Numerical solution of transport phenomena equations for nanofiltration membranes

Abstract

Nanofiltration is a membrane separation process whose driving force are pressure changes, which is used to remove ionic species from aqueous solutions. This technique lies between reverse osmosis and ultrafiltration membranes. Nanofiltration has received much attention in the last decades due to its applications in textile, paper, and food industries including water desalination. The separation mechanisms of nanofiltration membranes include steric and electric effects. Most of these membranes acquire an electrical charge when they come into contact with a polar medium due to the adsorption of charged species from the bulk to the pore walls, which leads to the favored transport of counter-ions and the exclusion of co-ions. The development of adequate mathematical models for nanofiltration is extremely important for better understanding of electrolyte transport phenomena and prediction of separation parameters. The transport of ions in charged membrane pores can be described by the system of Nernst–Planck, Poisson, and Navier–Stokes equations. In this work is modeled the behavior of ions, in our case NaCl, through a nanofiltration membrane with the system of differential equations of Nernst-Planck and Poisson in 1-D in transient. The system of partial differential equations is solved by means of numerical algorithms with the Wolfram Mathematica program. The results are compared with a more complex model (Poisson, Nernst-Planck and Navier Stokes in 2D) and a simpler model (Donnan-Steric-Partition-Dielectric exclusion model) to verify our work and later to be able to use the algorithm for more complex ionic systems