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Transport in Porous Media

, Volume 98, Issue 2, pp 323–347 | Cite as

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

  • Vincent Leroy
  • Benoît GoyeauEmail author
  • Jean Taine
Article

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

References

  1. Baillis, D., Sacadura, J.-F.: Thermal radiation properties of dispersed media: theoretical prediction and experimental characterization. J. Quant. Spectrosc. Radiat. Transf. 67(5), 327–363 (2000)CrossRefGoogle Scholar
  2. Bellet, F., Chalopin, E., Fichot, F., Iacona, E., Taine, J.: RDFI determination of anisotropic and scattering dependent radiative conductivity tensors in porous media: application to rod bundles. Int. J. Heat Mass Transf. 52(5–6), 1544–1551 (2009)CrossRefGoogle Scholar
  3. Carbonell, R.G., Whitaker, S.: Heat and mass transfer in porous media. Fundam. Transp. Phenom. Porous Media 82, 121 (1984)Google Scholar
  4. Chahlafi, M., Bellet, F., Fichot, F., Taine, J.: Radiative transfer within non Beerian porous media with semitransparent and opaque phases in non equilibrium: application to reflooding of a nuclear reactor. Int. J. Heat Mass Transf. 55(13–14), 3666–3676 (2012)CrossRefGoogle Scholar
  5. Consalvi, J., Porterie, B., Loraud, J.: A formal averaging procedure for radiation heat transfer in particulate media. Int. J. Heat Mass Transf. 45, 2755–2763 (2002)CrossRefGoogle Scholar
  6. Gomart, H., Taine, J.: Validity criterion of the radiative Fourier law for an absorbing and scattering medium. Phys. Rev. E 83(2), 1–8 (2011)CrossRefGoogle Scholar
  7. Gray, W.G.: A derivation of the equations for multi-phase transport. Chem. Eng. Sci. 30(2), 229–233 (1975)CrossRefGoogle Scholar
  8. Gusarov, A.V.: Homogenization of radiation transfer in two-phase media with irregular phase boundaries. Phys. Rev. B 77(14), 1–14 (2008)CrossRefGoogle Scholar
  9. Haussener, S., Lipiński, W., Petrasch, J., Wyss, P., Steinfeld, A.: Tomographic characterization of a semitransparent-particle packed bed and determination of its thermal radiative properties. J. Heat Transf. 131(7), 072701 (2009)CrossRefGoogle Scholar
  10. Haussener, S., Coray, P., Lipiński, W., Wyss, P., Steinfeld, A.: Tomography-based heat and mass transfer characterization of reticulate porous ceramics for high-temperature processing. J. Heat Transf. 132(2), 023305 (2010a)CrossRefGoogle Scholar
  11. Haussener, S., Lipiński, W., Wyss, P., Steinfeld, A.: Tomography-based analysis of radiative transfer in reacting packed beds undergoing a solid–gas thermochemical transformation. J. Heat Transf. 132(6), 061201 (2010b)Google Scholar
  12. Lipiński, W., Keene, D., Haussener, S., Petrasch, J.: Continuum radiative heat transfer modeling in media consisting of optically distinct components in the limit of geometrical optics. J. Quant. Spectrosc. Radiat. Transf. 111(16), 2474–2480 (2010a)CrossRefGoogle Scholar
  13. Lipiński, W., Petrasch, J., Haussener, S.: Application of the spatial averaging theorem to radiative heat transfer in two-phase media. J. Quant. Spectrosc. Radiat. Transf. 111(1), 253–258 (2010b)CrossRefGoogle Scholar
  14. Moyne, C.: Two-equation model for a diffusive process in porous media using the volume averaging method with an unsteady-state closure. Adv. Water Resour. 20(2–3), 63–76 (1997)CrossRefGoogle Scholar
  15. Petrasch, J., Haussener, S., Lipiński, W.: Discrete vs. continuous level simulation of radiative transfer in semitransparent two-phase media. J. Quant. Spectrosc. Radiat. Transf. 112, 1450–1459 (2011)CrossRefGoogle Scholar
  16. Petrasch, J., Wyss, P., Steinfeld, A.: Tomography-based Monte Carlo determination of radiative properties of reticulate porous ceramics. J. Quant. Spectrosc. Radiat. Transf. 105(2), 180–197 (2007)CrossRefGoogle Scholar
  17. Quintard, M., Kaviany, M., Whitaker, S.: Two-medium treatment of heat transfer in porous media: numerical results for effective properties. Adv. Water Resour. 20(2–3), 77–94 (1997)CrossRefGoogle Scholar
  18. Quintard, M., Ladevie, B., Whitaker, S.: Effect of homogeneous and heterogeneous source terms on the macroscopic description of heat transfer in porous media. Energy Eng. 2(January), 482–489 (2000)Google Scholar
  19. Quintard, M., Whitaker, S.: One- and two-equation models for transient diffusion processes in two-phase systems. Adv. Heat Transf. 23, 369 (1993a)CrossRefGoogle Scholar
  20. Quintard, M., Whitaker, S.: Transport in ordered and disordered porous media: volume-averaged equations, closure problems, and comparison with experiment. Chem. Eng. Sci. 48(14), 2537–2564 (1993b)CrossRefGoogle Scholar
  21. Quintard, M., Whitaker, S.: Transport in ordered and disordered porous media IV: computer generated porous media for three-dimensional systems. Transp. Porous Media 15(1), 51–70 (1994)CrossRefGoogle Scholar
  22. Quintard, M., Whitaker, S.: Theoretical analysis of transport in porous media. In: Vafai, K., Hadim, H.A. (eds.) Handbook of Heat Transfer in Porous Media, Chapter 1, pp. 1–52. Marcel Dekker, New York (2000)Google Scholar
  23. Taine, J., Bellet, F., Leroy, V., Iacona, E.: Generalized radiative transfer equation for porous medium upscaling: application to the radiative Fourier law. Int. J. Heat Mass Transf. 53(19–20), 4071–4081 (2010)CrossRefGoogle Scholar
  24. Taine, J., Iacona, E.: Upscaling statistical methodology for radiative transfer in porous media: new trends. J. Heat Transf. 134(3), 031012 (2012)CrossRefGoogle Scholar
  25. Tancrez, M., Taine, J.: Direct identification of absorption and scattering coefficients and phase function of a porous medium by a Monte Carlo technique. Int. J. Heat Mass Transf. 47(2), 373–383 (2004)CrossRefGoogle Scholar
  26. Torquato, S., Lu, B.: Chord-length distribution function for two-phase random media. Phys. Rev. E 47(4), 2950 (1993)CrossRefGoogle Scholar
  27. Valdés-Parada, F.J., Goyeau, B., Ochoa-Tapia, J.A.: Diffusive mass transfer between a microporous medium and an homogeneous fluid: jump boundary conditions. Chem. Eng. Sci. 61(5), 1692–1704 (2006)CrossRefGoogle Scholar
  28. Whitaker, S.: Coupled transport in multiphase systems: a theory of drying. Adv. Heat Transf. 31, 1–104 (1998)CrossRefGoogle Scholar
  29. Whitaker, S.: The Method of Volume Averaging. Kluwer Academic Publishers, Dordrecht (1999)CrossRefGoogle Scholar
  30. Wood, B.D., Quintard, M., Whitaker, S.: Jump conditions at non-uniform boundaries: the catalytic surface. Chem. Eng. Sci. 55(22), 5231–5245 (2000)CrossRefGoogle Scholar
  31. Zeghondy, B., Iacona, E., Taine, J.: Determination of the anisotropic radiative properties of a porous material by radiative distribution function identification (RDFI). Int. J. Heat Mass Transf. 49(17–18), 2810–2819 (2006a)CrossRefGoogle Scholar
  32. Zeghondy, B., Iacona, E., Taine, J.: Experimental and RDFI calculated radiative properties of a mullite foam. Int. J. Heat Mass Transf. 49(19–20), 3702–3707 (2006b)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Laboratoire EM2C, UPR CNRS 288Ecole Centrale ParisChatenay-MalabryFrance

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