(25a) Volume-Based Phase Equilibrium Calculations

Phase equilibrium calculations at pressure and temperature specifications, consisting in the minimization of the Gibbs free energy with respect to mole numbers, are the most commonly used and there are well documented in the literature. A very attractive alternative is given by the volume-based calculations, in which volume and mole numbers, which are the natural variables for pressure explicit EoS are treated as independent variables; pressure equality is an additional equation and there is no need to solve the EoS for volume. Volume-based phase equilibrium problems are bound and linear equality constrained optimizations. A major disadvantage is that, unlike in conventional calculations, the successive substitution method cannot be used, thus robust modified Newton iterations or quasi-Newton methods are required. Volume-based calculations are also more sensitive to the quality of the initial guess and imperatively require stability testing for initialization. A robust and efficient volume-based phase equilibrium calculation protocol (used in both stability testing and phase splitting) is presented. A modified Cholesky factorization (to ensure a descent direction) and a two-stage line search procedure are used in Newton iterations. For phase stability analysis, several tangent plane distance (TPD) functions derived from the Helmholtz free energy (in terms of molar densities and of mole numbers and volume) are considered, including a new modified TPD function. It is shown that the choice of independent variables is very important for both robustness and computational speed in Newton and quasi-Newton methods. Proper scaling of the Hessian highly improves robustness. A new method for flash calculations at isochoric-isotherm conditions is presented, as an unconstrained minimization of the Helmholtz free energy. Formal links are established between various volume-based and conventional methods, giving indications on relative convergence speed and suggesting how to pass from one method to another (important for rapidly modifying existing codes). A new density-based method for phase envelope construction in the molar density-temperature plane is presented and its application commented for usual and several types of unusual shaped phase envelopes. The effect of capillary pressure on phase equilibrium is taken into account by using a recently derived molar density based TPD function; the additional partial derivatives in the gradient and Hessian (in phase stability) and in the Jacobian (in phase envelope construction) have very simple expressions, unlike in conventional formulations, due to the explicit in volume nature of the interfacial tension model. Slightly different results between the proposed and conventional approaches including capillary pressure are explained in detail. The proposed methods are tested on many examples from the literature, for mixtures with various degrees of complexity, with emphasis on the vicinities of highly difficult conditions (stability test limit locus and spinodal for stability and critical point for phase splitting). Calculations are performed using refined grids in temperature-pressure and molar density-temperature planes (up to high pressures and in wide temperature ranges), and proved to be highly robust and faster than previous methods, with a low average number of iterations required for convergence for all mixtures. Any pressure explicit EoS can be used, only pressure, fugacity and their partial derivatives (simpler than their conventional counterparts) with respect to mole numbers and volume are required. Since the resolution of the EoS is avoided, the volume-based approach is particularly attractive for complex thermodynamic models.