A generalized solution method for the quasi‐chemical local composition equations

Abstract

The work of Abusleme and Vera based on the Quasi-Chemical Theory of Guggenheim in terms of groups is extended to multicomponent multigroup mixtures. The nonlinear equations arising from the theory are rearranged in a particular way that allows them to be solved using a generalized Newton-Raphson method. Initial estimates of the values of the non-random factors are obtained with a Taylor expansion around the random values. No more than five iterations are required for typical systems. Practical applications of the method are presented for ternary group systems.Le travail d'Abuslhne et Vera, bast sur la Thtorie quasi chimique de Guggenheim exprimte en termes de groupes, est ttendu a des mtlanges de plusieurs composts comportant plusieurs groupes. Les tquations non lintaires qui proviennent de cette Thtorie sont rt-arrangtes d'une manibre particulihre, qui permet de les rksoudre en employant une mtthode de Newton-Raphson gtntraliste. On obtient les estimations initiales des valeurs des facteurs non altatoires au moyen d'un dtveloppement en strie de Taylor autour des valeurs altatoires. Pour des systhes typiques pas plus de cinq inttrations ne sont ntcessaires. On prtsente des applications pratiques de la mtthode pour les systhmes a trois groupes.ince Guggenheim published his works on the Quasi-*A preliminary version of this paper was presented at the 34th Canadian Chemical Engineering Conference, Quebec City, October 1984.

New expressions for non-randomness in equation-of-state models

“…Thus, this particular contact will make a contribution to the sum in Eq. (2) equal to w 11 e −βε 11 + w 22 e −βε 22 + w 12 e −βε 12 (3) If there are M such bonds in total in the system, Ω T is the combined probability for the simultaneous occurrence of these M events, or…”

“…(48) and by the corresponding equation of the quasi-chemical approach [1,3], which, for convenience, is reproduced here…”

“…This approximation permitted the development of expressions for the well-known local compositions, which were deemed more appropriate than the bulk compositions for the evaluation of the total internal energy E. These expressions are analytical for binary mixtures only. Iterative procedures must be used for solving the equations for multicomponent mixtures [2,3]. Yan et al [4] have recently proposed expressions for local compositions in binary mixtures, based on their Monte Carlo simulation results.…”

A free-volume modification of GEM-QC to correlate VLE and LLE in polymer solutions

“…The non-randomness factor for unlike molecules, C kl , has been related to non-randomness factor for like-molecules using an interaction parameter through the following equation [23]:…”

“…Although, the method proposed by Panayiotou and Vera [18,22] can be used to solve the equations derived from the quasi-chemical approach for ternary and quaternary systems, this method cannot be generalized for multicomponent systems. Abusleme and Vera [23] proposed a generalized method to obtain the non-randomness factors using a Newton-Raphson method. Recently, Wang and Vera [24] proposed a model, GEM-QC, based on the renormalization of the Guggenheim canonical partition function, which meets both limits of total order and total disorder, while the original Guggenheim quasi-chemical model only considered the total disorder limit.…”