A local thermal non-equilibrium model for two-phase flows with phase-change in porous media F. Duval a , F. Fichot a, * , M. Quintard b a D epartement de Recherches en S ecurit e, Institut de RadioProtection et de S^ uret e Nucl eaire (IRSN), B.P. 3, 13115 Saint Paul Lez Durance, France b Institut de M ecanique des Fluides de Toulouse (IMFT), Avenue du Professeur Camille Soula, 31400 Toulouse, France Received 10 October 2002 Abstract The assumption of local thermal equilibrium for describing macroscopic heat transfer in a porous medium subjected to a liquid–vapor flow with phase change has been often investigated. Under certain circumstances, this assumption appears to be too restrictive and fails to be valid. In this paper, the method of volume averaging is used to derive a three-temperature macroscopic model considering local thermal non-equilibrium between the three phases. A closed form of the evaporation rate at the macroscopic level is obtained depending on the macroscopic temperatures and the effective properties. Six pore-scale closure problems are proposed, which allow to determine all the effective transport coefficients for representative unit cells. These closure problems are solved for simple unit cells and analytical results are presented in these cases. For these simplified unit cells, a comparison between averaged temperatures obtained from direct pore-scale simulations and averaged temperatures obtained from the three-equation model has been carried out for purely diffusive phase-change processes. A good agreement is obtained between the theory and the pore-scale calculations. This confirms the validity and the practical interest of the proposed approach. Ó 2003 Elsevier Ltd. All rights reserved. 1. Introduction Heat transfer and fluid flow with liquid–vapor phase change in porous media appears in a large number of situations of practical interest including drying processes [24], geothermal systems [71], heat-exchangers design [46], and nuclear safety analysis [45]. For the last ap- plication, there is a strong interest to understand and predict the conditions for which it is possible to cool a severely damaged reactor core. The macroscopic description of heat transfer in a porous medium subjected to a two-phase flow with phase change is often investigated by the use of a single temperature equation. While one-equation models have been proposed recently that do not make the assumption of local equilibrium ([48], in the case of no phase change), the one-equation models are generally based on this assumption. Here, local thermal equilibrium means that the macroscopic temperatures of the three phases are close enough so that a single temperature suffices to describe the heat transport process. Although the as- sumption of local thermal equilibrium is acceptable in many cases of unsaturated porous media with liquid– vapor phase change, particularly for most drying pro- cesses [69], the great simplicity of the one-equation model regarding the effective transport coefficients cer- tainly motivates its use. Nevertheless, the condition of local thermal equilibrium requires numerous constraints which have been investigated by several authors [55,58,64,68]. For instance, this condition of local equi- librium is no longer valid when the particles or pores are not small enough, when the thermal properties differ widely, or when convective transport is impor- tant. Moreover, when there is a significant heat gener- ation in any of the phases, the system will become rapidly far from local thermal equilibrium [38]. Finally, it must be noticed too that local thermal equilib- rium becomes uncertain for situations involving rapid * Corresponding author. Fax: +33-4-42-25-64-68. E-mail address: florian.fichot@irsn.fr (F. Fichot). 0017-9310/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2003.07.005 International Journal of Heat and Mass Transfer 47 (2004) 613–639 www.elsevier.com/locate/ijhmt