Modelling of combined Navier–Stokes and Darcy flows in crossflow membrane filtration

Abstract

Computer simulation of the flow field in crossflow membrane filtration in a porous tube and shell system depends on the imposition of permeable wall conditions on the surface of the inner tube. Porous wall conditions are often represented by the Darcy equation which relates the pressure gradients within a flow stream to the flow rates through the permeable walls of the flow domain. In crossflow filtration the feed stream which flows tangentially to the porous tube surface is modelled by the Navier–Stokes equations. These equations represent viscous laminar Newtonian flow. They can also be generalised to deal with non-elastic, non-Newtonian fluids. The existence of viscous stress terms in the Navier–Stokes equations, which are expressed in terms of second-order partial derivatives, makes the straightforward linking of these equations to the Darcy equation in a numerical solution scheme impossible. Therefore, in order to develop a fluid dynamical model for crossflow filtration, special techniques which resolve this difficulty must be used. In this paper first, various methods of linking the Navier–Stokes and the Darcy equations in a solution scheme are considered and the strength and weaknesses of these methods are discussed. Following this discussion the details of a novel method which is used to develop a robust, accurate and cost-effective finite-element simulation scheme for the combined Navier–Stokes/Darcy flows in crossflow filtration is presented.