405 After fusion, a feasibility-projection operator íD(. )ensures that the output adheres to process-level first principle and chemical constraints in eq [7-9]. Non-negativity of species concentrations for all the parameters in the input data and the system should satisfy the operational chemistry bounds (e.g., acidity, alkalinity, ionic balance): WT ≥0, ∀T (7) Where: WT = Decision variables (non-negative) ∀T = Constraint applies to all indexed variables pHmin ≤ pH(y) ≤ pHmax (8) where: pH(y) = Predicted pH from the hybrid model pHmin, pHmax = Allowed pH bounds • The projected hybrid output is therefore: NOP = íQ (N ℎ'K) (9) This NOPforms the calibrated digital twin, capturing both the structured predictability of first-principles models and the nonlinear, data-driven corrections required for real-plant fidelity. The digital twin supports high-confidence scenario analysis, parameter-sensitivity studies, and downstream optimization workflows. 3.3.5 Optimization Engine – Constraint-Aware Recommendation System The optimization engine utilizes the digital twin to compute optimal plant operating conditions under current process states such as ore grade, solvent change-over, agitation parameters, and hydrodynamic conditions. Internally, the engine queries the digital twin to estimate SX-recovery and impurity extraction outcomes for each candidate operating point. Optimization is performed under plant-design and operational constraints, ensuring recommendations remain feasible and aligned with equipment limitations, regulatory boundaries, and chemistry constraints already embedded in íD A weighted multi-objective formulation can be written as: ïTÅI ñ−21VQI(Ç)+22∑ Z * * (Ç)+23 ∅&I1)>' (Ç) ô (10) subject to: s*2H2D>1CH'>2(Ç)≤ s*,-5( sD 1 I 533(Ç)≤ sQ1I5,3>13572> 1õ-)$ (M)≤ 1õ(Ç,M)≤1õmax (M) Ç+,-)$ ≤Ç+ ≤Ç+,-5( NOP = íD(U(Ç)) Purity penalty term: ∅&I1)>' (Ç)= ∑ max( 0, * s*2H2D>1CH'>2(Ç)− s*,max) (11) Where: Ç = Decision/control variables 21, 22, 23 = Weight coefficients in the objective function VQI(Ç) = Copper recovery Z*(Ç) = Stage-wise current usage term ∅&I1)>' (Ç) = Purity penalty function NOP = Output of the digital twin F(u) = Physics model mapping from control inputs to process states íD (⋅) = ML correction or calibration operator
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