403 Cu²⁺ (aq) + 2HR (org) ⇌ CuR₂ (org) + 2H⁺ (aq) (1) D = K [HR]n / [H⁺]m (2) Where: D = Distribution coefficient of copper between organic and aqueous phase (–) K = Equilibrium constant for the extraction reaction (–) [HR] = Concentration of undissociated extractant (mol/L) n = Stoichiometric coefficient of extractant participating (typically 2 for hydroxime-type extractants) [H+] = Hydrogen ion concentration in the aqueous phase (mol/L) m = Reaction order with respect to hydrogen ions Equation (1) shows that D depends on extractant concentration and acidity, with n and m reflecting their stoichiometric roles Stagewise extraction efficiency is derived from equilibrium stage theory via Eq. (3), with R denoting the organic-to-aqueous phase ratio (O/A). To account for non-ideal mixing and finite contact time, a first-order approach-to-equilibrium correction is applied using residence time τ and rate constant kτ, as in Eq. (4), consistent with reaction-engineering heuristics [8]. Acidity is dynamically updated after each stage based on stoichiometric proton release (2 H⁺ per Cu, 3 H⁺ per Fe) and reported as sulfuric-acid equivalent for plant interpretation [7]; the IUPAC definition of pH and sulfuric-acid dissociation behavior underpin the mapping between [H⁺] and H₂SO₄ equivalents [5,6]. E = (D · R) / (1 + D · R) (3) Where: 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
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