Runge–Kutta Numerical Method Followed by Richardson’s Extrapolation for Efficient Ion Rejection Reassessment of a Novel Defect-Free Synthesized Nanofiltration Membrane

A defect-free, loose, and strong layer consisting of zirconium (Zr) nanoparticles (NPs) has been successfully established on a polyacrylonitrile (PAN) ultrafiltration substrate by an in-situ formation process. The resulting organic–inorganic nanofiltration (NF) membrane, NF-PANZr, has been accurately characterized not only with regard to its properties but also its structure by the atomic force microscopy, field emission scanning electron microscopy, and energy dispersive spectroscopy. A sophisticated computing model consisting of the Runge–Kutta method followed by Richardson extrapolation was applied in this investigation to solve the extended Nernst–Planck equations, which govern the solute particles’ transport across the active layer of NF-PANZr. A smart, adaptive step-size routine is chosen for this simple and robust method, also known as RK4 (fourth-order Runge–Kutta). The NF-PANZr membrane was less performant toward monovalent ions, and its rejection rate for multivalent ions reached 99.3%. The water flux of the NF-PANZr membrane was as high as 58 L · m(−2) · h(−1). Richardson’s extrapolation was then used to get a better approximation of Cl(−) and Mg(2+) rejection, the relative errors were, respectively, 0.09% and 0.01% for Cl(−) and Mg(2+). While waiting for the rise and expansion of machine learning in the prediction of rejection performance, we strongly recommend the development of better NF models and further validation of existing ones.