Analysis of solute deposition on semipermeable walls in the process of ultrafiltration

In this study, the fluid flow and species transfer are mathematically analyzed through a free-flow domain, where the walls are porous membranes. Two different porous wall boundary conditions are assumed for the flow filtration process; one is based on the permeable membranes, and the other is assumed to be a semipermeable membrane, where cross-flow velocity is assumed to evaluate the effective flow rate and pressure loss. The governing equations are continuity and momentum equations (Navier–Stokes equation) incorporated into the suction/injection velocity of the membranes along with the slip velocity at the walls. The solute extraction through the porous wall depends on the porosity structure, and it is controlled by both the permeability and the permeate velocity. To study the flow transport through the membrane, both the Permeability-Based Model and the Permeate Velocity-Based Model are employed. The solute mass flux boundary condition is employed along the flat porous membranes instead of a specified concentration boundary condition. The fluid is assumed to be a dense particulate mixture, and the nonsource boundaries are considered to extract the solute particles for the filtration of the fluid, applicable in the ultra-filtration process. The domain and the rate of convergence for the flow momentum equation are obtained by suitably opting for the auxiliary parameter and compared with the existing literature. The study elaborates on the behavior of the axial velocity variation and the skin friction effect, which influence the sample deposition. It is observed that by decreasing the diffusion coefficient, slip coefficient, and axial velocity, the deposition of concentration increases. Further, the concentration of the deposited solute layer directly depends on the permeate velocity of the wall, which is responsible for the variation in the values of local diffusion coefficient and trans-membrane pressure. This paper also provides notable ideas of the nonlinear coupling effect of concentration with different magnitudes of effective conductivities and a range of molecular diffusion coefficients. The relative significance of the pore hindrance for the diffusive transport and solute diffusivity are estimated to predict the hydrodynamic interactions that occurred by the wall and the flow domain.