Modified Bounded Homotopies in the Solving of Phase Stability Problems for Liquid–Liquid Phase-Splitting Calculations

Abstract

In this study, the modified bounded homotopies presented by Malinen and Tanskanen (Malinen, I.; Tanskanen, J. Modified bounded homotopies to enable a narrow bounding zone. Chem. Eng. Sci. 2008, 63, 3419) are investigated in order to solve phase stability analysis problems in liquidÀliquid equilibrium cases for phase-splitting calculations. The tangent-plane distance criterion is used to analyze the phase stability. The emphasis is on approaching the first root on the homotopy path. According to the observations, the bounding of the homotopy path with respect to the problem variables aids in the solving of a phase stability analysis problem. The main shortcomings of the modified bounded Newton, affine, and fixed-point homotopies are the starting point isolas, the existence of unfeasible solutions, and the convergence to only certain roots, respectively. The attraction domain of the global minimum was observed to be the largest with the fixed-point homotopy. Sequential usage of the fixed-point homotopy and starting points near pure components results in a robust algorithm for phase stability analysis and gives feasible initial estimates for the solving of phase splitting.

A Unifying Approach for the Calculation of Azeotropes and Pinch Points in Homogeneous and Heterogeneous Mixtures

“…However, any method for the determination of phase stability could be integrated, including global optimization methods for the determination of the minimum of Gibbs free energy. Refer to the articles of Kangas et al and Zhang et al for a review of global optimization approaches for phase equilibrium calculation. Similar to the application in optimization-based process design, the main reason for the implementation of the homotopy continuation approach is its computational efficiency, which is an operational necessity for the multitude of evaluations required for the pinch branch calculations.…”

Nonlinear Homotopic Continuation Methods: A Chemical Engineering Perspective Review

“…However, several authors have reported the failure of these criteria . As such, Kangas et al found that the fixed-point homotopy yields paths that converge to only certain solution vectors, which means that there are discontinuous homotopic paths in real domain. At this point, it is necessary to say that in other engineering fields Affine homotopy (Section 2.8.3) is placed in the same category of fixed-point homotopy .…”

“…Furthermore, the most novel approaches have been used in phase separation problems. Kangas et al used the bounded homotopy strategies of Malinen and Tanskanen to develop a robust method to calculate global minima. The general demeanor of the three classical homotopy forms was characterized based on their studies (Newton, FPN).…”

“…Kangas et al 71 identify the frequency of unfeasible roots for this homotopy and distinguish this situation from the isolas for the Newton form. However, no behavior that is distinct to isolas or infinity-opened paths has been reported as a cause of unfeasible roots.…”

Representación de los Equilibrios Líquidos Multifásicos de Sistemas Cuaternarios con los Modelos NRTL y UNIQUAC

“…Una excelente revisión sobre la aplicación de los métodos de convergencia local y de optimización global a la solución numérica del problema del equilibrio de fases isotérmico-isobárico, puede encontrarse en el trabajo presentado por Wakeham y Stateva (2004). En la última década, sólo algunos estudios han sido enfocados al cálculo de los equilibrios líquidos multifásicos con modelos de energía de Gibbs de exceso, entre los cuales se encuentran los reportados por Stateva et al (2000), García-Sánchez et al (2001), Solokhin et al (2002), Guo et al (2004), Denes et al (2006), Marcilla et al (2009) y Kangas et al (2011. Estos autores utilizaron diferentes estrategias numéricas para encontrar la solución del problema del flash isotérmico-isobárico de sistemas ternarios y/o cuaternarios que presentan múltiples fases líquidas.…”