Relating Critical and Limiting Fluxes to Metastable and Long-Term Stable Fluxes in Colloidal Membrane Filtration Through Collision-Attachment Theory

In membrane technology for water/wastewater treatment, the concepts of critical flux (JC) and limiting flux (JL) suggest the existence of a threshold flux below which no fouling occurs. However, their important roles on stable flux duration have not been sufficiently understood. This work adopts a collision-attachment approach to clarify the relationship of JC, JL to metastable (i.e., short-term stable) and long-term stable fluxes based on their dependence on initial flux (J0), foulant-clean-membrane energy barrier (Ef-m), and foulant-fouled-membrane energy barrier (Ef-f). When J0 is below JL, water flux remains stable over a long time even for the case of J0 over JC, thanks to the strongly repulsive Ef-f. At J0 > JL and J0 > JC, the water flux is unstable at the beginning of filtration, and the flux ultimately decreases to JL as the long-term stable flux. Under the condition of JL < J0 ≤ JC, an initial metastable flux appears owing to the high Ef-m, with longer metastable period observed at lower J0 and for more hydrophilic/charged membrane or colloids. Nevertheless, rapid flux decline occurs subsequently due to the energy barrier shifting to weak Ef-f, and the water flux eventually degenerates to JL in long-term fouling duration. Our results provide significant guidelines for fouling control strategies with respect to membrane design, feedwater pretreatment, and operational optimization.