Upscaling solute concentration transport equations of countercurrent dialyzer systems

The present paper aims at an upscaled description of solute concentration transport processes in countercurrent dialyzer systems by means of a porous media approach based on the volume averaging theory. In consideration of solute diffusion and ultrafiltration processes, a general set of upscaled solute concentration transport equations has been derived for the three phases in a hollow fiber dialyzer, namely, the blood, membrane and dialysate phases. Moreover, the corresponding closure problems, which can be solved to obtain the effective properties in the upscaled model, are proposed. For the one-dimensional case, analytical expressions of solute concentrations in both blood and dialysate phases are achieved, so as to elucidate the influences of ultrafiltration flow rate and distribution coefficient on both clearance enhancement and solute concentration variations along the length of dialyzer. The results show that both ultrafiltration flow rate and distribution coefficient have no significant effects on solute concentration variations along the length of dialyzer. Nevertheless, the clearance enhancement are found to have linear relationship with both ultrafiltration flow rate and distribution coefficient.