M A Badri

Y Favennec

P Jolivet

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

Conductive-radiative heat transfer within SiC-based cellular ceramics at high-temperatures: a discrete-scale finite element analysis

Keywords: Heat transfer, Conductive-radiative transfer, Open-cell cellular ceramics, Finite element analysis, Discrete-scale approach

Cellular ceramic materials possess many favorable properties that allow to develop efficient modern-day hightemperature thermal energy conversion systems and processes. The energy conversion within these porous media is governed by tightly coupled conduction-radiation physics. To efficiently design and optimize these systems, a comprehensive understanding of the conduction-radiation behavior within these materials becomes important. In this study, by performing large-scale numerical experiments, we analyze the conduction-radiation coupling characteristics within different (with respect to topology and porosity) SiC-based open-cell cellular ceramics surrounded by fictitious vacuum up to temperatures of 1800 K. To induct minimal approximations, our finite element simulations are based on a discrete-scale approach within which realistic discrete (porelevel) representations of the cellular ceramics are used as numerical media. The results presented in this study provide means to better understand the role of radiation in the coupled conduction-radiation physics within the ceramic samples. A detailed comparison on effectiveness of energy conversion is established for SiC-based full-scale cubic-cell, Kelvin-cell, and pseudo-random-cell ceramic structures which are at 80 % and 90 % porosity each. In conclusion, among the different standalone and full-scale ceramic samples, the Kelvin-cell structures at 90% porosity have proven to benefit the most from radiation coupling.

Introduction

Open-cell cellular ceramics belong to a class of smart materials that are often used as a key element for designing high-temperature engineering systems. Thermal insulation systems [START_REF] Ferrari | Sandwich structured ceramic matrix composites with periodic cellular ceramic cores: An active cooled thermal protection for space vehicles[END_REF], porous gas burners [START_REF] Fuessel | Advancement of cellular ceramics made of silicon carbide for burner applications[END_REF], volumetric solar power receivers [START_REF] Kribus | The promise and challenge of solar volumetric absorbers[END_REF], solar thermochemical systems [START_REF] Furler | Thermochemical CO2 splitting via redox cycling of ceria reticulated foam structures with dual-scale porosities[END_REF], hot gas filters [START_REF] Nacken | Development of a catalytic ceramic foam for efficient tar reforming of a catalytic filter for hot gas cleaning of biomass-derived syngas[END_REF], and heat exchangers [START_REF] Ortona | SiSiC heat exchangers for recuperative gas burners with highly structured surface elements[END_REF],

are few examples of such systems. Involvement of high-temperature physics implicates accurate prediction of heat transport as one of the primary factors for optimizing the global performance of these systems. As such, many researchers have focused on studying (numerically or experimentally) the behavior of open-cell cellular ceramics at high temperatures, especially at temperatures beyond 1200 K [START_REF] Mendes | Measurement and simplified numerical prediction of effective thermal conductivity of open-cell ceramic foams at high temperature[END_REF][START_REF] Dietrich | Thermal conductivity of ceramic sponges at temperatures up to 1000 • C, Special Topics & Reviews in Porous Media[END_REF][START_REF] Pelanconi | Thermal design, optimization and additive manufacturing of ceramic regular structures to maximize the radiative heat transfer[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]. Complementary to these researches, the current article focuses on analyzing coupled conductive-radiative heat transport of topologically different open-cell cellular ceramics up to 1800 K.

By definition, open-cell cellular ceramics are porous materials composed of an interleaved network of ligaments (the solid phase), which encompasses a void phase. Because of their intrinsic material properties, open-cell cellular ceramics offer many distinct advantages over other materials such as metals or polymers. They possess: high strength-to-weight ratio, high specific surface area, good flow-mixing capacity, high thermal shock resistance, and high resistance to chemical corrosion [START_REF] Colombo | Conventional and novel processing methods for cellular ceramics[END_REF]. These set of properties have made open-cell cellular ceramics an attractive choice for designing high-temperature compact systems, examples of which were provided in the preceding paragraph.

To improve the performance of high-temperature systems based on open-cell cellular ceramics, understanding the detailed energy transport involved within such systems is important. The energy transport within most of these devices are due to the coupling between the two main individual modes of heat transfer, conduction and radiation [START_REF] Fischedick | High temperature two-phase thermal conductivity of ceramic sponges with stagnant fluid-experimental results and correlation including thermal radiation[END_REF]. Because of their microstructure (pore size, wall thickness...), these materials have a complex multiscale texture which actively attenuates thermal transfer via both conduction and radiation. for temperatures beyond 1200 K, numerical approaches are decisive for estimating the conduction-radiation coupling via the effective thermal conductivity [START_REF] Tseng | Effect of foam properties on heat transfer in high temperature open-cell foam inserts[END_REF]. This is because such temperatures are beyond the working range of current-day experimental facilities (guarded hot-plate, transient plane source, transient hot-wire technique), often applied for the characterization of effective thermal conductivity [START_REF] Mendes | Measurement and simplified numerical prediction of effective thermal conductivity of open-cell ceramic foams at high temperature[END_REF][START_REF] Dietrich | Thermal conductivity of ceramic sponges at temperatures up to 1000 • C, Special Topics & Reviews in Porous Media[END_REF], or present hardly exploitable signals (laser flash technique) [START_REF] Coquard | Experimental investigations of the coupled conductive and radiative heat transfer in metallic/ceramic foams[END_REF].

To numerically study the conduction-radiation coupling behavior within porous media, and consequently in the open-cell cellular ceramics, two main approaches have been established: the continuum-scale approach and the discrete-scale approach. The continuum-scale approach consists in modeling the governing heat transfer equations within an equivalent continuous medium. The approach is often dubbed as the homogeneous phase approach [START_REF] Coquard | Homogeneous phase and multi-phase approaches for modeling radiative transfer in foams[END_REF]. As an input, it requires, beforehand, the knowledge of the effective thermophysical quantities (volume-averaged properties). These properties are either obtained from analytical/empirical relationships, or are computed [START_REF] Lipiński | Application of the spatial averaging theorem to radiative heat transfer in two-phase media[END_REF][START_REF] Whitaker | The Method of Volume Averaging[END_REF]. Given the volume-averaged properties, the computations are then carried out in representative elementary volumes, assuming thermal equilibrium between both solid and fluid phases. When thermal gradients exist, some authors have introduced a refined method called the multiphase approach that requires solving a set of two coupled radiative transfer equations [START_REF] Coquard | Homogeneous phase and multi-phase approaches for modeling radiative transfer in foams[END_REF][START_REF] Lipiński | Application of the spatial averaging theorem to radiative heat transfer in two-phase media[END_REF]. The reason that motivates the continuum-scale approach is its low computational cost, and that it avoids topological complexities of realworld porous problems. Undoubtedly, these methods can well predict the overall thermal transport behavior for the open-cell cellular ceramics. However, the calculation of the exact radiative fields or the exact temperature field within the volume of the open-cell cellular ceramics is beyond the capabilities of these methods. It may lead to misunderstanding the real role played by the solid network in terms of its radiative propagation.

The second approach, the one used in this communication, is called the discrete-scale or the pore-scale approach. This kind of approach involves direct pore-scale simulations, while using realistic topological data (tomographic or virtually generated) of the ceramic skeletons [START_REF] Mendes | Detailed and simplified models for evaluation of effective thermal conductivity of open-cell porous foams at high temperatures in presence of thermal radiation[END_REF][START_REF] Li | Integrated simulation of continuous-scale and discretescale radiative transfer in an open-cell foam made of semitransparent absorbing-scattering ceramics[END_REF]. The generalized heat equation laws are applied at the microstructure level, i.e, within ligaments and pores. Appealingly, these methods can provide critical information concerning the conduction-radiation coupling behavior within the bulk of the ceramics.

The method can also be used to identify the energy localization zones within the ceramics. Moreover, in [START_REF] Li | Pore-level numerical analysis of the infrared surface temperature of metallic foam[END_REF],

it was found that at high temperatures (> 1000 K), the discrete-scale approach yielded more accurate results than the continuum-scale approach. Currently, the method has been drawing considerable attention. However, a major bottleneck of this approach is its computational requirements.

In general, the continuum-scale approach remains popular for characterizing the conduction-radiation behavior of open-cell cellular ceramics. Moreover, for the sake of easiness, the conductive and the radiative contributions are determined separately [START_REF] Mendes | Detailed and simplified models for evaluation of effective thermal conductivity of open-cell porous foams at high temperatures in presence of thermal radiation[END_REF], in a decoupled fashion. One of the simplest ways for performing the radiative calculation is by incorporating the Rosseland diffusion approximation, which assumes that the porous medium can be considered optically thick [START_REF] Fischedick | High temperature two-phase thermal conductivity of ceramic sponges with stagnant fluid-experimental results and correlation including thermal radiation[END_REF][START_REF] Tseng | Effect of foam properties on heat transfer in high temperature open-cell foam inserts[END_REF][START_REF] Mendes | Experimental validation of simplified conduction-radiation models for evaluation of effective thermal conductivity of open-cell metal foams at high temperatures[END_REF][START_REF] Coquard | Conductive and radiative heat transfer in ceramic and metal foams at fire temperatures[END_REF]. Although this approximation allows a qualitative description of the prominent role of radiation when temperature rises, in many cases, the values can be ill-predicted when compared with experimental data [START_REF] Tseng | Effect of foam properties on heat transfer in high temperature open-cell foam inserts[END_REF][START_REF] Mendes | Detailed and simplified models for evaluation of effective thermal conductivity of open-cell porous foams at high temperatures in presence of thermal radiation[END_REF]. In fact, summing the conduction and radiation heat fluxes neglects the coupling effects, and such a simple addition is only applicable for weakly coupled conduction-radiation physics, i.e., at low temperatures.

Alternatively to this simplified superimposing, a more robust approach involves simultaneously considering the divergence both of the conductive and radiative fluxes. Such an approach becomes mandatory for obtaining feasible and reliable solutions. Consequently, in this case, one ends up solving a system of highly nonlinear equations that are tightly coupled. Chen et al. [START_REF] Tseng | Effect of foam properties on heat transfer in high temperature open-cell foam inserts[END_REF] used the spherical harmonics P 1 approximations to solve the radiative heat transfer and the system of equations was finite differenced. Mendes et al. [START_REF] Mendes | Measurement and simplified numerical prediction of effective thermal conductivity of open-cell ceramic foams at high temperature[END_REF] combined the discrete transfer method for the determination of the radiative intensity needed afterwards for estimating the divergence of the radiative flux and the finite volume method for the calculation of the coupled conduction-radiation heat flux. Coquard et al. [START_REF] Coquard | Experimental investigations of the coupled conductive and radiative heat transfer in metallic/ceramic foams[END_REF] used the finite volume method for the energy equation and the discrete ordinates method for the radiative transfer equation to approximate the coupled conductive-radiative physics, within the laser flash method. These numerical methods lead to satisfactory predictions of the temperature (globally), but, once again, since continuous scales were used, the prior accurate determination of both the effective thermal conductivity and of the thermal radiative properties was required. According to the choice of the methods (analytical versus numerical) for computing these effective quantities, some discrepancies can be found [START_REF] Mendes | Detailed and simplified models for evaluation of effective thermal conductivity of open-cell porous foams at high temperatures in presence of thermal radiation[END_REF][START_REF] Randrianalisoa | Thermal conductive and radiative properties of solid foams: Traditional and recent advanced modelling approaches[END_REF], therefore raising some questions on the accuracy of the continuum-scale approach. Furthermore, the calculation of the exact temperature fields within the volume of the cellular ceramics is beyond the capabilities of these methods whereas this information is crucial for understating the local behavior of other physical properties at high temperatures. Supporting this statement, for standalone radiation physics, in [START_REF] Loretz | Analytical modelling of the radiative properties of metallic foams: Contribution of X-ray tomography[END_REF] it was shown that the use of realistic tomographic representations for the open-cell porous materials yields results closer to the experimental measurements. It was also recently underlined in [START_REF] Hardy | 3D numerical modelling of the propagation of radiative intensity through a X-ray tomographied ligament[END_REF][START_REF] Cunsolo | Effects of ligaments shape on radiative heat transfer in metal foams[END_REF] that the real microstructure ligament geometries influence at large the radiative physics for the open-cell porous materials. Also, theoretical developments based on upscaling approaches for coupled heat transfer [START_REF] Leroy | Coupled upscaling approaches for conduction, convection, and radiation in porous media: theoretical developments[END_REF], emphasize on the precise prediction of local-scale energy transport which is possible by using discrete-scale approach.

The previous paragraph explains why the coupling between conduction and radiation directly within the 3D detailed geometries of the cellular ceramics is investigated [START_REF] Li | Integrated simulation of continuous-scale and discretescale radiative transfer in an open-cell foam made of semitransparent absorbing-scattering ceramics[END_REF][START_REF] Ferkl | Multiphase approach to coupled conduction-radiation heat transfer in reconstructed polymeric foams[END_REF][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]. These numerical advances are clearly driven by the increasing access to X-ray tomography experiments for determining the precised topology of realworld materials, and/or to the virtual generation of 3D realistic digitized images. For cellular ceramics, the typical size of cells arranged within these porous materials (∼ 0.5-2 mm) corresponds to the available spatial resolution of laboratory-scale tomographic setups (∼ 5-20 ➭m). This makes it easy to digitize real-world representative elementary volumes of materials, and perform uncoupled conductive and radiative numerical analysis on these volumes [START_REF] Randrianalisoa | Thermal conductive and radiative properties of solid foams: Traditional and recent advanced modelling approaches[END_REF][START_REF] Guévelou | Representative elementary volumes required to characterize the normal spectral emittance of silicon carbide foams used as volumetric solar absorbers[END_REF]. However, when conjugate heat transfers are being investigated, the present-day numerical approaches are limited to 3D images with modest volumes. This is due to important computational cost required to reproduce the underlying heat transfers [START_REF] Mendes | Detailed and simplified models for evaluation of effective thermal conductivity of open-cell porous foams at high temperatures in presence of thermal radiation[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]. Detailing and solving accurately discrete problems on these complex geometries requires meshes with large number of elements.

In this study we solve and analyze the conduction-radiation coupling (under vacuum) within discrete-scale ceramic samples operating at temperatures ranging from 1200 K to 1800 K. The finite element method is used to solve the underlying partial differential equations of the conduction-radiation physics. We focus our effort on SiC-based macroporous samples, this semiconducting material being intensively studied for designing high-temperature systems [START_REF] Scheffler | Cellular Ceramics: Structure, Manufacturing, Properties and Applications[END_REF][START_REF] Dietrich | Thermal conductivity of ceramic sponges at temperatures up to 1000 • C, Special Topics & Reviews in Porous Media[END_REF][START_REF] Guévelou | A simple expression for the normal spectral emittance of open-cell foams composed of optically thick and smooth struts[END_REF][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]. Our deterministic simulations are based on tightly coupled vectorial FE for solving the radiation physics and Galerkin FE for solving the conduction physics. Our vectorial FE solver [START_REF] Badri | Vectorial finite elements for solving the radiative transfer equation[END_REF][START_REF] Badri | High performance computation of radiative transfer equation using the finite element method[END_REF] was previously developed only for handling the radiation physics within participating media. In this article, we extend its capabilities to solve conduction-radiation physics efficiently.

The rest of the paper is as follows. Section 2 details the model of coupled conduction and radiation physics within cellular ceramics that are of interest to this study. Section 3 introduces the main numerical strategies, the variational forms, the implemented algorithm, and the parallel domain decomposition strategy. With the aim of understanding and characterizing conduction-radiation coupling within open-cell cellular ceramics, in section 4, we perform extensive numerical experiments on topologically different cellular ceramic materials. A detailed discussion on the behavior of these different porous ceramic samples is also included. The paper then ends with concluding remarks in section 5.

Model of coupled conduction and radiation

For the global energy transfer within a SiC-based open-cell cellular ceramic at high temperature, both conduction and radiation take place consequently within its solid and void phases. On one hand, the opaque solid phase (SiC ligament network) is subjected to heat conduction driven by large temperature gradients and restricted radiative sources on its boundaries. On the other hand, the transparent void phase around and inbetween the solid SiC ligaments is subjected to radiation. The coupling process between the two phases occurs in the vicinity of their shared boundary. Opaque materials allow for implementing such coupling conditions as all radiation received by the borders of an opaque medium travels negligible distances within the volume before being absorbed. This certainly is the case for SiC ligaments (microstructures), the material considered in this article, as shown in [START_REF] Hardy | 3D numerical modelling of the propagation of radiative intensity through a X-ray tomographied ligament[END_REF].

Before

E HCE = -∇ • λ∇T = 0 ∀x ∈ Ω s ⊂ R 3 , (1)
where T denotes the material temperature at spatial location x = (x, y, z) and λ denotes the homogeneous thermal conductivity of the material. Note that since the conduction-radiation coupling source term is restricted to the surfaces of the SiC ligament network, there is no source term is this equation.

We further use the steady-state discrete ordinates radiative transfer equations [START_REF] Fiveland | Discrete-ordinates solutions of the radiative transport equation for rectangular enclosures[END_REF] for solving the radiation physics involved, by semi-discretizing the integro-differential radiative transfer equation. From here on, this semi-discretized radiative transfer equation set will be referenced as RTE. The radiative problem then consists in searching for

N d scalar-valued functions {I m } N d m=1 : Ω v → R that satisfy a set of N d coupled equations, E RTE = {s m • ∇I m = 0} N d m=1 ∀x ∈ Ω v ⊂ R 3 , (2)
where N d is the total number of discrete directions (ordinates), I m is the mth discrete radiative intensity, and s m is the mth discrete direction vector. One could refer to [START_REF] Modest | Radiative Heat Transfer[END_REF][START_REF] Howell | Thermal Radiation Heat Transfer[END_REF] for detailed explanations on the radiative transfer equation (RTE), and to [START_REF] Ruan | Development of a finite element model for coupled radiative and conductive heat transfer in participating media[END_REF][START_REF] Ghattassi | Galerkin method for solving combined radiative and conductive heat transfer[END_REF] for the conditions applied on diverse boundaries. For the coupled conduction-radiation physics, we impose the following boundary conditions.

• A Dirichlet condition for E HCE is prescribed at both ends of the solid phase (the subscript 'D' stands for 'Dirichlet'):

T = T ∂ s,g- on ∂Ω s,D,-⊂ Γ; ∂Ω s,D,-= {x ∈ ∂Ω s , x < δ + min Ω s x ′ }, T = T ∂ s,g+ on ∂Ω s,D,+ ⊂ Γ; ∂Ω s,D,+ = {x ∈ ∂Ω s , x < -δ + max Ω s x ′ }. (3)
In this relationship, δ is a sufficiently small positive user-defined parameter so that the Dirichlet condition is applied on a boundary of sufficiently large enough area, i.e. |∂Ω s,D,± | > ǫ. T ∂ s,g+ and T ∂ s,g+ are the given hot and cold Dirichlet temperatures, respectively.

• E HCE is additionally supplied with the flux exchange (Neumann) condition, which takes into account the outgoing and the incoming radiation for the solid phase:

-λ∇T • n s = ε N d m=1 I m w m |s m • n v |✶ [Γ∩sm•nv>0] -ε σ n 2 v T ∂ s-v 4 on ∂Ω s \∂Ω s,D,± . (4)
In this equation, T ∂ s-v represents the solid-void interface temperature, ε denotes the emissivity of the material, σ denotes the Stefan-Boltzmann's constant, n v is the refraction index of the solid-void interface, w m is the quadrature weight such that w m = 4π/N d , and ✶ denotes the Heaviside step function which results in zero or one depending on the considered Boolean operation, i.e., ✶ [Γ∩sm•nv>0] equals to one if and only if Γ ∩ s m • n v > 0 and zero elsewhere. In this equation, the first term in the right-hand side denotes the incoming radiation flux (due to integration between the ligaments), and the second term denotes the emission loss.

• For the void phase in which E RTE is solved, the temperature on the boundary of the solid medium comes into play as the initial boundary condition (the subscript 'in' stands for 'input'):

{I m,in } N d m=1 = 1 π σ ε n 2 v T ∂ s-v 4 on Γ, s m • n v < 0. ( 5
)
This type of boundary condition is often called a diffused boundary condition.

• The specular reflection boundary condition is also prescribed for E RTE (the subscript 'R' stands for 'reflective'):

{I m,R } N d m=1 = {ρ N d j=1 δ m,j (n v )I j ✶ [Γ∩sj •nv>0] } N d m=1 on Γ, (6)
where, ρ denotes the surface reflectivity, ρ = (1 -ε), and δ m,j denotes the partition ratio coefficient. For details on δ m,j see [START_REF] Hardy | Specular reflection treatment for the 3D radiative transfer equation solved with the discrete ordinates method[END_REF].

• The vacuum boundary condition is also applied to the void phase. All the six outer boundaries, which form in fact the virtual bounding box of the ceramic, are considered radiatively cold. As such, radiation can pass on to the surroundings. For a standalone Kelvin cell, the various boundaries on which these conditions act are presented in fig. 1a.

Finite elements model

The coupled conduction-radiation physics is governed by the steady-state heat equation ( 1) and the steadystate discrete ordinates radiative transfer equation [START_REF] Fuessel | Advancement of cellular ceramics made of silicon carbide for burner applications[END_REF]. Dirichlet boundary conditions eq. ( 3), along with flux conditions eq. ( 4) are applied to the heat conduction problem. Eventually, the N d inflow conditions eq. ( 5) are applied to the radiation problem.

Before proceeding further, let us introduce Ω h s (resp. Ω h v ) as the finite element mesh of the solid (resp. void) phases, cf. fig. 1b. For the sake of visualization, we present a coarse solid mesh Ω h s (in gray) for a Kelvin cell surrounded by a clipped void phase mesh Ω h v (in red). It should be noted that the two meshes are conforming at the solid-void interface. This facilitates interpolation of different variables between both finite element spaces defined on each mesh.

Finite element modeling is used to solve E HCE . The variational form of the heat conduction problem, E h HCE , consists in searching T ∈ H 1 g (Ω h s ) such that:

E h HCE = Ω h s λ∇T • ∇v dx - ∂Ω h s ε N d m=1 I m w m |s m • n v |✶ [Γ∩sm•nv>0] v dx + ∂Ω h s σ ε n 2 s T T ∂ s-v 3 v dx = 0 ∀v ∈ H 1 (Ω h s ). (7)
Here, T ∂ s-v is the interface temperature, H 1 g (Ω h s ) is a subspace of H 1 (Ω h s ) with functions that satisfy Dirichlet conditions eq. ( 3).

The heat conduction problem E h HCE has been linearized by including the third-order term

T ∂ s-v 3 , with T ∂ s-v
assumed to be a known quantity when solving E h HCE . The second term in eq. ( 7) is the radiation gain, and the final term is the emission loss.

Finite elements are also used to solve E RTE within the void phase. More precisely, using a vectorial finite element space, whose advantages over traditional spaces are pointed out in [START_REF] Badri | Vectorial finite elements for solving the radiative transfer equation[END_REF][START_REF] Badri | Efficient finite element strategies for solving the radiative transfer equation[END_REF], it is possible to precondition and solve the discretized equation efficiently [START_REF] Badri | Preconditioned Krylov subspace methods for solving radiative transfer problems with scattering and reflection[END_REF].

To do so, let the vectorial radiative intensity I gather the set of N d unknown components of the radiative intensity in all prescribed discrete directions, such that I = (I 1 , I 2 , . . . , I N d ). In the same fashion, let the vector of directions S = (s 1 , s 2 , . . . , s N d ). Further, let the vectorial test function V be a vector of N d test functions,

such that V = (v 1 , v 2 , . . . , v N d ), with each v m being defined in a Sobolev space H 1 (Ω h v ), so that the vectorial test function is in the vectorial finite element space H 1 (Ω h v ) = H 1 (Ω h v ) × H 1 (Ω h v ) × • • • × H 1 (Ω h v ).
E h RTE = Ω h v (S • ∇V) ⊤ (γS • ∇I -I) dx + Γ h (R R : S • n v : I) ⊤ V dx + Γ h 1 π σ ε n 2 v T ∂ s-v 4 S • n v : ✶ [S•nv<0] ⊤ V dx = 0 ∀V ∈ H 1 (Ω h v ). (8)
In this equation, γ : Ω h v → R + is the SUPG (Streamline Upwind Petrov-Galerkin) [START_REF] Brooks | Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations[END_REF] stabilizing parameter chosen according to recommendations provided in [START_REF] Avila | Spatial approximation of the radiation transport equation using a subgrid-scale finite element method[END_REF]. T ∂ s-v is the known interface temperature. R R denotes the specular reflection operator (table) which is constructed from the knowledge of ρ(n v , n s , n v , s m ) and δ(n v , n s , n v , s m ) that denote the solid-void interface reflectivity and partition ratio coefficient, respectively.

These are parameters based on the partition ratio method proposed in [START_REF] Hardy | Specular reflection treatment for the 3D radiative transfer equation solved with the discrete ordinates method[END_REF]. In eq. ( 8), we also introduced the following notations, A ⊤ B = i A i B i and (A : B) i = A i B i , for the sake of conciseness.

The first integral of E h RTE deals with the transport of radiation within the bulk of the medium, the second integral deals with the physics of specular reflection on the shared solid-void interface, and the last integral (surface emission) is the influx boundary condition (diffused boundary condition).

The full steady-state problem gathering eqs. ( 7) and ( 8) is nonlinear. As such, in this study, starting from an initial guess, the two problems are solved successively one after the other, in a staggered fashion until the convergence is reached.

Algorithm 1 schematically presents the iterative algorithm to solve the conduction-radiation physics within the SiC structures. Lines 1 to 3 represent the initial setup phase of our algorithm which is done once. For the nonlinear solver (lines 4 to 13), a L ∞ norm test is used in terms of the change in the solution between iterates, with a tolerance ǫ = 10 -2 . Similar stopping criteria for the nonlinear iterations have also been suggested in [START_REF] Sakami | Application of the discrete ordinates method to combined conductive and radiative heat transfer in a two-dimensional complex geometry[END_REF][START_REF] Ruan | Development of a finite element model for coupled radiative and conductive heat transfer in participating media[END_REF]. Extract the solution on the boundary

Input
T ∂ s = T (x ∈ Γ) 7 if T ∂ s -T ∂ s-v ∞ < ǫ then return T (x ∈ Ωs), {Im(x ∈ Ωv)} N d m=1 8
Update the solution on the boundary

T ∂ s-v = T ∂ s 9
Interpolate on the boundary with

T ∂ v = P v s T ∂ s 10
Solve ERTE with given boundary conditions eqs. ( 5) and ( 6)

{Im(x ∈ Ωv)} N d m=1 11 Extract the radiative flux φ ∂ v (x ∈ Γ) = ε N d m=1 Imwm |sm • nv|✶ [Γ∩sm•nv>0]
12

Interpolate on the boundary with

φ ∂ v = P s v φ ∂ v 13 until convergence
Algorithm 1: iterative algorithm for the coupled conduction-radiation modeling.

At this stage, the finite element formulations to solve the coupled conduction-radiation physics involve a scalar finite element problem eq. ( 7) (HCE) and a vectorial finite element problem eq. ( 8) (RTE). Overall, the coupled finite element model requires the solution of a problem with N d + 1 fields. A high value of N d is needed if the geometry is complex and if the RTE is to be solved with high accuracy [START_REF] Hardy | 3D numerical modelling of the propagation of radiative intensity through a X-ray tomographied ligament[END_REF]. In this study, we have used

N d = 512
(second octahedral refinement), as such, the coupled finite element problem of conduction-radiation requires to solve a 513 field problem, with each field being defined at all spatial nodes.

To do so in a distributed-memory parallelism context, the global mesh (void or solid) Ω h is partitioned into

N p non-overlapping meshes {Ω h i } Np i=1 (
subdomains) using ParMETIS [START_REF] Karypis | Parmetis: Parallel graph partitioning and sparse matrix ordering library[END_REF]. A subdomain is then assigned to a single MPI process, and the global matrices for the RTE and the HCE may be assembled in a distributed fashion using eqs. ( 7) and ( 8), respectively. The open-source finite element kernel FreeFEM [START_REF] Hecht | New development in FreeFem++[END_REF] is used for the local matrix assemblies. The open-source library PETSc [START_REF] Balay | PETSc Web page[END_REF] is used for the parallel matrix assemblies and for solving the linear systems.

Numerical experiments

In this section, we detail and discuss the results of discrete-scale numerical experiments performed on different all of these geometries (the standalone cells or the full-scale ceramics) are assumed to be made out of highly dense 6H-SiC sintered ceramic. For this material, we assume ε = 0.9 and λ = 0.3 W cm -1 K -1 . This λ is in fact an average thermal conductivity constructed from the high-temperature experimental data for monocrystalline 6H-SiC, given in [START_REF] Müller | Experimental and theoretical analysis of the high temperature thermal conductivity of monocrystalline SiC[END_REF]. Overall, the description of all our tests is provided in table 1. All the tests (1A to 1D and 2A to 2F) were performed twice, once with the standalone conduction physics and once with the coupled conduction-radiation physics. For the standalone conduction simulations, E HCE was solved by using Dirichlet boundary conditions T ∂ s,g+ = 1800 K and T ∂ s,g-= 1200 K, on the hot and cold borders, respectively, see eq. ( 3), and with a null exchange flux everywhere else. For the coupled conduction-radiation simulations, E HCE and E RTE are solved in a coupled way according to the procedure explained in section 3.

For the boundary conditions, the same Dirichlet conditions are used, plus Neumann conditions from eq. ( 4) for E HCE . For E RTE , the boundary conditions given in eqs. ( 5) and ( 6) are applied. Next, as a numerical choice in the discretization of the RTE, the number of discrete ordinates, N d , has been chosen equal to 512.

To analyze the behavior of the upcoming numerical experiments, we introduce the following quantities.

• Radiative heat flux on the solid-void interface:

Q + r (x ∈ Γ h ) = N d m=1 I m (x)w m |s m • n v (x)|✶ [Γ h ∩sm•nv>0] .
Here, w m is the quadrature weight such that w m = 4π/N d . Q + r (x ∈ Γ h ) defines the net amount of radiative heat flux entering the solid interface along the direction of its normal vector

n s = -n v . Q + r (x ∈ Γ h ) can
be used as a metric for the absorbing quality of a structure.

• Total absorbed radiation on the ligament surface:

q Γ h = Γ h Q + r (x) dx.
This quantity measures the overall power of the sample to absorb incoming radiation.

• Radiative density in the void phase:

G(x ∈ Ω h v ) = N d m=1 I m (x)w m .
Here, G(x ∈ Ω v ) defines the total radiative energy (photon concentration) at a given point x in Ω h v .

• z-averaged temperature T z , i.e, mean temperature for a given xy-plane located at z = ẑ:

T z (x ∈ Ω h s , z = ẑ) = Ω h s T (x)✶ [z=ẑ] dx Ω h s ✶ [z=ẑ] dx .
This quantity is the plane-averaged scalar temperature that is often used in the cellular ceramics literature to assess the thermal behavior of porous media.

• Volumetric surface of the structure:

S v = Γ h dx Ω h s dx + Ω h v dx .
This quantity measures the surface area of ligaments per unit of volume. S v is often directly linked to volume-averaged properties of porous materials, like the effective extinction coefficient β eff or the effective thermal conductivity λ eff , cf. [START_REF] Cunsolo | Radiative properties of irregular open cell solid foams[END_REF].

Setup

Topology generation. The primary challenge in discrete-scale simulations of open-cell cellular ceramic materials is detailing their complex topologies. To tackle the problem of topology generation for these ceramics, which is inherent of deterministic methods like the FEM, we have used an in-house virtual material generator genMat1 [START_REF] Guévelou | A simple expression for the normal spectral emittance of open-cell foams composed of optically thick and smooth struts[END_REF]. Based on the marching cubes [START_REF] Lorensen | Marching cubes: A high resolution 3D surface construction algorithm[END_REF], genMat is able to generate topologies of various porous materials with standalone cells and open-cell cellular ceramics.

The topologies of the standalone SiC cells used in tests 1A to 1D and the topologies of the full-scale SiC-based ceramics used in tests 2A to 2F are presented in fig. 2 and fig. 3 respectively. As mentioned, the standalone cells are 0.5 cm in size, while the full-scale ceramic samples are 2.5 cm.

Meshing. The voxel-based surface meshes from genMat contain millions of surface nodes, as is the case for most X-ray micro-tomography surface meshes. Being voxel-based, these meshes are well suited for ray-tracing algorithms based on random walks. However, constructing Delaunay meshes from these surface meshes was not possible due to memory constraints. To deal with this problem, topology reconstruction through mesh adaption based on a tuned Hausdorff metric [START_REF] Dapogny | Three-dimensional adaptive domain remeshing, implicit domain meshing, and applications to free and moving boundary problems[END_REF] was used. This operation is performed by the open-source library Mmg [START_REF] Dobrzynski | Anisotropic Delaunay mesh adaptation for unsteady simulations[END_REF]. Finally, using the re-meshed surface topologies, the volumetric meshing is performed by another open-source package, Gmsh [START_REF] Geuzaine | Gmsh: A 3-D finite element mesh generator with built-in pre-and postprocessing facilities[END_REF]. This involves creating two tetrahedral meshes for both the solid and the void phases. Additional details on the topology and the mesh generations are given in [START_REF] Badri | A parallel implicit mixed-FEM solution for complex domain radiative transfer problems using immersed meshes[END_REF]. To motivate reproducibility of our results, the standalone ceramic cell meshes, among other data, are provided as supplementary resources to this paper. Solver specifications. For solving E RTE , the Jacobi-preconditioned BiCGSTAB method is used, with a convergence tolerance on the relative residual set to 10 -6 . As specified earlier, in all test cases, N d = 512

discrete directions are used. The number of unknowns for the RTE linear systems is given in the last column of table 1. For solving E HCE , an exact LU factorization is computed by MUMPS [START_REF] Amestoy | MUMPS: a general purpose distributed memory sparse solver[END_REF] (interfaced with PETSc).

More advanced preconditioners could have been used, like multigrid methods, but the number of unknowns for this field, reported in the 7th column of table 1, is rather small. For all test cases, only few nonlinear iterations are needed to reach convergence, as can be seen from the 6th column of table 1. LIGER, a cluster hosted at ICI, Centrale Nantes, France, was used to perform the numerical simulations.

100 MPI processes were used to solve the standalone cell simulations (tests 1A to 1D) and 1,000 MPI processes were used to solve the full-scale ceramic simulations (tests 2A to 2F). These problems are solved in few minutes for each case.

The largest simulation of this study (90 % porosity Kelvin-cell ceramic structure) consists in solving a system of 0.52 billion unknowns for the radiation coupled to a system of 0.37 million unknowns for the conduction.

For this test case, it took 9 iterations for the nonlinear solver to converge. This emphasizes the role of efficient solution techniques for discrete-scale conduction-radiation simulations.

Note that our radiation solver has been validated thoroughly using the method of manufactured solutions [START_REF] Badri | Vectorial finite elements for solving the radiative transfer equation[END_REF][START_REF] Badri | High performance computation of radiative transfer equation using the finite element method[END_REF], and against other deterministic or Monte-Carlo benchmark results [START_REF] Badri | Efficient finite element strategies for solving the radiative transfer equation[END_REF]. In particular, we verified that our solver is able to produce accurate radiative heat fluxes on surfaces. Besides these validation tests, in [START_REF] Badri | Efficient finite element strategies for solving the radiative transfer equation[END_REF], we verified our conduction-radiation solver against experimental results provided in [START_REF] Coquard | Experimental investigations of the coupled conductive and radiative heat transfer in metallic/ceramic foams[END_REF]. For this test, thermograms from the laser flash method applied to different metallic open-cell foams were analyzed and compared up to T = 673 K. We showed that our solver predict well the experimental temperatures with relative errors between 2 % and 5 %.

Results and discussions

The coupled conduction-radiation temperature fields for the 90 % porosity SiC cubic-and Kelvin-cell samples are presented in fig. 4a. Based on the location from the hot Dirichlet boundary ∂Ω h s,g+ , the temperature within the cell ligaments quasi-linearly decreases from 1800 K to 1200 K. To investigate the heat flow characteristics, in fig. 4b, we present a heatline visualization from the coupled conduction-radiation simulations of the 90 % porosity standalone cubic-and Kelvin-cell structures (tests 1B and 1D). This represents the net heat flow as it passes through the SiC ligaments. A heatline is similar to a streamline, except that it visualizes the net energy flow caused due to heat transfer situations, c.f. [START_REF] Deng | Numerical visualization of mass and heat transport for conjugate natural convection/heat conduction by streamline and heatline[END_REF][START_REF] Costa | Unified streamline, heatline and massline methods for the visualization of two-dimensional heat and mass transfer in anisotropic media[END_REF]. Heatlines, here, are constructed by numerical integration (performed in ParaView [START_REF] Ayachit | The ParaView Guide: A Parallel Visualization Application[END_REF]) of the heat flux vector (-λ dT / dx, -λ dT / dy, -λ dT / dz) ⊤ . Heat energy flows through four distinct, but symmetric due to geometry, paths via the four central ligaments of the cubic cell. For the Kelvin cell, due to its topology, a heatline bifurcation happens in the central region. Heatlines for the 80 % porosity standalone cell structures (tests 1A and 1C not shown here) had similar characteristics. Visualization of the radiative heat flux fields Q + r (x ∈ Γ h ) for the 90 % porosity standalone SiC cells is presented in fig. 5. For the cubic cell in fig. 5a, the inner corners of the cells near the hot border receive the maximum radiative heat flux, max(Q + r (x ∈ Γ h )) = 1.69 W cm -2 . We can observe how the inner faces of the SiC cubic-cell ligaments receive the higher amount of radiation, while the outer parts remain comparatively cold.

The high radiative heat fluxes on the inner faces of the ligaments are due to the radiation energy interaction inbetween the ligaments. Indeed, due to the high-temperature radiation, energy is emitted by a ligament in different directions. Depending on its path, it either encounters a neighboring ligament (radiation interaction) or escapes directly to the surrounding via one of the six vacuum boundaries. Concerning the radiation interaction, a part of the energy, proportional to ε, gets absorbed by the ligament while the other part is reflected to other directions. A similar scenario exists for the SiC Kelvin cell, see fig. 5b. However, for this case, the maximum radiative heat flux max(Q + r (x ∈ Γ h )) = 2 W cm -2 is higher than for the SiC cubic cell. Similar results were observed for the 80 % porosity SiC cubic-and Kelvin-cell structures. The Kelvin cell has a higher max(Q + r (x ∈ Γ h )) than the cubic cell. It was also observed that more radiative heat flux is injected into the Kelvin cell than into the cubic cell. The SiC Kelvin cell has a more absorbing structure than the SiC cubic cell. From the Kirchhoff's principle, the Kelvin cell has a more emitting structure than the cubic cell. By this discussion, one could also say that, at a particular porosity (90 % or 80 %), the radiation plays a more prominent role in the SiC Kelvin-cell structures than in the SiC cubic-cell structures. Overall, the SiC Kelvin-cell structure maximizes the radiation gains compared to the SiC cubic-cell structure.

0 1.69 Flux Q + r [W cm -2 ]
(a) cubic cell

0 2.85 Flux Q + r [W cm -2 ]
(b) Kelvin cell Radiation densities G(x ∈ Ω h v ) for 90 % porosity standalone SiC cubic and Kelvin cells are displayed in fig. 6.

It clearly shows that the radiation energy diffusion is greater in the SiC Kelvin cell than in the SiC cubic cell.

First, the Kelvin cell has a higher value of S v , see table 1. As a result, more energy is being emitted to the void region leading to more radiation interaction between the ligaments. Second, the Kelvin cell has a greater surface, so there is more possibility for photons to be reflected and propagated further in the void region and ultimately interact with other ligaments. Eventually, the inner void region of the cubic cell has a greater volume. Thus, the greater area for effective radiation within the Kelvin cell increases the probability of interaction between the ligaments. Consequently, we observe the absorbing power q Γ h of the 90 % (resp. 80 %) porosity standalone SiC Kelvin cell is 2.4 (resp. 1.4) times higher than that of the cubic cell, see table 2. These results complement the conclusion from the last paragraph that the standalone SiC Kelvin cell maximizes radiation gains compared to the standalone SiC cubic cell. To quantify the effect of the coupling, we also plot the difference in T z , denoted ∆T z , in fig. 8. The plots show that:

• T z for all topologies is under-predicted if radiation is not considered. This suggests a tight coupling of conduction and radiation, at least for the 80 % and 90 % porosities and for the used temperature range;

• T z varies the most in the central region of cells (∆T z has a parabolic profile). This variation is greater in the Kelvin cell than in the cubic cell. Particularly, the 90 % (resp. 80 %) porosity Kelvin cell has a maximum increase of T z of 46 K (resp. 30 K);

• both 80 % and 90 % porosity cubic-cell simulations performed with conduction-radiation physics led to a parabolic-like increase in T z over the conduction alone physics. However, the porosity change did not show any effect in the magnitude of T z . A maximum variation of ∼ ±2 K was observed. This insensitivity of the In fig. 9, we display the temperatures on the ligament networks of the full-scale SiC ceramics at 90 % porosity. These temperatures were obtained from conduction-radiation simulations (tests 2B, 2D, and 2F). Based on the location from the hot Dirichlet boundary, the temperature quasi-linearly decreases from 1800 K to 1200 K. , and total radiation energy absorbed q Γ in W, for the standalone cells and the full-scale ceramic structures.

In the same figures (figs. 12 to 14), we also present the radiative density profiles on two orthogonal void planes. Among the deterministic ceramic structures, the density distribution reaches its maximum in the central regions, while it decreases towards the lateral faces. This lateral decrease in the radiative energy was not observed in the random-cell ceramic samples.

In table 1 values of the volumetric surface S v for the cubic-, Kelvin-, and random-cell ceramic samples are provided. These values suggest that among the three different ceramic topologies examined, the Kelvin-cell topology has a greater surface area to emit/absorb radiation. Compared to the 90 % (resp. 80 %) porosity cubic-cell ceramic, the S v of the Kelvin-cell ceramic is 1.49 (resp. 1.52) times larger. Similarly, in comparison to the 90 % (resp. 80 %) porosity random-cell ceramic, the S v of the Kelvin-cell ceramic is 1.26 (resp. 1.37) times larger. As a consequence, in table 2 we observe higher values of q Γ h for the Kelvin-cell ceramic samples, in comparison to the cubic-cell or random-cell samples, suggesting higher absorbing capabilities of the Kelvin-cell ceramic samples. Precisely, the value of q Γ h for the 90 % porosity Kelvin-cell ceramic sample is 1.9 (resp. 1.5) times more than that of the 90 % porosity cubic-cell (resp. random-cell) ceramic sample. At 80 % porosity, the value of q Γ h for the Kelvin-cell ceramic sample is 1.8 (resp. 1.3) times more than that of the cubic-cell (resp. random-cell) ceramic sample. As for the standalone SiC cells, an analysis of conduction-radiation coupling effects within different ceramic samples is given in fig. 15 using T z . From these plots, observe that a porosity change from 90 % to 80 % has no effect when only conduction is considered. For the coupled conduction-radiation simulations, a porosity change from 90 % to 80 % has a clear effect on T z . As expected, due to higher probabilities of interactions of radiation between ligaments, T z is higher in the central region of the ceramics for the coupled simulations.

To quantify the effect of the coupling, we also plot the difference in T z , denoted ∆T z , in fig. 16. The plots show that:

• ∆T z for the 90 % porosity ceramic samples is overall higher than for the 80 % porosity ceramic samples.

A similar qualitative observation could be derived from analyzing the volumetric surface data S v provided in table 1. For instance, by using analytic laws dependent on S v and φ proposed in [START_REF] Cunsolo | Radiative properties of irregular open cell solid foams[END_REF] for the Kelvin-cell ceramic samples, β eff for the 80 % and 90 % porosities is equal to 3.13 cm -1 and 1.78 cm -1 , respectively.

This suggests that radiation propagates more in the 90 % porosity samples than in the 80 % porosity samples. The same observation holds for the random-cell ceramic samples, see table 2 for the values of β eff calculated according to the analytical formulations provided in [START_REF] Guevelou | Evolution of the homogenized volumetric radiative properties of a family of α-SiC foams with growing nominal pore diameter[END_REF];

• with respect to the topology, the radiation is most effective within the Kelvin-cell ceramic samples as it reported the maximal increase in ∆T z , followed by the random-cell and then the cubic-cell ceramic samples; • Table 2 shows for all simulations max(∆T z ), as well as the ordinates L max(∆T z ) that maximizes the difference. It is the greatest for the 90 % porosity Kelvin-cell ceramic sample, max(∆T z ) = 140 K. For all ceramic samples, the radiation propagation lengths increase when switching from 80 % to 90 % porosity It is also possible to discuss the conduction-radiation results observed for the random-cell ceramics by using the conduction-to-radiation parameter N = λ eff β eff (4σT 3 ref ) [START_REF] Modest | Radiative Heat Transfer[END_REF], also referred as the Stark or the Planck number as defined by Howell et al. [START_REF] Howell | Thermal Radiation Heat Transfer[END_REF]. This parameter can be used if we consider that an upscaling approach may be applied with confidence [START_REF] Whitaker | The Method of Volume Averaging[END_REF][START_REF] Guevelou | Evolution of the homogenized volumetric radiative properties of a family of α-SiC foams with growing nominal pore diameter[END_REF]. By using the analytical formulations proposed in [START_REF] Öchsner | Cellular and Porous Materials: Thermal Properties Simulation and Prediction[END_REF] for the 80 % (resp. 90 %) porosity random-cell sample, λ eff is equal to 0.02 (resp. 0.01) W cm -1 K -1 . These λ eff and β eff from table 2 lead to N equal to 0.29 (resp. 0.07). As found in the literature, for both porosities, N ≪ 10.

This highlights again that radiation plays a more important role than conduction [START_REF] Modest | Radiative Heat Transfer[END_REF]. Because the number of nonlinear iterations, k in table 1, is higher for solving the conduction-radiation problem with the 90 % than with the 80 % porosity random-cell ceramic, the coupling is likely tighter. A similar phenomenon when dealing with decreasing values of N was observed in [START_REF] Mishra | Analysis of conduction and radiation heat transfer in a 2-D cylindrical medium using the modified discrete ordinate method and the lattice Boltzmann method[END_REF].

To examine various properties and physical behaviors of macroporous cellular materials, the Kelvin cellrandom cell analogy is often used to efficiently deal with multiphysics [START_REF] Contento | The prediction of radiation heat transfer in open cell metal foams by a model based on the Lord Kelvin representation[END_REF]. As such, Kelvin-cell structures are often used as representatives of random-cell structures, and numerical or experimental tests are performed on Kelvin-cell structures. To gauge the accuracy of such an assumption, we can apply here the same Kelvin-cellrandom-cell analogy to our results from tests {2C,2E} and {2D,2F}. The analogy is true when only considering conduction, at least for our porosity range from 80 % to 90 %. Minor temperature variations (±9 K) are present when comparing results at a constant porosity, see left-hand side fig. 17. However, the coupled conductionradiation simulations show greater variations in temperatures, so that the Kelvin-cell-random-cell analogy does not hold. Note that max(∆T z ) for the 90 % porosity Kelvin-cell ceramic is 1.5 times higher that for the random-cell ceramic. It also suggests that the heating power of SiC Kelvin-cell ceramics is better than the one of random-cell ceramics within similar regimes (porosity and heating conditions).

Conclusion

In this paper, a finite element model was established to solve the coupled conduction-radiation physics within

SiC-based open-cell cellular ceramics operating at temperatures ranging from 1200 K to 1800 K. In particular, the vectorial finite element model was used for approximating the radiation physics, i.e., solving the radiative transfer equation, which was nonlinearly coupled to the standard finite element model for approximating the conduction physics, i.e., solving the heat conduction equation. These models were assembled together using a finite element domain-specific language, FreeFEM, and then solved with PETSc.

Specific numerical experiments were designed with the aim of highlighting the role of conduction-radiation coupling within the SiC-based open-cell cellular ceramic structures. More precisely, in these numerical experiments, the cubic-, Kelvin-, and random-cell cellular ceramic structures at both 80 % and 90 % porosities were investigated. Besides these full-scale simulations, the standalone cubic and Kelvin cells were also investigated.

To ensure comprehensive physics-based simulations compared to the conventional continuum-scale approaches that average the volumes, the geometries, and the properties, our numerical simulations were based on a discrete-scale approach involving realistic geometries with microstructure properties. Due to its important computational cost, the discrete-scale approach has mostly been used for modest mesh sizes when using deterministic radiation models. We presented results using detailed open-cell cellular ceramic geometries thanks to the efficient use of distributed-memory parallelism.

Radiative transfer, that forms a dominant part of conduction-radiation energy exchange within the SiC ceramics, was analyzed by means of radiation density and radiation heat flux, and by calculating the total absorbed radiation by the ceramic ligaments. It was revealed that the Kelvin-cell topology is favorable for maximizing the radiation gains when compared to the cubic-and the random-cell topology.

The data from our numerical experiments provides means for better understanding the role of radiation in coupled conduction-radiation physics within cellular ceramics. Our numerical data suggests that within the standalone or full-scale SiC structures operating at temperatures ranging from 1200 K to 1800 K, the radiation plays a dominant role. As such, it cannot be neglected while modeling such scenarios. Considering only conduction leads to an under prediction of the real temperature within the SiC cell ligaments. This effect was even more pronounced in the full-scale ceramics than in the standalone cells. To quantify the effect of the coupling, we compared the difference in mean temperatures.

The numerical experiments also suggest that the Kelvin-cell ceramics are, indeed, good representatives of random-cell ceramics when only considering conduction. However, the same cannot be said for coupled conduction-radiation, for which there are important mean temperature differences between the Kelvin-cell ceramics and the random-cell ceramics. Our results are furthermore coherent with upscaling approaches based on the use of the Stark or the Planck number -which is also known as the conduction-to-radiation parameter.

Changes in porosity had a considerable effect on the conduction-radiation coupling for all the SiC structures except the standalone SiC cubic cell. We observed that changing the porosity from 80 % to 90 % increased the radiative transfer, leading to a tighter coupling. As a side effect, the number of nonlinear iterations needed by our solver to reach convergence slightly increased.

The role of convection within macroporous ceramics has yet to be determined. Also, extending our numerical experiments to quantify the effect of conduction-radiation in semi-transparent materials, which adds refractions alongside reflections, remains a challenge. This numerical work could be extended to applied studies where cellular materials are used as host supports for conversion energy processes occurring at temperatures higher than 1000 K.

  Such a framework explains why engineers and researchers, interested in innovative design of open-cell cellular ceramics, use the time-consuming and expensive trial-and-error iterative design cycle. The reduction of this development Nomenclature of the associated manufacturing cost motivates numerical modeling strategies aiming at accurate predictions of coupled conduction-radiation behavior of open-cell cellular ceramics. Additionally,

  different boundary conditions (b) void and solid phase meshes

Figure 1 :

 1 Figure 1: boundary conditions and meshes for a standalone Kelvin cell.

  SiC-based open-cell cellular ceramic structures and on the related standalone SiC cells. To study the pore-level characteristics of conduction-radiation coupling in tests 1A to 1D, we analyze the standalone cubic and Kelvin cells at 80% and 90% porosity each. Structurally, the cubic cells are 0.5 cm in length, and the Kelvin cells are 0.5 cm in diameter. These cells are constructed with ligaments that are ∼ 0.1 cm in diameter for the 80% porosity structures and ∼ 0.069 cm for the 90% porosity structures. To study the full-scale open-cell cellular ceramic structures, in tests 2A to 2F, we analyze the cubic, Kelvin, and pseudo-random structures at 80% and 90% porosity each. At this level, let us precise that the term "pseudo-random" means that the position in space of a cell is not totally random but affected by the spatial arrangement of the neighboring cells. For simplicity, we now only use the term random instead of pseudo-random. Structurally, the cubic-and Kelvin-cell ceramic samples are constructed by joining together 125 standalone cubic and Kelvin cells, i.e., 5 cells per spatial directions, 5 × 5 × 5 = 125. Thus, cubic-and Kelvin-cell ceramic samples (tests 2A, 2B, 2C, and 2D) at length 2.5 cm are 5 times larger. The random-cell ceramic samples use the same length of 2.5 cm. The ligaments of

Figure 2 :

 2 Figure 2: standalone cell topologies.

Figure 3 :

 3 Figure 3: full-scale ceramic topologies.

Figure 4 :

 4 Figure 4: coupled conduction-radiation simulation results for the standalone SiC cubic and Kelvin cells at 90 % porosity.

Figure 5 :

 5 Figure 5: radiative heat flux on clipped surfaces of 90 % porosity standalone SiC cells.

Figure 6 :

 6 Figure 6: combined solution field visualization for the 90 % porosity standalone SiC cells: temperature field visualized for the solid part and radiative density field visualized for the orthogonal void cross-sections.

Figure 7 :

 7 Figure 7: T z for conduction and conduction-radiation simulations for the standalone SiC cubic (left) and Kelvin (right) cells.

standaloneFigure 8 :

 8 Figure 8: ∆T z for the standalone SiC cubic (left) and Kelvin (right) cells.

  (a) cubic cells (b) Kelvin cells (c) random cells

Figure 9 :

 9 Figure 9: temperature fields for the 90 % porosity full-scale ceramic samples.

Figure 10 :

 10 Figure 10: coupled conduction-radiation heatlines arising from a line source on the hot Dirichlet boundary for 90 % porosity ceramic samples.

Figure 11 :

 11 Figure 11: conduction (green) and conduction-radiation (red) heatlines from a point source on the hot Dirichlet boundary for the SiC random-cell ceramic sample at 90 % porosity.

Figures 12 to 14

 14 Figures 12 to 14 display truncated views of the net radiative heat fluxes Q + r (x ∈ Γ h ) on the surfaces of the 90 % porosity SiC cubic-, Kelvin-, and random-cell ceramic structures. Among the deterministic ceramic structures, i.e., with cubic and Kelvin cells, the latter has the highest max(Q + r (x ∈ Γ h )) = 0.29 W cm -2 . Like for standalone SiC cells, higher values of Q + r (x ∈ Γ h ) are observed in the vicinity of the hot Dirichlet boundary and they decrease as one moves away from this boundary.

Figure 12 :

 12 Figure 12: combined solution field visualization for the SiC cubic-cell ceramic sample at 90 % porosity. Right: temperature field visualized for the solid part and radiative density field visualized for the orthogonal void cross-sections. Left: radiative heat flux field visualized for the truncate solid phase boundary.

Figure 13 :

 13 Figure 13: combined solution field visualizations acquired from conduction-radiation simulation of the SiC Kelvin-cell ceramic samples at 90 % porosity. Right: temperature field visualized for the solid part and radiative density field visualized for the orthogonal void cross-sections. Left: radiative heat flux field visualized for the solid part boundary.

Figure 14 :

 14 Figure 14: combined solution field visualizations acquired from conduction-radiation simulation of the SiC random-cell ceramic samples at 90 % porosity. Right: temperature field visualized for the solid part and radiative density field visualized for the orthogonal void cross-sections. Left: radiative heat flux field visualized for the solid phase boundary.

Figure 15 :

 15 Figure 15: T z for conduction and conduction-radiation simulations for the cubic-cell (left), Kelvin-cell (middle), and random-cell (right) ceramics.

Figure 16 :

 16 Figure 16: ∆T z for the cubic-cell (left), Kelvin-cell (middle), and random-cell (right) ceramics.

BFigure 17 :

 17 Figure 17: T z for the conduction (left) and the coupled conduction-radiation (right) simulations for the full-scale SiC-based Kelvinand random-cell ceramics.

  further describing the mathematical model that governs the conduction-radiation physics within the open-cell ceramics, let us denote the open bounded set Ω s ⊂ R 3 (resp. Ω v ⊂ R 3 ) representing the solid (resp. void) phase domain. The solid (resp. void) phase boundary is denoted ∂Ω s (resp. ∂Ω v ). The solid-void interface is denoted Γ = ∂Ω s ∩ ∂Ω v . Also, let n s (resp. n v ) denote the outward unit vector normal to the solid

	(resp. void) phase.
	In this article, the conduction physics within the open-cell ceramics is solved using the steady-state heat
	conduction equation (HCE), which we denote E HCE . The conduction problem then consists in searching for a
	scalar-valued function T : Ω s → R such that:

  With the use of vectorial finite elements, the resulting variational form is then a single equation, while the use of scalar finite element schemes results in N d equations. The corresponding vectorial FE variational problem, E h

	in searching I ∈ H 1 (Ω h v ) such that:	RTE , consists

Table 1 :

 1

	Test	Type	Cell	φ	Sv	k	unknowns HCE	unknowns RTE
	1A	standalone	cubic	80	7.14	4	29,985	23,064,576
	1B	standalone	cubic	90	5.39	5	18,248	25,138,176
	1C	standalone	Kelvin	80	8.46	7	33,549	26,625,024
	1D	standalone	Kelvin	90	5.91	9	20,629	29,829,632
	2A	full-scale	cubic	80	4.44	6	267,544	231,302,656
	2B	full-scale	cubic	90	3.51	7	201,787	263,202,304
	2C	full-scale	Kelvin	80	6.77	8	388,020	338,101,760
	2D	full-scale	Kelvin	90	5.24	9	374,928	517,838,848
	2E	full-scale	random	80	4.94	7	587,203	389,615,616
	2F	full-scale	random	90	4.13	10	449,117	414,130,176

characteristics of the standalone SiC cell and the full-scale SiC ceramic numerical tests. For the mentioned properties read φ in %, Sv in cm -1 , k is total number of nonlinear iterations for solving the coupled conduction-radiation problem.

Table 2 :

 2 maximum change max(∆T z ) in K, distance of max(∆T z ) to the hot boundary L max(∆T z ) in cm, effective extinction coefficient β eff in cm-1

			standalone SiC cells			full-scale SiC cellular ceramics
		cubic cells	Kelvin cells	cubic cells	Kelvin cells	random cells
		80 %	90 %	80 %	90 %	80 %	90 %	80 %	90 %	80 %	90 %
	L max(∆T z )	0.25	0.25	0.25	0.25	1.03	1.07	1.06	1.15	1.11	1.13
	max(∆T z )	21.97	22.58	27.40	41.63	41.70	64.52	74.35	140.54	50.48	92.00
	q Γ h β eff	0.37 -	0.41 -	0.52 -	0.99 -	2.87 -	3.98 -	5.29 3.13	7.57 1.78	4.09 1.92	5.03 0.96

genMat: a C++ and Qt based software developed collaboratively by LTeN & IUSTI, CNRS laboratories, France.

Acknowledgments

The author M. A. Badri would like to thank IRT Jules Verne, Bouguenais, France (Institute in Research and Technology in Advance Manufacturing Technologies for Composite, Metallic, and Hybrid Structures) for funding his research during his whole Ph.D. thesis. The company Mersen, Gennevilliers, France, is also acknowledged for its final financial contribution. The authors would like to thank Arnaud Biallais from SATT Ouest Valorisation, France (currently working at Pixel Group, France) for his kind help with the genMat software. The authors also thank the Institute of Intensive Computation, Centrale Nantes, France, for the access to LIGER. M. A. Badri, Y. Favennec, and B. Rousseau would like to warmly thank Distinguished Professor Emeritus Michael F. Modest, University of California, Merced, USA, and Professor Sophia Haussener, EPFL, Lausanne, Switzerland, for fruitful and valuable discussions on the subjects of radiative transfer and porous material radiation. Finally, the French Research Network ACCORT (GDR CNRS 3438), funded by CNRS, is also thanked for providing high-level framework for scientific exchanges.