Simultaneous solution of the very large set of nonlinear and algebraic equations that describe a reactive distillation column is difficult, especially with the high degree of non-linearity attributable to reaction kinetics. The relaxation method is efficient and robust in solving this large set of equations. This method is used to calculate steady-state mole fractions and temperature profiles throughout the column. In general, relaxation methods use the equilibrium stage model equations in unsteady-state form and integrate them numerically until the steady-state solution is found (all time derivatives = 0). Liquid holdups on trays are assumed constant, that is, instantaneous hydraulics. The net reaction rate for component j on tray n in the reactive zone is given by where Vj is the stoichiometric coefficient of component j.

The steady-state vapor and liquid rates are constant through the stripping and rectifying sections because equimolal overflow is assumed. However, these rates change through the reactive zone because of the exothermic reaction. The heat of reaction vaporizes some liquid on each tray in that section; therefore, the vapor rate increases up through the reactive trays and the liquid rate decreases down through the reactive trays.

The dynamic component balances for the column are reflux drum:

d(xD, jMp) dt rectifying and stripping trays:

— L„+1X„+1, j + V„-1y„-1,j — Lnxn, j — V„y„,j (3.36)

Ln+1 Xn+1, j + Vn—1yn—1, j Lnxn,j Vnyn,j + Rn, j (3.37)

feed trays:

: Ln+1Xn+1,j + Vn—1yn—1,j — LnXn,j — Vnyn,j + Rn,j + FnZn,j (3.38)

With the equimolal overflow assumption mentioned above, all of the vapor rates (Vn) throughout the stripping section are equal to VS, and all of the liquid rates (Ln) are equal to LS. Analogously, all Vn beginning from the top feed tray throughout the rectifying section and total condenser are VR, and all Ln are equal to LR.

The vapor-liquid equilibrium is assumed ideal. Column pressure P is optimized for each case. With pressure P and tray liquid compositions xn,j known at each point in time on each tray, the temperature Tn and the vapor compositions yn,j can be calculated. This is a bubblepoint calculation and can be solved by a Newton-Raphson iterative convergence method. 