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 Mg²⁺ rejection, the relative errors were, respectively, 0.09% and 0.01% for Cl⁻ and Mg²⁺. 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.

Titanium (Ti) nanoparticles (NPs) were successfully seeded on the platform of a polyacrylonitrile (PAN) ultrafiltration (UF) membrane previously coated with bio-glue (a co-deposition of dopamine hydrochloric bicarbonate buffer having undergone pyrocatechol deprotonation). The tools in vogue, especially field emission scanning electron microscopy (FESEM), energy dispersive spectroscopy (EDS), and atomic force microscopy (AFM), have made it possible to fully characterize the structure of the new organic-inorganic nanofiltration (NF) membrane, namely NF_PAN_Ti. A soft computing model has been applied to make commonplace the complex and implicit extended Nernst–Planck equations that govern the transport of ions through NF membranes. Euler’s numerical method was applied with a small step-size and the results obtained were very interesting. The filtration velocity approach of GUEROUT-ELFORD-FERRY helped to estimate the average pore size of NF_PAN_Ti to r_p = 0.538 nm. A six-day test carried out on NF_PAN_Ti demonstrated its long-term stability and showed a steady-rejection rate of 89.3% of MgCl₂ salt and permeate flux of 56 L·m⁻²·h⁻¹. The Euler’ numerical method corroborated perfectly the experimental findings since the relative error was found to be very low at 0.33% for Cl⁻ and 0.09% for Mg²⁺ (RE << 0.1). These practical prediction tools may henceforth help in the choice and calibration of next-generation NF membranes’ synthesis.