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Abstract We derive a class of exact solutions for Stokes flow in infinite and semi‐infinite channel geometries with permeable walls. These simple, explicit, series expressions for both pressure and Stokes flow are valid for all permeability values. At the channel walls, we impose a no‐slip condition for the tangential fluid velocity and a condition based on Darcy's law for the normal fluid velocity. Fluid flow across the channel boundaries is driven by the pressure drop between the channel interior and exterior; we assume the exterior pressure to be constant. We show how the ground state is an exact solution in the infinite channel case. For the semi‐infinite channel domain, the ground‐state solutions approximate well the full exact solution in the bulk and we derive a method to improve their accuracy at the transverse wall. This study is motivated by the need to quantitatively understand the detailed fluid dynamics applicable in a variety of engineering applications including membrane‐based water purification, heat and mass transfer, and fuel cells. 

Short-term microfiltration (MF) fouling is commonly abated by periodically reversing the flow to remove foulants that weakly adhered to the membrane. Strong oxidants (i.e., chlorine) can be added to hydraulic backwash water to augment its efficacy—a process called chemically enhanced backwashing (CEB). Herein, we report a rigorous mathematical model for constant flux MF incorporating hydraulic backwashing and CEB, and validate it with laboratory data obtained using untreated and alum-coagulated water from the Foss Reservoir in Oklahoma, USA. We implemented an optimal control procedure and used it to predict MF behavior long past experimental timescales. We identified a frequency threshold beyond which the necessary transmembrane pressure (TMP) reached an asymptotic value, indicating a pseudo steady-state, periodic solution to the model when coupling hydraulic backwashing with CEB. We report differences in TMP saturation values and timescales by simulating transient MF of untreated and pretreated water. Numerical simulations revealed that the operating flux could be increased 10-fold after pretreatment (compared with raw water) before reaching the maximum manufacturer-recommended pressure for the hollow-fibers. The predicted higher flux and extended duration between cleaning-in-place demonstrated advantages of coagulation pretreatment under hydraulic backwashing and CEB. Model observations could guide decision making for CEB timing and frequency. 

One tool in efforts to tackle the ever growing problem of water scarcity is municipal wastewater reclamation to produce drinking water. Microfiltration (MF) is a central technology for potable reuse because it is highly effective in removing pathogenic protozoa, bacteria, and other colloids and for reverse osmosis pretreatment. However, as microfiltered materials accumulate at the membrane surface, its productivity is reduced requiring periodic removal of foulants. A mathematical model of MF is described in the context of hollow fiber filtration that focused on optimizing constant flux operation with backwashing. Design curves were also proposed for determining backwash timing. The model analysis is evaluated against real-world MF fouling for membranes that range in age from a few weeks to three years, observed at the world’s largest water reuse facility operated by the Orange County Water District. The presented model compares well with the full-scale operational data, and model parameters accurately capture variations in fouling kinetics with membrane age, providing clues to changes in optimal regeneration timing and frequency as membrane performance declines over long time scales.