Mathematical simulation of a reflux mass-transfer process

Abstract

We study a numerical-analytic method of solving an initial-boundary value problem for a quasilinear system of differential equations of parabolic type with initial condition given by the Dirac delta function. One figure. Bibliography: 6 titles.Reflux mass transfer processes (ion exchange, adsorbtion, isotopic exchange, and so forth) taking place in reflux columns are widely applied in ecology, medicine, and many areas of industry. They have been successfully used to solve very diverse problems: water purification, syrups, amino acids, antibiotics, waste solutions of different manufacturing processes, separation of mixtures of closely related materials, and the like [ 1].The basic problems of the mathematical description of these processes can be solved using one-dimensional continuous models, for example, the nonequilibrium diffusion model studied in [2], which takes account of diffusion, convection, and nonlinear interphase exchange of a reflux two-phase mass transfer process.Analysis of reflux mass-transfer processes using the nonequilibrium diffusion model involves solving an initialboundary-value problem for a quasi-linear system of two differential equations of parabolic type.. An effective method of determining the parameters of a reflux two-component (for example, liquid-gas or ionite-solution) mass-transfer column is the pulse method [1]. It is based on the study of an artificially created pulse perturbation in the concentration of the component being transferred in the column. The response curves that characterize the distribution of concentration of the component at different times or the time-dependence of the concentration in a cross section of the column, captured experimentally or computed theoretically, give the needed information to solve the inverse problem of determining the parameters of the column.In the mathematical formulation of the problem the initial condition under pulsed introduction of the enriched mixture into the column is singular and is given by the Dime delta-function. The given specifics of the initialboundary-value problem determines the choice of the proposed numerical-analytic method of solving it.The idea of the method is to construct an approximate closed-form solution for the initial period of time and to extend the computations by the finite-difference method [3] on a grid that adapts to the character of the behavior of the solution of the problem, which has large gradients caused by the singularity of the initial condition.Mathematical statement of the problem. The system of equations of a reflux mass-transfer process in the case of nonequilibrium diffusion model of longitudinal transfer and the difference model of interphase exchange has the form [2] dgN L rgN _ •2N X ~mL--~-= ---~+ lsr --~-+ Kt.[ f (n)-N],(1)where N and n are the relative concentrations of the component in the liquid and the gas respectively; Xt. and X6 are the fractions of a unit volume occupied by the liquid and the gas; m/. and m e are the total volume concentrations of interchanging components in t...