A Componentwise Convex Splitting Scheme for Diffuse Interface Models with Van der Waals and Peng--Robinson Equations of State

Abstract

This paper presents a componentwise convex splitting scheme for numerical simulation of multicomponent two-phase fluid mixtures in a closed system at constant temperature, which is modeled by a diffuse interface model equipped with the Van der Waals and the Peng--Robinson equations of state (EoS). The Van der Waals EoS has a rigorous foundation in physics, while the Peng--Robinson EoS is more accurate for hydrocarbon mixtures. First, the phase field theory of thermodynamics and variational calculus are applied to a functional minimization problem of the total Helmholtz free energy. Mass conservation constraints are enforced through Lagrange multipliers. A system of chemical equilibrium equations is obtained which is a set of second-order elliptic equations with extremely strong nonlinear source terms. The steady state equations are transformed into a transient system as a numerical strategy on which the scheme is based. The proposed numerical algorithm avoids the indefiniteness of the Hessian matrix arisi...