## W Vn y Ln 1xin

Note that wi is always constant in each column section. We will assume constant molar flows (L = Ln = Ln_i and V =Vn = Vn + i) and, assuming constant relative volatility, the VLE relationship is:

aixi

divide an equation for 0k with the one for 0j, the following expression appears:

aixin

We divide eqn [38] by V, multiply with the factor ai/(ai — 0), and take the sum over all components:

aiwi

The parameter 0 is free to choose, and the Underwood roots are defined as the values of 0 which make the left-hand side of eqn [40] unity, i.e.:

The number of values 0 satisfying this equation is equal to the number of components.

Most authors usually use a product composition or component recovery (r) in this definition, e.g. for the top (subscript T) section or the distillate product (subscript D):

but we prefer to use w since it is more general. Note that use of the recovery is equivalent to using net component flow, but use of the product composition is only applicable when a single product stream is leaving the column. If we apply the product recovery, or the product composition, the defining equation for the top section becomes:

Stage-to-Stage Calculations

L X «xn + 1 " (« — 0) V X («i — 0) " X ^n

and we note the similarities with the Fenske and Kremser equations derived earlier. This relates the composition on a stage (n) to a composition on another stage (n + m). The number of independent equations of this kind equals the number of Underwood roots minus 1 (since the number of equations of the type as in eqn [44] equals the number of Underwood roots), but in addition we also have X xi = 1. Together, this is a linear equation system for computing xi,n#m when xi,n is known and the Underwood roots are computed from eqn [41].

Note that so far we have not discussed minimum reflux (or vapour flow rate), thus these equations hold for any vapour and reflux flow rates, provided that the roots are computed from the definition in eqn [41].

### Some Properties of the Underwood Roots

Underwood showed a series of important properties of these roots for a two-product column with a reboiler and condenser. In this case all components flows upwards in the top section (wiT > 0), and downwards in the bottom section (wijB < 0). The mass balance yields: wi;B = wi;T — wi;F where wi;F = Fzi. Underwood showed that in the top section (with Nc components) the roots (0) obey:

And in the bottom section (where wi>n = wi;B < 0) in general we have a different set of roots denoted computed from VB = X i [«w^« — =

Xi[«i( — ri,B)zj/(<Xi-iA)] =Xi[«i( -(1 — ri,D))zi/(«i —iA)]

which obey:

> «1 > ^2 > «2 > ^3 > «3 > ^ > ^n

With the definition of 0 from eqn [41], eqn [40] can be simplified to:

This equation will be valid for any of the Underwood roots, and if we assume constant molar flows and

Note that the smallest root in the top section is smaller than the smallest relative volatility, and the largest root in the bottom section is larger than the largest volatility. It is easy to see from the defining equations that VT p rn 0i p « and similarly

When the vapour flow is reduced, the roots in the top section will decrease, while the roots in the bottom section will increase, but interestingly Underwood showed that 0i > ^ + 1. A very important result m by Underwood is that for an infinite number of stages

Then, at minimum reflux, the Underwood roots for the top ($) and bottom sections coincide. Thus, if we denote the common roots (0), and recall that VT — VB = (1 — q)F, we obtain the following equation for the common roots (0) by subtracting the defining equations for the top and bottom sections:

1. Compute the common root (0j) for which aj > 0j > a +1 from the feed equation: (1 — q) =

2. Compute the minimum energy by applying 0j to the definition equation:

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