Water and solute transport in multimembrane systems

The solute and water transport across membranes, eventually coupled to chemical reactions, is of fundamental importance in biology, biotechnology and chemical engineering. In all these fields, a membrane can be seldom considered as a single elementary barrier for diffusion and permeation. Usually, series arrays of barriers (including unstirred layers) are involved and one or more of the membranes in the array my be composed by transporting elements arranged in parallel. A recursive algorithm has been developed, in order to write the transport equations across a n membranes and unstirred layers array, starting from locally linear Kedem-Katchalsky practical equations, as suggested by Patlak, Goldstein and Hoffman. The integration of these equations across a barrier gives the solute concentration on one side as a function of its concentration on the other side and of the solute and volume flows. The volume flow turns out to be a linear function of the hydrostatic and osmotic pressure differences across the barrier. The non-linear solute concentration function may be obtained for any membrane in the array and the recursive substitution of the (i-l)-th concentration in the i-th expression allows to obtain the (n+l)-th concentration in the “right” external solution as a function of the concentration in the first (“left”) solution and of the solute and volume flows. From this equation, the solute flow equation may be obtained. By substitution of the concentrations in the volume flow equation, a non-linear relationship is obtained between the volume flow and its external driving forces, hydrostatic and osmotic pressure differences. From these equations it is possible to obtain the overall phenomenological coefficients of the array, which appear to be a generalization of the ones already obtained for a two membrane system.