where: pH(y) = Predicted pH from the hybrid model pHmin, pHmax = Allowed pH bounds • The projected hybrid output is therefore: = ( ℎ ) (9) This forms 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 A weighted multi-objective formulation can be written as: [− 1 ( ) + 2 ∑ ( ) + 3 ∅ ( ) ] (10) subject to: ( )≤ , ( )≤ , ( )≤ ( , )≤ max ( ) , ≤ ≤ , = ( ( )) Purity penalty term: ∅ ( ) = ∑ max( 0, ( )− ,max) (11) Where: = Decision/control variables 1, 2, 3 = Weight coefficients in the objective function ( ) = Copper recovery ( ) = Stage-wise current usage term ∅ ( ) = Purity penalty function = Output of the digital twin F(u) = Physics model mapping from control inputs to process states (⋅) = ML correction or calibration operator The optimizer produces recommendations that maximize Cu recovery, minimize impurity transfer maintain electrolyte purity, stabilize raffinate Cu and pH trajectories, operate strictly
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