Predicting nanofiltration performance: A Runge-Kutta numerical framework for ion-specific rejection and flux-driven selectivity inversion in single and mixed salts

This study systematically investigates the synergistic effects of operational parameters, membrane properties, and ionic characteristics on ion rejection in nanofiltration (NF). The primary objectives are to (1) model ion transport mechanisms via the extended Nernst-Planck equation, (2) quantify rejection for key ions (K⁺, Br⁻, Cl⁻, SO₄²⁻, PO₄³⁻) across variable fluxes (10⁻⁷–10⁻⁵ m³/m²/s) and pore radii (0.4–5 nm), and (3) elucidate competitive ion interactions in multicomponent systems. A novel numerical model integrating the extended Nernst-Planck equation with a fourth-order Runge-Kutta analysis was developed. The model was applied to both single (KBr, KCl, K₂SO₄, K₃PO₄) and mixed salt solutions to calculate ion concentrations and rejection rates under defined boundary conditions. The analysis yielded three critical advances: (i) Cation rejection (K⁺, up to 99%) consistently exceeded anion rejection in single salts due to charge interactions. (ii) In mixtures, Br⁻ achieved anomalously high rejection (80%) attributable to its optimal ionic radius and dynamic membrane charge effects. (iii) Competitive ion displacement induced negative rejection for Cl⁻ and PO₄³⁻, highlighting previously unreported flux-driven selectivity inversions. This work establishes a predictive link between membrane design (pore size), operational flux, and ion-specific properties (charge, radius) across different environments. The findings demonstrate that tailored membrane optimization must account for competitive ion transport, advancing the rational design of high-efficiency NF systems for selective ion removal.