Countercurrent solvent extraction is solved stage by stage with the Kremser equation, exactly like the tray-count method used for distillation but applied to a liquid-liquid distribution coefficient D instead of a vapour-liquid K-value. For a linear equilibrium (organic-phase concentration = D · aqueous-phase concentration, valid at the dilute Sc loadings here) and N ideal countercurrent stages:
E = D · (O/A) extraction factor
Recovery = (E^(N+1) − E) / (E^(N+1) − 1) Kremser equation
x_N = C_feed · (1 − Recovery) raffinate concentration
D itself is chemistry, not a free parameter: the cation-exchange extractant used for Sc (D2EHPA-type) releases H⁺ as it binds Sc³⁺, so rising free acidity pushes the equilibrium back toward the aqueous phase — D falls roughly as Kd/(Kd+acidity). Selectivity sets how much more strongly the extractant favours Sc³⁺ over the co-leached Fe³⁺/Al³⁺, which is run through the same Kremser formula with its own, smaller D to get the unwanted impurity carryover.
- Staircase — the classic McCabe-Thiele construction: horizontal steps hit the equilibrium line y=D·x, vertical steps hit the operating line (mass balance, slope 1/(O/A)); each full step is one theoretical stage, and the count matches the stage slider.
- Stage profile — the aqueous-phase Sc concentration measured after each stage, falling from the feed value toward the raffinate value x_N computed above.
- Strip — a single back-extraction stage with fresh dilute acid; more strongly loaded organic (higher D) is intrinsically harder to strip, modelled as a falling strip distribution ratio.