The Chemical Engineering Journal, 32 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA (1986) 15 - 19 15 A Matrix Solution to Linearized Reaction-Diffusion Boundary Value Problems MACIEJ STARZAK and STANISLAW LEDAKOWICZ Institute of Chemical Engineering, E6di Technical University, Ul. Wdlczafiska 175, 90 - 924 L6di (Poland) (Received December 6,1984; in final form March 21,1985) ABSTRACT zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA two-point boundary value problem (BVP) has been developed. zyxwvutsrqponmlkjihgfedcbaZYXWVU The balance equations of multicomponent diffusion with chemical reactions of linearized kinetics valid for diluted species have been given in the vector-matrix form. A very con- venient matrix exponential solution to the two-point boundary value problem originating from the film theory has been proposed. 2. PROBLEM STATEMENT An illustrative example taken from the a bsorp tion-desorp tion process occurring in fermentation broths has been analy zed according to the approach developed. 1. INTRODUCTION In multicomponent diffusion-reaction sys- tems one is frequently faced with a physical situation the mathematical description of which involves the solution of a set of non- linear coupled differential equations [ 1,2]. In order to obtain an analytical or approxi- mate analytical solution of the problem the non-linear term of reaction kinetics is usually linearized, e.g. by the Taylor series expansion or by the Hikita and Asai integral approxima- tion [3,4], or the pseudo-first order assump- tion is applied [5]. However, in spite of these linearization procedures the difficulties involved in solving the set of balance equa- tions are often not removed. If the system involves two linearly independent chemical reactions only it is necessary to solve a charac- teristic equation of an order higher than two. Moreover, integral forms of linearly depen- dent balance equations introduce non- homogeneity into the resulting differential equations, which in consequence, need the application of tedious methods of solution (e.g. the method of variation of parameters). To overcome these problems a simple matrix method of solution of the second order linear Let us consider a problem of (n + 1) com- ponent diffusion with m simultaneous chemical reactions between two boundaries. For the sake of simplicity we assume that the n components are present in very small con- centrations, so that the mole fraction of the (n + 1)th species is close to unity. Such a situation is often encountered during absorp- tion of gases into liquid film, where the solvent is in great excess. In this case an effec- tive diffusivity, according to Wilke’s defini- tion [6] is equal to the binary diffusion coefficient of the ith species through the (n + 1)th inert component and the convective mass fluxes Ni are identical to the diffusion fluxes Ji , i.e. dci Ji=Ni=-Di,n+lz (i = 1, . . . . n) (1) Hence, the following system of diffusion- reaction balance equations may be formulated D$ = -vR(c) (2) where x is the diffusion length coordinate, c is an n-dimensional vector of species concen- trations, D is an n X n diagonal matrix of diffusivities, v is an n X m stoichiometric matrix (with the convention of Vii> 0 for products) and R is an m-dimensional vector of the chemical reaction rates. As we restrict our analysis to linear kinetic systems the reaction rate term R(c) may be expressed in the following matrix form R(c) = Kc (3) where K is an m X n matrix of reaction rate constants. The diffusivity matrix D is non- 0300-9467/86/$3.50 @ Elsevier Sequoia/Printed in The Netherlands