Chemical Engineering Journal 138 (2008) 307–332 Diffusion and reaction in three-phase systems: Average transport equations and jump boundary conditions E. Morales-Z´ arate a , F.J. Vald´ es-Parada a , Benoˆ ıt Goyeau b , J.A. Ochoa-Tapia a,∗ a Divisi´ on de Ciencias B ´ asicas e Ingenier´ ıa, Universidad Aut´ onoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186 Col. Vicentina, M´ exico 09340, D.F., Mexico b Laboratoire FAST, Universit´ e Pierre et Marie Curie, CNRS, Bˆ at 502, Campus Universitaire, F91405 Orsay, France Received 1 March 2007; received in revised form 26 May 2007; accepted 30 May 2007 Abstract 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. © 2007 Elsevier B.V. All rights reserved. Keywords: Volume averaging; Closure problem; Jump condition; Local equilibrium; One-equation model; Liquid membrane 1. Introduction This study deals with the analysis of diffusion and chemical reaction in a system composed by three phases (Fig. 1) where the external phase (γ -phase) contains dispersed drops called membrane phase (μ-phase), themselves containing small dispersed droplets (σ -phase). This system is similar to double emulsions which are used in many extraction processes such as hydrocarbons fractioning [1,2], recuperation of rare component ions [3], recovery of metals [4], purification of fatty esters [5], elimination of contaminants in aqueous streams [6], and the concentration of pharmaceuticals [7]. The so-called liquid surfactant membrane has been used for lactic acid extraction [8] and to explore enzymatic reactions [9]. Moreover, the study of gas dispersion and mass exchange between bubbles and emulsion phases, including interfacial mass transfer, (with and without chemical reaction) is essential in order to model mass transfer in fluidized beds [10,11]. In addition, a clear study of the rheology in double emulsion systems has been recently, performed by Pal [12]. These extraction processes involve the transport of a solute of interest (species A) from the external phase (γ -phase) to the droplets (σ -phase). The transport is based in the difference of solubility of the several phases and is increased by means of a reversible chemical reaction in the μ-phase while an irreversible reaction takes place in the σ -phase. This type of membrane separation represents a relatively new unit operation which, ultimately, is expected to replace a significant proportion of conventional separation processes [13]. Unlike classical process such as distillation, extraction, and crystallization, membrane separation generally does not involve phase transition and therefore requires lower energy consumption. Theoretical studies of diffusion and reaction in double emulsions have been carried out [1,3,14–24] but most of the works have been focused in the solution of the differential equations. Most of the models are based in intuitive considerations that could lead to rough approximations and inaccurate interpretation of experimental results. In their large majority, these models implicitly consider average equations where macroscopic quantities are not explicitly related to local values and therefore prediction of the effective ∗ Corresponding author. Tel.: +52 55 5804 4648; fax: +52 55 5804 4900. E-mail address: jaot@xanum.uam.mx (J.A. Ochoa-Tapia). 1385-8947/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cej.2007.05.054