Adaptive Random Search: A Promising Method for Determining the Stability of Mixtures

Abstract

Mixtures stability analysis is a calculation procedure to determine whether a mixture will exist as a homogeneous phase or as multiple phases in equilibrium, for an established condition. The mathematical problems related to this analysis generally represent a great challenge with respect to the use of optimization methods, due to the occurrence of great number of local minima. The Adaptive Random Search (ARS) method is a stochastic global optimization method that has revealed superior to other stochastic methods such as genetic algorithm and the Particle Swarm Optimization. In the present work the hybridization of the ARS method with an algorithm for the solution of nonlinear equation systems is proposed, aiming to achieve the following characteristics: i) capability of covering the whole search space without being attracted to local minima, ii) small dependence on the initial guess, and iii) low computational cost. The methodology for stability analysis using hybrid ARS was implemented for three different equations of state - Soave-Redlich-Kwong (SRK), Peng-Robinson (PR) and PC-SAFT (Perturbed Chain – Statistical Associating Fluid Theory) - and tested on some mixtures for which data are available in the literature. The results of the original and hybrid versions of the ARS method were compared to those obtained with deterministic methods: a quasi-Newton method with BFGS approximation of the Hessian and with the DIRECT, which is another deterministic global search method that has been used in stability analysis problems. The hybrid ARS method allowed determining the correct thermodynamic condition (i.e., stable or unstable) of all tested mixtures. Additionally, in the cases where multiple local minima were present, this method has provided more than one solution in a single run, which is very useful for the solution of the phase equilibrium problem. In this sense, the composition values obtained by solving the stability analysis with the hybrid ARS were used to initialize the algorithm that solves Rachford-Rice equation (flash problem). Furthermore, the hybrid ARS method was able to validate all the solutions found when solving flash problem, for both stable and unstable mixtures.