Concentration polarisation in tubular membranes — a numerical approach

A theoretical investigation of particle deposition onto a permeable surface of a tubular membrane is presented. The mass transport mechanisms are mathematically expressed using the two-dimensional convective diffusion equation. A numerical scheme is presented to solve the two-dimensional convective diffusion equation at the steady state for the case of nonuniform permeation velocity. This equation is solved numerically using a finite difference method. The numerical prediction of mass transfer in the mass boundary requires the use of a very dense grid. The concentration profiles along the membrane surface and the mass boundary layer are predicted. The effect of the Reynolds number, the wall Reynolds number and the Schmidt number were investigated. Correlations for the concentration boundary layer thickness *c/D = 2(z/D)0.33 (Re Sc)!0.33 Rew!0.3 (1!0.4377 Sc!0.0018 Rew!0.1551), and for the Sherwood number Sh = 1.230 [(D/z) ReSc]0.33 (1 + 0.010 Re!0.125 Sc1.055 Rew1.132) based on the predicted values of the solute concentration profiles, are proposed, in the operating condition ranges 300 < Re < 1000, 0.02 < Rew < 0.3 and 600 < Sc < 3200.

In this paper the solute equation has been mathematically expressed using the twodimensional convective diffusion equation in a cylindrical coordinate system and was solved using a finite difference scheme. The numerical model developed here successfully predicts the fundamental mechanisms involved in flux decline behaviour during crossflow filtration. The axial concentration profiles underline the important influence of the membrane length. It is apparent from the results presented that Reynolds wall number is a very important parameter that controls particle deposition onto a permeable surface of membranes. The Reynolds wall number influences the solute concentration profile and also controls the concentration polarisation along the tubular membrane. The polarisation concentration under a wide range of operating conditions can be analysed in terms of the concentration boundary layer thickness. A correlation for the concentration boundary layer thickness based on the predicted values of the solute concentration profiles was obtained: This equation is valid in the operating condition ranges of 300 < Re < 100, 0.02 < Rew <0.3, 600 < Sc < 0.3 and 0 < z/D < 100. This correlation shows that for higher Schmidt numbers, the increase of the Reynolds wall number leads to a weak concentration polarisation, and for a low Schmidt number, the increase of the Reynolds wall number does not allow for a decrease in the concentration polarisation. Additionally, for a given wall Reynolds number, higher values of both the Schmidt number and Reynolds axial numbers induce a decrease of the concentration polarisation. A new correlation was proposed for the Sherwood number to take into account the Reynolds wall number influences: This equation is valid in the operating condition ranges of 300 < Re < 1000, 0.02 < Rew < 0.3, 600 < Sc < 3200 and 5 < z/D < 100. These correlations can be used to calculate the value of the mass transfer coefficient from Sherwood relations.