Enhanced Parameter Estimation for Liquid-Liquid Equilibrium Using Homotopy Continuation as Stability Constraints

Parameter estimation for the activity models such as NRTL and UNIQUAC for liquid-liquid equilibrium (LLE) is a nontrivial task, and this is due to the complex nature of these models and the need to uniquely fit the binary interaction parameters that predict phase behaviour and compositions similar to the experimental ones while ensuring satisfaction of the stability criterion and other conditions. To achieve this task, there is a need for a robust formulation that will capture all the necessary constraints and a good algorithm that will proffer the right solution. This study formulated an LLE problem by introducing the homotopy continuation to create discretised points for tracking the tie-lines to ensure phase stability. Other constraints were also included in this formulation to avoid fictitious phase prediction. The algorithm was applied to several LLE problems, including binary, ternary, and quaternary, for isothermal and non-isothermal LLE systems. The algorithm tackled all the problems, and the estimated parameters predicted stable phases with existent behaviour and compositions with very small discrepancies from the measured compositions. Though the constraint may increase the computational effort, it was beneficial, and the formulation also avoids the back and forth of a posteriori check.