Julien Sorel Djeumegni
email: julien.sorel.djeumegni@univ-poitiers.fr

Myriam Lazard

Vital Le Dez

Hervé Thierry

Tagne Kamdem

= {Ω ∈ ℝ 2 / |Ω| ≤ 1} (𝑒 ⃗ 𝑥 , 𝑒 ⃗ 𝑦 ) unit vectors of 𝑥, 𝑦 directions 𝐺 volumic incident radiation (𝑊𝑚 -3 ) Conductive-radiative heat transfer in a 2D semi-transparent medium with a square centered obstacle

Keywords: conductive-radiative, semi-analytical, obstacle, semi-transparent medium Subscripts

The present paper deals with an exact semi-analytical formulation of a combined conductiveradiative heat transfer, applied to a two-dimensional semi-transparent medium carrying a square centered obstacle. The gray participating medium with black boundaries absorbs, emits but does not scatter radiation. One intends to evaluate the temperature and radiative heat flux distributions within the semi-transparent medium. The radiative transfer equation has been solved using an exact analytical expansion of Bickley-Naylor and Altaç angular integrated Bickley-Naylor functions, then solved numerically with Gauss quadrature. Energy equation has been directly discretized and approximated numerically using the centered finite differences method and consequently the dimensionless temperature has been obtained after an iterative scheme. The results of radiative quantities obtained have been verified with benchmark with an excellent agreement, both for simple and complex geometries. Simulations have been performed to obtain results for different sizes of the centered obstacle and the optical thickness. The effects of the conduction-radiation parameters, discrete directions and the size of the obstacle have also been investigated.

Introduction

The conductive-radiative heat transfer in participating medium keeps getting attention of researchers, because it is encountered in various engineering applications such as heat transfer in the semi-transparent media, glass fabrication, fibrous thermal insulation, porous materials, industrial furnaces [START_REF] Yuen | Analysis of combined conductive-radiative heat transfer in a two dimensional rectangular enclosure with a gray medium[END_REF][START_REF] Kim | Analysis of combined conductive and radiative heat transfer in a two dimensional rectangular enclosure using the discrete ordinates method[END_REF][START_REF] Lee | Two-dimensional combined conduction and radiation heat transfer: comparison of the discrete ordinates method and the diffusion approximation methods[END_REF]. Hereafter, thermal radiation investigated in the present paper is applied within a cavity of complex or irregular shape at high level of temperature. Radiation contributes in the heat transfer because of absorption and emission processes inside the participating medium.

Up to now, an exact analytical solution of radiative transfer equation (RTE) is difficult to find for multi-dimensional complex geometries, but remains in strong demand to solve various thermal problems [START_REF] Kim | Analysis of combined conductive and radiative heat transfer in a two dimensional rectangular enclosure using the discrete ordinates method[END_REF]. Numerous studies dealing with combined conduction-radiation heat transfer have already been examined, whose implementation was realized in the case of simple and complex geometries by using numerical techniques. Mostly, numerical methods used included various schemes such as the zonal method, the Monte Carlo method, the spherical harmonic method, the discrete ordinates method (DOM), the finite-volume method (FVM), the finite element method (FEM), the ray tracing/nodal-analyzing method (RTNAM) and the Discontinuous Galerkin method (DG).

In simple geometries, such as square enclosures containing semi-transparent medium, Yuen and Takara [START_REF] Yuen | Analysis of combined conductive-radiative heat transfer in a two dimensional rectangular enclosure with a gray medium[END_REF] considered a combined conductive-radiative heat transfer in a two dimensional enclosure with a gray medium. The authors applied a numerical procedure, based on tabulated values from a class of generalized exponential integral function 𝑆 𝑛 (𝑥), known to be particularly accurate and efficient in the case of pure radiation [START_REF] Yuen | Analysis of radiative equilibrium in a rectangular enclosure with gray medium[END_REF][START_REF] Yuen | Analysis of radiative equilibrium in a rectangular enclosure with gray medium: cases with diffusely reflecting surfaces and heat flux boundary conditions[END_REF]. Numerical data of temperature and flux fields were tabulated as benchmark solutions of the coupled problem, since exact analytical solutions are unable to obtain except for restricted one-dimensional problems. The authors successfully shown that for some ranges of optical thicknesses and conduction-radiation parameters, the diffusion approximation commonly used in practical engineering problems yields to significant errors on radiative quantities. Kim and Baek [START_REF] Kim | Analysis of combined conductive and radiative heat transfer in a two dimensional rectangular enclosure using the discrete ordinates method[END_REF] studied the same problem by using the DOM combined to a diamond spatial scheme for the RTE, whereas the energy equation was discretized with a central finite differences scheme. Their results were compared with the literature in good agreement. Lee and Viskanta [START_REF] Lee | Two-dimensional combined conduction and radiation heat transfer: comparison of the discrete ordinates method and the diffusion approximation methods[END_REF] investigated a combined conductive-radiative heat transfer by comparing the solutions in the case of a two-dimensional semi-transparent medium. The authors used the DOM and the Rosseland diffusion approximation to solve the RTE. The authors claimed that, the results predicted by the DOM are in good agreement compared with those based on the 1D integral formulation but suffers from the numerical smearing. In the other hand, the diffusion approximation greatly underpredicts the radiative quantities; especially for optically thin medium. Mishra et al. [START_REF] Mishra | Application of the lattice Boltzmann method for solving the energy equation of a 2-D transient conduction-radiation problem[END_REF] used the Lattice Boltzmann Method (LBM) to solve the conductionradiation energy equation inside a two-dimensional rectangular enclosure. The authors examined the case of a gray absorbing, emitting and scattering participating medium, enclosed inside a single cavity without inner obstacle. The FVM and LBM where applied to solve the energy equation, while the RTE was solved by using the collapse dimension method (CDM). The results of dimensionless temperature distributions were plotted, and the number of required iterations to reach steady-state solutions between the LBM and the FVM where compared. It was observed that the LBM converged faster than FVM in case of coupled radiation-diffusion. In fact, researchers such as Sakami et al. [START_REF] Sakami | Application of the discrete ordinates method to combined conductive and radiative heat transfer in two-dimensional complex-geometry[END_REF], Rousse et al. [START_REF] Rousse | Numerical predictions of two dimensional conduction, convection and radiation heat transfer-I[END_REF], Mahapatra et al. [START_REF] Mahapatra | Analysis of combined conduction and radiation heat transfer in presence participating medium by development of hybrid method[END_REF], Luo and Shen [START_REF] Luo | Numerical method of the ray tracing-node analyzing method for solving 2-D transient coupled heat transfer in a rectangular semi-transparent medium[END_REF], Lazard [START_REF] Lazard | Heat transfer in a semi-transparent parallelogram shaped medium[END_REF], Ghattassi et al. [START_REF] Ghattassi | Galerkin method for solving combined radiative and conductive heat transfer[END_REF], Keshtkar [START_REF] Keshtkar | Investigation of transient conduction-radiation heat transfer in a square cavity using combinaison of LBM and FVM[END_REF], have developed similar analysis with excellent results compared to benchmarks solutions, when the size of the obstacle is set to be zero.

For complex or irregular geometries shapes such as rectangular shaped geometries, Asllanaj et al. [START_REF] Asllanaj | Transient Radiation and conduction heat transfer in a gray absorbing-emitting medium applied on two-dimensional complex-shaped domains[END_REF] analysed a transient conduction-radiation combined heat transfer for a gray absorbingemitting medium in a 2D cavity modelled with unstructured triangular meshes. The RTE has been solved with both a new FVM based on vertex scheme and a modified exponential scheme. Amiri et al. [START_REF] Amiri | Combined conductive and radiative heat transfer in an anisotropic scattering participating medium with irregular geometries[END_REF] studied a combined conductive-radiative heat transfer problem, applied to irregular geometries containing an anisotropic scattering participating medium. The authors used the FVM to solve the energy equation and the DOM to solve the radiative transfer equation. The results in terms of dimensionless temperature and radiative flux were compared to benchmark with a good accuracy. They also kindly discussed the influence of conduction-radiation parameters and optical properties on the values of radiative quantities. Lari and Gandjalikhan Nassab [START_REF] Lari | Analysis of combined radiative and conductive heat transfer in three-dimensional complex geometries using blocked-off method[END_REF] analysed the combined radiative-conductive heat transfer in 3D complex geometries by using a block-off method. As applied in CFD even in two-dimensions, the FVM was used to solve the gas energy equation while the DOM was used to solve the RTE in the absorbing-emitting and linear anisotropic scattering medium. The results of the model were validated, and compared to the literature then, applied to analyze the effects of conduction-radiation parameters and radiative properties on the temperature and flux fields in the medium. Talukdar et al. [START_REF] Talukdar | Modelling of conductionradiation in porous medium with blocked-off region approach[END_REF] discussed on a new technique to perform combined conduction-radiation heat transfer in a complex porous structure. FVM and block-off methods based on Cartesian coordinates were used to evaluate the effective thermal conductivity for different temperature, emissivity, thermal conductivity of the solid and absorption coefficient of the fluid. The results of dimensionless temperature and heat flux distributions were compared to those obtained when using FLUENT 6.3, and remained in good agreement. Zare and Gandjalikhan Nassab [START_REF] Zare | Gandjalikhan Nassab, combined radiative-conductive heat transfer in twodimensional complex geometries with variable thermal conductivity[END_REF] investigated the radiative-conductive heat transfer problem with a variable thermal conductivity inside 2D complex geometries. In fact, the FVM was used to solve the energy equation whereas the discrete ordinates method has been used to solve the RTE, by deducing the temperature and heat flux distributions in the medium. The blocked-off procedure in Cartesian coordinates has also been employed to treat both straight and curvilinear boundaries of the concerned geometries. The effects of a variable thermal conductivity, the non-uniformity of boundary heat fluxes due to the isotherm obstacle, both on temperature and heat flux distributions have been analysed and carefully discussed. Keshtkar and Amiri [START_REF] Keshtkar | Numerical simulation of radiative-conductive heat transfer in an enclosure with an isotherm obstacle[END_REF] simulated the combined radiative-conductive heat transfer in a 2D enclosure containing an isotherm rectangular obstacle located at the cavity's center. The RTE and energy equation was solved by using the DOM and LBM respectively. The authors discussed how the radiative properties, the conduction-radiation parameters, and the isotherm obstacle modify the temperature and heat flux distributions. Studies involving coupled radiation-convection inside rectangular enclosures with obstacles [START_REF] Addini | Combined mixed convection and radiation heat transfer in an obstacle wall mounted lid-driven cavity[END_REF][START_REF] Saravanan | Coupled thermal radiation and natural convection heat transfer in a cavity with a heated plate inside[END_REF] are also available, for which similar methods as examined previously were implemented.

From the literature survey dealing with the coupled conductive and radiative heat transfer within multi-dimensional complex geometries, mostly numerical methods have been developed to solve the problem. Moreover, the aim of the present paper is to establish an exact semi-analytical solution of the conductive-radiative heat transfer problem, applied to a 2D absorbing and emitting medium with a centered square obstacle. Exact expressions of the incident radiation from the RTE as developed in the case of radiation heat transfer [START_REF] Djeumegni | Modeling of radiative heat transfer in a gray semi-transparent medium with internal fluid cavity limited by black boundary surfaces[END_REF][START_REF] Djeumegni | Radiative heat transfer in a 2D semitransparent gray medium with a centered inner square cavity[END_REF] are set and coupled with energy equation by using a centered finite scheme in order to derive the temperatures and heat flux distributions. An adequate numerical algorithm to solve the considered conductive-radiative problem is given and performed to obtain the numerical results of temperature and heat flux.

Geometry

One considers in the Fig. 1 a the semi-transparent medium enclosed in a two dimensional square cavity of length 𝐻, containing a square obstacle of length ℎ at a center position. From Djeumegni et al. [START_REF] Djeumegni | Radiative heat transfer in a 2D semitransparent gray medium with a centered inner square cavity[END_REF], analytical expressions of the radiative quantities examined in this paper are those derived for the case 𝐻 = 3ℎ. Modelling involving geometries for which 𝐻 < 3, 3ℎ < 𝐻 ≤ (2 + √5)ℎ and 𝐻 > (2 + √5)ℎ are not developed here for conciseness. However, some numerical Modelling of the combined heat transfer is done in the same plane (𝑥, 𝑦), implying no variation of radiative quantities along the 𝑧-axis.

The incoming intensity depends of the direction vector Ω ⃗⃗⃗ = ( 𝑐𝑜𝑠𝜑𝑠𝑖𝑛𝜃 𝑠𝑖𝑛𝜑𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃

), with 𝜑 ∈ [0,2𝜋] and 𝜃 ∈ [0, 𝜋]. So, in the plane (𝑥, 𝑦) the third component 𝑐𝑜𝑠𝜃 is not taken in consideration but must not be forgotten in the integrated radiative quantities.

Mathematical formulation

2.1.Conductive heat transfer

The energy equation for a transient combined conductive-radiative heat exchange without external heat source is written:

𝜌𝐶 𝑝 𝜕𝑇(𝑡,𝑠) 𝜕𝑡 = -∇ ⃗ ⃗⃗ . (𝑞 ⃗ 𝑟 (𝑡, 𝑠) + 𝑞 ⃗ 𝑐 (𝑡, 𝑠)), (2)
where 𝜌 is the mass density of the medium and 𝐶 𝑝 its specific heat. The different thermo-physical properties are assumed constant and the conductive heat flux is given from the Fourier's law as:

𝑞 ⃗ 𝑐 (𝑡, 𝑠) = -𝑘 𝑐 ∇ ⃗ ⃗⃗ 𝑇(𝑡, 𝑠) , (3)
where, 𝑘 𝑐 is the thermal conductivity of the participating medium.

On the other hand, the radiative source term is expressed by:

𝑆 𝑟 (𝑡, 𝑠) = -∇ ⃗ ⃗⃗ . 𝑞 ⃗ 𝑟 (𝑡, 𝑠) , (4)
whence Eq. ( 2) is rewritten:

𝜌𝐶 𝑝 𝜕𝑇(𝑡,𝑠) 𝜕𝑡 -𝑘 𝑐 ∆𝑇(𝑡, 𝑠) = 𝑆 𝑟 (𝑡, 𝑠), (5)
for (𝑡, 𝑠) ∈ [0, 𝜍] × {𝒟} , in addition 𝑇(0, 𝑠) = 𝑢 0 (𝑠), for 𝑠 ∈ {𝒟}.

where 𝑢 0 is the temperature field at the boundary. From Eq. (4), temperature distribution 𝑇(𝑡, 𝑠) remains the major unknown. Hence, to overcome with this issue, the radiative source term 𝑆 𝑟 (𝑡, 𝑠) in term of temperature is needed solve Eq. ( 5), so it will be computed from the RTE in the next subsection.

2.2.Radiative Heat transfer

In [START_REF] Djeumegni | Radiative heat transfer in a 2D semitransparent gray medium with a centered inner square cavity[END_REF], it has been discussed that the present geometry (𝐻 = 3ℎ) reveals the existence of sixteen sub-domains, divided into four squares and twelve triangles as illustrated in Fig. 2-(b), where the different expressions of the useful radiative quantities to evaluate have not the same analytical expressions. Assuming a uniform grid representation of the domain, the latter one is divided into 𝑁 𝑥 × 𝑁 𝑦 cells characterized either by their corresponding space coordinates (𝑥 𝑖 , 𝑦 𝑗 ) or their nodes labelled (𝑖, 𝑗) in order to discretize the needed equations. Therefore, any point 𝑀 𝑖,𝑗 of the participating medium is located by its discrete coordinates (𝑥 𝑖 , 𝑦 𝑗 ) = [(𝑖 -1)∆𝑥, (𝑗 -1)∆𝑦], with a constant step length following 𝑥-axis given by ∆𝑥 = 𝑥 𝑖 -𝑥 𝑖-1 , while the one for 𝑦-axis is ∆𝑦 = 𝑦 𝑗 -𝑦 𝑗-1 .

Radiation from a given point inside the semi-transparent medium that reaches any point 𝑀 ∈ {𝒟} follows a path length supported by the direction vector Ω ⃗⃗⃗ . In the present paper, as illustrated in the Therefore, following a similar procedure developed in [START_REF] Djeumegni | Modeling of radiative heat transfer in a gray semi-transparent medium with internal fluid cavity limited by black boundary surfaces[END_REF][START_REF] Djeumegni | Radiative heat transfer in a 2D semitransparent gray medium with a centered inner square cavity[END_REF], all the locations of the calculation points in the medium are characterized by their Cartesian coordinates (𝑥 𝑝 , 𝑦 𝑞 ). One has 𝑥 𝑝 -∆𝑥 ≤ 𝑥 𝑖 ′ ≤ 𝑥 𝑝 + ∆𝑥 and 𝑦 𝑞 -∆𝑦 ≤ 𝑦 𝑗 ′ ≤ 𝑦 𝑞 + ∆𝑦, where 𝑥 𝑝 = (𝑝 -2)∆𝑥, and 𝑦 𝑞 = (𝑞 -2)∆𝑦.

Since, 𝑥 𝑖 ′ and 𝑦 𝑗 ′ are respectively linked to 𝑥 𝑖 and 𝑦 𝑗 , one obtains the integers (𝑝, 𝑞) in terms of (𝑖, 𝑗) as:

{ 𝑝 = 𝐸[(𝑖 + 1) ± (𝑗 -1)𝛿 𝑚 𝑡𝑎𝑛𝜑] 𝑞 = 𝐸[(1 -𝛿 𝑚 )(𝑗 -1) + 2] (6)
where, 𝐸(𝑎) denotes the integer part of the real 𝑎.

The RTE in its general form for a unit refractive index is written:

where, 𝜍 is a given time, 𝑘 𝑎 denotes the absorption coefficient of the medium, 𝑐 = 2.997930 10 8 𝑚𝑠 -1 in the vacuum and 𝐼 𝑏 [𝑇] is the radiation intensity of the blackbody, depending of transient temperature in the medium and expressed by:

𝐼 𝑏 [𝑇] = 𝜎 𝐵 𝑇 4 𝜋 , (9)
where, 𝜎 𝐵 = 5.6698 × 10 -8 𝑊. 𝑚 2 . 𝐾 -4 is the Stefan-Boltzmann constant. Among the main applications in the semi-transparent media, the radiative propagation time is much faster compared to the thermal response of the medium, consequently

1 𝑐 𝜕𝐼(𝑡,𝑠,Ω) 𝜕𝑡 ≈ 0.
Then, although temperature may vary slowly with time in the medium, the radiation is considered at steady state.

The incoming radiation, for a given direction of propagation Ω ⃗⃗⃗ is given by:

𝐼(𝑡, 𝑠, Ω) = 𝜎 𝐵 𝑇 𝑟𝑒𝑓 4 𝜋 𝑒 -𝑘 𝑎. 𝛿(𝑠,Ω) + 𝑘 𝑎. 𝜎 𝐵 𝜋 ∫ 𝑇 4 𝛿 0 (𝑡, 𝑠 ′ )𝑒 -𝑘 𝑎. 𝛿 ′ (𝑠 ′ ,Ω) 𝑑𝛿 ′ , ( 10
)
where the boundary is at a prescribed temperature 𝑇 𝑟𝑒𝑓 . The incident radiation at any point 𝑀(𝑥, 𝑦) ∈ {𝒟} at a given instant 𝑡 defined as:

𝐺(𝑡, 𝑠) = ∫ 𝐼(𝑡, 𝑠, Ω)𝑑Ω Ω=4𝜋 , (11)
and the corresponding radiative heat flux is written:

𝑞 ⃗ 𝑟 (𝑡, 𝑠) = ∫ 𝐼(𝑡, 𝑠, Ω)Ω ⃗⃗⃗ 𝑑Ω Ω=4𝜋 . ( 12
)
The radiative flux field divergence is classically given by:

𝑆 𝑟 (𝑡, 𝑠) = 𝑘 𝑎 (𝐺(𝑡, 𝑠) -4𝜋𝐼 𝑏 [𝑇(𝑡, 𝑠)]) . ( 13
)
Therefore, in order to combine radiation and conduction, incident radiation must be evaluated from radiative transfer equation.

Incident radiation

The semi-analytical formulation of incoming radiation developed actually will enable to combine it with energy equation given at Eq. ( 5), since they are strongly coupled by the incident radiation 𝐺(𝑡, 𝑠) and the temperature distribution 𝑇(𝑡, 𝑠).

From Eq. ( 10), at instant t, when only the bottom boundary surface is hot at imposed temperature 𝑇 𝑟𝑒𝑓 , and the others are cold, the incoming radiation is expanded in (𝑒 ⃗ 𝑥 , 𝑒 ⃗ 𝑦 ) plane by:

𝐼(𝑡, 𝑥, 𝑦, 𝜃, 𝜑) = 𝜎 𝐵 𝑇 𝑟𝑒𝑓 4 𝜋 𝑒 -𝑘 𝑎 { 𝑦 𝑠𝑖𝑛𝜑𝑠𝑖𝑛𝜃 } + 𝑛 2 𝑘 𝑎 𝜎 𝐵 𝜋 ∫ 𝑇 4 (𝑡, 𝑥 ′ , 𝑦 ′ )𝑒 -𝑘 𝑎 { 𝑦-𝑦 ′ 𝑠𝑖𝑛𝜑𝑠𝑖𝑛𝜃 } 𝑑𝑦 ′ 𝑠𝑖𝑛𝜑𝑠𝑖𝑛𝜃 𝑦 ′ =𝑦 𝑦 ′ =0 , (14)
 ∀(𝑡, 𝑥, 𝑦) ∈ ([0, 𝜍]) × (]0, 𝐻-ℎ 2 [ × ]0, 𝐻-ℎ 2 [),
The transient incident radiation at a time t is deduced using the relation:

𝐺(𝑡, 𝑥, 𝑦) = 2 ∫ ∫ 𝐼(𝑡, 𝑥, 𝑦, 𝜃, 𝜑)𝑠𝑖𝑛𝜃𝑑𝜃𝑑𝜑 𝜃= 𝜋 2 𝜃=0 𝜑 2 𝜑 1 , (15)
with,

𝜑 𝑠 1 = 𝑡𝑎𝑛 -1 { 𝑥 𝑦 } and 𝜑 𝑠 2 = 𝑡𝑎𝑛 -1 { 𝐻-𝑥 𝑦 }.
Following a similar procedure proposed in [START_REF] Djeumegni | Modeling of radiative heat transfer in a gray semi-transparent medium with internal fluid cavity limited by black boundary surfaces[END_REF][START_REF] Djeumegni | Radiative heat transfer in a 2D semitransparent gray medium with a centered inner square cavity[END_REF], a first set of Bickley-Naylor functions [START_REF] Altaç | Integrals involving Bickley and Bessel function in radiative transfer and generalized exponential integral function[END_REF] is introduced and set by:

𝐾𝑖 𝑛 (𝑢) = ∫ 𝑒 -𝑢 𝑠𝑖𝑛𝜃( 𝑠𝑖𝑛𝜃) 𝑛-1 𝑑𝜃 𝜃= 𝜋 2 𝜃=0 , 𝑛 ∈ ℕ, 𝑢 ∈ ℝ + . ( 16
)
In order to eliminate the variable 𝜃 in the first integrals of Eq. ( 14), The Altaç functions [START_REF] Altaç | Integrals involving Bickley and Bessel function in radiative transfer and generalized exponential integral function[END_REF] are introduced to eliminate the 𝜑 dependence in Eq. ( 10) and expressed for 𝑛 ∈ ℕ, 𝑢 ∈ ℝ + by:

𝐵𝑖𝑠 𝑛 (𝑢, 𝜃) = ∫ 𝐾𝑖 𝑛 ( 𝑢 𝑐𝑜𝑠𝜑 ) (𝑐𝑜𝑠𝜑) 𝑛-2 𝜑=𝜃 𝜑=0 𝑑𝜑 . ( 17
)
For any point 𝑀(𝑥, 𝑦) located in the square sub-domain sharing the left and south boundary surfaces of the participating medium, the exact dimensionless expression of the incident radiation 𝐺 𝑖,𝑗 * for rays covering aperture 𝜑 𝑠 1 + 𝜑 𝑠 2 is calculated. Its discretized form when using a regular grid with square cells (∆𝑥 * = ∆𝑦 * ) is then given by:

𝐺 𝑖,𝑗 * = 1 2𝜋 {𝐵𝑖𝑠 2 (𝑘 𝑎 (𝑗 -1)∆𝑦 * , 𝑡𝑎𝑛 -1 { 𝑖-1 𝑗-1 }) + 𝐵𝑖𝑠 2 (𝑘 𝑎 (𝑗 -1)∆𝑦 * , 𝑡𝑎𝑛 -1 { 𝑁-𝑖+1 𝑗-1 })} , + 𝑘 𝑎 2𝜋 (𝑗 -1)∆𝑦 * 𝑡𝑎𝑛 -1 { 𝑖-1 𝑗-1 } ∑ 𝜔 𝑙 𝑁 𝜑 𝑙=1 ∑ 𝜔 𝑚 𝑐𝑜𝑠𝜑 𝑙 1 𝑀 𝑚=1 𝑇 𝑘,𝑝,𝑞 * 4 𝐾𝑖 1 ( 𝑘 𝑎 (𝑗-1)∆𝑦 * 𝛿 𝑚 𝑐𝑜𝑠𝜑 𝑙 ) , + 𝑘 𝑎 2𝜋 (𝑗 -1)∆𝑦 * 𝑡𝑎𝑛 -1 { 𝑁-𝑖+1 𝑗-1 } ∑ 𝜔 𝑙 𝑁 𝜑 𝑙=1 ∑ 𝜔 𝑚 𝑐𝑜𝑠𝜑 𝑙 1 𝑀 𝑚=1 𝑇 𝑘,𝑝,𝑞 * 4 𝐾𝑖 1 ( 𝑘 𝑎 (𝑗-1)∆𝑦 * 𝛿 𝑚 𝑐𝑜𝑠𝜑 𝑙 ) (18)
with,

𝜑 𝑙 1 = 𝛽 𝑙 × 𝑡𝑎𝑛 -1 { 𝑖-1 𝑗-1 } and 𝜑 𝑙 2 = 𝛽 𝑙 × 𝑡𝑎𝑛 -1 { 𝑁-𝑖+1 𝑗-1
} where, 𝛽 𝑙 is the angular abscissa, 𝜔 𝑙 and 𝜔 𝑚 are the Gauss weights, 𝑘 is a time index, and 𝑁 𝜑 denotes the order of the Gauss quadrature used to approximate the respective integrals.

, {]2, 𝑁 𝑥 -1[ × ]2, 𝑁 𝑦 -1[}\ {]2, 𝐸 ( 𝐻-ℎ 2∆𝑥 )[ × ]2, 𝐸 ( 𝐻-ℎ 2∆𝑦
)[}, the associated equations, are not presented there for conciseness, but can be denoted as (𝐺 𝑙 * ) 𝑖,𝑗 , where 𝑙 ∈ {1,2, … 20} represents the number of subdomains that contribute to the combined heat transfer.

Once 𝐺 is calculated for all the grid's cells including the boundary surfaces, Eqs. [START_REF] Ghattassi | Galerkin method for solving combined radiative and conductive heat transfer[END_REF][START_REF] Keshtkar | Investigation of transient conduction-radiation heat transfer in a square cavity using combinaison of LBM and FVM[END_REF] are used to compute the radiative heat flux vector 𝑞 ⃗ 𝑟 and the radiative source 𝑆 𝑟 = -∇. ⃗⃗⃗⃗ 𝑞 ⃗ 𝑟 .

2.3.Combined conductive-radiative heat transfer

Inserting the radiative source 𝑆 𝑟 from (13) into Eq. ( 5) leads to the general energy equation which depends on the incident radiation at the calculation point:

{ 𝜌𝐶 𝑝 𝜕𝑇(𝑡,𝑠) 𝜕𝑡 -𝑘 𝑐 ∆𝑇(𝑡, 𝑠) = 𝑘 𝑎 (𝐺(𝑡, 𝑠) -4𝜋𝐼 𝑏 [𝑇(𝑡, 𝑠)]), 𝑇(0, 𝑠) = 𝑣 0 (𝑠), (19)
for

(𝑡, 𝑠) ∈ [0, 𝜍] × {𝒟},
where, 𝑣 0 denotes the temperature field at a boundary surface for 𝑡 = 0. Eq. ( 18) is modified and converted into dimensionless form, by using some dimensionless parameters detailed in Appendix A. Consequently, one replaces each of dimensionless parameters set above in Eq. [START_REF] Keshtkar | Numerical simulation of radiative-conductive heat transfer in an enclosure with an isotherm obstacle[END_REF]

where, 𝜂 = 𝑘 𝑎 2 𝐻 2 /𝑁 𝑝𝑙 . When 𝜕𝜉 ∞, the steady state is reached and one obtains:

∆𝑇 𝑙 * (𝑠 * ) = 𝜂 (𝑇 𝑙 * 4 (𝑠 * ) -𝐺 𝑙 * (𝑠 * )) = -𝜂𝑆 𝑙 𝑟 * , (21)

Semi-analytical solution of combined conductive-radiative heat transfer

Several numerical methods already exist to solve this problem dealing with combined radiation and conduction. In this paper, one implements a discrete finite differences scheme regarding its simplicity and accuracy on results.

One discretizes uniformly the time domain 𝜉 ≥ 0 by setting ∆𝜉 = 𝜉 𝑘+1 -𝜉 𝑘 with 𝜉 𝑘 = 𝑘 ∆𝜉 for any 𝑘 ∈ [0, 𝑁 𝑡 ], where 𝑁 𝑡 is the maximum number of time steps.

 Temperature distribution

The temperature distribution is evaluated using a finite difference method and Dirichlet boundary condition applied to the dimensionless energy equation. At time index 𝑘, and at node labelled (𝑖, 𝑗), Eq. ( 19) is rewritten as:

( 𝜕𝑇 𝑙 * 𝜕𝜉 ) 𝑖,𝑗 𝑘 -[( 𝜕 2 𝑇 𝑙 * 𝜕𝑥 * 2 ) 𝑖,𝑗 𝑘 + ( 𝜕 2 𝑇 𝑙 * 𝜕𝑦 * 2 ) 𝑖,𝑗 𝑘 ] = 𝜂[(𝐺 𝑙 * ) 𝑖,𝑗 𝑘 -(𝑇 𝑙 * 4 ) 𝑖,𝑗 𝑘 ] (22)
The left side of Eq. ( 22) is approximated by a finite differences method. The temperature is then approximated by a first order time forward difference method and set as:

( 𝜕𝑇 𝑙 * 𝜕𝜉 ) 𝑖,𝑗 𝑘 ≈ (𝑇 𝑙 * ) 𝑘+1 -(𝑇 𝑙 * ) 𝑘 ∆𝜉 + 0(∆𝜉) (23)
where, ∆𝜉 = 𝜉/𝑁 𝑡 .

The second order of the partial differential equation, which represents the spatial variation of temperature in the (𝑥, 𝑦) reference is also given when using a central finite differences scheme as:

( 𝜕
where, ∆𝑥 * = ∆𝑥/𝐻 and ∆𝑦 * = ∆𝑦/𝐻 . Substituting Eq. (23-25) into Eq. ( 22), implies:

+ 𝜂∆ * 2 [(𝐺 𝑙 * ) 𝑖,𝑗 𝑘 -(𝑇 𝑙 * 4 ) 𝑖,𝑗 𝑘 ] (27)
The stability condition of this previous equation is ensured by 𝐹 0 * ≤ 1/4, while the maximum number of uniform cells to used is set by 𝑁 𝑥 = 𝑁 𝑦 = 𝑁 𝑚𝑎𝑥 ≤ 𝐸 (

Practically, the non-linearity of Eq. ( 28) is analyzed using a Taylor series truncated at the linear term following the relation:

(𝑇 𝑙 * 4 ) 𝑖,𝑗 𝑘+1 = (𝑇 𝑙 * 4 ) 𝑖,𝑗 𝑘 + 4(𝑇 𝑙 * 3 ) 𝑖,𝑗 𝑘 [(𝑇 𝑙 * ) 𝑖,𝑗 𝑘+1 -(𝑇 𝑙 * ) 𝑖,𝑗 𝑘 ] (29)
However, for one labelled subdomain of the participating medium, the dimensionless temperature appearing in Eq. ( 27) as the unknown variable, will be evaluated in the next section by numerical computation. A Crank-Nicolson scheme can also be involved for time discretization [START_REF] Ghattassi | Galerkin method for solving combined radiative and conductive heat transfer[END_REF] and provides accurate solutions. Once the dimensionless temperature distribution 𝑇 𝑙 * is deduced in a subdomain labelled by 𝑙, it is used to calculate the conductive and the total heat flux distributions in the medium: thus, an iterative scheme is designed and proposed at Section 3. Consequently, one can deduce the dimensionless temperature distribution in the participating medium from the contribution of all the labelled sub-domains such that:

𝑇 * (𝜉, 𝑠 * ) = ∑ 𝑇 𝑙 * (𝜉, 𝑠 * ) 20 1 (30)

 Heat flux distribution

The total heat flux in the medium 𝑞 ⃗ (𝑡, 𝑠) = 𝑞 𝑥 𝑖 ⃗ + 𝑞 𝑦 𝑗 ⃗ is deduced from the summation of both conductive heat flux 𝑞 ⃗ 𝑐 (𝑡, 𝑠) and radiative heat flux 𝑞 ⃗ 𝑟 (𝑡, 𝑠). Similarly with the expressions of radiative heat flux developed in Ref. [START_REF] Altaç | Integrals involving Bickley and Bessel function in radiative transfer and generalized exponential integral function[END_REF] when 𝐻 = 3ℎ , one has the relation below:

𝑞 ⃗ (𝑡, 𝑠) = -𝑘 𝑐 ∇ ⃗ ⃗⃗ 𝑇(𝑡, 𝑠) + ∑ 𝑞 ⃗ 𝑙 𝑟 20 𝑙=1 , ( 31
)
where the radiative heat flux is calculated at internal points with the temperature equation Eq. ( 28) using the relation:

𝑞 ⃗ 𝑙 𝑟 = ∫ 𝐼 𝑙 (Ω)|Ω ⃗⃗⃗ . 𝑛 ⃗⃗|𝑑Ω Ω=4𝜋 , 𝑙 = 1, … 20 (32)
where 𝑛 ⃗⃗ denotes the outward normal to the considered boundary surface.

In a dimensionless form, the total heat flux is set along 𝑥 and 𝑦 directions as:

𝑞 𝑥 * = -
where 𝑞 𝑥 𝑟 and 𝑞 𝑦 𝑟 represent the expressions of the radiative heat flux along 𝑥 and 𝑦 axis; 𝜏 𝑥 and 𝜏 𝑦 denote the optical thickness of the medium along the two 𝑥 and 𝑦 directions.

Numerical results

Suitable calculations dealing with the combined conductive-radiative heat transfer in a 2D semitransparent medium, with a centered obstacle of length ℎ have been carried out in the present paper. Now, one intends to evaluate the accuracy of the present method by displaying numerical results both at steady and transient states. One applied it to a simple (ℎ = 0) and complex (ℎ ≠ 0) geometries. All calculations have been performed with a computer DELL, Intel(R) Core(TM) i7-8700 CPU@ 320 3.19 GHz, using a FORTRAN software.

The numerical computation follows an iterative scheme and the main steps are the following ones:

a. Initialize the temperature 𝑇 0 in the whole medium and at the boundary surfaces. b. Solve RTE by using the exact semi-analytical method for each subdomain labelled by 𝑙 at each point located by (𝑖, 𝑗) following the algorithm:  Compute the incident radiation 𝐺 𝑖,𝑗 * from Eq. ( 18) at radiative equilibrium  Evaluate the radiative heat flux 𝑞 ⃗ 𝑟 and the radiative source 𝑆 𝑟 . c. Solve the energy equation Eq. ( 20) to obtain dimensionless temperature distribution 𝑇 * from Eq. ( 28). The number of required iterations depends on the conduction radiation parameter, the quadrature and the number of cells choice. When running the studied cases, the results of 𝐺 𝑖,𝑗 * have been obtained with less than 500 iterations with the same criterion indicated at step d. Contrariwise for the results of 𝑇 𝑖,𝑗 * , less than 100 iterations have been useful at transient state to reach the steady one.

Simple geometry : ℎ = 0

The concerned geometry consists in a square cross section of 𝐻 × 𝐻 = 1 × 1 𝑚 2 containing an absorbing, emitting but non-scattering medium. The thermal conductivity 𝑘 𝑐 is constant, all the boundary surfaces are black (𝜀 = 1). The south boundary surface is hot, maintained at a reference constant temperature 𝑇 𝑟𝑒𝑓 = 𝑇 𝑠 = 100 𝐾 and the other one are cold at 𝑇 𝑁 = 𝑇 𝑊 = 𝑇 𝐸 = 𝑇 𝑟𝑒𝑓 /2 ; temperature in the medium is initially imposed to be cold at 𝑇 0 = 𝑇 𝑟𝑒𝑓 /2. With these conditions, dimensionless temperature distributions, 𝑇 * = 𝑇/𝑇 𝑟𝑒𝑓 and heat fluxes 𝑞 * = 𝑞/4𝜎 𝐵 𝑇 𝑟𝑒𝑓 4 are computed by an iterative process until convergence, then displayed and discussed.

Steady-state

The problem has already been investigated by several authors at steady-state such as Yen and Takara [START_REF] Yuen | Analysis of combined conductive-radiative heat transfer in a two dimensional rectangular enclosure with a gray medium[END_REF], Kim and Baek [START_REF] Kim | Analysis of combined conductive and radiative heat transfer in a two dimensional rectangular enclosure using the discrete ordinates method[END_REF], Sakami et al. [START_REF] Sakami | Application of the discrete ordinates method to combined conductive and radiative heat transfer in two-dimensional complex-geometry[END_REF], Ismail and Salinas [START_REF] Ismail | gray radiative conductive 2D modeling using discrete ordinates method with multidimensional spatial scheme and non-uniform[END_REF], where each of them used different methods such as FVM, finite element method (FEM), and DOM. When using the present method in the 2D enclosure, Fig. 3-(a) shows the evolution of the dimensionless temperature 𝑇 * along the centerline position 𝑥/𝐻 = 0.5. The results have been obtained with the conductionradiation parameters 𝑁 𝑝𝑙 ∈ {0.01, 0.1, 1}, 𝐻 = 1𝑚, a unit absorption coefficient 𝑘 𝑎 = 1 𝑚 -1 corresponding to a unit optical thickness (𝜏 = 𝑘 𝑎 𝐻) have also been validated with the numerical solutions proposed by Mishra et al. [START_REF] Mishra | Application of the lattice Boltzmann method for solving the energy equation of a 2-D transient conduction-radiation problem[END_REF] at steady state conditions. The subsequent computation has been done with a choice of 21 × 21 rectangular grids to facilitate the validation of results. The same Fig. 3-(a) shows a great matching between the dimensionless temperature distribution given by Mishra et al. [START_REF] Mishra | Application of the lattice Boltzmann method for solving the energy equation of a 2-D transient conduction-radiation problem[END_REF], taken as benchmark results and the present one. One can argue on a good The respective values assigned to the conduction-radiation parameter 𝑁 𝑝𝑙 = 𝑘 𝑐 𝑘 𝑎 /(4𝜎𝑇 𝑟𝑒𝑓 4 ) exist to show the influence of both conduction and radiation on the temperature distribution in the participating medium. When 𝑁 𝑝𝑙 = 1, conduction predominates, hence only the hot boundary surface contribute substantially to heat transfer. The solution obtained remains similar with the case of pure conduction seen as the main mode of heat transfer. When 𝑁 𝑝𝑙 = 0.1, neither conduction nor radiation predominates one over the other. Consequently, when the Planck number 𝑁 𝑝𝑙 decreases and reaches 0.01, the values of temperature increase but stay less than the case of pure radiation. In this condition, radiation heat transfer predominates over conduction. Sakami et al. [START_REF] Sakami | Application of the discrete ordinates method to combined conductive and radiative heat transfer in a two-dimensional complex geometry[END_REF], Rousse et al. [START_REF] Rousse | Numerical Predictions of two-dimensional conduction, convection and radiation heat transfer. II. Validation[END_REF], and Talukdar et al. [START_REF] Talukdar | Conduction-radiation interaction in 3d irregular enclosures using the finite volume method[END_REF] have already proven this analysis. At steady state, one can also observe on Fig. 3-(a) that, as 𝑘 𝑎 increases, the participating medium does not enable radiations coming from boundary surfaces to penetrate in the medium. Fig. 3-(b) illustrates the effect of absorption coefficient 𝑘 𝑎 on the dimensionless temperature profile at 𝑥 = 𝐻/2. One can note that when 𝑘 𝑎 increases and reaches a large value such as 𝑘 𝑎 = 5 𝑚 -1 for the present case, the shape of 𝑇 * displayed becomes similar to a straight line, whereas in the opposite direction one can observe a curvilinear shape. For this purpose, it means that the medium becomes thick and the energy generated by radiation is very quickly dissipated by absorption process. When 𝑘 𝑎 = 0.1 𝑚 -1 , the medium still participates weakly to radiative heat transfer, whereas from up to 𝑘 𝑎 = 1 𝑚 -1 there is a strong participation. In this section, the numerical results of 𝑇 * are compared with the literature as well as the effect of cell numbers on 𝑇 * . The CPU time obtained for different values of 𝑁 𝑝𝑙 and 𝑀 𝜃 × 𝑀 𝜑 is also presented.

Tabulated data of conductive-radiative quantities

Table 3

Computational time to obtain a converged solution of 𝑇 * at ℎ = 0, 𝑘 𝑎 = 1𝑚 -1 , 𝑁 𝑥 × 𝑁 𝑦 = 251 × 251.

Table 1 highlights the values of 𝑇 * that correspond to the absorbing-emitting medium computed when the optical thickness takes a value 𝜏 = 1. The results are presented at three locations along the centerline 𝑥 = 𝐻/2 for three values of the conduction-radiation parameter 𝑁 𝑝𝑙 and compared with those reported in the literature [START_REF] Yuen | Analysis of combined conductive-radiative heat transfer in a two dimensional rectangular enclosure with a gray medium[END_REF] and [START_REF] Wu | Transient two-dimensional radiative and conductive heat transfer in a scattering medium[END_REF][START_REF] Mishra | computational efficiency improvements of the radiative transfer problems with or without conduction-a comparison of the collapsed dimension method and the discrete transfer method[END_REF][START_REF] Mishra | solving transient conduction and radiation heat transfer problems using the lattice Boltzmann method and the finite volume method[END_REF]. For all aspect ratios in the LBM+DTM, LBM+DOM, LBM+FVM with 6 × 12 rays used for numerical integrations, there is not a significant difference on results obtained using the present method compared with the literature. In this case where we are dealing with a simple geometry (without obstacle), the advantage to be noted is related to the ease of handling and computing the semi-analytical solutions of the problem.

Using the present method, Table 2 presents the effects of cell numbers on dimensionless temperature 𝑇 * set at three locations, 𝑦/𝐻 = 0.2, 0.5 and 0.7 along the centerline 𝑥 = 𝐻/2. From the same table, 𝑀 𝜃 × 𝑀 𝜑 = 6 × 6 directions have been used, and there is no significant variation on results obtained for small and large values of cell numbers.

Table 3 shows the computational time taken to obtain the converged results of 𝑇 * at steady state using the present method. The number of cells used in computation is 𝑁 𝑥 × 𝑁 𝑦 = 251 × 251. For 𝑁 𝑝𝑙 = 1, there is dominance of conduction compared to radiation and since conduction is a local phenomenon, the results converge faster with almost all the values of 𝑀 𝜃 × 𝑀 𝜑 involved in the table. In contrast to 𝑁 𝑝𝑙 = 0.01, there is dominance of radiation compared to conduction, and convergence on results is reached with a difference of 15 s with 𝑀 𝜃 × 𝑀 𝜑 = 6 × 6; 6 s with 𝑀 𝜃 × 𝑀 𝜑 = 5 × 5 and 2 s with 𝑀 𝜃 × 𝑀 𝜑 = 4 × 4 respectively. The more the number of directions 𝑀 𝜃 × 𝑀 𝜑 increases, the more the CPU time also increases. From these tabulated results on can assert with good confidence, the accuracy and fastness of the method expanded.

Numerical simulation of conductive-radiative quantities

In this section, the numerical simulations of dimensionless temperature and heat flux are presented for different values of the conduction-radiation parameter. 34), can also be used to deduce the evolution of heat flux at the boundaries as already discussed in [START_REF] Ghattassi | Galerkin method for solving combined radiative and conductive heat transfer[END_REF][START_REF] Amiri | Combined conductive and radiative heat transfer in an anisotropic scattering participating medium with irregular geometries[END_REF][START_REF] Mishra | Performance evaluation of four radiative transfer methods in solving multi-dimensional radiation and/or conduction heat transfer problems[END_REF]. The authors have shown in 1-D, the heat flux by radiation and/or conduction at the hot wall of the 2-D enclosure.

Transient state

In this section, the results of 𝑇 * obtained with the present method are compared with the literature and the numerical simulations of the dimensional temperature distribution at different times 𝜉 are presented. converge suitably with those existing in the literature. According to the present study, some investigations can also be done using Dirichlet and Robin boundary conditions [START_REF] Ghattassi | Galerkin method for solving combined radiative and conductive heat transfer[END_REF]. Indeed, the In this section, three simulations of the transient dimensionless temperature distribution are presented for a simple geometry (𝐻 = 1𝑚). In this case, one used a dimensionless time step ∆𝜉 = 10 -6 and 𝑁 𝑥 × 𝑁 𝑦 = 251 × 251 cells. This choice of parameters has been done in order to respect the condition of stability 𝐹 0 ≤ 1/4 in Eq. ( 27) and to display suitable simulations. The choice of 𝑁 𝑝𝑙 = 0.1 is only because neither radiation nor conduction predominates in the process of heat transfer. However, the simulations show that heat spreads through the medium as time increases. For 𝜉 = 0.001, the program takes 58 s to generate the results, whereas for 𝜉 = 0.005, it takes 4 min 17 s and for 𝜉 = 0.015 it takes 14 min 5s. Therefore, even if one increases or decreases the value of 𝑁 𝑝𝑙 , the proposed simulations would have kept their symmetrical shapes, but with different CPU time to obtain the results.

Numerical verification

Numerical simulations

Finally, it should be noted that the computational time needed to reach convergence using the present method as shown in Table 3, increases when the conduction-radiation parameter 𝑁 𝑝𝑙 decreases. So, in contrast to Ghattassi et al. [START_REF] Ghattassi | Galerkin method for solving combined radiative and conductive heat transfer[END_REF] where the Galerkin method has been used, less important the contribution of the radiative part, a more quickly steady state was reached.

3.2.Complex geometry: ℎ ≠ 0

In this section, one intends to present the numerical results of dimensionless temperature and heat flux involved for the conductive-radiative heat transfer. Computation has been performed at steady and transient state when 𝐻 = 1𝑚, to show how heat transfer propagates in the semi-transparent medium. The predicted dimensionless temperature 𝑇 * and heat flux distributions have been displayed and discussed.

Steady state

In this sub-section, the impact of the obstacle, the cell numbers, the discrete directions and the conduction-radiation parameter are presented and discussed for 𝐻 = 3ℎ. Some numerical simulations of dimensionless temperature 𝑇 * distribution are also given for 𝐻 = 3ℎ, 𝐻 = (2 + √5)ℎ and 𝐻 = 8ℎ.

[, the medium doesn't participate to heat transfer that is why the profile is segmented into two parts because of the centered obstacle. On the other hand Fig. 8-(b) suitably shows the effect of 𝑁 𝑝𝑙 on the dimensionless temperature profile at 𝑥/𝐻 = 0.5. Higher the parameter 𝑁 𝑝𝑙 is and higher the decrease of 𝑇 * and the computation time is optimized. At the center line position, when 𝑦/𝐻 ≥ 1/3, the obstacle blocks any particle carrying heat from the participation and releases at 𝑦/𝐻 ≥ 2/3, that is why there is an empty space when 𝑦/𝐻 ∈ ] Fig. 9 indicates how the obstacle modifies the temperature profile along the centerline position of the medium. For optically thick medium (𝑘 𝑎 = 5 𝑚 -1 ), the heat blockage effect is significant, the medium behaves like a solid semi-transparent medium, and the CPU time elapsed to reach the convergence has been 18min 51s. On the other hand, when the medium becomes relatively thin (free path), the obstacle does not have a considerable influence on the heat propagation, the CPU time has decreased to 15min 14s and 13min 7s when 𝑘 𝑎 = 1 𝑚 -1 and 𝑘 𝑎 = 0.1 𝑚 -1 respectively.

Tabulated data of conductive-radiative quantities

Table 6

Computational time to obtain a converged solution of 𝑇 * for 𝐻 = 3ℎ, 𝐻 = (2 + √5)ℎ, at 𝑘 𝑎 = 1𝑚 -1 , 𝑁 𝑥 × 𝑁 𝑦 = 251 × 251.

Table 4 shows how the cell numbers modify quantitatively the values of 𝑇 * in the medium. So, higher the cell numbers are, more the results are approximated accurately. The series expansions of 𝐵𝑖𝑠 𝑛 (𝑢, 𝜃), with 𝑢 ∈ [0, 𝑘 𝑎 𝐻] and 𝜃 ∈ ]0, 𝜋 2

[ remain accurate up to the value of 𝑢 ≤ 8. Then, one should be careful on the choice of the number of terms in the series above this limit because of the round-off errors; why a double precision programming is recommended to evaluate the results expected [START_REF] Altaç | Integrals involving Bickley and Bessel function in radiative transfer and generalized exponential integral function[END_REF]. Table 5 shows the effects of discrete directions in the computation of 𝑇 * values. One can note that, when 𝑦/𝐻 = 0.9 there is no a significant change on results when the discrete directions variate. This is because heat transfer by radiation, becomes weak when 𝑦/𝐻 ≥ 0.9 and 𝑁 𝑝𝑙 = 0.1. In contrast, there is a slight change on results when one reaches close to the hot boundary surface. Table 6 gives prominence to the fact that, as the dimensions of the square obstacle increase, the CPU time also increases. This is because the number of sub-domains where the integral solutions of the radiative transfer equations have been calculated and computed had increased. So, one can also note that the CPU time is greater in the case of complex geometry compared with the one of simple geometry (ℎ = 0) and as 𝑁 𝑝𝑙 increases, the CPU time increases slightly for this complex geometry. Numerical simulations shown at Fig. 12 have been performed when the external surfaces of the centered obstacle are kept hot 𝑇 𝑟𝑒𝑓 , but the boundary surfaces of the 2D enclosure are maintained cold, at the initial temperature in the medium 𝑇 0 . Absorption coefficient is 𝑘 𝑎 = 1𝑚 -1 , corresponding to a unit optical thickness (𝜏 = 𝑘 𝑎 𝐻), with 𝐻 = 1𝑚. In this case, the size of the centered obstacle is small and for 𝑁 𝑝𝑙 = 0.1, neither conduction nor radiation predominates one on the other. The curves describing the set of hot points in the participating medium follow concave profiles around the centered obstacle. The number of the hot points concerned variates with the conduction-radiation parameter. Therefore, more 𝑁 𝑝𝑙 increases, better the number of hot points is. When 𝑁 𝑝𝑙 increases and reaches a high value, the number of hot points also increases and reaches a threshold; because only conduction predominates at this moment in the process of heat transfer.

With regard to the suitable previous simulations, the complexity of the geometry makes increase the number of sub-domains where the semi-analytical solutions are calculated. Therefore, each given geometry corresponds to different kind of equations modelling the problem and the smaller the dimensions of the obstacle, the greater the difficulty on simulations. By the way, as compared with the numerical methods like DOM, the present one doesn't suffer from ray effect when one solves the radiative transfer equation [START_REF] Djeumegni | Radiative heat transfer in a 2D semitransparent gray medium with a centered inner square cavity[END_REF] before combining it to the energy equation.

Effect of the obstacle on conduction-radiation temperature

The aim of this subsection is to present how the obstacle modifies quantitatively heat transfer in the participating medium. Therefore, Table .5 illustrates it with the corresponding values of 𝑇 * for ℎ = {0, √5 -2, 1/3}𝐻 along the y-direction of the 2D enclosure. Thus, the choice of 𝑁 𝑝𝑙 = 0.1 is made because there is a good interaction between conduction and radiation heat transfer. So, there is a slight difference on the values of 𝑇 * obtained when ℎ = 0, compared with those of ℎ ≠ 0, because of blockage of rays in the medium. When, ℎ = (√5 -2)𝐻,and ℎ = 𝐻/3, the radiation contributes less because the size of the obstacle becomes more important, then some rays from top, bottom, left and right sides of the participating medium are blocked. Consequently, more the size of the obstacle decreases more the values of 𝑇 * converges to those computed with ℎ = 0. If one continues this investigation for low value of 𝑁 𝑝𝑙 , some results already established for pure radiation [START_REF] Djeumegni | Radiative heat transfer in a 2D semitransparent gray medium with a centered inner square cavity[END_REF] can be used. On the other hand, if 𝑁 𝑝𝑙 takes large values, radiation heat transfer becomes ineffective and the obstacle does not affect heat transfer, because conduction heat transfer is a local phenomenon.

Table.7

Effect of the obstacle on conductive-radiative dimensionless temperature, 𝑁 𝑝𝑙 = 0.1, 𝑘 𝑎 = 1𝑚 -1 , 𝑁 𝑥 = 𝑁 𝑦 = 251 at the centerline position 𝑥/𝐻 = 0.5. When 𝑁 𝑝𝑙 ∈ {0.01,0.1, 1}, one can observe through the shape of the curves shown in Fig. 13 that one gets closer to the steady state as the time increases. Unfortunately, the steady state is not reached there during the same elapsed time, because the more 𝑁 𝑝𝑙 decreases, the faster it is reached. In this case dealing with complex geometry, the obstacle blocks rays coming from north, south, est and west sides of the 2-D enclosure. When the value of 𝑁 𝑝𝑙 decreases and facilitates the predominance of radiation, all the rays will not fully participate in the heat transfer because of the obstacle which will block them. Consequently, in the same conditions when the values of 𝑁 𝑝𝑙 decrease, the obstacle modifies quantitatively and qualitatively the field of conduction-radiation temperature in the semi-transparent medium.

Transient state

3.2.2.1.Numerical simulations

The numerical simulations presented concern the geometries 𝐻 = 3ℎ, and 𝐻 = (2 + √5)ℎ. The dimensionless step time ∆𝜉 = 10 -6 and 𝑁 𝑥 = 𝑁 𝑦 = 251 have been used to respect the condition of stability and to display suitable simulations.

 Case: 𝐻 = 3ℎ When 𝑁 𝑝𝑙 = 1 and 𝜉 = {1, 5, 20} × 10 -3 , heat spreads slowly from the south boundary to the north one of the participating medium but keeps it symmetry with respect to the line 𝑦 = 0.5.

Steady state takes considerable time to reach when 𝑁 𝑝𝑙 = 1, because of conduction predominance.

When 𝑁 𝑝𝑙 = 0.1 there is a faster propagation of heat between 0.001 < 𝜉 ≤ 0.005 compared with the case of 𝑁 𝑝𝑙 = 1. This is because radiation spreads fastly there during a short time; furthermore, when 𝜉 > 0.02 one tends towards steady state. When 𝑁 𝑝𝑙 = 0.01, radiation imposes that heat propagates very quickly by deflecting the obstacle until reaching the north boundary in a very short time. Consequently, at 𝜉 = 0.005 steady state is already reached compared with those cases where 𝑁 𝑝𝑙 takes the values of 1 and 0.1. These proposed numerical simulations also approve the behavior of the 1D numerical results presented in Fig. 13. The behavior of the simulations here is not far from those presented on Fig. 16, but the obstacle is smaller and enables more rays travel from south to north boundaries. There is a slight problem of symmetry there because it becomes difficult to mesh the geometry when the size of the obstacle is small.

With regard to the deep analysis carried out, although the present method uses the exact calculation of the integral form of radiative transfer equation, it provides suitable results for optically thick medium [START_REF] Djeumegni | Radiative heat transfer in a 2D semitransparent gray medium with a centered inner square cavity[END_REF]. Nevertheless, for the combined conductive-radiative heat transfer, when 𝑁 𝑝𝑙 becomes close to zero with an optical thickness 𝜏 ≥ 5 the present method leads to slight errors in computations.

Conclusion

An investigation on the combined conduction-radiation heat transfer in 2D semi-transparent medium with a centered obstacle has been carried out. Exact semi-analytical and then, numerical formulation of the problem has been presented. Modelling of radiation heat transfer has been established similarly with a ray tracing method adapted for the present geometry, and energy equation have been computed with the centered finite differences method. Conduction-radiation temperature and the total heat flux have been calculated, simulated then displayed both for simple and complex geometries. The effect of the conduction radiation parameter, the cell numbers, the discrete directions and the size of the cavity have been discussed. Numerical results of dimensionless temperature when there is no obstacle in the medium have been compared with the literature in good accuracy, both at steady and transient states. Several simulations of the semitransparent medium have also been proposed when there is variation of the conduction-radiation parameter and the size of the obstacle. The present method used to solve the conductive-radiative problem in complex geometry, have been realized with satisfactory computation time and without ray effect from radiation. Without some difficulties encountered with the use of regular grids in this irregular geometry, the method is promising to solve the conductive-radiative heat transfer in complex geometries. Later, one intends to combine convection-conduction and radiation of real gases or fluid flow filled in a similar geometry with diffusely reflecting boundaries.

Fig. 1 .

 1 Fig.1. Section of the semi-transparent medium in (𝑥, 𝑦) reference.

Fig. 2 -

 2 (a), only the bottom surface of the square section is hot at temperature 𝑇 𝑟𝑒𝑓 and the other surfaces are set to be cold. As shown Fig.2-(a), the radiation travelling from the bottom boundary surface up to the point 𝑀 follows the path of length 𝛿 such that 𝑀 𝑠 𝑀 ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ = 𝛿Ω ⃗⃗⃗ . Similarly, any point on the same ray is characterized by a path length 𝛿 ′ given by 𝑀′𝑀 ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ = 𝛿 ′ Ω ⃗⃗⃗ . In the reference plane (𝑥, 𝑦), one has 𝛿 = -𝑦/𝑐𝑜𝑠𝜑𝑠𝑖𝑛𝜃 and 𝛿 ′ = (𝑦 ′ -𝑦)/𝑐𝑜𝑠𝜑𝑠𝑖𝑛𝜃. The spatial relation linking any emissive point 𝑀′ within the participating medium and the calculation point 𝑀 on a same path for a given propagation direction, strongly depends on the angle 𝜑 and its positions 𝛿 𝑚 ∈ [0,1]. One needs to calculate the temperature and the radiative flux at each node (𝑖, 𝑗), hence, 𝑥 𝑖 ′ -𝑥 𝑖 = ±𝛿 𝑚 𝑦 𝑗 𝑡𝑎𝑛𝜑 and 𝑦 𝑗 ′ = (1 -𝛿 𝑚 )𝑦 𝑗 where (𝑥 𝑖 ′ , 𝑦 𝑗 ′ ) ∈ {𝒟}.

Fig. 2 .

 2 Fig. 2. (a) Ray propagation within the semi-transparent medium, (b) mesh grid within the semi-transparent medium.

  When a point 𝑀(𝑥, 𝑦) belongs to the other triangles or square subdomains delimited by broken red lines in Fig.2-(a), incident radiation have different analytical expressions due to the geometry in the medium. For any point in the domain such that

  d. Repeat steps c until convergence with the criterion 𝑀𝑎𝑥 [

  agreement with the literature according to the maximum relative error between the two shapes estimated at

Fig. 3 .

 3 Fig.3. Dimensionless temperature at = 𝐻/2 , ℎ = 0: (a) 𝑁 𝑝𝑙 ∈ {1, 0.1, 0.01} and 𝑘 𝑎 = 1𝑚 -1 , (b) 𝑘 𝑎 ∈ {5, 1, 0.1}𝑚 -1 and 𝑁 𝑝𝑙 = 0.1.

Fig. 4 .

 4 Fig.4. Dimensionless temperature distribution 𝑇 * : 𝜏 = 1, 𝜀 = 1, ℎ = 0, 𝑁 𝑝𝑙 ∈ {0.01,0.1,1}.

Fig. 4

 4 Fig.4highlights three main numerical simulations of dimensionless temperature 𝑇 * in a semitransparent medium enclosed in a 2D enclosure when there is no obstacle (ℎ = 0) and for the values of 𝑁 𝑝𝑙 = 0.01, 0.1 and 1. The aim is to show the behavior of 𝑇 * with the conductionradiation parameter 𝑁 𝑝𝑙 . The respective simulations have been displayed at steady state; it enables to observe the variation of 𝑇 * during a long time until convergence. 𝑇 * decreases from hot to cold points belonging the semi-transparent medium. Thus, when the value of 𝑁 𝑝𝑙 increases, the set of

Fig. 5 .

 5 Fig.5. Dimensionless conductive-radiative heat flux 𝑞 𝑦 * following y-direction for 𝑁 𝑝𝑙 ∈ {0.01, 0.1, 1}, 𝜀 = 1, 𝑘 𝑎 = 1𝑚 -1 , ℎ = 0.

Fig. 5

 5 Fig.5shows simulations of the dimensionless conductive-radiative heat flux following 𝑦-direction computed for 𝑁 𝑥 = 𝑁 𝑦 = 251, 𝜀 = 1, 𝑘 𝑎 = 1𝑚 -1 , and 𝑁 𝑝𝑙 ∈ {0.01, 0.1,1}. When, 𝑁 𝑝𝑙 = 0.01 conduction in the medium remains ineffective compared to radiation. Hence, the simulation takes quietly a view similar to Fig.8in the Ref.[START_REF] Djeumegni | Radiative heat transfer in a 2D semitransparent gray medium with a centered inner square cavity[END_REF] when 𝜏 = 1. At once 𝑁 𝑝𝑙 starts to increase from 0.01 to 1, energy transfer becomes straightly dominated by conduction heat transfer due to the hot boundary surface. Regardless the difference on magnitude of each simulation presented, there is a symmetry with respect to the centerline position 𝑥/𝐻 = 0.5. The numerical data of conductiveradiative heat flux obtained to display Fig.5using Eq. (34), can also be used to deduce the evolution of heat flux at the boundaries as already discussed in[START_REF] Ghattassi | Galerkin method for solving combined radiative and conductive heat transfer[END_REF][START_REF] Amiri | Combined conductive and radiative heat transfer in an anisotropic scattering participating medium with irregular geometries[END_REF][START_REF] Mishra | Performance evaluation of four radiative transfer methods in solving multi-dimensional radiation and/or conduction heat transfer problems[END_REF]. The authors have shown in 1-D, the heat flux by radiation and/or conduction at the hot wall of the 2-D enclosure.

Fig. 6 . 5 .

 65 Fig.6. Dimensionless temperature 𝑇 * in the participating medium when ℎ = 0, 𝑘 𝑎 = 1𝑚 -1 at different instant, with 𝑁 𝑝𝑙 ∈ {0.01,1, 0.1} for (a), (b) and (c) respectively.

  .1, 0.5, 1.5, 4, 28} × 10 -2 Steady state (𝒄) use of the thermal Robin boundary conditions requires the knowledge of the convective exchange coefficient ℎ 𝑐 of the wall; the surrounding temperature 𝑇 ∞ and the Biot number 𝐵 𝑖 = ℎ 𝑐 𝐻 𝑘 𝑐 and one can neglect the gain and losses by radiation for sake of complexity on equations to expand.

Fig. 7 .

 7 Fig.7. Transient dimensionless temperature distribution 𝑇 * : 𝜏 = 1, 𝜀 = 1, ℎ = 0, 𝑁 𝑝𝑙 = 0.1, 𝜉 ∈ {1,5,15} × 10 -3 .

Fig. 8 .

 8 Fig.8. Dimensionless temperature profile when 𝐻 = 3ℎ, 𝑘 𝑎 = 0.1𝑚 -1 : for (a) 𝑁 𝑝𝑙 = 1, 𝑥/𝐻 ∈ {0.2, 0.5, 0.7}, for (b) 𝑁 𝑝𝑙 ∈ {0.1, 0.5, 1}.

Fig. 9 .

 9 Fig.9. Dimensionless temperature 𝑇 * along the centerline position, 𝑥 = 𝐻/2 for 𝐻 = 3ℎ, 𝑘 𝑎 ∈ {0.1,1,5}𝑚 -1 and 𝑁 𝑝𝑙 = 0.1, 𝑁 𝑥 × 𝑁 𝑦 = 251 × 251, 𝜀 = 1.

Fig. 10 .

 10 Fig.10. Dimensionless temperature distribution 𝑇 * and total heat flux 𝑞 𝑦 * : 𝐻 = 3ℎ, south boundary surface is hot, 𝑘 𝑎 = 1𝑚 -1 , 𝑁 𝑥 = 𝑁 𝑦 = 251, 𝑁 𝑝𝑙 ∈ {0.01, 0.1, 1}.

Fig. 10 𝑁Fig. 11 .

 1011 Fig.10illustrates the effect of conduction-radiation parameter on the dimensionless temperature and the total heat flux fields, when 𝐻 = 3ℎ, 𝑘 𝑎 = 1𝑚 -1 and 𝑁 𝑥 = 𝑁 𝑦 = 251. The results of 𝑇 * presented inform that, more 𝑁 𝑝𝑙 increases, less heat transfer propagates in the participating medium. Similar simulations have been shown in the case of radiation heat transfer[START_REF] Djeumegni | Radiative heat transfer in a 2D semitransparent gray medium with a centered inner square cavity[END_REF], but with different physical profiles. When 𝑁 𝑝𝑙 = 0.01, there are more hot points in the participating medium, but those hot points decrease when 𝑁 𝑝𝑙 increases because of the medium that strongly absorbs radiations. Thus, similar observations are made at Fig.4in the sub-section 3.1.1, except the fact; there is an obstacle in this case. The results of 𝑞 𝑦 * show that, when 𝑁 𝑝𝑙 = 0.01, heat transfer in the medium is ensured by radiation. This analysis remains in accordance with Eq. (34), and the profile is very close to Fig.5obtained when there is no obstacle, but with different data because some rays have been blocked. When 𝑁 𝑝𝑙 reaches the unit value, conduction heat transfer takes over, mostly the hot south boundary surface contributes to energy transfer, and the wall heat fluxes can be deduced.

Fig. 12 .

 12 Fig.12. Dimensionless temperature distribution 𝑇 * : 𝐻 = 8ℎ, external boundary surfaces of the centered obstacle are hot, 𝑘 𝑎 = 1𝑚 -1 , 𝑁 𝑥 = 𝑁 𝑦 = 251, 𝑁 𝑝𝑙 ∈ {0.01, 0.1, 1}.

Fig. 13 .

 13 Fig.13. Transient dimensionless temperature 𝑇 * in the participating medium when 𝐻 = 3ℎ, 𝑘 𝑎 = 1𝑚 -1 at different instant 𝜉,with 𝑁 𝑝𝑙 ∈ {0.01,0.1, 1} for (a), (b) and (c) respectively.

Fig. 14 . 29 Fig. 15 .Fig. 16 .

 14291516 Fig. 14. Transient dimensionless temperature distribution 𝑇 * for 𝐻 = 3ℎ, south boundary surface is hot, 𝑘 𝑎 = 1𝑚 -1 , 𝑁 𝑥 = 𝑁 𝑦 = 251, 𝑁 𝑝𝑙 = 1, 𝜉 = {1, 5, 20} × 10 -3 .

30 Fig. 17 .Fig. 18 .Fig. 19 .Fig. 17 ,

 3017181917 Fig. 17. Transient dimensionless temperature distribution 𝑇 * for 𝐻 = (2 + √5)ℎ, south boundary surface is hot, 𝑘 𝑎 = 1𝑚 -1 , 𝑁 𝑥 = 𝑁 𝑦 = 251, 𝑁 𝑝𝑙 = 1, 𝜉 = {1, 5, 20} × 10 -3 .

  , that characterizes the transient heat transfer by conduction and radiation within each subdomain located by the label 𝑙 as:

	{ 𝜕𝑇 𝑙 * (𝜉,𝑠 * ) 𝜕𝜉	-∆𝑇 𝑙 * (𝜉, 𝑠 * ) = 𝜂 (𝐺 𝑙 * (𝜉, 𝑠 * ) -𝑇 𝑙 * 4 (𝜉, 𝑠 * )) = 𝜂𝑆 𝑙 𝑟 * ,
		𝑇 𝑙 * (0, 𝑠 * ) = 𝑣 𝑙,0 * (𝑠 * ),

  consequently the steady state is achieved and similar calculations yield to the same solution:

	(𝑇 𝑙 * ) 𝑖,𝑗 𝑘+1 =	1 4	{(𝑇 𝑙 * ) 𝑖-1,𝑗 𝑘	+ (𝑇 𝑙 * ) 𝑖+1,𝑗 𝑘	+ (𝑇 𝑙 * ) 𝑖,𝑗-1 𝑘	+ (𝑇 𝑙 * ) 𝑖,𝑗+1 𝑘	}
	+ 𝜂∆ * 2 [(𝐺 𝑙 * ) 𝑖,𝑗 𝑘 -(𝑇 𝑙 * 4 ) 𝑖,𝑗 𝑘+1 ]

1 √4∆𝜉 + 1), where 𝐸 denotes integer part of the real concerned. When the dimensionless time 𝜉 ∞, then 𝐹 0 * 1/4;

Table 1

 1 Comparison of dimensionless temperature 𝑇 * at steady state for 𝑥 = 𝐻/2 at location 𝑦/𝐻 ∈ {0.3, 0.5, 0.7}, for 𝜏 = 1.0.

	𝑁 𝑝𝑙	𝑦/𝐻 Wu and Ou [30]	Yuen and Takara [1]	Mishra et al [31]	Mishra and Roy [32]	Present
						FDM-FVM	LBM-FVM
		0.3	0.733	0.737	0.737	0.737	0.737	0.733
	1.0	0.5	0.630	0.630	0.630	0.630	0.630	0.630
		0.7	0.560	0.560	0.564	0.564	0.564	0.563
		0.3	0.760	0.763	0.759	0.759	0.759	0.755
	0.1	0.5	0.663	0.661	0.663	0.664	0.663	0.661
		0.7	0.590	0.589	0.594	0.596	0.596	0.591
		0.3	0.791	0.807	0.789	0.782	0.783	0.789
	0.01	0.5	0.725	0.726	0.725	0.726	0.725	0.726
		0.7	0.663	0.653	0.666	0.676	0.677	0.662

Table 2

 2 Effects of cell numbers on dimensionless temperature 𝑇

	𝑁 𝑥 × 𝑁 𝑦	𝑦/𝐻 = 0.2	𝑦/𝐻 = 0.5	𝑦/𝐻 = 0.7
	81 × 81	0.81224	0.65977	0.58941
	101 × 101	0.81387	0.65965	0.58935
	125 × 125	0.81400	0.65956	0.58879
	251 × 251	0.81542	0.66171	0.59134

* at 𝑥 = 𝐻/2, 𝑁 𝑝𝑙 = 0.1, and 𝑘 𝑎 = 1𝑚 -1 , ℎ = 0 using quadrature 𝑀 𝜃 × 𝑀 𝜑 = 6 × 6.

Table 4

 4 Effect of cell numbers on dimensionless temperature 𝑇 * at 𝑥/𝐻 = 0.5, 𝐻 = 3ℎ, 𝑘 𝑎 = 0.1𝑚 -1 ,

		𝑁 𝑝𝑙 = 0.1.
		𝑇 *
	𝑁 𝑥 × 𝑁 𝑦	𝑦/𝐻 = 0.2	𝑦/𝐻 = 0.7	𝑦/𝐻 = 0.9
	51 × 51	0.8247	0.6063	0.5369
	151 × 151	0.8260	0.6120	0.5372
	251 × 251	0.8260	0.6110	0.5364

Table 5

 5 Effect of discrete directions on dimensionless temperature 𝑇 * at 𝑥/𝐻 = 0.5, 𝐻 = 3ℎ 𝑘 𝑎 = 0.1𝑚 -1 ,

		𝑁 𝑝𝑙 = 0.1, 𝑁 𝑥 × 𝑁 𝑦 = 201 × 201.
		𝑇 *
	𝑀 𝜃 × 𝑀 𝜑	𝑦/𝐻 = 0.2	𝑦/𝐻 = 0.7	𝑦/𝐻 = 0.9
	4 × 4	0.8266	0.6123	0.5374
	5 × 5	0.8266	0.6124	0.5374
	6 × 6	0.8235	0.6123	0.5374

Acknowledgment

Institut Pprime, University of Poitiers, CNRS and ENSMA UPR 3346 in France full supported this work.