Simultaneous development of velocity and concentration profiles in reverse osmosis systems

Abstract

A mathematical analysis for the simultaneous development of the velocity and concentration profiles in a reverse osmosis system consisting of two parallel flat membranes is presented. This analysis, based on boundary layer theory, takes into account the nonlinear effects created by the fact that the water flux produced varies in the longitudinal direction. The initial velocity and concentration profiles are assumed to be uniform and the analysis is resctricted to two dimensional, steady state laminar flows with constant physical properties. The momentum and diffusion equations, coupled through both the boundary conditions and the convective terms in the differential equations, are solved simultaneously using the approximate integral method. The analysis is applicable to both the entrance region and the fully developed region at large distances from the conduit inlet. Data are presented on the concentration build-up at the wall for a wide range of parameters and typical results for the water flux produced and productive capacity are given. Since the concentration profile at any value of x + is given as a simple polynomial, the results can be used conveniently to study the relaxation or attenuation, of concentration polarization as a Graetz type problem. Reasonable agreement is observed between the results of the present work and the limited experimental results available in the literature. However, it is shown that some of the reported experimental data were probably significantly influenced by natural convection. It is shown that the extent of concentration polarization may be significantly different depending on whether the velocity profile in the stream entering the membrane section is parabolic or flat. An explanation is offered for the somewhat surprising fact that the water flux is greater and the polarization is less in fully developed flow than when the velocity profile is flat at the system entrance. This effect will be most pronounced in the highly productive membranes which probably will be developed in the future.