# Coupled Upscaling Approaches For Conduction, Convection, and Radiation in Porous Media: Theoretical Developments

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## Abstract

This study deals with macroscopic modeling of heat transfer in porous media subjected to high temperature. The derivation of the macroscopic model, based on thermal non-equilibrium, includes coupling of radiation with the other heat transfer modes. In order to account for non-Beerian homogenized phases, the radiation model is based on the generalized radiation transfer equation and, under some conditions, on the radiative Fourier law. The originality of the present upscaling procedure lies in the application of the volume averaging method to local energy conservation equations in which radiation transfer is included. This coupled homogenization mainly raises three challenges. First, the physical natures of the coupled heat transfer modes are different. We have to deal with the coexistence of both the material system (where heat conduction and/or convection take place) and the non-material radiation field composed of photons. This radiation field is homogenized using a statistical approach leading to the definition of radiation properties characterized by statistical functions continuously defined in the whole volume of the porous medium. The second difficulty concerns the different scales involved in the upscaling procedure. Scale separation, required by the volume averaging method, must be compatible with the characteristic length scale of the statistical approach. The third challenge lies in radiation emission modeling, which depends on the temperature of the material system. For a semi-transparent phase, this temperature is obtained by averaging the local-scale temperature using a radiation intrinsic average while a radiation interface average is used for an opaque phase. This coupled upscaling procedure is applied to different combinations of opaque, transparent, or semi-transparent phases. The resulting macroscopic models involve several effective transport properties which are obtained by solving closure problems derived from the local-scale physics.

## Keywords

Coupled heat transfer Porous medium Radiation Upscaling Macroscopic modeling## List of Symbols

## Roman Symbols

- \(A\)
Specific area of the porous medium

- \(\fancyscript{A}^\mathrm{m}\)
Fluid-solid interface or interfacial area within \(\fancyscript{V}^\mathrm{m}\)

- \(\fancyscript{A}^\mathrm{R}\)
Fluid-solid interface or interfacial area within \(\fancyscript{V}^\mathrm{R}\)

- \(\mathbf{{b}}_{\gamma _1 \gamma _2}\)
Closure variables for the deviation temperature fields (problems I and II)

- \(c_{p\gamma }\)
Heat capacity per unit mass of the \(\gamma \)-phase

- \(g_\gamma \)
Scattering asymmetry parameter

- \(h\)
Effective heat transfer coefficient

- \(I\)
Radiation intensity

- \(\mathbb{K }_{\gamma _1\gamma _2}\)
Effective thermal diffusion-dispersion tensor

- \(k\)
Thermal conductivity

- \(L\)
Macroscopic system typical size

- \(l_\gamma \)
Typical local-scale size of phase \(\gamma \)

- \(n\)
Refractive index

- \(\mathbf{{n}}_{\gamma _1 \gamma _2}\)
Normal unit vector from the \(\gamma _1\)-phase to the \(\gamma _2\)-phase

- \(P, \fancyscript{P}\)
Energy generation rate per unit volume

- \(\mathbf{{p}}_\gamma , \mathbf{{p}}_{\gamma _1\gamma _2}\)
Effective property associated with heterogeneities of the average radiative sourceterm

- \(\mathbf{{q}}\)
Energy flux vector

- \(\mathbf{{r}}\)
Position vector

- \(r^\mathrm{m}\)
Size of the averaging volume

- \(r^\mathrm{R}\)
Size of the radiative averaging volume

- \(r_\gamma \)
Closure variables for the deviation temperature fields (problem IV)

- \(s, s^{\prime }\)
Curvilinear abscissa along a ray

- \(S\)
Source term in the GRTE

- \(s_\gamma \)
Closure variables for the deviation temperature fields (problem III)

- \(T\)
Temperature

- \(\mathbf{{u}}_{\gamma _1 \gamma _2}\)
Macroscopic pseudo-convective transport vector

- \(\mathbf{{u}}\)
Direction unit vector

- \(\fancyscript{V}^\mathrm{R}_\gamma \)
Volume of the \(\gamma \)-phase within \(\fancyscript{V}^\mathrm{R}\)

- \(\fancyscript{V}^\mathrm{m}_\gamma \)
Volume of the \(\gamma \)-phase within \(\fancyscript{V}^\mathrm{m}\)

- \(\mathbf{{v}}\)
Velocity

- \(\fancyscript{V}^\mathrm{m}\)
Averaging volume

- \(\fancyscript{V}^\mathrm{R}\)
Radiative averaging volume

## Greek Symbols

- \(\alpha , \alpha _\gamma \)
Coefficient relating the radiation source term to its average

- \(B_\gamma \)
Generalized extinction coefficient at equilibrium

- \(\beta _\gamma \)
Extinction coefficient at equilibrium

- \(\varPi _\gamma \)
Volume fraction of the \(\gamma \)-phase

- \(K\)
Generalized absorption coefficient at equilibrium

- \(\kappa _\gamma \)
Absorption coefficient at equilibrium

- \(\lambda \)
Wavelength

- \(\Sigma _\gamma \)
Generalized scattering coefficient

- \(\sigma _\gamma \)
Scattering coefficient

- \(\sigma \)
Stefan-Boltzmann constant

- \(\varphi \)
Energy flux

- \(\rho _\gamma \)
Density

- \(\xi _\gamma , \xi _{\gamma _1 \gamma _2}\)
Macroscopic distribution coefficient

- \(\varTheta \)
Homogenized temperature (radiation model)

- \(\nu \)
Radiation frequency

## Subscripts

- \(f\)
Fluid

- \(s\)
Solid

- \(w\)
Wall

- \(t\)
Tomography

- \(\gamma , \gamma _1, \gamma _2\)
General index for a phase (\(f, s\) or \(w\))

## Superscripts

- eff
Effective

- cd
Conductive

- R
Radiative

- sc
Scattering

- ext
Extinction

- e
Emission

- a
Absorption

- l
Leaving a boundary (radiation)

- \((j)\)
\(j\)-th perturbation order

## Special notations

- \(\langle {\cdot }\rangle ^\mathrm{m}\)
Superficial volume averaging operator

- \(\langle {\cdot }\rangle ^{\mathrm{m}\gamma }\)
Intrinsic volume averaging operator related to the \(\gamma \)-phase

- \(\langle {\cdot }\rangle ^{\mathrm{R}\gamma }\)
Radiative intrinsic volume averaging operator related to the \(\gamma \)-phase

- \(\langle {\cdot }\rangle ^{\fancyscript{A}^\mathrm{R}}\)
Radiative interfacial averaging operator

- \(\widetilde{\cdot }\)
Deviation

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