Our work presents algebraic models for liquid-liquid extraction, a process that enables the recovery of REEs while also purifying wastewater.3 The models are grounded in equilibrium thermodynamics and use distribution coefficients, extractant and complexing agent concentrations, and pH to simulate single- and multi-stage extraction systems, including cross-current and counter-current setups. Implemented in Python, the model is used to solve both determined and overdetermined systems, enabling the analysis of how different parameters affect solute recovery.
The accompanying plots show stage-wise recovery. While the cross-current scheme is more efficient for extraction because the solvent is refreshed at each stage, the counter-current scheme is more efficient in solvent usage. In case studies4 in which we implemented the models, Y and Er achieve ~100% recovery by stage three, while Yb shows lower overall extraction but uses the solvent more efficiently per unit recovery. Comparing the stages, after the third stage, the marginal gain might not justify the additional cost.
The framework has been validated against peer-reviewed experimental datasets, showing strong agreement with reported final concentrations. Extension bs of the approach might support more sustainable process design by aiming to maximize solute transfer efficiency while minimizing solvent and resource use. In addition to the current steady-state model, we aim to model the processes in a simulation software to compare results, and, in the future, to integrate dynamic optimization to support real-time process control. This algebraic model serves as a practical tool for REE, wastewater, and broader resource recovery applications in sustainable process systems engineering.
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