Calculation of phase equilibria for multi-component mixtures using highly accurate Helmholtz energy equations of state

To test the thermodynamic stability and to determine the equilibrium phase compositions in case the original phase is found unstable is one of the greatest challenges associated with calculating thermodynamic properties of multi-component mixtures. The minimization of the tangent plane distance function is a widely used method to check for stability, while different approaches can be chosen to minimize the Gibbs energy in order to find the phase equilibrium. While these two problems have been applied to several different thermodynamic models, very little work has been published on such algorithms using multi-parameter Helmholtz energy equations of state. In this work, combined stability and flash calculation algorithms at given pressure and temperature (p,T), pressure and enthalpy (p,h), and pressure and entropy (p,s) are presented. The algorithms by Michelsen et al. (1982, 1982, 1987) are used as basis and are adapted to multi-parameter Helmholtz energy models. In addition, a robust and sophisticated density solver is proposed which is necessary for the calculation of properties from the Helmholtz energy model at given state variables other than temperature and density. All partial derivatives necessary to solve the isothermal, isenthalpic and isentropic flash problems using numerical methods based on the Jacobian matrix are derived analytically and given in the supplementary material to this article. Results for some multi-component systems using the GERG-2008 model (Kunz and Wagner, 2012) are shown and discussed.