Mathematical Framework for Combined Modeling of Liquid–Liquid Extraction and Phase Separation in Surfactant-Containing Systems

This study introduces a new, integrated mathematical framework that simultaneously addresses liquid–liquid extraction (LLX) and liquid–liquid phase separation (LLPS) in surfactant-containing systems using a population balance approach. Unlike conventional models that treat extraction and phase separation as distinct stages, the proposed framework unifies both processes to capture the dynamics of droplet interaction, coalescence, and breakage under surfactant influence. By incorporating population balance equations, the model explicitly accounts for how surfactants alter droplet size distributions and interfacial phenomena─both enhancing extraction through increased surface area for mass transfer and delaying separation by stabilizing emulsions and reducing coalescence rates. A comprehensive mixer–settler simulation is employed to quantify the trade-off between improved extraction efficiency and the resulting challenges in phase separation as a function of surfactant concentration. Model predictions, which closely match experimental observations, indicate that optimizing surfactant levels is essential for balancing these competing effects. This dual focus on extraction and phase separation represents a significant advancement in modeling surfactant-containing systems, offering a predictive tool for optimizing both droplet dynamics and overall process performance on an industrial scale.