Improved Two-Sided Tangent Plane Initialization and Two-Phase-Split Calculations

Abstract

The initialization procedure proposed by Michelsen for locating stationary points in the tangent plane distance function was applied to several cases considered as challenging in the literature concerning phase equilibrium calculations. For most of the systems studied in the literature, such a simple procedure is enough to detect the phase instability; thus it is unnecessary to resort to more robust, more elaborated, and more time-consuming initialization schemes. For the conditions and systems where problems were found, a simple procedure was proposed to extend Michelsen's original scheme with the aim of increasing the probability of obtaining a negative value of the tangent plane distance function. Two algorithms for phase-split calculations were used. Both algorithms use a relaxation parameter to increase their convergence regions. No difficulties were found to calculate the equilibrium distributions in contrast with several convergence problems reported, in which other approaches were applied to the same two-phase systems.

Application of stochastic algorithms for parameter estimation in the liquid–liquid phase equilibrium modeling

“…Comparison of results presented in Table 2 with those presented in the literature [2,53] allows us to conclude that the algorithm employed to phase equilibrium calculations with phase stability analysis is reliable, since very well agreement was found between two sources. Besides, several systems presented by Cañas-Marin et al [48] were tested and the results obtained were identical to those reported by the authors (not included in the paper for brevity). Table 3 presents the systems used in this work and the related literature source.…”

“…As pointed out by Michelsen [47], usually, one of these initial estimations will converge to the trivial solution and the other one to the desired minimum for a specification in the heterogeneous region. However, as presented by Cañas-Marin et al [48] this strategy can be failed in solving phase stability problems for certain systems and specific conditions. Thus, as proposed by Cañas-Marin et al [48], the following convex interpolative strategy is used in this work:…”

“…However, as presented by Cañas-Marin et al [48] this strategy can be failed in solving phase stability problems for certain systems and specific conditions. Thus, as proposed by Cañas-Marin et al [48], the following convex interpolative strategy is used in this work:…”

“…Variables y i,1 and y i,2 are obtained from Eqs. (14) and (15), where kp i is the partition coefficient used for initialization and can be estimated by the Wilson correlation [48]:…”

Multiphase equilibrium calculations using a reduction method

“…Phase stability analysis, consisting in the minimization of the tangent plane distance (TPD) function [3] plays a key role in the initialization of multiphase flashes. While for two-phase flash calculation the stability is tested at most twice (Michelsen's two-side initialization [2]), in a multiphase context multiple initial guesses are required for phase stability [2,4,5]. If conventional variables are used (the minimum is sought in the compositional space) a nonlinear system of nc  (np À 1) equations must be solved.…”