Analysis of non-Darcian eects on temperature dierentials in porous media A. Mara®e, K. Vafai * Department of Mechanical Engineering, The Ohio State University, Columbus, OH 43210, USA Received 30 June 2000; received in revised form 8 March 2001 Abstract Forced convection ¯ow through a channel ®lled with a porous medium is investigated analytically. A non-thermal equilibrium, two-equation model is utilized to represent the ¯uid and solid energy transport. The Darcy±Forchheimer± Brinkman model is used to represent the ¯uid transport within the porous medium. Analytical solutions are obtained for both ¯uid and solid temperature ®elds incorporating the eects of various pertinent parameters such as the Biot number, the thermal conductivity ratio, the Darcy number and the inertial parameter. The present analytical solution for the two-equation model is validated against the exact solution for the one-equation model available in the literature as well as the analytical solution for the non-thermal equilibrium case based on a Darcian ¯ow ®eld. Error maps for the validity of one-equation model are established for various physical conditions taking into account the Darcy and in- ertial parameters as well as the Biot and the thermal conductivity ratio of ¯uid to solid phases. It is shown that the Darcy number and the inertial parameter have a lesser in¯uence in establishing the validity of the local thermal equilibrium assumption. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Porous media; Local thermal non-equilibrium; Forced convection; Non-Darcian eects 1. Introduction Forced convective heat transfer in porous media has been the subject of many recent studies due to numerous practical applications such as nuclear waste repository, energy storage units, electronic cooling, thermal insula- tion, packed bed heat exchangers, heat pipes, drying technology, catalytic reactors, petroleum industries and geothermal systems. The assumption of local thermal equilibrium is widely used in many of these applications. However, this assumption breaks down when a sub- stantial temperature dierence exists between the solid and the ¯uid phases. More recently, local thermal non- equilibrium has received considerable attention due to its pertinence in applications where such a temperature dierential exists between the solid and the ¯uid phases. The work of Vafai and Tien [1] was one of the early attempts to account for the boundary and inertia eects in the momentum equation for a porous medium. They found that the momentum boundary layer thickness is of the order of  K=e p . Vafai and Thiyagaraja [2] pre- sented analytical solutions for the velocity and tem- perature ®elds for the interface region using the Brinkman±Forchheimer±extended Darcy equation. They considered three fundamental types of the inter- face namely, the interface between two porous media, the interface between a porous medium and a ¯uid layer and the interface between a porous medium and an impermeable medium. Amiri and Vafai [3] employed a general ¯uid ¯ow model and a two-phase energy equa- tion to investigate the forced convective heat transfer within a channel with constant wall temperature. They included the eects of variable porosity and thermal dispersion in their analysis and error maps for assessing the importance of various simplifying assumptions that are commonly used were established in their work. Lee and Vafai [4] employed the non-thermal equi- librium model to investigate the forced convective ¯ow International Journal of Heat and Mass Transfer 44 2001) 4401±4411 www.elsevier.com/locate/ijhmt * Corresponding author. Present address. Department of Mechanical Engineering, University of California, A363 Bourns Hall, Riverside, CA 92521-0425, USA. Tel.: +1-909- 787-2135; fax: +1-909-787-2899. E-mail address: vafai@engr.ucr.edu K. Vafai). 0017-9310/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII:S0017-931001)00099-0