(694i) Upscaling Multi-Components Two-Phase Flow Dispersion with Partitioning Coefficient

Mass transfer and fluid flow with liquid-vapour phase exchange in porous media appears in a large number of situation such as nuclear safety devices, absorption processes, transport in petroleum reservoirs, aquifer contamination, desalination processes by way of distillation...

This paper comes in the continuity of the works of Quintard and Whitaker (1994) [1] and of Coutelieris and al. (2006) [2]. In the first paper, a macroscopic model describing a binary mixture is obtained from the pore-scale equations using a volume averaging upscaling method. This theory is extended in the second paper to multicomponent mixtures with a partitioning equilibrium condition at the interface between an immobile non-aqueous liquid phase and a flowing aqueous phase. In this paper, we extend these theoretical results to the case of a two-phase, multicomponent system. The two phases are mobile. In addition, the upscaling methodology is improved with respect to the interface movement.

The pore-scale problem under consideration corresponds to mass transfer of a component A in a two-phase flow system. The liquid (denoted β-phase) and its vapour (γ-phase) flow through a porous medium (σ-phase). It is assumed that the chemical potential equilibrium at the interface is linearised to a partitioning relationship (Henry's law). We have for component A,

The volume averaging method is applied to the microscopic equations written above in order to develop a macro-scale model for homogeneous porous media. We consider local mass non-equilibrium, i.e., averaged concentrations are not necessarily linked by the equilibrium interface relationship Eq.4. The resulting macroscopic equations involve dispersion tensors, additional convective transport terms and a mass exchange term that are influenced by the mass transfer process. All these effective coefficients can entirely be determined by three closure problems solved at the unit-cell scale. A closed form of the total mass transfer rate is also proposed. We obtain the following two equations which are linked by a mass transfer coefficient

In these equations, ε_i denotes the volume fraction of the i-phase and the intrinsic volume average of a Φ_i variable is defined as

where V_i is the volume occupied by the i-phase, and in order to clarify the notation, the average quantities are denoted as Ω_iA=iA>ⁱ and V_i=v_i>ⁱ.

The closed form of the mass exchange term of the A-component depends on the effective variables u_γβ, u_{β γ} and α which are calculated from the closure problems

and we can calculate the total mass transfer rate by

To validate the approach, we consider the case of the falling film along a vertical fixed wall. In this case, the closure problems can be solved analytically. The obtained effective dispersion coefficients are in good agreement with the falling film theory results and the mass exchange coefficient corresponds to the Lewis and Whitman double film theory. Moreover, we compared with success the macro-scale model with a direct numerical simulation (see Fig.1) of the pore-scale model.

References

[1] Michel Quintard and Stephen Whitaker. Convection, dispersion, and interfacial transport of contaminants: Homogeneous porous media. Advances in Water Resources, 17(4):221-239, 1994.

[2] F.A. Coutelieris, M.E. Kainourgiakis, A.K. Stubos, E.S. Kikkinides, and Y.C. Yortsos. Multiphase mass transport with partitioning and inter-phase transport in porous media. Chemical Engineering Science, 61(14):4650-4661, 2006.

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