Membrane mass transport modeling with the periodic boundary condition

Abstract

A two-dimensional simulation of membrane permeation and separation in laminar cross-flow configuration has been developed for cases with the periodic boundary condition (PBC) applied at the upstream (inlet) and downstream (outlet) faces of a repeating flow domain. The periodic boundary condition was applied through decoupled contributions of both solvent and solute fluxes. The upstream boundary conditions were scaled from the downstream boundary conditions after accounting for solute and solvent losses via permeation. The simulation incorporates fluid flow, solute convection–diffusion and separation, variable viscosity and density, as well as the effect of osmotic pressure on the permeation rate. The methodology developed here represents a novel theoretical advance. The commercial CFD software FLUENT ® was used for this project. The utility of the PBC approach is that it enables significant reductions in computational requirements. Only a small portion of a larger domain need be treated to quantitatively infer information about a substantially larger system. For example, it could be applied to the repeating volume between crossing filaments in spacer filled channels of spiral wound or plate and frame modules, or to a section of a long closed channel such as in plate and frame modules or in tubular membranes. The plausibility of the model was demonstrated in two dimensions for the case of solute separation in the ultrafiltration regime under laminar flow in a channel. First, the velocity and solute concentration profiles were simulated for a 1 m long module. These results were then replicated, using 20 cm sections, and the periodic boundary condition methodology, applied at various positions along the channel length.