A Rigorous Mathematical Proof of the Area Method for Phase Stability

Abstract

The paper introduces new developments of the original AREA method. A rigorous mathematical proof that the equilibrium points are the only ones which satisfy the maximum AREA criterion in the case of a two-component, two-phase system is given for the first time. A rigorous proof that the maximum AREA criterion is a necessary but not a sufficient condition for equilibrium in the case of an N-component, two-phase system is given also for the first time in the paper. Two test examples which reinforce the validity of the theoretical results obtained are presented and discussed.

A Web Service Infrastructure and its Application for Distributed Chemical Equilibrium Computation

“…The numerical method most often used to calculate an equilibrium distribution is based on minimizing a system's Gibbs free energy function with constraints. Several techniques for Gibbs energy minimization have appeared in the literature such as the tangent line/plane procedure suggested by Michelsen [1,2], the maximum area method developed by Eubank et al [3] and Elhassan et al [4,5], and the equal area method of Eubank and Hall [6], Shyu et al [7,8], and Hanif et al [9,10].…”

“…The minimum stationary point of (1) will be the vector of species molar values where vanishes. Differentiating (1), we obtain (3) From the isothermal, isobaric Gibbs-Duhem equation we know that (4) and so we seek the unique vector such that (5) where…”

“…is the number of moles of species and is the total number of moles in the equilibrium composition. Solving (5) amounts to solving a nonlinear constrained minimization problem. The traditional numerical method used for solving optimization problems [11] of this type is the method of Lagrange multipliers using an iterative Newton-Raphson technique for solving the resulting set of nonlinear equations.…”

Constructing Binary and Ternary Phase Diagrams on the Basis of Phase Stability Analysis

“…There have been many attempts documented in the literature to address the issues with the flash calculation described above. Typical recent examples include GLOPEQ, 3 the area method, 4 the homotopy-continuation algorithm, 5 the advanced flash algorithm by Lucia et al, 6 etc. These examples utilize the Gibbs energy as the equilibrium criterion.…”

Calculation of Liquid−Liquid Equilibrium Based on the Global Stability Analysis for Ternary Mixtures by Using a Novel Branch and Bound Algorithm: Application to UNIQUAC Equation

“…This has widely been used in all kinds of calculations of phase equilibria, especially in critical point region. Recently, this method has obtained many further developments. − Smith et al obtained the necessary and sufficient conditions for stability of the system where chemical reaction may or may not take place, using the Karush−Kuhn−Tucker optimality conditions as the basis of their consideration. Sun and Seider 2 used a two-stage method and employed a homotopy algorithm in an attempt to find all stationary points; however, this method could still fail for certain cases.…”