Calculation of ternary liquid‐liquid equilibrium data using arc‐length continuation

Liquid‐liquid extraction is an important separation technique used in several chemical engineering processes. When experimental data are unavailable, reliable prediction of liquid‐liquid equilibrium (LLE) data is essential for the optimal design of these processes. This paper describes a new methodology based on the application of the arc‐length continuation numerical technique in order to compute LLE data using non‐random two liquid or universal quasi‐chemical activity coefficient models. This technique is straightforward to implement in Mathematica©. In addition, the arc‐length method avoids the main disadvantage of root finding methods, such as Newton's method, for which the convergence often depends on the proper choice of an initial guess. Another advantage of the proposed approach, based on arc‐length continuation, is that all LLE data are generated in one single calculation. Several case studies, involving the three main types of ternary systems (ie, Type I, II and 0), illustrate the proposed method. For type I, we study the following three ternary systems: (1) water‐ethanol‐benzene, (2) 5‐hydroxymethylfurfural‐1‐butanol‐water, and (3) methanol‐benzene‐water. We have selected, for type II systems, the following mixtures: (1) isobutyl alcohol‐cyclohexane‐water, (2) methanol‐cyclohexane‐water, and (3) 4‐methyl‐2‐pentanone‐2‐butanol‐water. For type 0 systems, we have chosen a mixture composed of DMSO‐THF‐water. Finally, for each case, our predicted results are benchmarked against the corresponding experimental data. This paper proposes a new method for the prediction of Liquid‐Liquid Equilibrium data using arc‐length continuation. The method, implemented in MATHEMATICA©, is tested considering several case studies, involving the three main types of ternary systems (ie, Type I, II, and 0). For each case, the predicted results are benchmarked against the corresponding experimental data.