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Research Papers

Upscaling Statistical Methodology for Radiative Transfer in Porous Media: New Trends

[+] Author and Article Information
Jean Taine1

e-mail: estelle.iacona@ecp.fr CNRS, UPR288 Laboratoire d’Énergétique Macroscopique et Moléculaire, Combustion (EM2C) École Centrale Paris Bâtiment Péclet, Grande Voie des Vignes 92295 Châtenay-Malabry Cedex, Francejean.taine@ecp.fr

Estelle Iacona

e-mail: estelle.iacona@ecp.fr CNRS, UPR288 Laboratoire d’Énergétique Macroscopique et Moléculaire, Combustion (EM2C) École Centrale Paris Bâtiment Péclet, Grande Voie des Vignes 92295 Châtenay-Malabry Cedex, Francejean.taine@ecp.fr

1

Corresponding author.

J. Heat Transfer 134(3), 031012 (Jan 13, 2012) (10 pages) doi:10.1115/1.4005133 History: Received July 16, 2010; Revised May 31, 2011; Published January 13, 2012; Online January 13, 2012

The morphology of a porous medium is now generally known from X and γ ray tomography techniques. From these data and radiative properties at the pore scale, a homogenized medium associated with a porous medium phase is exhaustively characterized by radiative statistical functions, i.e., by a statistical cumulative extinction distribution function, absorption, and scattering cumulative probabilities and a general scattering phase function. The accuracy is only limited by the tomography resolution or the geometrical optics validity. When this homogenized medium follows the Beer’s laws, extinction, absorption, and scattering coefficients are identified from these statistical functions; a classical radiative transfer equation (RTE) can then be used. In all other cases, a generalized radiative transfer equation (GRTE) is directly expressed from the radiative statistical functions. When the homogenized medium is optically thick at a spatial scale such as it is practically isothermal, the radiative transfer can simply be modeled from a radiative Fourier’s law. The radiative conductivity is directly determined by a perturbation technique of the GRTE or RTE. An accurate validity criterion of the radiative Fourier’s law has recently been defined. Some paths for future research are finally given.

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Copyright © 2012 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Mullite foam sample for catalytic combustion; typical pore size: 0.5 mm

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Figure 2

Axial cross section of a small-sized model of rod bundle of a nuclear reactor core; intact system (above), degraded system (below); images built from γ ray tomography data [39]; typical rod diameter: 8 mm; porosity of the intact system: 0.56

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Figure 3

Different cases of extinction; transparent, and opaque phases at the pore scale (M: source point, I: impact point, r: reflected, a: absorbed rays)

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Figure 4

Different cases of extinction; transparent, and semitransparent phases at the pore scale (M: source point, I: impact point, r: reflected, a: absorbed, s: scattered, t: transmitted rays)

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Figure 5

Mullite sample of Fig. 1: differential intrusion results measured by mercury porosimetry; the largest pores are only detected from tomography data [24]

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Figure 6

Extinction cumulative distribution function of DOTS Gext parametrized by the medium porosity Π: 0.18, 0.26, 0.37, 0.48, 0.56, 0.65, 0.72, 0.78, 0.82 (from left to right); calculated in Ref. [30] by using the model of [23]

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Figure 7

Normalized extinction coefficient β+ , equal to β+ /βot , issued from the RDFI approach for DOTS, and associated standard deviation ɛβ+ versus medium porosity Π: □; generalized extinction coefficient at equilibrium B+ , equal to B/βot : × [30]

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Figure 8

Extinction cumulative distribution function of DOTS Gext (Π = 0.65); exact: –; computed from βRDFI : - -; from βot : - · -; from B: - &nbsp;… -; from β01 : Δ; from β12 : □; from β23 : ○; [30]

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Figure 10

Degraded rod bundles (see Fig. 2): extinction cumulative distribution function associated with the azimuthal angle φ in a cross section (1.5°: +, 16.5°: ×, 31.5°: *, 43.5°: □) [31]; corresponding cumulative distribution function over all φ values (thick line); βref  = A/(πΠ)

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