Dispersion in a slit with crossflow filtration through a porous wall

Crossflow filtration is a separation technique where fluid flows tangentially across a membrane or porous media, reducing clogging by sweeping away retained particles and allowing continuous filtration. Dispersion of solute matter in crossflow filtration plays an essential role in the separation performance of many industrial processes. The current work aims to generalize the Taylor dispersion theory and study the solute transport in a slit–porous medium system with a crossflow. The solute is depleted by the porous medium at the slit top porous wall or the interface between the slit and the porous medium, where the continuity of the solute concentration and mass flux is applied. The solute is simultaneously transported axially by the main flow along the slit and vertically by the crossflow perpendicular to the slit. Using the Reynolds decomposition and cross-sectional averaging techniques, the generalized reduced-order model for the advection-dispersion solute transport in the slit–porous medium system with the presence of a crossflow is established, where the effective velocity of the main flow and the effective dispersion coefficient are obtained. The analysis of the results reveals that the nondimensional Taylor dispersion coefficient scales with the Peclet numbers for the crossflow and the main flow, respectively, as DT∼Pev−5/3 (when Pev≥20) and DT∼Peu2. The proposed theoretical model, along with the findings of this study, paves the way for the fundamental study of dispersion in more complex systems.