E = Extraction efficiency (–) D = Distribution coefficient (–) R = Organic-to-aqueous flow ratio (O/A) (–) F = 1 - e (−kτ · τ), τ = Vmixer / Combined flow (4) Where: F = Fraction extracted in the mixer (–) kτ = Mass-transfer rate constant (s⁻¹) τ = Mixer residence time (s) Vmixer= Mixer volume (m³) Combined flow = Total aqueous + organic inflow (m³/s) Equations (3)– (4) relate extraction efficiency to phase ratio and mass-transfer kinetics The solvent‑extraction framework represents the process as a sequence of extraction, phase‑settling, and stripping steps that collectively transfer copper from the aqueous feed to the loaded organic and subsequently regenerate the extractant to produce a copper‑rich electrolyte [1,4]. Material balances are solved stage‑by‑stage across this sequence, allowing estimation of aqueous and organic compositions at the outlet of each extraction and stripping step. This approach yields raffinate conditions after extraction, organic loading ahead of stripping, and the final electrolyte composition after regeneration. Model outputs include the evolution of Cu and Fe concentrations in both phases, pH profile, sulfuric‑acid balance expressed as H₂SO₄ equivalent, as well as extraction and stripping efficiencies and overall copper recovery. These results provide a mechanistic basis for calibration, sensitivity assessments, and process optimization across different circuit configurations. While the formulation captures the essential first principle governing metal–extractant equilibria and acid–base interactions in SX systems, certain simplifying assumptions are used. These include fixed stripping performance and heuristic adjustments to represent mass‑transfer limitations. Enhancing the model with equilibrium isotherms, rate‑based kinetic descriptions, and detailed mass transfer formulations would further strengthen its predictive Capability and applicability to more complex circuits [1,4]. 3.3 Hybrid Approach — Training Workflow (Hybrid Twin Calibration) 3.3.1 Inputs and baseline first principle simulation. Plant inputs (PLS Composition, O/A by stage, flow rates, temperatures, mixer volumes, and extractant concentration etc.) are provided to the first principle -based SX model. Stagewise predictions—raffinate Cu/Fe, organic loading, pH/H₂SO₄ profile, and strip outputs—are computed using Equations (1)–(4) (chelation equilibria, distribution ratios, single-stage extraction fraction, and approach-to-equilibrium corrections). 3.3.2 Residual Model – Learning Systematic Mismatch: The residual model ( ) designed to learn the structured mismatch Δy between plant measurements and first principle -based model outputs. The model is provided with a comprehensive feature vector X, which includes plant operating variables, selected outputs from the first principle solver (predicted pH, flows, PLS compositions), and contextual metadata such
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