Statistical modelling of radiative transfer within non-Beerian effective phases of macroporous media

International audienceA purely statistical modelling of radiative transfer is applied to statistically anisotropic and sometimes non-homogeneous macroporous media, in which the geometrical optics laws are assumed to be valid. It is focused on the physical features of media characterised, after statistical homogenisation, by non-Beerian effective phases such that the extinction law is not exponential. Approaches that have been developed for media with Opaque and Transparent phases (OT) are generalised to media with Opaque and Semi-Transparent phases (OST), Semi-Transparent and Transparent phases (STT) and two Semi-Transparent phases (ST2). Many assumptions of previous papers are justified or invalidated. Limitations of the models are introduced. Non-Beerian effective phases are exhaustively characterised by accurate radiative statistical functions: Extinction cumulative distribution functions, scattering cumulative probabilities and general scattering phase functions. Key studied phenomena are the correlations that occur in radiative transfer: i) Between transmission, from source points to interfacial points, and interfacial extinction; ii) In OT and OST cases, between interfacial emission, the following transmission and interfacial extinction; iii) In the case of non-diffuse reflection or transmission law, between an interfacial incident intensity and an interfacial scattered intensity, and between this scattered intensity and the following transmission and extinction. In the case of a diffuse reflection or transmission law, the two first types of correlations are taken into account within a non-Beerian effective phase by a unique Generalised Radiative Transfer Equation (GRTE) that is expressed in terms of the radiative statistical functions and also includes the boundary conditions. The GTREs associated with OT, OST, STT and ST2 media are detailed. In the case of a non-diffuse reflection or transmission law, the three types of correlations are taken into account for a non-Beerian effective phase by specific GRTEs. These GRTEs are associated with the successive elementary paths from emission to final absorption after multiple scattering events. Under specific validity conditions, the GRTE of a homogeneous effective phase of an OT medium degenerates into a classical RTE and a radiative Fourier law can be applied within this effective phase. In the case of an OST or a ST2 medium, this model can only be applied if the homogeneous effective phases are characterised by the same temperature field. A radiative Fourier law is never valid for homogeneous effective phases of a STT medium or for a non-homogeneous effective phase of any type. A non-homogeneous effective phase, with strong volume fraction gradients, is characterised by radiative statistical functions that also depend on the coordinates of the source points. All the previous models of radiative properties are generalised by using a global transmissivity, product of the volume fraction at the source point by the transmissivity deduced from the extinction cumulative distribution function. Finally, the spatial limitations of the use of a GRTE or a radiative Fourier law are enlightened from a comparison of all the spatial scales involved in the characterisation of the radiative statistical functions with the thickness of the radiative boundary layer of the porous medium and with the spatial resolution of the temperature field