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The Isothermal Flash Problem: New Methods for Phase Split Calculations summarises two main approaches at the time, minimisation and gradient descent. The maths is presented clearly, and the results are clear for the cases used.
The Isothermal Flash Problem. Part 1. Stability a method of deciding if a phase is stable. If the single phase system is stable, then no iterations are required, if it’s unstable then you still obtain an accurate initial estimate for initialising iterations
The Isothermal Flash Problem. Part 2. Phase-Split Calculation a flash algorithm involving Broyden acceleration steps whilst far from the critical point, and Newton steps as you approach. I don’t recommend working from this paper, later papers detail methods better.
The Negative Flash allowing negative vapour fractions in when solving the Rachford-Rice relation in the inner loop of a flash problem. It also shows that this corresponds to a saddle point in the Gibbs energy surface.
Mixing and Combining Rules theoretical boundary conditions, the Huron-Vidal Model and the Wong-Sandler Model
Original Wong-Sandler Mixing Rule Paper a composition dependent mixing rule
Reformulation of Wong-Sandler Mixing Rule the Wong-Sandler mixing rule to improve smoothness of transition between boundary conditions