Validity criterion of the radiative Fourier law for an absorbing and scattering medium

Abstract

For radiative heat transfer applications, in particular in homogenized phases of porous media, an exhaustive and accurate validity criterion of the radiative Fourier law, depending only on the logarithmic derivative of the temperature field and an effective absorption coefficient, accounting for possible multiple scattering phenomena, has been established for a semitransparent medium. This effective absorption coefficient is expressed as a function of the absorption coefficient, the albedo, and the scattering asymmetry parameter. The criterion can be applied to semitransparent media that do not follow Beer's laws related to extinction, absorption, and scattering.

Comments over homogenisation scales for interfacial emission and scattering by a divided medium: Beerian and non Beerian behaviours

“…This is due to the fact that the transmissivity between s and s is strongly correlated with extinction within [s , s + ds ] but also with a previous type of distribution function. The successive extinction distribution functions associated with interfacial sources G If the validity conditions of the radiative Fourier's law [13] are fulfilled, i.e. when the medium is quasi isothermal at the scale of D, a radiative transfer only occurs at a spatial scale larger than D. The discretised elements [s", s" + ds"] of the previous analysis have now sizes larger than D and are non correlated.…”

“…Nevertheless, Chalhafi et al [7] have shown by a more precise approach that this phase is strongly non Beerian. As this last model is only applied in the conditions of validity of the radiative Fourier's law [12,13], the GRTE degenerates into a classical Beerian radiative transfer equation: The use of G ext instead of G S ext and of a generalised absorption coefficient at equilibrium for modeling interfacial emission is then pertinent. In the case of the work of Zeghondy et al [2] applied to a real mullite foam, of STT type, the globally homogenised medium is also Beerian with a excellent approximation: It justifies the use of G ext and p for the characterisation of internal and external scattering.…”

Radiative transfer within non Beerian porous media with semitransparent and opaque phases in non equilibrium: Application to reflooding of a nuclear reactor

“…The GRTE, developed in Ref. [1,21] for possibly non Beerian phases of this kind of porous media, is directly based on G ext (s − s, θ, ϕ), P sc,ν (s − s, θ, ϕ), n w,ν (θ, ϕ) and the scattering phase function p w (θ i , ϕ i , θ sc , ϕ sc ). For optically thick media, the GRTE rigorously degenerates into a classical RTE, characterized by generalized extinction, absorption and scattering coefficients at equilibrium B w (θ, ϕ), K w,ν (θ, ϕ) and Σ w,ν (θ, ϕ) given by…”

“…By using a perturbation technique of the GRTE, a radiative conductivity tensor has been directly determined by the same authors for non Beerian porous media with opaque and transparent phases. An accurate validity criterion of the associated Fourier's law has also been recently defined [21].…”

Extending the diffusion approximation to the boundary using an integrated diffusion model

“…(15). In conjunction with proper boundary conditions, 7,14,15 which may limit the ultimate level of accuracy, the governing equation Eq. (9) is ready to be solved.…”