Diffusion and reaction in three-phase systems: Average transport equations and jump boundary conditions

A macroscopic modeling of diffusion and chemical reaction in double emulsion systems using the method of volume-averaging is presented. In this three-phase system, chemical reaction takes place in the drops and membrane phases (ω-region) while passive diffusion is considered in the continuous external phase (η-region). First, a generalized one-equation model, free of the usual length scale constraints, is derived in order to describe the solute transfer in both homogeneous regions and in the ω–η inter-region. The up-scaling in the ω-region is based in the local mass equilibrium assumption between the two phases. Equations in both homogeneous regions are deduced from the generalized one-equation model. Then, the jump boundary condition at the dividing surface is derived and associated closure problems are established in order to calculate the jump coefficients.