Implementation of the DSPM model using a commercial finite element system

A solution procedure based on the use of the finite element method (FEM) was developed to solve the Donnan steric-partioning pore model (DSPM) for a nanofiltration membrane coupled with a boundary layer. The numerical solution of nanofiltration models like DSPM is usually carried out by combining an integration method like Runge-Kutta with an iterative procedure that starts from guessed permeate concentrations values. However, this approach may have problems of lack of convergence. As an alternative procedure, a finite element system is used to easily implement the transport equation for the membrane and the boundary layer with Donnan-steric conditions in both membrane sides. To test the solution method, an analytical solution was developed for the transport of an uncharged solute across a membrane with boundary layer. Afterwards, the method was applied to the calculation of binary and ternary ionic systems and its performance was compared with that obtained with the classical iterative method. The developed methodology benefits from the accuracy of FEM and the representation and analysis capabilities of the software used.