An algorithm for three-phase equilibrium calculations

Abstract

A new algorithm is developed for performing liquid-liquid equilibrium, three phase bubble-point and three-phase equilibrium flash calculations. Material-balance and equilibrium equations are rearranged suitably and solved by the Newton-Raphson method. The method of Shah [1] is used for testing liquid stability and for generating initial values of the component moles required in the liquid-liquid equilibrium calculation. A procedure for producing initial values of the variables required in three-phase equilibrium flash is proposed. The partial derivatives of vapor-liquid K-factors required in the method are calculated using the procedure proposed by Wu and Bishnoi [2]. The reliability and efficiency of the proposed algorithm are illustrated by solving a number of example problems. The liquid solution properties are calculated using the UNIQUAC model and the vapor-phase properties are calculated using the virial equation of state. Scope—A relible and efficient algorithm to perform equilibrium calculations for coexisting three phases is required in the design and simulation of process in the chemical and petroleum industry. Heidemann [3], Gautam and Seider [4], Soares et al. [5] and Mehra [6] have developed calculation method based on the minimization of total Gibbs free energy with material-balance constrains. Alternatively, solution of material-balance and equilibrium equations using iterative schemes has been suggested by Fournier and Boston [7], Michelsen [8] and Nghiem and Heidemann [9]. Since the equilibrium and material-balance equations are highly nonlinear, the reliability and efficiency of a solution method depend on the arrangement of the equations and selection and the “good” initial values of iterative variables. The algorithm proposed in this paper combines, in a suitable manner, the material-balance and equilibrium equations and solves the resulting equations and the summation equations for component moles using the Newton-Raphson method. The partial derivatives of vapor-liquid K-factors required in the method are calculated by expressing the K-factor as aproduct of three functions to separate the effects of temperature, composition and volatility characteristics, as suggested by Wu and Bishnoi [2]. Schemes are presented to perform liquid-liquid equilibrium, three-phase bubble-point and three-phase equilibrium flash calculations. A phase is tested for thermodynamic stability before performing the phase equilibrium calculations. Heidemann [3], Gautam and Seider [4], Shah [1] and Michelsen [10] have discussed the problem of phase stability. The stability test of Shah [1], When applied to liquid mixtures, produces estimates of split factor and component mole fractions for the two liquid phases. Hence in the proposed algorithm the method of Shah [1] is used for the stability test of the liquid phase and for generating initial values of the component moles required in the liquid-liquid equilibrium calculation. The vapor-phase stability is tested by performing vapor-liquid isobaric-isothermal flash calculations. For three-phase calculations, a procedure is formulated to determine whether are present or absent and to generate good initial guesses for the Newton-Raphson method. The proposed algorithm is used to solve a wide variety of problems. The liquid solution properties are calculated using the UNIQUAC mode [11] and the vapor-phase properties are calculated using the virial equation of state truncated after the second virial coefficient. Conclusions and Significance—An algorithm is developed for liquid-liquid equilibrium and three-phase bubble-point and flash calculations. The algorithm solves the material-balance and equilibrium equations by the Newton-Raphson method and it is found to be reliable and efficient even when the liquid mixture is near its plait point or near the liquid-liquid equilibrium boundary curve. A procedure is formulated for generating initial values for the three-phase equilibrium calculations. The procedure determines whether phases are present or absent. This allows the Newton-Raphson method to be advantageously to solve the equations for known combinations of coexisting phases, with good initial guesses, and without the difficulties caused by disappearing phases. The algorithm and the procedure were found to be reliable and efficient in solving a wide variety of problems.