Thermodynamically consistent modeling and simulation of multi-component two-phase flow with partial miscibility

Abstract

A general diffuse interface model with a realistic equation of state (e.g. Peng-Robinson equation of state) is proposed to describe the multi-component two-phase fluid flow based on the principles of the NVT-based framework which is a latest alternative over the NPT-based framework to model the realistic fluids. The proposed model uses the Helmholtz free energy rather than Gibbs free energy in the NPT-based framework. Different from the classical routines, we combine the first law of thermodynamics and related thermodynamical relations to derive the entropy balance equation, and then we derive a transport equation of the Helmholtz free energy density. Furthermore, by using the second law of thermodynamics, we derive a set of unified equations for both interfaces and bulk phases that can describe the partial miscibility of two fluids. A relation between the pressure gradient and chemical potential gradients is established, and this relation leads to a new formulation of the momentum balance equation, which demonstrates that chemical potential gradients become the primary driving force of fluid motion. Moreover, we prove that the proposed model satisfies the total (free) energy dissipation with time. For numerical simulation of the proposed model, the key difficulties result from the strong nonlinearity of Helmholtz free energy density and tight coupling relations between molar densities and velocity. To resolve these problems, we propose a novel convex-concave splitting of Helmholtz free energy density and deal well with the coupling relations between molar densities and velocity through very careful physical observations with a mathematical rigor. We prove that the proposed numerical scheme can preserve the discrete (free) energy dissipation. Numerical tests are carried out to verify the effectiveness of the proposed method. specification of pressure, temperature, and mole fractions does not always determine the equilibrium state of the system uniquely. In addition, in compositional fluid simulation, the pressure is not known a-priori and there exists no intrinsic equation for the pressure. This leads to the complication of constructing the pressure evolution equation for application of the NPT-based framework [25].In order to resolve the issues of the NPT-based framework, an alternative framework [19] has been developed, and it uses the moles, volume, and temperature (the so-called NVT-based framework) as the primal state variables. The NVT-based framework is initially applied to the phase-split computations of multi-component fluids at the constant moles, volume and temperature (also called NVT flash computation) in [19], and for the subsequent research progress of the NVT flash computation, we refer to [10,11,15,20]. Recently, the NVT-based framework has been successfully applied in the numerical simulation of compositional two-phase flow in porous media [25]. Very recently, in [16], we have extended the NVT-based framework to the pore scale and developed a diffuse interface model to simulate multi-component t...

A Novel Energy Factorization Approach for the Diffuse-Interface Model with Peng--Robinson Equation of State

“…On the basis of the thermodynamic fundamental laws and realistic equations of state (e.g. PR-EoS), general diffuse interface models for compressible multi-component two-phase flows have been proposed in [16] for isothermal fluids and [17,18] for non-isothermal fluids.…”

Review on Dynamic Van der Waals Theory in two-phase flow

“…For the numerical simulation of the proposed model, the key difficulty is due to the strong nonlinearity of the Helmholtz free energy density and the tight coupling between the molar density and the velocity. In order to solve these problems, they proposed a novel convex and concave splitting of Helmholtz free energy density and deal well with the coupling relationship between molar density and velocity by very careful physical observation Kou and Sun, 2016a).…”

“…The framework begins by assuming that the free energy function of the square gradient term is included in the order parameter (for example, the population density and the component concentration field) Kou and Sun, 2016a). From this function, the spatial distribution of these order parameters in the interface area is generated through an energy minimization theorem.…”

Stabilized Energy Factorization Approach for Allen–Cahn Equation with Logarithmic Flory–Huggins Potential

“…We observe that the stabilization approach is effective for the double well potential, but it works not well for the logarithmic potential probably because of more complicate nonlinearity of it. In [28,32], a nonlinear stabilization approach has been proposed for the Peng-Robinson equation of state (PR-EOS) [39] that is one of the most useful tools in petroleum industry and chemical engineering. For the logarithmic Flory-Huggins potential, the instability will occur and numerical results may be out of normal range when the energy parameter takes a large value.…”

“…For the logarithmic Flory-Huggins potential, the instability will occur and numerical results may be out of normal range when the energy parameter takes a large value. Inspired by the approach in [28,32], we will incorporate the nonlinear stabilization term for the logarithmic Flory-Huggins potential to ensure the symmetric positive definiteness and discrete maximum principle of the resultant scheme in the case with a large energy parameter.…”