# Application Of The Theta Method Of Convergence To Distillation Columns In Which Chemical Reactions Occur

The 0 method of convergence, which was originally formulated for the solution of distillation problems in which chemical reactions did not occur, is extended to include the case where one or more chemical reactions occur on each stage of a distillation column. The computational time of the 0 method, which is one of the fastest known methods for solving distillation problems, is not significantly increased by the inclusion of one or more chemical reactions per stage. In the development which follows, the 0 method of convergence is applied to distillation columns in which one chemical reaction occurs on each plate. The formulation of the algorithm is followed by the solution of several numerical examples.

In recent years, several calculational procedures have been proposed for solving distillation problems in which a chemical reaction occurs on each plate. Suzuki, et al.24 used a procedure based on Muller's method; whereas, Jelinek and Hlavâcek14 used a relaxation method. Komatsu and Holland17 and Nelson19 proposed different formulations of the Newton-Raphson method.

### Formulation of the 9 Method of Convergence

Consider the case of an existing column such as the one shown in Fig. 2-3. Suppose that the specifications are taken to be: (1) the total number of stages and the location of the feed plate, (2) the complete definition of the feed (the total-flow rate, composition, and thermal condition), (3) the reflux rate, (4) the distillate rate, (5) the column pressure, and (6) the type of condenser (total or partial). On the basis of this set of specifications, it is desired to find the resulting compositions of the distillate and bottom products.

The equations required to describe this column are developed in the order in which they are solved sequentially in the proposed calculational procedure. On the basis of assumed temperature and L/V profiles, the material balances, the physical equilibrium relationships, and the chemical rate expressions (or chemical equilibrium expressions) are solved for the moles of each component which reacts per stage per unit time and for the component-flow rates. A formulation of the Newton-Raphson method is used.

The remainder of the calculational procedure is analogous to that proposed for distillation columns without chemical reactions. After the component-mate-rial balances have been solved for the moles reacted and the component-flow rates, a 0 multiplier is found that places the column in overall material balance and in agreement with the specified value of the distillate rate D. Next, new sets of compositions are computed, and these are used to find a new set of temperatures by the Kb method. On the basis of these temperatures and the most recent sets of compositions, a new set of total-flow rates is found by use of the enthalpy balances and the total material balances. The enthalpy balances are stated in the constant-composition form.

Vapor-Liquid Equilibrium Relationships

The equilibrium relationship for stage j and component i [the first expression of Eq. (5-1)] is readily restated in the following form

where the absorption facto is defined in the usual way

Material Balances

The equations are formulated first for the case where a single reaction aA + bB<±cC + dD (8-3)

occurs in the liquid phase on each stage. In the analysis which follows, this reaction is restated in the following equivalent form vaA + vhB + vcC + vnD = 0 (8-4)

where

The contents on each plate are assumed to be perfectly mixed.

Let rjj denote the moles of A which disappear by reaction per unit time for any plate j. Then the mass balance for component / for a typical interior stage j (j + If - If N) is given by

where

Mi = molecular weight of component i Sji = vtfj

Since Mi is common to each term of Eq. (8-5), the material balance for component i on stage j reduces to

Use of the vapor-liquid equilibrium relationship ljt = An Vji to eliminate the liquid flow rates from Eq. (8-6) yields

The complete set of component-material balances for the conventional column may be represented by the following matrix equation

where the tridiagonal matrix A, and the column vectors f, and v, contain the elements shown beneath Eq. (2-18). The column vector v\ is of the form n = [h r\2 • • rjsV

The total material balance enclosing a typical plate j is found by summing each member of Eq. (8-6) over all components i to give i + Lj-1 — Vj — Lj + Aj = 0 (8-9)

where c c