An Algorithm for Phase Determination in Multiphase Equilibrium

Efficient and robust methods for phase determination in multiphase-multicomponent equilibrium are key tools for chemical processes simulation. The most widely used methods for equilibrium calculations are generally based on Gibbs free energy minimization or on simultaneous solving of the material balance and equilibrium equations. Both techniques exhibit numerical challenges due to the involvement of highly nonlinear functions/equations and their efficiencies depend on initialization procedures since the number and type of phases present are not known a priori. In this work, an equation-solving algorithm is proposed for determination of the number and type of phases, and their corresponding fractions and compositions, in vapor-liquid equilibrium, for given temperature, pressure and overall composition. The algorithm follows a decision flowchart that, based on actual pressure and the system's phase change pressure values at fixed temperature and overall compositions, determines the number and type of equilibrium phases present and the corresponding set of equations describing material balances and phase equilibria. Assumption on a bound for the number of equilibrium phases only need to be done for the calculation of the bubble point pressure. A solution is first exhaustively searched for the case of two liquid phases; if no solution is found, then a single liquid phase is considered for bubble point calculation. The robustness of the algorithm is based on the facts that the two-phase (vapor-liquid and liquid-liquid) and three-phase (vapor-liquid-liquid) isothermal flash calculations are performed only when a physically meaningful solution is guaranteed to exist, and good initial guesses for their corresponding solutions are obtained from a previous calculation of a phase change pressure. The algorithm was implemented using the Patel-Teja Equation of State and the Wong-Sandler mixing rules for calculation of thermodynamic properties; and numerical solutions of the resulting nonlinear equation sets were obtained using homotopy-based methods. Applications of the algorithm to typical examples for two-phase and three-phase equilibrium calculations are presented along with comparisons against other calculation and initialization methods in the literature. Description of the algorithm The proposed algorithm follows the decision flowchart shown in Fig. 1., that based on actual pressure (P) and the system's phase change pressure values (dew pressure (Pd), bubble pressure (Pb), and secondary dew pressure (Pds)) at fixed temperature (T) and overall compositions (zi ∀i = 1,..., N), determines the number and type of equilibrium phases present and the corresponding set of equations describing material balances and phase equilibria. The main equations used by the algorithm are summarized below. The description of the notation is given in a separate section at the end. The fugacity coefficients can be calculated using the Patel-Teja Equation of State (Patel and Teja, 1982) and the Wong-Sandler mixing rules (Wong and Sandler, 1992). Fig 1. Algorithm’s flowchart.