Liquid-Liquid Extraction

When designing liquid-liquid extraction columns, there are a variety of methods available to calculate the number of equilibrium stages needed. If it is assumed that the liquid carrier and liquid solvent are not mutually soluble, then a McCabe-Thiele style construction method can be used.

If a liquid solute is dissolved in a liquid carrier it can be removed by the addition of a liquid solvent. The equilibrium distribution (partition) coefficient of the solute in the two phases can be calculated from the solubility coefficients as, $$K_{D,A}=\exp\left(\frac{v_A(\delta_A-\delta_C)^2- v_A(\delta_A-\delta_S)^2}{RT}\right)$$ The equilibrium concentration of solute in the solvent, $y_A$, and the carrier, $x_A$, is then given by, $$y_A=K_{D,A}x_A$$ As with scrubbing/stripping the mass ratio of the solute can be used to allow the system to by solved graphically, $$X_A=\frac{x_A}{x_C}=\frac{x_A}{1-x_A}$$ $$Y_A=\frac{y_A}{y_S}=\frac{y_A}{1-y_A}$$ A mass balance over the column can be given by, $$CX_0+SY_{T+1}=CX_T+SY_1$$ $$C(X_0-X_T)=S(Y_1-Y_{T+1})$$ $$\frac{C}{S}=\frac{Y_1-Y_{T+1}}{X_0-X_T}$$ where $C$ is the carrier flow rate and $S$ is the solvent flow rate.

The graph below shows an example of the McCabe-Thiele style method for liquid-liquid extraction. The key parameters can be varied to help understand how they affect the design.