Numerical Results for the Effective Flow and Thermal Properties of Idealized Graphite Foam

Abstract

To simulate the heat transfer performance of devices incorporating high-conductivity porous materials, it is necessary to determine the relevant effective properties to close the volume-averaged momentum and energy equations. In this work, we determine these effective properties by conducting direct simulations in an idealized spherical void phase geometry and use the results to establish closure relations to be employed in a volume-averaged framework. To close the volume-averaged momentum equation, we determine the permeability as defined by Darcy’s law as well as a non-Darcy term, which characterizes the departure from Darcy’s law at higher Reynolds numbers. Results indicate that the non-Darcy term is nonlinearly related to Reynolds number, not only confirming previous evidence regarding such behavior in the weak inertia flow regime, but demonstrating that this is generally true at higher Reynolds numbers as well. The volume-averaged energy equation in the fluid phase is closed by the thermal dispersion conductivity tensor, the convecting velocity, and the interfacial Nusselt number. Overall, it has been found that many existing correlations for the effective thermal properties of graphite foams are oversimplified. In particular, it has been found that the dispersion conductivity is not well characterized using the Péclet number alone, rather the Reynolds and Prandtl numbers must be considered as separate influences. Additionally, the convecting velocity modification, which is not typically considered, was found to be significant, while the interfacial Nusselt number was found to exhibit a nonzero asymptote at low Péclet numbers. Finally, simulations using the closed volume-averaged equations reveal significant differences in heat transfer when employing the present dispersion model in comparison to a simpler dispersion model commonly used for metallic foams, particularly at high Péclet numbers and for thicker foam blocks.

A new approach to digital generation of spherical void phase porous media microstructures

“…The present model is first compared to the unit-cube model analyzed by DeGroot and Straatman [5]. Because both approaches use idealized geometry, contrasting the two highlights only the effects of using a randomized geometry over a uniform one.…”

“…This so-called unit-cube model is easy to generate and analyze digitally, however it does not capture the effect of bubble size variation, and it exhibits uncharacteristic properties due to the axis alignment of the bubbles. When used for studying trends in convective/conductive heat transfer and fluid flow, the unit cube model is sufficient, as predictions of permeability, inertial drag coefficients, and Nusselt number correlations were shown to be reasonable when the flow direction was oriented 45°from all principle planes [4,5]. Kumar et al [6] performed Computational Fluid Dynamics (CFD) analyses involving the periodic tetrakaidecahedron structure (Kelvin cell) relevant to open-cell foams.…”

Improved PISO algorithms for modeling density varying flow in conjugate fluid–porous domains

“…Following the derivations in [6] and the closure modeling in [17] and [18], the equations governing the flow of fluid and heat in isotropic, heterogeneous porous media, consisting of a fluid phase α and a solid phase β are presented. The mass conservation for fluid flow is…”

“…In Eqs. (2) to (5) the closure terms suggested in [17,18] were adopted and local thermal non-equilibrium between the two phases was assumed. This is to illustrate that the so-called one-domain approach [19] is taken, solving the described equations for the entire fluid-porous domain, rather than splitting the domain into parts which are governed by multiple sets of equations and imposing interfacial boundary conditions in places where they merge.…”

“…The expression for the update of mass flux follows from the pressure equation (16): (17) where, again, the pressure gradient at the face is directly computed using the Rhie-Chow assumption from Eq. (15).…”