Effects of Solid-to-Fluid Conductivity Ratio on Thermal Convection in Fluid-Saturated Porous Media at Low Darcy Number

This study presents experimental data on the effects of the solid-to-fluid thermal conductivity ratio on natural convective heat transfer in a fluid-saturated porous medium heated from below. Argon is used as the saturating fluid, while a bed of glass, steel, or aluminum spheres constitutes the solid porous matrix. Emphasis is placed on attaining high Rayleigh numbers while maintaining low Darcy numbers (⁠$5.68×10−8≤Da≤5.22×10−7$⁠). At low modified Rayleigh numbers (⁠$Ra*$⁠) corresponding to the Darcy regime, the Nusselt number is independent of the medium conductivity. As $Ra*$ increases and the system transitions into the Forchheimer regime, the data diverge, with Nusselt numbers decreasing with increased thermal conductivity ratio at a fixed $Ra*$⁠. This nonintuitive result is shown to be the result of the traditional choice of $Ra*$ and Da as the controlling parameter since the heat transfer coefficient appears independent of the conductivity ratio. Scaling arguments are used to identify transition points between the regimes, which yield the transition criterion $Ra*$ ∼ Pr_p, where Pr_p is the modified Prandtl number. When the data are expressed by scaling with Pr_p, it is shown that the data for multiple parameter combinations collapse onto a single curve, which also agrees well with some theoretical predictions. In light of this finding, the data from available literature are assessed, and it is proposed that deviations from theory are likely the result of the strong porous medium condition (low Da) not being satisfied.