(262v) Modeling Phase Equilibria Using Numerical Algebraic Geometry

Accurate prediction of phase equilibrium is a fundamental aspect of applied chemical engineering thermodynamics. However, the resulting equations are often highly nonlinear and difficult to solve. Objective functions may present many local minima and trivial solutions, with multiphase and multicomponent systems being especially challenging. As a result, traditional optimization methods are sometimes wholly insufficient for solving these problems, or require unreasonably high quality initialization. Here, we demonstrate the use of homotopy continuation [and more general numerical algebraic techniques] as a powerful tool in solving these difficult problems. In particular, we outline a parameter free method for calculating phase equilibria from Helmholtz-explicit fundamental equations of state, requiring no use of auxiliary equations. Additionally, we show that polynomial homotopy continuation may be powerfully used to enumerate all existent critical points of multicomponent mixtures. These results serve to demonstrate the power of numerical algebraic geometry across a broad range of classical thermodynamic problems.