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

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

The solution to the familiar eigenvalue problem is routinely taught in undergraduate linear algebra courses, but it is rare that the generalized eigenproblem 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 -order polynomial.

A·ξ=λI·ξ

A·ξ=λB·ξ

th

n

Consider a mixture of six hydrocarbons (5 mole % , 10 mole % , 15 mole % , 30 mole % , 30 mole % , and 10 mole % ). Assume that this mixture is ideal, and that the constant relative volatilities of the components , , , , , and are equal to 11.5198, 4.3934, 2.5522, 2.0836, 1.1641, and 1.0000. All relative volatilities are computed with 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.

C

2

C

3

i-

C

4

n-

C

4

i-

C

5

n-

C

5

C

2

C

3

i-

C

4

n-

C

4

i-

C

5

n-

C

5

n-

C

5

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)=F-ϕ

N

c

∑

i=1

α

i

z

i

α

i

V

min

N

c

∑

i=1

α

i

x

i,dist

α

i

Once the values for equation (1) are found, then 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 . This equation can be rewritten as the polynomial

ϕ

i

V

min

f(ϕ)=-ϕ-(1-q)

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∑

i=1

α

i

z

i

α

i

g(ϕ)=(-ϕ)-(1-q)(-ϕ)

N

c

∑

i=1

N

c

∏

j=1,i≠j

β

i

γ

i

N

c

∏

i=1

β

i

γ

i

which for has distinct real roots. For , is a root, together with -1 real roots, all of which can be found using Mathematica's built-in function NSolve.

q≠1

N

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q=1

ϕ=∞

N

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It has recently been suggested [1] that the roots of can also be obtained by finding the eigenvalues of a generalized eigenvalue problem (): , where the eigenvalues are given by

f(ϕ)

GEP

(B-ϕΓ).ξ=0

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

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

α

i

z

i

ψ=1-q

β

i

z

i

Because has rank , there are also infinite eigenvalues. The solution of such a problem is readily accomplished using Mathematica's built-in function Eigensystem.

Γ

p<+1

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c

(+1)-p

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The Demonstration plots in blue and indicates all roots by colored dots. You can set the value of the feed quality to a value between 0 and 1. The numerical values of the finite eigenvalues found are identical to those found using NSolve.

f(ϕ)

q