A simulation of mass transport across series arrays of membranes with chemical reaction-coupled solute flow

Both in biology and technology, membranes can be seldom treated as single, thin and linearly behaving barriers. Mass transport across thick membranes, as first suggested by Patlak, Goldstein and Hoffman (1), can be described integrating across the membrane thickness the local, linear, practical equations by Kedem and Katchalsky (2), which have been recently shown to derive directly fom the local energy dissipation function of the membrane (3). When the heat flow associated to chemical reactions or to a temperature gradient can be neglected, the presence of a solute "active" transport can also be accounted for (1,4). In the present paper we extend to n membranes, by means of a recursive procedure, the previous treatment of a series array of few membranes (1, 4) and derive some parameters characterizing the transport properties of the complex barrier. This task has been performed analytically, with the help of the symbolic computation program REDUCE, obtaining a non-linear correlation between the flows and their driving forces, depending on the volume flow and on the solute concentration of the transported solution. The nonlinearity appears to be a consequence of the solute accumulation in the inner compartments of the array. The classical linear law by Darcy is a limiting case of our volume flow equation when only pure solvent is transported. Around the volume and solute flow equations we have written a program in Pascal allowing the simulation of a series array of up to 10 membranes and unstirred layers, assimilated to non-selective membranes. The results of the simulation are in agreement with experimental data obtained using complex biological barriers like epithelia. 1. C.S. Patlak, D.A. Goldstein, J.F. Goldstein: J. Teor. Biol. 5, 426-442 (1963); 2. O. Kedem, A. Katchalsky: Biochim. Biophys. Acta 27, 229-246 (1958); 3. F. Celentano, G. Monticelli: Local Practical Equations for Heat and Mass Transport Driven by Temperature Gradients, Proc. Europe-Japan Congr. Membranes and Membrane Proc., Stresa, June 18-22 1984, in press; 4. G. Monticelli, F. Celentano: Bull. Math. Biol. 45, 1073-1096 (1983).