## Underwood System of Equations

The Underwood system of equations can be obtained from the conditions of componentwise material balance and of phase equilibrium in the cross-section of constant concentration zones of each section. For example, the following expression can be obtained from the equation of componentwise material balance at the contour, embracing the part of top section from the cross-section in the zone of constant concentration to the output of top product, taking into consideration the conditions of phase equilibrium between the incoming vapor flow and the outgoing liquid flow in this zone:

Designating p = L/VKn and summing up by all the components, we obtain one of the main equations of Underwood:

The analogous equation for the bottom section is as follows:

The main achievement of Underwood consists in the proof of equality of parameters 0 and f in Eqs. (5.1) and (5.2) in the mode of minimum reflux. Sum up these equations is following main equation:

i where q is a portion of liquid in the feed and 6 is a common parameter (root) of Eqs. (5.1) and (5.2).

For the set composition of the top product and for the set reflux number, the number of roots of Eq. (5.1) equals that of the components in top product (k):

0 < <1 < ak < <2 < ak-1 < ■■■ < a2 < <pu < a1

Similarly, the number of roots of Eq. (5.2) equals that of components in the bottom product (m):

am < f 1 < am-1 < ■■■ < a2 < fm-1 < a1 < f m

Correspondingly, the number of roots of Eq. (5.3) is less by one than that of the components, present in the top and in the bottom products (i.e., the number of distributed components).

The Underwood equation system determines separation product compositions and internal liquid and vapor flows in the sections for the set values of two parameters, characterizing the separation process. The reflux number R and withdrawal of one of the products D/F or recoveries of some two components into the top product fi = di / fi and f j = dj/ fj, etc., can be chosen as such two parameters. For example, at direct split of three-component ideal mixture 1(2) : (1)2,3 (here the top product contains component 1 and small admixture of component 2 and the bottom product contain components 2,3 and small admixture of component 1), Eq. (5.3) has only one common for both section root a2 <6 < a1. If f 1 and f 2 are set, then d1 and d2 can be defined and V"1" can be obtained from Eq. (5.1). The rest of internal flows in the column section can be defined with the help of the material balance equations.

In a more general case, when there are several distributed components, it is necessary to obtain from Eq. (5.3) the common roots for two sections. After the substitution of each of these roots into Eq. (5.1) or (5.2), we obtain the system of linear equations relatively to di and Vrmin or bi and Vsmin, the solution of which determines separation product compositions and internal vapor and liquid flows in the column sections. In addition, one can find the compositions of equilibrium phases in the cross-sections of constant concentration zones (i.e., stationary points of sections trajectories bundles).

The main problem in solving the Underwood equation system, as it was shown in Shiras et al. (1950), is the correct determination of the list of distributed components at two specified parameters. At the wrong setting of this list, the solution of the equation system leads to unreal values of di and bi for some components (di > fi or di < 0).

In this case, it is necessary to correct the list of distributed components, referring those components, for which unreal values of di or bi were obtained, to undistributed ones.