Calculation of chemical and phase equilibrium based on stability analysis by QBB algorithm: application to NRTL equation

Abstract

The Gibbs tangent plane criterion is of crucial importance to confirm the reliability of the solution obtained for the chemical and phase equilibrium (CPE) problem, and it consequently facilitates the search for the true equilibrium state if a postulated solution is thermodynamically unstable. However, the nonconvex and nonlinear natures of the thermodynamic models, which are the necessary conditions in order to describe the chemical and phase equilibrium problems, make the application of the deterministic global optimization techniques to minimize the tangent plane distance function (TPDF) become very difficult. In this paper, a general quadratic underestimation function based branch and bound (QBB) algorithm is developed by the construction of a rigorous underestimator for TPDF, which is the sum of the original convex part in the TPDF and a quadratic underestimation function of the generic nonconvex part upon the basis of the maximal eigenvalue estimation of its interval Hessian matrix. The linear constraints, especially the simplex feasible region, provide the novel compact partition in the above framework and mathematically guarantee the ε-global convergence of this global optimization algorithm together with the rigorous underestimator obtained by interval analysis. The phase equilibrium compositions are then calculated by the Newton–Raphson method. The preliminary results for three ternary systems with up to two or three phases described by NRTL activity coefficient equation showed that the novel QBB algorithm could solve the global stability problem effectively.