A combined optimization algorithm for stability, phase, and chemical equilibria

This work is concerned with the simultaneous determination of stability and chemical and phase equilibrium at a specified temperature, pressure, and overall composition by minimizing the Gibbs free energy of the system. An algorithm has been developed which combines the two major classes of the phase separation procedure, the stability analysis and the calculation of equilibrium phase compositions. Numerical results on a variety of test problems show that by a reformulation of the Gibbs free energy function based on the selection of a major reference phase, a correct equilibrium solution is specified when excess phases exist. This procedure is utilized to indicate the potential for the formation of new phases. In addition, the use of the Lagrange multipliers of the test phases as variables serves to identify accurately the phase boundaries and to indicate possible spurious solutions. If a phase is found to be absent at equilibrium, the magnitude of its associated Lagrange multiplier indicates the potential for the formation of this phase at alternate temperatures or pressures. A nonlinear programming procedure is utilized in this study to provide the Lagrange multipliers simultaneously with the solutions of phase compositions. This is shown to be an effective procedure to calculate the phase stability variables.;The algorithm has been extended to combined chemical and phase equilibrium. Theory and numerical results show that a correct chemical equilibrium solution is obtained. The Lagrange multiplier for the test phase is utilized to indicate the presence and absence of the phase for the chemical and phase equilibrium problem. This study proposes a single unified optimization formulation, with the Gibbs free energy function and the corresponding constraints expressed in terms of the extents of the independent reactions, which automatically enables the solution of phase equilibrium conditions with or without chemical reactions. The algorithm has been applied to a number of non-ideal multiphase systems involving vapor-liquid-liquid equilibrium and complex phase equilibria, including the methanol synthesis process. Several new features to facilitate the application of this approach have been implemented in the algorithms of this study, including the use of a powerful variable metric optimization algorithm, the introduction of reference and test phases, new procedures for the correct phase identification of roots of the equation of state, non-ideality tests to minimize unnecessary calculations, and tests to carefully establish proper input variables for the test phases.