Solution of the First Underwood Equation Formulated as a Generalized Eigenvalue Problem

​
feed quality
0.5
root loci
root list
The solution to the familiar eigenvalue problem
A·ξ=λI·ξ
is routinely taught in undergraduate linear algebra courses, but it is rare that the generalized eigenproblem
A·ξ=λB·ξ
is discussed, often because of a lack of suitable physical examples where such problems arise. In this Demonstration we show how a familiar distillation problem in undergraduate chemical engineering can be reformulated as a generalized eigenvalue problem to find the roots of an
th
n
-order polynomial.
Consider a mixture of six hydrocarbons (5 mole %
C
2
, 10 mole %
C
3
, 15 mole %
i-
C
4
, 30 mole %
n-
C
4
, 30 mole %
i-
C
5
, and 10 mole %
n-
C
5
). Assume that this mixture is ideal, and that the constant relative volatilities of the components
C
2
,
C
3
,
i-
C
4
,
n-
C
4
,
i-
C
5
, and
n-
C
5
are equal to 11.5198, 4.3934, 2.5522, 2.0836, 1.1641, and 1.0000. All relative volatilities are computed with
n-
C
5
taken as a reference component. This mixture is fed to a distillation column with one overhead stream and one bottom stream. The Fenske–Underwood–Gilliland procedure is a well-known shortcut for estimating the number of theoretical equilibrium trays in the column, assuming a constant molar overflow. The shortcut method is often used as a reference step before undertaking a more rigorous analysis.
The first and second Underwood equations, given by equations (1) and (2), have to be solved in order to evaluate the minimum reflux ratio.
F(1-q)=
N
c
∑
i=1
α
i
F
z
i
α
i
-ϕ
(1)
V
min
=
N
c
∑
i=1
α
i
D
x
i,dist
α
i
-ϕ
(2)
Once the
ϕ
i
values for equation (1) are found, then
V
min
is determined using equation (2). The difficulty arises in a distillation problem where non-key components are distributing. Then you have to find all the roots of the function
f(ϕ)=
N
c
∑
i=1
α
i
z
i
α
i
-ϕ
-(1-q)
. This equation can be rewritten as the polynomial
g(ϕ)=
N
c
∑
i=1

N
c
∏
j=1,i≠j
(
β
i
-
γ
i
ϕ)-(1-q)
N
c
∏
i=1
(
β
i
-
γ
i
ϕ)
(3),
which for
q≠1
has
N
c
distinct real roots. For
q=1
,
ϕ=∞
is a root, together with
N
c
-1
real roots, all of which can be found using Mathematica's built-in function NSolve.
It has recently been suggested [1] that the roots of
f(ϕ)
can also be obtained by finding the eigenvalues of a generalized eigenvalue problem (
GEP
):
(B-ϕΓ).ξ=0
, where the eigenvalues are given by
det[B-ϕΓ]=0
and where
B=
β
1
0
0
…
0
1
0
β
2
0
…
0
1
0
0
β
3
…
0
1
…
…
…
…
…
…
0
0
0
…
β
N
1
1
1
1
…
1
ψ
,
N
c
=6
,
Γ=
γ
1
0
0
…
0
0
0
γ
2
0
…
0
0
0
0
γ
3
…
0
0
…
…
…
…
…
…
0
0
0
…
γ
N
0
0
0
0
…
0
0
,
γ
i
=1/(
α
i
z
i
)
,
ψ=1-q
, and
β
i
=1/
z
i
.
Because
Γ
has rank
p<
N
c
+1
, there are also
(
N
c
+1)-p
infinite eigenvalues. The solution of such a problem is readily accomplished using Mathematica's built-in function Eigensystem.
The Demonstration plots
f(ϕ)
in blue and indicates all roots by colored dots. You can set the value of the feed quality
q
to a value between 0 and 1. The numerical values of the finite eigenvalues found are identical to those found using NSolve.