B Rousseau
email: benoit.rousseau@univ-nantes.fr

S Ouchtout

L Cangémi

Y Favennec

J Vicente

F Enguehard

IMPACT OF THE EXTINCTION BEHAVIOUR OF SIC LATTICES ON THEIR CONDUCTIVE-RADIATIVE HEAT TRANSFERS

Keywords: Cellular lattices, Radiative Distribution Function Identification, non Beerian behaviour, conduction-radiation coupling

Refractory architected ceramics constitute a class of highly porous media which gains in importance for enginnering high-temperatures applications. From a thermal modelling viewpoint, one of the main challenges, is to finely describe the transport of thermal radiation which plays a major role in the dtermination of their thermal performances. Such a consideration is today crucial for developing topology optimization processes in order to define the best 3D geometries for a given set of objectives. To conduct these methodologies, it is important to quickly solve the radiative transfer equation at the continuous scale while taking into account as accurately as possible the meso-texture desbribing the 3D solid network. This goes back to assume that the physical statements governing this equation are valid, the porous architectures having to behave as an equivalent Beerian medium. However, when non Beerian behaviour is highlighted, the latter framework needs to be revised. To go one step further with this issue, numerical 3D geometries with a regular arrangement of cubic cells are generated with an homemade software. The analysis of their extinction cumulative distribution function curves obtained with the Radiative Distribution Function Identification method allows us to check whether or not the attenuated thermal radiation follows a Beerian behaviour. The effects of the textural parameters will then be discussed. Finally, preliminary practical considerations will be given for describing their conductive radiative behaviour when they are enclosed between a hot and a cold plate through both a continuous and the discrete scale methodologies, using a stabilized vectorial Finite Element Method solver.

INTRODUCTION

In the field of carbon-free heat generation, there is a growing interest in the design of compact and longlife high temperature energy systems such as gas-to-gas heat exchangers [START_REF] Bayaniahangar | Toward extreme high-temperature supercritical CO2 power cycles: Leakage characterization of ceramic 3D-printed heat exchangers[END_REF], volumetric solar receivers [START_REF] Heisel | Digital design and 3D printing of innovative SiC architectures for high temperature volumetric solar receivers[END_REF] (VSR) and radiant tube inserts [START_REF] Pelanconi | Thermal design, optimization and additive manufacturing of ceramic regular structures to maximize the radiative heat transfer[END_REF], among others. They are mostly based on architected refractory ceramics (p ∼75-95%, cell size ∼0.1-10 mm) which can be depicted by a continuous ligament network (the solid phase) delimiting open cells in which a fluid can flow. When silicon carbide (SiC) is used as the constitituve material [START_REF] Heisel | Digital design and 3D printing of innovative SiC architectures for high temperature volumetric solar receivers[END_REF][START_REF] Pelanconi | Application of ceramic lattice structures to design compact, high temperature heat exchangers: material and architecture selection[END_REF], the whole 3D architecture exhibits outstanding properties such as high specific surface area, good flow-mixing capacity, high thermal shock resistance and high resistance to chemical corrosion. The rapid development of additive manufacturing (AM) extends furthermore the possibilities to elaborate a large range of 3D geometries going from regular lattice structures with different types of unit cells, triply periodic minimal surface-based structures up to more classical irregular strut-based structures. In order to propose a topology optimization process aiming at the definition of the best 3D architectures for prescribed thermal objectives, the detailed modelling of their thermal behaviour is essential [START_REF] Pelanconi | Thermal design, optimization and additive manufacturing of ceramic regular structures to maximize the radiative heat transfer[END_REF][START_REF] Ouchtout | Finite element framework for modeling conducto-radiative transfers within heterogeneous media at both discrete and continuous scales[END_REF]. In particular, one of the main challenges, is to be able to exactly describe the role of thermal and/or solar radiation in the heat balance of the system for both unsteady or steady regimes. To date, two main routes can be used to determine the temperature and/or heat flux fields within 3D structures [START_REF] Ouchtout | Finite element framework for modeling conducto-radiative transfers within heterogeneous media at both discrete and continuous scales[END_REF][START_REF] Avila-Marin | Modelling strategies for porous structures as solar receivers in central receiver systems: A review[END_REF][START_REF] Petrasch | Discrete vs. continuum-scale simulation of radiative transfer in semitransparent twophase media[END_REF][START_REF] Xia | Integrated simulation of continuous-scale and discrete-scale radiative transfer in metal foams[END_REF]. The first class of methods, at the discrete mesoscopic scale, allows a treatment of the radiative heat exchange directly at the scale of the individual constituents (here the struts and walls constituting the basic cells). Several parallelized approaches have been developed for conducting these approaches : stabilized vectorial Finite Element schemes [START_REF] Ouchtout | Finite element framework for modeling conducto-radiative transfers within heterogeneous media at both discrete and continuous scales[END_REF], cell centered Finite Volume models with ray tracing [START_REF] Kumar | A numerical method based on domain decomposition to solve coupled conduction-radiation physics using parallel computing within large porous media[END_REF][START_REF] Perraudin | Numerical quantification of coupling effects for radiation-conduction heat transfer in participating macroporous media: Investigation of a model geometry[END_REF], hybrid schemes combining motion of Brownian walkers in the solid opaque phase and ray tracing in the fluid phase [START_REF] Vignoles | A hybrid random walk method for the simulation of coupled conduction and linearized radiation transfer at local scale in porous media with opaque solid phases[END_REF], and Monte Carlo strategies based on integral formulations [START_REF] Caliot | Combined conductive-radiative heat transfer analysis in complex geometry using the Monte Carlo method[END_REF]. The second class of method, at the continuous macroscopic scale, requires to rigorously solve the radiative transfer equation as long as the assumptions (randomness, homogeneity and continuity) that govern its statement are valid. Let us clarify that an architected ceramic can be viewed as an equivalent domain absorbing, emitting and scattering the thermal radiation. These latter physical phenomena are, in fact, induced by the light-matter interaction at the scale of the ligaments. If the equivalent medium is optically thick, the radiative transfer equation can be reduced to a simpler diffusion equation by applying, for example, the spherical harmonics method at the order 1 (the P1 approximation) or, even more crudely, the Rosseland approximation [START_REF] Avila-Marin | Modelling strategies for porous structures as solar receivers in central receiver systems: A review[END_REF]. Whatever the precision sought, these approaches can only be applied if the volume radiative properties (extinction, absorption and scattering coefficients, scattering phase function) are identified from an equivalent semitransparent continuous Beerian medium. The term Beerian stands, here, for an extinction law characterised by an exponential function of the optical thickness. This statement changes when spatial correlations occur meaning that the equivalent semitransparent medium becomes not easily homogeneisable. In general, the samples are considered as Beerian media [START_REF] Pelanconi | Thermal design, optimization and additive manufacturing of ceramic regular structures to maximize the radiative heat transfer[END_REF][START_REF] Tseng | Effect of foam properties on radiative properties of opencell silicon carbide foams[END_REF][START_REF] Avila-Marin | Experimental study of innovative periodic cellular structures as air volumetric absorbers[END_REF] without an in-depth investigation of their ability to extinct radiation being really carried out. By using the Radiative Distribution Function Identification method, Taine et al. have shown that a dispersion of overlapping transparent spheres in an opaque continuous hosting medium deviates from the Beerian behavior when its volume fraction, 𝑓𝑓 𝑣𝑣 , becomes important (𝑓𝑓 𝑣𝑣 > 0.4) [START_REF] Taine | Generalized radiative transfer equation for porous medium upscaling: Application to the radiative Fourier law[END_REF]. Such a synthetic medium can be used to represent a statistically isotropic foam. More recently, Ouchtout et al. have computed the temperatures fields within two SiC-based regular cubic lattices (𝑝𝑝 = 0.91, 𝑑𝑑 𝑛𝑛𝑛𝑛𝑛𝑛 = 0.16 mm, 𝐿𝐿 = 0.38 and 0.76 mm) by solving a conductive-radiative numerical scheme with stabilized vectorial finite element methodologies applied respectively at both the continuous scale and the discrete scale. Both the temperature profiles are closely identical [START_REF] Ouchtout | Finite element framework for modeling conducto-radiative transfers within heterogeneous media at both discrete and continuous scales[END_REF]. Since the continuous approach assumes for the radiative part that the Beer law is valid, it can be concluded that this cubic lattice exhibits a Beerian behaviour. Such a survey indicates that a study investigating the radiative behavior of synthetic lattices as a function of variable textural features (spatial repartition, shape and size of cells, size of the computing domain) is still relevant. To go one step further with these issues, a set of virtual regular SiC-based cubic cell structures with prescribed textural features will be designed with the GenMat software (C++, Qt) [START_REF] Guévelou | Representative elementary volumes required to characterize the normal spectral emittance of silicon carbide foams used as volumetric solar absorbers[END_REF]. Then, the Radiative Distribution Function Identification method will be applied to compute the extinction cumulative distribution functions of the whole set of numerical samples. This will allow us to provide preliminary practical criteria for defining the radiative behaviours (non Beerian or Beerian) of the samples. A set of extinction coefficients will be determined and their impacts on the modelling of the conductive-radiative behaviour of the cubic cells structures will be discussed.

NUMERICAL METHODOLOGIES

3D structures generation

In this work, a homemade 3D structures generator, called GenMat (C++, Qt) [START_REF] Guévelou | Representative elementary volumes required to characterize the normal spectral emittance of silicon carbide foams used as volumetric solar absorbers[END_REF], has been used to provide the whole set of numerical samples. The generator allows the numerical elaboration of regular and irregular 3D structures with prescribed textural features (porosity, nominal pore diameter). For the regular structures, a simplified four-steps process is applied. The centers of all cells are firstly created. Secondly, the nodes, edges, faces of all cells are defined, and a structuration loop adds solid voxels between each apex, forming a thin skeleton. Thirdly, as for irregular foams, the ligaments can be shaped with a spherical or a square structuring element. Lastly, the desired porosity is reached by creating a distance map converted in grey levels. Once the 3D binarized structures are generated, a robust marching cube process allows to mesh the fluid/solid interface. GenMat proposes an export module for recording surfacic meshes.The software also contains modules which provide the volume fraction of the void phase i.e. the porosity, 𝑝𝑝, and the volumetric surface, 𝐴𝐴 of each numerical sample.

Radiative Distribution Function Identification method

In this work, the Radiative Distribution Function Identification (RDFI) method, originally proposed by Tancrez and Taine [START_REF] Tancrez | Direct identification of absorption and scattering coefficients and phase function of a porous medium by a Monte Carlo technique[END_REF], will be applied to determine the volumetric radiative properties of the 3D architectures. This method has been implemented in the GenMat framework [START_REF] Guevelou | Evolution of the homogenized volumetric radiative properties of a family of α-SiC foams with growing nominal pore diameter[END_REF]. More precisely, the RDFI is a collision-based Monte Carlo Ray Tracing method assuming that for each local interaction between a ray and the solid matter, the geometrical optic approximation is valid. It means that the mean characteristic size of the ligaments and of the cells constituting the generated 3D structures must be much larger than the incident wavelength i.e. the Mie size parameter is greater than 1. The RDFI method consists in launching a huge number of rays from points 𝑀𝑀 that are uniformly distributed in the fluid phase surrounding the ligaments. Their directions are randomly chosen within the entire 4𝜋𝜋 steradian sphere. Once a ray hits a solid phase element at the point 𝐼𝐼, the latter is considered as totally extinguished and the path length to collision, 𝑠𝑠 𝑒𝑒 = |𝑀𝑀𝐼𝐼| is computed [START_REF] Tancrez | Direct identification of absorption and scattering coefficients and phase function of a porous medium by a Monte Carlo technique[END_REF]. For a large number of rays, 𝑁𝑁 𝑅𝑅 , a set of path length 𝑠𝑠 𝑒𝑒,𝑗𝑗 values are obtained from which the corresponding normalized extinction distribution function, 𝐹𝐹 𝑒𝑒 (𝑠𝑠) is computed :

𝐹𝐹 𝑒𝑒 (𝑠𝑠) = 1 𝑁𝑁 𝑅𝑅 ∑ 𝛿𝛿�𝑠𝑠 -𝑠𝑠 𝑒𝑒,𝑗𝑗 � 𝑁𝑁 𝑅𝑅 𝑗𝑗=1 (4)
where 𝑠𝑠 is the path length variable and 𝛿𝛿 the Dirac delta distribution. The extinction cumulative distribution function 𝐺𝐺 𝑒𝑒 (𝑠𝑠) is then obtained by integrating 𝐹𝐹 𝑒𝑒 (𝑠𝑠) :

𝐺𝐺 𝑒𝑒 (𝑠𝑠) = ∫ 𝐹𝐹 𝑒𝑒 (𝑠𝑠 ′ )𝑑𝑑𝑠𝑠 ′ 𝑠𝑠 0 (5)
To decide whether the extinguished ray is absorbed or scattered, a random number, 𝜂𝜂, uniformly distributed over [0,1] is selected and compared to the local reflectivity, 𝜌𝜌, that can be given by the Fresnel law (if the interacting fluid/solid interface is considered as optically smooth). If 𝜂𝜂 > 𝜌𝜌 the ray is considered as absorbed and otherwise scattered. This allows to determine the absorption and the scattering cumulative probabilities with a similar numerical process as for 𝐺𝐺 𝑒𝑒 (𝑠𝑠). If the studied equivalent medium is Beerian, the quantity �1 -𝐺𝐺 𝑒𝑒 (𝑠𝑠)� shows a typical Beer-Lambert exponential decay behaviour expressed by:

1 -𝐺𝐺 𝑒𝑒 (𝑠𝑠) = 𝑒𝑒 -𝛽𝛽𝑠𝑠 = 𝐼𝐼(𝑠𝑠) 𝐼𝐼 0 (6)
where 𝐼𝐼(𝑠𝑠) is the radiant intensity at the distance 𝑠𝑠 and 𝐼𝐼 0 is the impinging intensity at 𝑠𝑠 = 0 from a collimated beam. In that situation, the extinction coefficient 𝛽𝛽 of the equivalent Beerian medium can be evaluated by a fit of the numerically evaluated �1 -𝐺𝐺 𝑒𝑒 (𝑠𝑠)� function.

Models of conductive radiative transfers

Before describing the two approaches used in this work and described in [START_REF] Ouchtout | Finite element framework for modeling conducto-radiative transfers within heterogeneous media at both discrete and continuous scales[END_REF], let us specify that the global computational domain is depicted by a cube sandwhiched, on two of its opposite faces, by a hot opaque wall maintened at 𝑇𝑇 ℎ𝑛𝑛𝑜𝑜 (for 𝑥𝑥 = 𝑥𝑥 𝑛𝑛𝑚𝑚𝑛𝑛 ) and a cold opaque wall (for 𝑥𝑥 = 𝑥𝑥 𝑛𝑛𝑚𝑚𝑚𝑚 ) with 𝑇𝑇 = 𝑇𝑇 𝑐𝑐𝑛𝑛𝑐𝑐𝑐𝑐 .

Continuous-scale coupled model

For the continuous-scaled coupled model, the standard steady-state form of the radiative transfer equation, which describes the transport of the radiative intensity, 𝐼𝐼(𝒙𝒙, 𝒔𝒔), within a Beerian semitransparent medium is used :

𝒔𝒔. ∇𝐼𝐼(𝒙𝒙, 𝒔𝒔) + 𝛽𝛽(𝒙𝒙)𝐼𝐼(𝒙𝒙, 𝒔𝒔) -𝜎𝜎(𝒙𝒙) ∫ 𝐼𝐼(𝒙𝒙, 𝒔𝒔 ′ )Φ(𝒔𝒔 ′ , 𝒔𝒔)𝑑𝑑𝒔𝒔 ′ 4𝜋𝜋 -𝜅𝜅(𝒙𝒙) 1 𝜋𝜋 𝜎𝜎 𝐵𝐵 𝑛𝑛 2 𝑇𝑇 4 = 0 (7)
Here, 𝒙𝒙 = (𝑥𝑥, 𝑦𝑦, 𝑧𝑧) ∈ Ω where Ω is the spatial domain of interest, 𝒔𝒔 ∈ S 2 is the direction of propagation where S is the solid angle space, 𝛽𝛽(𝒙𝒙) = 𝜅𝜅(𝒙𝒙) + 𝜎𝜎(𝒙𝒙) is the extinction coefficient with 𝜅𝜅(𝒙𝒙)the absorption coefficient and 𝜎𝜎(𝒙𝒙) the scattering coefficient, 𝜎𝜎 𝐵𝐵 is the Stefan-Boltzmann constant, 𝑚𝑚 is the real part of the complex index of refraction of the medium and 𝑇𝑇 the temperature. Φ(𝒔𝒔, 𝒔𝒔 ′ ) is the scattering phase function which is depicted in this work by the Henyey-Greenstein function. In the present study, the spatial

domain of interest Ω is a cube i.e. Ω = {(𝑥𝑥, 𝑦𝑦, 𝑧𝑧)|𝑥𝑥 ∈ [0, 𝐿𝐿], 𝑦𝑦 ∈ [0, 𝐿𝐿], 𝑧𝑧 ∈ [0, 𝐿𝐿]}.
To provide closure to the model, the following boundary conditions are used for inward directions 𝒔𝒔. 𝒏𝒏 < 0:

𝐼𝐼(𝒙𝒙, 𝒔𝒔) = 𝜀𝜀 1 𝜋𝜋 𝜎𝜎 𝐵𝐵 𝑛𝑛 2 𝑇𝑇 4 ∀𝑥𝑥 = {𝑥𝑥 𝑛𝑛𝑚𝑚𝑛𝑛 , 𝑥𝑥 𝑛𝑛𝑚𝑚𝑚𝑚 } (8) 𝐼𝐼(𝒙𝒙, 𝒔𝒔) = (1 -𝜀𝜀)𝐼𝐼(𝒙𝒙, 𝒔𝒔 ′ ) ∀𝑥𝑥 = ]𝑥𝑥 𝑛𝑛𝑚𝑚𝑛𝑛 , 𝑥𝑥 𝑛𝑛𝑚𝑚𝑚𝑚 [ ( 9
)
where 𝜀𝜀 is the total emissivity of the two emissive walls at 𝑥𝑥 𝑛𝑛𝑚𝑚𝑛𝑛 (fixed at 𝑇𝑇 ℎ𝑛𝑛𝑜𝑜 ) and 𝑥𝑥 𝑛𝑛𝑚𝑚𝑚𝑚 (fixed at 𝑇𝑇 𝑐𝑐𝑛𝑛𝑐𝑐𝑐𝑐 ). Furthemore 𝒔𝒔 ′ is deduced from 𝒔𝒔 by using the Householder rotation matrix. The Eq. ( 9) depicts periodic boundary conditions with specular reflection for the four other faces of the cube. The knowledge of 𝐼𝐼(𝒙𝒙, 𝒔𝒔) allows to compute the integrated intensity 𝐺𝐺(𝒙𝒙) defined by :

𝐺𝐺(𝒙𝒙) = ∫ 𝐼𝐼(𝒙𝒙, 𝒔𝒔)𝑑𝑑𝒔𝒔 4𝜋𝜋 (10)
This latter quantity is injectetd in the steady-state heat energy balance which writes :

∇. (-𝜆𝜆(𝒙𝒙)∇𝑇𝑇) + 𝜅𝜅(𝒙𝒙)4𝜋𝜋𝐼𝐼 𝑏𝑏 (𝑇𝑇) -𝜅𝜅(𝒙𝒙)𝐺𝐺(𝒙𝒙) = 0 (11)
with 𝐼𝐼 𝑏𝑏 (𝑇𝑇) = 1 𝜋𝜋 ⁄ 𝜎𝜎 𝐵𝐵 𝑛𝑛 2 𝑇𝑇 4 . The boundary conditions for solving Eq. ( 11) are the following :

𝑇𝑇 = 𝑇𝑇 ℎ𝑛𝑛𝑜𝑜 ∀𝑥𝑥 = {𝑥𝑥 𝑛𝑛𝑚𝑚𝑛𝑛 }, 𝑇𝑇 = 𝑇𝑇 𝑐𝑐𝑛𝑛𝑐𝑐𝑐𝑐 ∀𝑥𝑥 = {𝑥𝑥 𝑛𝑛𝑚𝑚𝑚𝑚 }, -∇𝑇𝑇. 𝒏𝒏 = 0 ∀𝑥𝑥 = ]𝑥𝑥 𝑛𝑛𝑚𝑚𝑛𝑛 , 𝑥𝑥 𝑛𝑛𝑚𝑚𝑚𝑚 [ (12)

Discrete-scale coupled model

The discrete-scale coupled model consists in considering a solid phase domain, Ω 𝑠𝑠 , immersed in a fluid phase domain, Ω 𝑓𝑓 . Ω 𝑠𝑠 ∪ Ω 𝑓𝑓 forms the whole computational domain Ω. This description corresponds well to domains defined from the 3D architectures given by GenMat. The solid (resp. fluid) phase boundary is given by ∂Ω 𝑠𝑠 (resp. ∂Ω 𝑓𝑓 ) and the solid-fluid interface is denoted by Γ = ∂Ω 𝑠𝑠 ∩ ∂Ω 𝑓𝑓 . Let us define 𝒏𝒏 𝒔𝒔 (resp. 𝒏𝒏 𝒇𝒇 ) the outward unit vector normal to the solid (resp. fluid) phase. The approach has already been detailed elsewhere [START_REF] Ouchtout | Finite element framework for modeling conducto-radiative transfers within heterogeneous media at both discrete and continuous scales[END_REF][START_REF] Badri | Conductive-radiative heat transfer within SiC-based cellular ceramics at high-temperatures: A discrete-scale finite element analysis[END_REF] and is now decribed briefly. First the steady-state heat conduction equation is solved in the solid phase which must be optically thick. The energy transfer between the solid phase and the void phase occurs at the boundary only. The conduction problem consists in searching a scalar function 𝑇𝑇 𝑠𝑠 verifying :

∇. (-𝜆𝜆 𝑠𝑠 ∇𝑇𝑇 𝑠𝑠 ) = 0 ∀𝑥𝑥 ∈ Ω 𝑠𝑠 ( 13
)
where 𝜆𝜆 𝑠𝑠 is the thermal conductivity of the solid phase. Similarly to the continuous-scale model, the temperature is prescribed on the opposite parts of the solid phase.

𝑇𝑇 𝑠𝑠 = 𝑇𝑇 ℎ𝑛𝑛𝑜𝑜 on ∂Ω 𝑠𝑠,𝐷𝐷,-⊂ Γ with ∂Ω 𝑠𝑠,𝐷𝐷,-= {𝑥𝑥 ∈ ∂Ω 𝑠𝑠 , 𝒙𝒙 < 𝛿𝛿 + 𝑚𝑚𝑚𝑚𝑛𝑛 Ω 𝑠𝑠 𝒙𝒙 ′ } (14)

𝑇𝑇 𝑠𝑠 = 𝑇𝑇 𝑐𝑐𝑛𝑛𝑐𝑐𝑐𝑐 on ∂Ω 𝑠𝑠,𝐷𝐷,+ ⊂ Γ with ∂Ω 𝑠𝑠,𝐷𝐷,+ = {𝑥𝑥 ∈ ∂Ω 𝑠𝑠 , 𝒙𝒙 < 𝛿𝛿 + 𝑚𝑚𝑚𝑚𝑥𝑥 Ω 𝑠𝑠 𝒙𝒙 ′ } (15)
In these relationships, the subscript 𝐷𝐷 stands for Dirichlet and 𝛿𝛿 is a sufficiently small positive parameter so that the Dirichlet condition is applied on a boundary of sufficiently large enough area. The boundary condition over Γ are flux continuity relations taking into account both the emission loss and the incoming flux radiation:

-𝜆𝜆 𝑠𝑠 ∇𝑇𝑇 𝑠𝑠 . 𝒏𝒏 𝒔𝒔 = 𝜀𝜀 𝑠𝑠 𝜎𝜎 𝐵𝐵 𝑛𝑛 𝑠𝑠 2 𝑇𝑇 𝑠𝑠 4 -𝜀𝜀 𝑠𝑠 ∫ 𝐼𝐼 𝑓𝑓 𝒔𝒔.𝒏𝒏 𝒇𝒇 >0 𝒔𝒔. 𝒏𝒏 𝒇𝒇 𝑑𝑑𝒔𝒔 [START_REF] Guévelou | Representative elementary volumes required to characterize the normal spectral emittance of silicon carbide foams used as volumetric solar absorbers[END_REF] where 𝑛𝑛 𝑠𝑠 is the index of refraction of the solid phase and 𝑇𝑇 𝑠𝑠 the temperature on Γ. Furthermore the radiation problem is only solved in the fluid phase (considered here as purely transparent) where the steady state radiative transfer equation is reduced to :

𝒔𝒔. ∇𝐼𝐼 𝑓𝑓 = 0 ∀𝑥𝑥 ∈ Ω 𝑓𝑓 (17)
Since thermal exchanges can occur on Γ, an inlet boundary condition is defined (the " � " symbol stands for inlet condition):

𝐼𝐼 ̂(𝒙𝒙, 𝒔𝒔) = 𝜀𝜀 𝑠𝑠 1 𝜋𝜋 𝜎𝜎 𝐵𝐵 𝑛𝑛 2 𝑇𝑇 4 + (1 -𝜀𝜀 𝑠𝑠 ) 𝐼𝐼(𝒙𝒙, 𝒔𝒔 ′ ) on Γ with 𝒔𝒔. 𝒏𝒏 𝒇𝒇 < 0 ( 18
)
The problem is closed by considering on the two ends of Ω 𝑓𝑓 , the two hot and cold prescribed temperatures, yielding the relationship:

𝐼𝐼(𝒙𝒙, 𝒔𝒔) = 𝜀𝜀 1 𝜋𝜋 𝜎𝜎 𝐵𝐵 𝑛𝑛 2 �𝑇𝑇 ℎ𝑛𝑛𝑜𝑜 � 4 ∀𝑥𝑥 = {𝑥𝑥 𝑛𝑛𝑚𝑚𝑛𝑛 } and 𝐼𝐼(𝒙𝒙, 𝒔𝒔) = 𝜀𝜀 1 𝜋𝜋 𝜎𝜎 𝐵𝐵 𝑛𝑛 2 �𝑇𝑇 𝑐𝑐𝑛𝑛𝑐𝑐𝑐𝑐 � 4 ∀𝑥𝑥 = {𝑥𝑥 𝑛𝑛𝑚𝑚𝑚𝑚 } (19)

RESULTS AND DISCUSSION

In the following, the room-temperature complex refractive index of silicon carbide at the wavelength of 4 µm, is affected to the solid skeleton of each numerical 3D structure [START_REF] Coquard | Homogeneous phase and multi-phase approaches for modeling radiative transfer in foams[END_REF]. It ensures that their ligaments can be considered as optically thick, their lowest diameter being fixed at 30 µm. For the sake of clarity, the acronym CCS means cubic cell structures. To complete their denomination, one adds an information regarding the nominal pore diameter, 𝑑𝑑 𝑛𝑛𝑛𝑛𝑛𝑛 , the number of cells, 𝑁𝑁 𝑐𝑐 , and the size of the domain, 𝐿𝐿. CCS_160_1_380 corresponds therefore to a cubic cell structure with 𝑑𝑑 𝑛𝑛𝑛𝑛𝑛𝑛 = 160 µm, 𝑁𝑁 𝑐𝑐 = 1 and 𝐿𝐿 = 380 µm.

Influence of the textural parameters of cubic cell structures on their extinction behaviour

Five cubic cell structures with a growing cell number (𝑁𝑁 𝑐𝑐 = 1 3 , 3 3 , 5 3 , 7 3 , 9 3 ) with same size (𝑑𝑑 𝑛𝑛𝑛𝑛𝑛𝑛 = 160 µm) are firstly created with GenMat. Table 1 gives their respective porosity and volumetric surface after having performed the final porosity tuning process as explained in section 2.1. The numerical samples are described in Fig. 1 Table 1 The analysis of their two-point autocorrelation functions indicate that the influence of the lattice periodicity weakens when the size of the computational domain becomes important. Fig. 2) shows their extinction cumulative distribution functions 𝑙𝑙𝑛𝑛�1 -𝐺𝐺 𝑒𝑒 (𝑠𝑠)� as a function of 𝑠𝑠, obtained for 𝑁𝑁 𝑅𝑅 = 10 6 .

Fig. 1 3D images of (1) CCS_160_1_380 (2) CCS_160_27_760, (3) CCS_160_125_1040 ( 4) CCS_160_343_1330 (5) CCS_160_729_1710

Special cares have been taken to define a suitable shooting zone for computing 𝑙𝑙𝑛𝑛�1 -𝐺𝐺 𝑒𝑒 (𝑠𝑠)� i.e., not too small to be representative, and not too large to limit outgoing rays. The curves clearly deviate from what is expected for a Beerian behaviour (a perfect linear profile is expected in this case). It is noted that the deviations are all the less proncunced when the size of the computational domain become large.

To go one step further, each extinction cumulative distribution function curve has been nicely reproduced using the Levengberg-Marquardt scheme by the following expression:

𝑦𝑦(𝑠𝑠) = -𝑚𝑚�𝑒𝑒 𝑏𝑏𝑠𝑠 -1� (20)
The results of the fitting process are shown for for CCS_160_1_380 and CCS_160_27_760 in Fig2b).

In Gomez-Garcia et al, the authors proposed a quite similar expression for analysing the radiation propagation within a stack of SiC-based square grids [START_REF] Gomez-Garcia | Numerical analysis of radiation propagation in a multi-layer volumetric solar absorber composed of a stack of square grids[END_REF]. Table 1 lists the values of 𝑚𝑚 and 𝑏𝑏 for the five numerical samples. Going from CCS_160_1_380 up to CCS_160_729_1710, it can be evidenced that the lowest value of 𝑏𝑏 are required to confer a Beerian behaviour to the set of numerical 3D architectures.

It is worth of notify that if 𝑏𝑏𝑠𝑠 is lower than 1, 𝑦𝑦(𝑠𝑠)~-𝑚𝑚𝑏𝑏𝑠𝑠 and then the extinction curve becomes Beerian. By using Eq. ( 6) and Eq. ( 20), one can deduce the following relationship 𝛽𝛽(𝑠𝑠) = 𝑦𝑦(𝑠𝑠) 𝑠𝑠 ⁄ . For small values of 𝑠𝑠, the exponential term can be approximated by its Taylor expansion and in this case, 𝛽𝛽 𝑚𝑚𝑠𝑠 ∼𝑚𝑚𝑏𝑏.

Here, 𝛽𝛽 𝑚𝑚𝑠𝑠 represents the asymptotic value of 𝛽𝛽(𝑠𝑠). This latter quantity is different from 𝛽𝛽 𝑛𝑛𝑛𝑛 = 𝐴𝐴 4𝑝𝑝 ⁄ which is the value of the extinction coefficient for a pure beerian medium at the optically thin limit. With GenMat, one can also check by fixing 𝐿𝐿, and by progressively increasing 𝑑𝑑 𝑛𝑛𝑛𝑛𝑛𝑛 , that the deviation to the Beerian behaviour become more accentuated. On the other side, by keeping 𝑁𝑁 𝑐𝑐 constant and decreasing 𝑑𝑑 𝑛𝑛𝑛𝑛𝑛𝑛 so that 𝐿𝐿 is homothetically reduced, the deviation to the Beerian behaviour appears more and more important. In conclusion, it can be highligted that for regular cubic cell architectures constituted of an opque solid network a large value of 𝐿𝐿 combined with a low value of 𝑑𝑑 𝑛𝑛𝑛𝑛𝑛𝑛 must exhibit an extinction feature closed to the one that can be expected from a Beerian medium.

Influence of the textural parameters of cubic cell structures on their conductiveradiative behaviour

In this last part, the different extinction coefficients reported in Table 1 for CCS_160_1_380 and CCS_160_27_760 are used when the continuous-scale (CS) coupled approach described in section 2.3. is applied. Let us recall that this work is focused on solid opaque phases (SiC-based consolidated networks) immersed within a transparent fluid phase. This suggests that their corresponding homogeneized extinction coefficients must be equal to those corresponding to their fluid phase (here 𝛽𝛽(𝑥𝑥) and 𝛽𝛽 𝑚𝑚𝑠𝑠 ) as demonstrated Baillis et al. in Ref. [START_REF] Coquard | Homogeneous phase and multi-phase approaches for modeling radiative transfer in foams[END_REF] for metallic foams constituted of opaque skeletons immersed in air.

.

Fig 3 a)

Average temperature along the axis of propagation with (1) the DM approach (2) the CS approach with 𝛽𝛽 𝑛𝑛𝑛𝑛 (3) the CS approach with 𝛽𝛽 𝑚𝑚𝑠𝑠 (4) the CS approach with 𝛽𝛽(𝑥𝑥) for CCS_160_1_380 b) Average temperature along the axis of propagation with (1) the DM approach (2) the CS approach with 𝛽𝛽 𝑛𝑛𝑛𝑛 (3) the CS approach with 𝛽𝛽 𝑚𝑚𝑠𝑠 (4) the CS approach with 𝛽𝛽(𝑥𝑥) for CCS_160_27_76

In addition to the CS approach, the discrete-scale (DM) coupled approach, also exposed in section 2.3, is directly used with the two 3D structures. Concerning the stabilized vectorial Finite Element Method framework required to solve the set of equations for the two approaches and the post-processing method developed for extracting the temperature profiles, full details are given in Ref [START_REF] Ouchtout | Finite element framework for modeling conducto-radiative transfers within heterogeneous media at both discrete and continuous scales[END_REF][START_REF] Badri | Conductive-radiative heat transfer within SiC-based cellular ceramics at high-temperatures: A discrete-scale finite element analysis[END_REF]. As previsouly evoked, we solve the conductive-raditive problem pionneringly defined by Grosh and Vistankta for a 1D geometry where a semi-transparent medium is sandwhiched between two isothermal parallel opaque plates with different temperatures (𝑇𝑇 ℎ𝑛𝑛𝑜𝑜 = 2222 K and 𝑇𝑇 𝑐𝑐𝑛𝑛𝑐𝑐𝑐𝑐 = 1111 K) [START_REF] Viskanta | Heat transfer by simultaneous conduction and radiation in an absorbing medium[END_REF]. Let us explain that for the DM approach, 𝜀𝜀 𝑠𝑠 = 0.9 and 𝜆𝜆 𝑠𝑠 = 30 W.m -1 .K -1 . In this work, the curves given by the DM method are considered as the reference data (see curves 1 in Fig. 3a and Fig. 3b). For the two numerical samples, the injection of 𝛽𝛽 𝑛𝑛𝑛𝑛 in Eq. ( 7) gives temperature profiles pretty closed to those computed through the DM approach. However, when 𝛽𝛽 𝑚𝑚𝑠𝑠 and 𝛽𝛽(𝑥𝑥) are used in the CS approach for CCS_160_1_380 and CCS_160_27_760, it can be emphasised that the temperature profiles present stronger differences. Indeed, the injection of 𝛽𝛽 𝑚𝑚𝑠𝑠 and 𝛽𝛽(𝑥𝑥) in the Eq. ( 7) tends to increase the conduction-to-radiation parameter (i.e. the Planck number) which results in an unexpected conductive behaviour. The differences with the reference profiles (∆𝑇𝑇 𝑛𝑛𝑚𝑚𝑚𝑚 ∼ 300 K in CCS_160_1_380) are the most important at more than three quarters of the propagation of the radiation in the two porous structures. Moreover, the use of 𝛽𝛽(𝑥𝑥) tends to accentuate the conductive transfer and to decrease the radiative one.

CONCLUSION

This work combines the numerical generation of cubic lattices constituted of SiC, the determination of their extinction cumulative distribution functions and the computation of the their conductive-radiative b) a)

behaviour respectively with a FEM-based continuous-scale and a discrete scale approaches. All the numerical samples can be considered as non-Beerian and their extinction cumulative distribution functions can be described with an exponential growth law as a function of the variable path length. It can be underlined that the injection of the exctintion coefficient, obtained at the optically thin limit in the framework of the Radiative Distribution Function Identification method, in the continuous scale approach give results quite similar to what one can be directly computed on the 3D images through the discrete scale approach. For regular macroporous 3D architectures, more works are required to finely connect their extinction coefficients (and their possible spatial variation) for each considered phase (fluid and solid) with the exact corresponding effective extinction coefficient required to solve the radiative transfer equations at the macroscopic scale.

Fig2 a )

 a Fig2 a) Extinction cumulative distribution function: (1) CCS_160_1_380 (2) CCS_160_27_760, CCS_160_125_1040 (4) CCS_160_343_1330 (5) CCS_160_729_1710 b) (1) Extinction cumulative distribution function for CCS_160_1_380 (2) Fitted curve with Eq. 20 for CCS_160_1_380 (3) -𝛽𝛽 𝑚𝑚𝑠𝑠 𝑠𝑠 for CCS_160_1_380 (4) Extinction cumulative distribution function for CCS_160_27_760 (5) Fitted curve with Eq. 20 for CCS_160_27_760 (6) -𝛽𝛽 𝑚𝑚𝑠𝑠 𝑠𝑠 for CCS_160_27_760 (7) -𝛽𝛽 𝑛𝑛𝑛𝑛 𝑠𝑠 for CCS_160_27_760


Alexandra Gruss
email: alexandra.gruss@inrae.fr

A FAS solution to a DEAD case

CshA mutant cold sensitivity is a membrane problem

. This SCFA to BCFA ratio is critical to membrane fluidity. Fatty acid extractions showed that SCFA to BCFA ratios were elevated in the cshA mutant compared to the WT strain. Less PDH (or more BCFA precursor production) in tested suppressor mutants restored this ratio to that of the WT, so that the membrane would regain fluidity. This highly documented study identifies pdh mRNA as the essential degradosome target for cold survival, and highlights the intimate connection between membrane state and central metabolism via acetyl-CoA.

Why are FASII genes targets for cshA suppressors?

Most cshA suppressors mapped to FASII or associated pathways. Mutations in FASII-related genes all presumably slow down phospholipid synthesis. Khemici and colleagues generalized this observation by showing that subinhibitory amounts of triclosan, which inhibits the FASII pathway protein FabI, suppressed cshA cold sensitivity. Their study also revealed that fakA, encoding a fatty acid kinase, was a cshA suppressor hotspot that corrected the SCFA to BCFA ratio. FakA depletion leads to free fatty acid accumulation that may modify S. aureus regulation and FASII activity and alter acetate utilization [START_REF] Demars | Redirection of Metabolism in Response to Fatty Acid Kinase in Staphylococcus aureus[END_REF][START_REF] Ericson | Role of Fatty Acid Kinase in Cellular Lipid Homeostasis and SaeRS-Dependent Virulence Factor Expression in Staphylococcus aureus[END_REF]. How the slowdown of FASII synthesis adjusts fatty acid membrane composition and fluidity is a new open question arising from this work.

Do all cshA mutant suppressors relate to the SCFA to BCFA ratio and membrane fluidity?

The selection for cshA cold-tolerant suppressors identified loci seemingly unrelated to membrane biogenesis. However, underlying links may exist. For example, mutations in ndhF, encoding a membrane protein involved in the electron transport chain, would lead to reduced respiration. Consequently higher intracellular NADH might lower PDH activity, which uses NAD as cofactor. Also, suppression by mutating RNA polymerase cofactor gene rpoE may have pleiotropic effects on expression of numerous membrane-related functions [START_REF] Weiss | The delta subunit of RNA polymerase guides promoter selectivity and virulence in Staphylococcus aureus[END_REF]. Mutations that impact lipoteichoic acid production and lipoprotein maturation may not directly affect phospholipid composition but could still affect membrane fluidity. Using the SCFA to BCFA ratio as readout to understand cshA suppressors as done here will be a valuable means of characterizing these "unknown" suppressors.

New lines of exploration arising from the DEAD-membrane connection

The approach and findings reported by Khemici and colleagues provide a conceptual scaffold for discovering factors involved in the balance between central metabolism and the membrane. For example, anaerobic growth, which uses Pfl rather than Pdh to produce acetyl-CoA, might generate different conditions for cshA suppression. S. aureus incorporates exogenous unsaturated fatty acids, as enriched in host tissues [START_REF] Kenanian | Permissive Fatty Acid Incorporation Promotes Staphylococcal Adaptation to FASII Antibiotics in Host Environments[END_REF], which might fluidify membranes and suppress cshA cold sensitivity, and idem for fluidifying solvents. Cold-sensitivity of cshA is suppressed in solid, but not liquid medium, raising the question of cell-to-cell contact, nutrient availability, and growth phase, in modulating membrane properties.

PDH and its product, acetyl-CoA, were identified here as the connecting link between cshA mutant cold survival and membrane lipid composition, as a means of adjusting membrane fluidity. Remarkably, all PDH subunits in S. aureus are prominent components of extracellular membrane vesicles [START_REF] Tartaglia | Staphylococcus aureus Extracellular Vesicles Elicit an Immunostimulatory Response in vivo on the Murine Mammary Gland[END_REF]. Membrane blebbing could be a rapid means of removing or shuttling enzymes. Given the present study, a far-fetched but testable possibility is that PDH enrichment in extracellular vesicles might deplete intracellular PDH and favor BCFA synthesis and cell membrane fluidity. This epigenetic regulation from "without" could be an economical and rapid way to adjust membrane properties.

The membrane as a bacterial thermometer was demonstrated by several studies in the de Mendoza lab. The rigid or fluid state of membrane lipids is transmitted to membrane proteins by affecting their conformation and/or localization [START_REF] Inda | A lipid-mediated conformational switch modulates the thermosensing activity of DesK[END_REF][START_REF] Sastre | Membrane fluidity adjusts the insertion of the transacylase PlsX to regulate phospholipid biosynthesis in Gram-positive bacteria[END_REF]. In S. aureus, membrane fluidity affects expression of virulence factors [START_REF] Lopez | Host-derived fatty acids activate type VII secretion in Staphylococcus aureus[END_REF]. Khemici and colleagues' studies add CshA-degradosome turnover of pdh mRNA as another type of response to the membrane thermometer. The signal leading to pdh mRNA degradation remains to be investigated, but one possibility is that a membrane-associated degradosome component, RNase Y [START_REF] Khemici | Decay-Initiating Endoribonucleolytic Cleavage by RNase Y Is Kept under Tight Control via Sequence Preference and Sub-cellular Localisation[END_REF], acts as a sensor-relay of the membrane state.

The branched-chain α-keto acid dehydrogenase complex (referred to as BKD or BFM), synthesizes branched-chain acyl-CoA, the preferred competitors of acetyl-CoA in S. aureus [START_REF] Qiu | Crystal structure and substrate specificity of the beta-ketoacyl-acyl carrier protein synthase III (FabH) from Staphylococcus aureus[END_REF]. Interestingly, BKD is a PDH complex homolog. Contrary to pdh mRNA degradation at low temperature, mRNA of the BKD analog in Bacillus subtilis is cold-stabilized [START_REF] Nickel | Cold induction of the Bacillus subtilis bkd operon is mediated by increased mRNA stability[END_REF]. It will be exciting to determine whether the S. aureus BKD complex is regulated by the degradosome under conditions where fluidity may be deleterious, e.g., at high temperature.

Altogether, this original and highly documented study brings a new clarification to the way S. aureus coordinates fundamental metabolic functions. The original conclusions and insight provided in this work will be valuable in gaining a holistic understanding of how cells respond to stress.

Fig 1 .

 1 Fig 1. Schematic model of how CshA, the DEAD-box helicase, modifies the membrane state in cold conditions. A cold stress signal (snowflake) is transmitted via the membrane (at top), leading to membrane changes (at bottom). (A) cshA mutant. pdh mRNA accumulates in the cold. Greater PDH activity increases acetyl-CoA pools relative to competing branched-chain acyl-CoA substrates, favoring synthesis of straight-chain (rigid) fatty acids. Membrane rigidity accounts for cold sensitivity. (B) WT strain. The RNA degradosome comprising CshA is active and degrades pdh mRNA, which limits acetyl-CoA production. Less acetyl-CoA relative to branched-chain acyl-CoA favors BCFA synthesis, which fluidifies the membrane. Numerous cshA suppressor mutations restore the balance between straight and BCFA [1]. Products leading to rigid membranes are blue; those leading to fluid membranes are red. Black wavy lines represent pdh mRNA; dashed line indicates mRNA is degraded. BC, branched-chain; BCFA, branched-chain fatty acid; BKD, branched-chain αketo acid dehydrogenase; PDH, pyruvate dehydrogenase; WT, wild type. https://doi.org/10.1371/journal.pgen.1008842.g001

PLOS Genetics | https://doi.org/10.1371/journal.pgen.1008842 July 30, 2020

Acknowledgments

AG gratefully acknowledges P. Bouloc (CNRS, France) for critical reading of this review, and colleagues J. Anba-Mondoloni, P. Wongdontree, A. Pathania, and K. Gloux (Micalis) for frequent discussions on the roles of lipids in bacterial physiology.

Work in AG group is supported by funding from the Fondation pour la Recherche Medicale (DBF20161136769) and the French Agence Nationale de la Recherche (StaphEscape project ANR-13001038).