Non-Darcy natural convection in high porosity metal foams M.S. Phanikumar a, * , R.L. Mahajan b a Departments of Civil and Environmental Engineering and Geological Sciences, Michigan State University, East Lansing, MI 48824, USA b CAMPMode (Center for Advanced Manufacturing and Packaging of Microwave, Optical and Digital Electronics), Department of Mechanical Engineering, University of Colorado, Boulder, CO 80309, USA Received 23 March 2001; received in revised form 16 February 2002 Abstract We present numerical and experimental results for buoyancy-induced flows in high porosity metal foams heated from below. A Brinkman–Forchheimer-extended Darcy flow model and a semi-heuristic two-equation energy model obtained by relaxing the local thermal equilibrium (LTE) assumption are adopted. Experiments conducted under natural convection conditions for the same configuration are used to test the numerical model and the validity of the thermal equilibrium assumption for metal foams. Aluminum foam samples of different pore sizes (5–40 PPI) and porosities (0:89 6 e 6 0:97) are used to illustrate the effects of metal foam geometry on heat transfer. In addition, several metal foam–fluid combinations (aluminum–air, carbon–air, aluminum–water, and nickel–water) are used to study the heat transfer enhancement relative to the base case in which there is no metal foam but only a heated plate. Thermal dispersion effects and the effects of Darcy number on heat transfer are reported. Our results indicate that the thermal non-equilibrium model provides a superior description of heat transfer in metal foams, especially in the presence of fluid–porous interfaces. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Porous media; Metal foams; Natural convection; Enhancement; Local thermal non-equilibrium 1. Introduction Flow and transport at the interface between a porous medium and a clear fluid are of interest in a variety of engineering applications as well as in the environment. Solidification processes, thermal insulation, heat pipes, the interaction of groundwater with surface water and solute exchange in the hyporheic zone are some such examples. This paper deals with fluid flow and heat transfer in metal foams in the presence of fluid–porous interfaces. High porosity metal foams (e > 0:85) have gained attention in recent years as potentially excellent candidates for meeting the high thermal dissipation de- mands in the electronic industry. The mechanisms that contribute to the enhanced heat transfer include heat conduction in the metal foam matrix (whose conduc- tivity is usually several orders of magnitude higher compared to the fluid conductivity), and thermal dis- persion in the fluid at high velocities. The dispersion conductivity accounts for the effects of pore-level hy- drodynamics on the macroscopic transport and essen- tially represents the enhanced mixing due to the presence of the solid phase. The well-known Darcy’s law is based on a balance between the pressure gradient and the viscous forces and breaks down for high velocities when inertia terms are no longer negligible. Non-Darcy effects become particularly important in metal foams as the fluid moves in tortuous paths and eddies are shed behind the solid fibers in the interstitial pore volume. The resulting pressure drop across the medium and the increased mixing (or disper- sion) accounts for an increase in the net transport. Ear- lier efforts to quantify the effects of dispersion were mostly confined to packed beds. More recent studies for forced convection have shown an increase in heat transfer with the inclusion of thermal dispersion [1–5]. Jiang et al. [6] found that, if thermal dispersion effects are International Journal of Heat and Mass Transfer 45 (2002) 3781–3793 www.elsevier.com/locate/ijhmt * Corresponding author. Tel.: +1-517-353-4366; fax: +1-517- 353-8787. E-mail addresses: phani@msu.edu (M.S. Phanikumar), ma- hajan@spot.colorado.edu (R.L. Mahajan). 0017-9310/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII:S0017-9310(02)00089-3