Calculation of Liquid−Liquid Equilibrium Based on the Global Stability Analysis for Ternary Mixtures by Using a Novel Branch and Bound Algorithm:  Application to UNIQUAC Equation

Abstract

In this paper, a novel branch and bound algorithm based on the idea of Horst (Math Program. 1976, 10, 312) was developed to minimize the tangent plane distance function (TPDF) of the stability analysis problem where nonideal liquid phases are described by the UNIQUAC activity coefficient equation. Some simple transformations were used to construct the DC (i.e., difference of two convex functions) programming of TPDF that does not rely on any newly introduced parameters except the inner ones of the UNIQUAC equation. A compact partition of the feasible space and a valid underestimation function of the TPDF depending on the linear characteristics of the stability analysis problem were developed to guarantee the new branch and bound algorithm obtaining the -global convergence to the global solution of the TPDF. In comparison with the algorithms presented by Sun and Seider (Fluid Phase Equilib. 1995, 103, 213) and Floudas (AlChE J. 1995, 41, 1798), the novel branch and bound algorithm is slightly slower; however, it can provide complete reliability for solving the phase stability analysis problem. Liquid-liquid equilibrium compositions were then calculated by the Newton-Raphson method (Fluid Phase Equilib. 1982, 9, 21) on the basis of the global minimum of TPDF. The preliminary calculation results for three ternary systems showed that the novel branch and bound algorithm can solve the global stability problem effectively.

Multilocal Programming and Applications

“…Naturally, a compromise must be made between efficiency and reliability, analyzing the probability of obtaining the global minimum within a reasonable computational effort. On the other hand, a branch and bound algorithm has been used with several thermodynamic models [59,65]. These authors claim that it can effectively solve the global stability problem, but only a few studies have been carried out.…”

Robust Dynamic Simulation of Three-Phase Reactive Batch Distillation Columns

“…However, such methods are known to fail in some cases because good initial values for the heterogeneous flash calculation cannot always be obtained . To overcome this shortcoming, global methods based on either global optimization (McDonald and Floudas, Yushan and Zhizhong) or interval methods (Tessier et al) exploit sufficient stability criteria, such as the tangent plane criterion of Michelsen . Although these global approaches guarantee the detection of any phase split, they have a high computational demand and require specialized model formulations and solvers.…”

A Combined Algorithm for Stability and Phase Equilibrium by Gibbs Free Energy Minimization

“…Other complex procedures have been introduced to solve for the global solution of the tangent function, including the differential geometry approach of Wasylkiewicz et al, 9 optimization with multiple initial guesses, 10 and the branch-and-bound algorithm of Yushan and Zhihong. 11 Gupta et al 12 proposed a method for the simultaneous determination of stability and chemical and phase equilibria. In this approach, a reference phase is introduced that is known to be present at equilibrium.…”

“…This approach proved to be successful in indicating the phase behavior of systems such as methane−hydrogen sulfide−carbon dioxide with the Peng−Robinson and Soave equation of states and of other mixtures including toluene−water−hydrogen with the NRTL and UNIQUAC activity coefficient models. Other complex procedures have been introduced to solve for the global solution of the tangent function, including the differential geometry approach of Wasylkiewicz et al, optimization with multiple initial guesses, and the branch-and-bound algorithm of Yushan and Zhihong …”