3.1 Digital Twin The digital twin for the solvent extraction (SX) circuit is implemented as a hybrid model that integrates a first principle-based core (Section 3.2) with a data-driven residual learner and a compensation layer. Two complementary workflows operationalize this hybrid twin. The Hybrid model enables interpretable, constraint-consistent, and plant-specific predictions that support operator decision-making. 3.2 First principle -based modelling for SX The SX model is grounded in first-principles chemistry and stagewise mass balances to capture mechanistic copper transfer and acidity evolution across extraction and stripping. The governing equilibrium for copper chelation is expressed as Eq. (1), where HR denotes the hydroxyoxime extractant species; the reaction dictates both metal partitioning and proton release— two H⁺ per mole of Cu—leading to progressive acidity increase and known impacts on selectivity and kinetics [1,2,7]. Iron (III) is modeled analogously with higher proton stoichiometry, influencing Fe co-loading under varying pH [2]. The distribution ratio D is computed using Eq. (2), with exponents (Cu: n=2, m=2; Fe: n=3, m=3) consistent with chelation stoichiometry and slope analysis [2,3]. 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:
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