Algebraic Geometric Method for Calculating Phase Equilibria from Fundamental Equations of State

Computing the saturation properties from highly accurate Helmholtz equations of state can be challenging for many reasons. The presence of multiple Maxwell loops often results in incorrect solutions to equations defining fluid-phase coexistence. Near the critical point, the same equations also become ill-conditioned. As a consequence, without highly accurate initial guesses, it is difficult to avoid the trivial solution. Here, we propose an algorithm applying the technique of Newton homotopy continuation to determine the coexistence curve for all vapor-liquid equilibrium conditions between the triple and critical points. Importantly, our algorithm is entirely convergence-parameter free, does not rely on the use of auxiliary equations, requires no initial guesses, and can be used arbitrarily close to the critical point. It is also fully generalizable to arbitrary equations of state, only requiring that they be locally analytic away from the critical point. We demonstrate that the method is capable of handling both technical and reference quality fundamental equations of state, is computationally inexpensive, and is useful in both evaluating individual state points and plotting entire fluid-phase envelopes.