# Conjugate conduction-radiation heat transfer of non-gray gases in a 2D combustion chamber using ray tracing method and SLW correlated model

Julien Sorel Djeumegni, Vital Le Dez, Myriam Lazard

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Julien Sorel Djeumegni, Vital Le Dez, Myriam Lazard. Conjugate conduction-radiation heat transfer of non-gray gases in a 2D combustion chamber using ray tracing method and SLW correlated model. International Communications in Heat and Mass Transfer, 2026, 172 (4), pp.110527. 10.1016/j.icheatmasstransfer.2026.110527. hal-05493836

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# Conjugate conduction-radiation heat transfer of non-gray gases in a 2D combustion chamber using ray tracing method and SLW correlated model

Julien Sorel Djeumegni<sup>a,b,*</sup>, Vital Le Dez<sup>c</sup>, Myriam Lazard<sup>c</sup>

## ARTICLE INFO

**Keywords:**

Conduction-radiation

Non-gray gases

SLW correlated model

Semi-analytical method

Bickley-Naylor function

Planck-mean

## ABSTRACT

The present study examines combined conduction-radiation heat transfer in non-gray gases within a two-dimensional rectangular domain, representing a section of a combustion chamber. The participating gases considered are 10%CO₂/N₂, 20%H₂O/N₂ and their corresponding mixture, whereas N₂ is treated as non-participating. A correlated SLW model is developed to evaluate the Planck-mean absorption coefficients by summing the products of discrete gray absorption coefficients and their associated weights. The radiative properties of the non-gray gases are obtained over their absorption spectral bands using data from HITEMP2010, while thermal conductivities are computed using the Sutherland relation. An exact semi-analytical formulation based on ray tracing, Bickley-Naylor functions, and Gauss quadrature is employed to solve the radiative transfer equation (RTE) using Planck-mean absorption coefficients. The energy equation is discretised using a finite-difference approach with a Crank-Nicolson scheme. The effects of gas thermal properties, gas temperature, the number of discrete gray gases, gas species, and mixture ratios on thermal quantities are analysed in detail. Numerical simulations of dimensionless conduction-radiation temperature and total heat flux are presented.

## 1. Introduction

The conjugate conduction-radiation heat transfer in non-gray gases at high temperatures, remains of significant interest for industrial applications, such as combustion chambers in aircraft, aerospace, and rocket engines [1]. Variations in gas spectral radiative properties with temperature and geometry strongly affect heat transfer rates [2]. The problem involves radiative heat transfer by carbon dioxide (CO₂), water vapor (H₂O) and Nitrogen (N₂), combined with conduction along solid walls [3]. Accurate evaluation of absorption coefficients and thermal conductivities is required, making computationally efficient, reliable, and compatible solution methods essential for addressing this combined thermal problem.

Over the past several years, various models have been proposed for evaluating the absorption coefficients of non-gray gases, each offering different levels of accuracy. The most empirical and accurate approach is the line-by-line (LBL) model [4,5], which divides the absorption spectrum into very narrow bands (Δη ∈ [2.10⁻⁴, 2.10⁻²] cm⁻¹), treated as monochromatic rays. It relies on spectroscopic databases such as HITEMP [6], HITRAN [7], and CDSD for CO₂ [8]. Nevertheless, the LBL

model requires extensive computations and high computer resources. To reduce computational effort, narrow-band models [9] were developed, dividing the spectrum into intervals of Δη ∈ [5, 50]cm⁻¹ to estimate absorption coefficients. These models account for line overlap through line spacing, intensity, and half-width variations. While narrow-band models improve efficiency, their computational cost remains significant. Global models have therefore been introduced to further reduce computational time while computing non-gray gas absorption coefficients [10–13]. Their formulations are based on the absorption spectral bands of the respective gases [14]. Hottel and Sarofim [15] introduced the Weighted-Sum-of-Gray-Gas Model (WSGGM), widely used in engineering applications and considered accurate for temperatures between 600 K and 2400 K [16]. Modest [11] proposed reordering the wave numbers of non-gray gases into a set of discrete gray gases. Building on this, Denison and Webb [14] developed the Spectral-Line-Based WSG (SLW) model, which has proven highly accurate in combination with gray gas models. Pearson et al. [17,18] further developed a correlation of the absorption-line blackbody distribution function (ALBDF) to improve the SLW model of Denison and Webb [14]. The Full-Spectrum Correlated-k (FSK) [19] and the multi-scale Full-Spectrum Correlated-k (MSFSK) models [20] have also been proposed, employing

<footnote><sup>a</sup> Nantes Université, ONIRIS, CNRS, GEPEA, UMR-6144, 37 Bd de l'université BP406, 44602 Saint-Nazaire, France</footnote>

<footnote><sup>b</sup> Université de Perpignan, PROMES-CNRS, UPR 8521, Tecnosud, Rambla de la thermodynamique, 66100 Perpignan, France</footnote>

<footnote><sup>c</sup> Institut Pprime, Université de Poitiers, CNRS, ENSMA, UPR 3346, 2 Rue Pierre Brousse, Batiment B25, TSA 41105, 86073, Poitiers Cedex 9, France</footnote>

<footnote>* Corresponding author at: Nantes Université, ONIRIS, CNRS, GEPEA, UMR-6144, 37 Bd de l'université BP406, 44602 Saint-Nazaire, France.
E-mail address: julien.djeumegni@univ-nantes.fr (J.S. Djeumegni).</footnote>
Nomenclature

a<sub>g</sub> weighted coefficients of discrete gray gases

a<sup>loc</sup><sub>g</sub> weighted coefficients of discrete gray gases at local temperature

b<sub>0</sub> prescribed wall temperature (K)

C<sub>abs</sub> prescribed absorption cross section (m<sup>2</sup>/mol)

C<sub>max</sub> maximum absorption cross section (m<sup>2</sup>/mol)

C<sub>min</sub> minimum absorption cross section (m<sup>2</sup>/mol)

C̃<sub>g</sub> supplemental absorption cross section (m<sup>2</sup>/mol)

C̃<sub>g</sub><sup>loc</sup> supplemental absorption cross section at local temperature (m<sup>2</sup>/mol)

C̃<sub>g</sub><sup>ref</sup> supplemental absorption cross section at reference temperature (m<sup>2</sup>/mol)

C<sub>η</sub> spectral absorption cross section (cm<sup>2</sup>/molecule)

C<sup>Ω</sup> unit disk of R<sup>2</sup>: C<sup>Ω</sup> = {Ω ∈ R<sup>2</sup>/|Ω| ≤ 1}

c<sub>0</sub> prescribed dimensionless wall temperature

($\vec{e}_x, \vec{e}_y$) unit vectors of x, y directions

F<sup>*</sup><sub>N</sub> Fourier number

g gray gas number

G volumic incident radiation (Wm<sup>-3</sup>)

G<sup>*</sup> dimensionless incident radiation

L, H dimensions of the geometry section along x and y direction (m)

(i,j,p,q) cells numbering

I incoming radiation intensity (Wm<sup>-2</sup>Sr)

I<sub>b</sub> black body radiation intensity (Wm<sup>-2</sup>Sr)

I<sub>w</sub> radiation intensity at wall (Wm<sup>-2</sup>Sr)

k<sub>η</sub> spectral absorption coefficients (cm<sup>-1</sup>)

k<sub>g</sub> discrete absorption coefficient (m<sup>-1</sup>)

k<sup>loc</sup><sub>g</sub> discrete absorption coefficient at local temperature (m<sup>-1</sup>)

K<sub>ln</sub> Bickley-Naylor functions

l, m angular and spatial numbering quadrature

N<sub>g</sub> number of discrete gray gases

M<sub>φ</sub>, N<sub>θ</sub> number of angular quadrature

N<sub>x</sub>, N<sub>y</sub> number of cells of the mesh grid

N<sub>pl</sub> conduction-radiation parameter

P<sub>T</sub> total pressure at which gas characteristics are evaluated (Pa)

q̃<sup>r</sup> radiative flux vector (Wm<sup>-2</sup>)

q̃<sup>r*</sup> dimensionless radiative flux vector

q̃<sup>c</sup> conductive heat flux (Wm<sup>-2</sup>)

q̃<sup>c*</sup> dimensionless conductive heat flux

q̃<sup>t*</sup> dimensionless conductive-radiative heat flux

q̃<sup>t*</sup><sub>x</sub> x-component of the dimensionless conductive-radiative heat flux

q̃<sup>t*</sup><sub>y</sub> y-component of the dimensionless conductive-radiative heat flux

R<sub>u</sub> ideal gas constant (P<sub>a</sub>m<sup>-3</sup>/molK)

S̃<sup>r</sup> radiative source vector (Wm<sup>-3</sup>)

s curvilinear abscissa (m)

T<sub>E</sub>, T<sub>N</sub>, T<sub>W</sub>, T<sub>S</sub> prescribed temperature at East, North, West, and South walls (K)

T radiation temperature (K)

T<sub>0</sub> initial temperature in the medium (K)

T<sub>hot</sub>, T<sub>cold</sub> hot and cold temperature at wall (K)

T<sup>ref</sup>, T<sup>loc</sup> reference and local temperature (K)

T<sub>b</sub> temperature of the blackbody (K)

T<sub>w</sub> temperature at wall (K)

T<sup>*</sup> temperature at which gas characteristics are calculated (K)

u dimensionless temperature in the medium

u real number at which E(u) is evaluated

Y<sub>s</sub> chemical ratio of absorbing gases

Greek symbols

κ<sub>c</sub> thermal conductivity (Wm<sup>-1</sup>K<sup>-1</sup>)

κ<sub>p</sub> Planck-mean absorption coefficients (m<sup>-1</sup>)

Δx, Δy space step along x and y respectively (m)

Δx<sup>*</sup>, Δy<sup>*</sup> dimensional space step along x and y respectively

∇, Δ Gradient and Laplacian operators respectively

β<sub>l</sub> angular abscissa

σ<sub>B</sub> Stefan-Boltzmann constant (5.67 10<sup>-8</sup>Wm<sup>-2</sup>K<sup>-4</sup>)

φ, θ azimuthal and zenith angle of unit vector Ω̃

Ω̃ unit radiation propagation vector

∇.q̃<sub>r</sub> radiative flux divergence(Wm<sup>-3</sup>)

δ radiative path length from boundary to a point in the medium

η wave number (m<sup>-1</sup>)

Δη spectral interval (m<sup>-1</sup>)

ω<sub>l</sub>, ω<sub>m</sub> Gaussian weights

δ<sub>l</sub>, δ<sub>m</sub> spatial and angular abscissa

φ<sub>l</sub> azimuthal angle at position l

ε<sub>N</sub>, ε<sub>S</sub>, ε<sub>E</sub>, ε<sub>W</sub> unit emissivity at North, South, East and West walls

Subscripts

∂<sub>Σ</sub> bounded domain of geometry sections

{Σ<sub>i</sub>} participating domain of geometry sections R<sup>2</sup>

{Σ} entire domain of geometry sections R<sup>2</sup>

Γ<sub>E</sub>, Γ<sub>N</sub>, Γ<sub>W</sub>, Γ<sub>S</sub> East, North, West, South walls of the geometry section

absorption coefficient distribution functions (ADFFG) computed from high-resolution HITEMP [6] and HITRAN databases [7]. Because the FSK and SLW models depend on the reference gas thermodynamic state, rank-correlated versions (RC-FSK and RC-SLW) [21] were developed to overcome this limitation. Among these global models, the SLW model demonstrates excellent compatibility for simulating non-gray radiative heat transfer problems using either numerical or analytical methods. Consequently, the ALBDF correlation in the SLW model [17,18] makes it particularly suitable for coupling with conduction-radiation governing equations.

Studies investigating absorption coefficient models for non-gray

gases, combined with conduction-radiation heat transfer in multidimensional geometries remain scarce in the literature. Greif R. [22] carried out an empirical investigation of radiative and conductive energy transfer considering variable gas properties, including temperature and frequency dependence. The study considers a one-dimensional slab where heat transfer occurs through gas absorption, emission, and conduction, enabling evaluation of the 1D energy flux. Thermal conductivity and absorption coefficients were assumed to follow a power-law dependence on temperature and the gray gas. Picket fence, and modified picket fence approximations were also examined. Furthermore, Sun Y. et al. [23] analysed non-gray gases, combined with conduction-
radiation heat transfer using finite volume method (FVM) and the SLW in a two-dimensional rectangular geometry. The authors used the SLW method to evaluate absorption coefficients of carbon dioxide, water vapor and carbon monoxide. The FVM has been applied to solve radiative transfer and energy equations, enabling an evaluation of temperature distribution in the medium and heat fluxes at walls. Accuracy of each individual gas shows similar temperature profiles and heat fluxes at walls, with small relative errors. Subsequently, Atashafrooz et al. [24] investigated conduction-radiation heat transfer in a complex enclosure through combined gray and non-gray simulations based on the FSK method. The blocked-off method was used to convert the inclined recess in the complex enclosure into an equivalent regular rectangular surface. The spectral absorption coefficients of the gas mixture were determined using the HITRAN 2008 database. Simulations of the non-gray medium were performed using the full-spectrum k-distribution (FSK) approach, whereas the gray medium was modeled using the Planck-mean absorption coefficient. Therefore, discrete ordinates method (DOM) and FVM have been respectively applied to compute radiation transport and energy equations respectively. The effects of surface emissivity on conductive, radiative, and total heat fluxes were also presented, demonstrating good agreement with expected results. Similar research considering non-gray gases combined with radiation heat transfer can be followed from Refs. [25, 26]. Additionally, ray-tracing has been recently applied to model a conductive-radiative heat transfer at the particle scale in pebble-bed [27], and to evaluate the radiative thermal conductivity of particles [28]. Therefore, among these research work in 2D geometries, most numerical methods have been applied to solve the radiative problem, leading to a significant source of computational errors.

Indeed, in engineering applications involving combustion, exact solutions for conduction-radiation heat transfer in non-gray gases within two-dimensional burner sections, are still needed. Despite investigations of the combined conduction-radiation of gray semi-transparent medium inside a rectangular section with a squared centered obstacle by ray tracing method [29], no research has yet followed a similar methodology in case of non-gray gases inside a rectangular geometry. Because combustion processes occur at high temperature, an SLW correlated model is then proposed for computations of the Planck-mean absorption coefficient, from the ALBDF [17,18]. The use of Planck-mean absorption coefficients in computations leads to accurate results, low computational time and memory resources as required. Additionally, the compatibility in combining the ray tracing method and the Planck-mean absorption coefficient to solve the conjugated thermal problem in a 2D geometry motivates the present research work.

This study aims to characterise thermal properties of carbon dioxide, water vapor, and their mixture with the SLW-correlated model, and combine the results with the conduction-radiation equations within the 2D furnace section. The Planck-mean absorption coefficients of non-gray gases are computed from the updated ALBDF [17,18], whereas their corresponding thermal conductivities are computed following Sutherland relation [30]. Exact semi-analytical solutions of incident radiation are obtained using ray tracing method, additionally with specific Bickley

Naylor functions [31–33] and Gauss quadrature. The temperature and total heat flux distributions are obtained after solving the respective governing equations yielded from the conjugate conduction-radiation problem. Therefore, numerical results of thermal quantities are then highlighted following curve shape, tabulated data and suitable numerical simulations.

## 2. Modeling

### 2.1. Geometry

Let's consider the rectangular section of $L \times H$ shown on Fig. 1-(a), which denotes the two-dimensional section of the combustion chamber. There is no variation of radiative quantities along the z-direction of the geometry. The propagation of the gas flame considers both conduction and radiative heat transfer. Along the $(x,y)$ coordinates, the participating domain $\{\mathcal{D}_i\}$ for the conjugate problem is defined as shown in Eq. (1):

$$ \{\mathcal{D}_i\} = \{\forall(x,y) \in [0,L[ \cap [0,H[\}. \quad (1) $$

The black boundaries ($\epsilon = 1$), denoted as $\partial\mathcal{D}_e \in \{\Gamma_E \cup \Gamma_W \cup \Gamma_N \cup \Gamma_S\}$, represent the East, West, North and South walls, each maintained at a prescribed constant temperature $T_E, T_W, T_N$ and $T_S$.

For any $(x,y)$ within the entire domain, $\{\mathcal{D}\} \in \partial\mathcal{D}_e \cup \{\mathcal{D}_i\}$. The gas flame is initially at temperature $T_0$ and the z-axis is assumed infinite, leading to thermal quantities that are invariant along this axis. Referring to Fig.1-(b), the gaseous semi-transparent medium is discretised into an isothermal mesh grid of uniform cells $\Delta x \times \Delta y$, where $\Delta x = L/(N_x - 1)$ and $\Delta y = H/(N_y - 1)$. The integers $N_x$ and $N_y$ represent the total number of nodes along the x- and y-axes, respectively formed by the mesh grid. Thermal quantities are evaluated at point $M(x,y)$, which are discretized into nodes positions $(x_i, y_j)$ for numerical calculations, with $(i,j) \in [2, N_x - 1] \times [2, N_y - 1]$. The medium is composed of non-gray gases such as carbon dioxide ($CO_2$), water vapor ($H_2O$) or their mixture, which are commonly encountered in exhaust engineering applications [15,34]. Because thermal radiation is closely linked to the geometry, an analysis of ray propagation within the domain is performed before addressing the conduction-based energy equation.

### 2.2. Ray tracing

The ray-tracing method enables to evaluate path lengths of radiation reaching any calculation point $M(x,y)$, contributing to solve the radiative transfer equation.

Subsequently, radiation propagates along a specified direction vector $\vec{\Omega} = \begin{pmatrix} \cos\varphi\sin\theta \\ \sin\varphi\sin\theta \\ \cos\theta = 0 \end{pmatrix}$, where $\varphi$ and $\theta$ represent the azimuth and zenith angles respectively, such that $\varphi \in [0, 2\pi]$ and $\theta \in [0, \pi]$. Accordingly, radiation at any point $M(x,y)$ is the sum of contributions from rays emitted by the West, North, South, and East

<figure><img src="image_3.png" /><figcaption>Fig. 1. 2D section of the combustion chamber with non-gray gases $10\%CO_2 + 20\%H_2O$: (a) ray path length from radiation to point M, (b) mesh grid.</figcaption></figure>
walls, propagating along two optical paths: $\overline{M_{\partial \mathcal{S}^i} \overline{M}} = \delta \overline{\Omega}$ and $\overline{M'_{\partial i} \overline{M}} = \delta \overline{\Omega}$. Let design the mathematical formulation of the two kinds of ray path lengths, considering the radiation contribution from the East wall. Therefore, the radiation pathway propagating from the East wall into the medium follows the relation: $\delta = (L-x)/\cos(\varphi_{E1,2})\sin\theta$ and $\delta' = (x'-x)/\cos(\varphi_{E1,2})\sin\theta$ in the $(x,y)$ reference coordinates. Consequently, because temperature distribution is determined iteratively from radiative equilibrium, each radiative source point $M(x',y) \in \{\mathcal{S}^i\}$ in the medium depends on the azimuth angle $\varphi_{E1,2} = (\varphi_{E1}, \varphi_{E2})$, as illustrated on Fig.1-(a), and the weighted positions $\delta_m \in [0, 1]$. Specifically, the coordinates are given by: $x'_i = x_i + (L-x_i)\delta_m$ and $y'_j = y_j + (L-x_i)\delta_m \tan(\varphi_{E1,2})$. Due to the following relations, $x'_i \in [x_p - \Delta x, x_p + \Delta x]$ and $y'_j \in [y_q - \Delta y, y_q + \Delta y]$, the discretised integer position $(p, q)$ defined by, $p = E\left[(i + \frac{1}{2}) + (N_x - i - \frac{1}{2})\delta_m\right]$ and $q = E\left[(j + \frac{1}{2}) + (N_x - i - \frac{1}{2})\delta_m \tan(\varphi_{E1,2})\right]$ is derived. The illustrated method is applied to all walls over the azimuthal range $\varphi \in [0, 2\pi]$, to compute the corresponding radiative quantities followed.

## 2.3. SLW correlated model

The fraction of blackbody energy contained within specific portions of the spectrum is described by the absorption-line blackbody distribution function (ALBDF), denoted as $F(C_{abs}, T_b, T_g, P_T, Y_s)$, as explained in Refs. [17, 18]. This function is computed from high-resolution absorption coefficient spectra and depends on the absorption cross section $C_\eta$, the blackbody temperature $T_b$, the gas temperature $T_g$, the total pressure $P_T$, and the molar fraction of the gas species $Y_s$. It represents the weighted fraction $a_g$ of the absorption coefficient spectrum, weighted by the Planck emissive power, such that the absorption cross section remains below a prescribed value $C_{abs}$. The ALBDF is defined by Eq. (2):

$$F(C_{abs}, T_b, T_g, P_T, Y_s) = \frac{1}{\sigma_B T^4} \int_{\{\eta: C_\eta(t_g, P_T, Y_s) \le C_{abs}\}} E_{b\eta}(\eta, T_b) d\eta, \quad (2)$$

where, the subscript g denotes the $g^{th}$ spectral interval, and the integration is performed over the entire spectrum $\eta$. Considering the updated HITEMP-2010 database, ALBDF computations [17] are carried out over spectral intervals $\Delta\eta$ and compared with prescribed values of $C_{abs}$. The Planck blackbody intensities within each interval are summed and normalized by the total blackbody emissive power at the reference temperature $T_b$. An updated correlation proposed by Pearson et al. [18], improves accuracy by applying a least-squares fit to the original hyperbolic tangent formulation of the ALBDF introduced by Denison and Webb [14], combined with a reference-state approach. Absorption cross-section functions of $CO_2$ and $H_2O$ at atmospheric pressure are provided in Appendix A, as part of the present correlation. To account for temperature variations in high-temperature gas processes, the classical reference approach of Denison and Webb [14] was further enhanced by Darbandi and Abrar [35] through a modified reference temperature, which is adopted in the present work. Consequently, discrete gray gas absorption coefficients and their corresponding weights, derived from both the correlated ALBDF and the modified reference approaches, are incorporated into the radiative transfer equation solver.

According to the LBL model, the SLW correlated model partitions the full absorption wavelength range into a set of absorption cross section $\tilde{C}_g$ for $0 \le g \le N_g$. The supplemental absorption coefficient is selected between its minimum value $\tilde{C}_0 = C_{min}$ and maximum value $\tilde{C}_{N_g} = C_{max}$ as defined in Ref. [18] following the Eq. (3):

$$\tilde{C}_g = C_{min} \left( \frac{C_{max}}{C_{min}} \right)^{\frac{g}{N_g}}, 0 \le j \le N_g \quad (3)$$

where, $N_g$ is the number of discrete gray gases. The black-body temperature used with the modified reference approach is considered as the local temperature, $T_b = T^{loc}$, and the gas temperature as the reference temperature, $T_g = T^{ref}$ to compute absorption coefficients $k_g = k_g^{loc}$ of discrete gray gases, following Eq. (4):

$$k_g = NY_s (\tilde{C}_{g-1} - \tilde{C}_g)^{\frac{1}{2}}, \quad (4)$$

with $N = \frac{p_T}{R_u T_g}$, and $R_u$ defining the ideal gas constant. Then, the modified absorption cross section follows the relation, $\tilde{C}_g = \tilde{C}_g^{loc} = \tilde{C}_g^{ref}$. The weighted gray gas coefficients $a_g = a^{loc}$ are computed based on the selected absorption cross sections, such as defined in Ref. [35] and given by Eq. (5):

$$a_g = F(\tilde{C}_g^{ref}, T_g = T^{ref}, T_b = T^{loc}) - F(\tilde{C}_{g-1}^{ref}, T_g = T^{ref}, T_b = T^{loc}) \quad (5)$$

The computation of radiative properties in the present SLW correlated method is performed using the updated correlated kinetic equations, as developed by Pearson et al. [18] and presented in Appendix A. A new set of coefficients are added to better represent the ALBDF at a constant pressure of 1atm. These coefficients related to carbon dioxide doesn't depend on mole fraction, unlike those for water vapor, which affect the model because of self-broadening. The coefficients in the predictive model are valid for temperatures between 400K $\le T_g$, $T_b \le 3000K$, and pressures between 0.1atm $\le P \le 50atm$. The use of $10^{-4}m^2/mol$ for $C_{min}$ for both $CO_2$ and $H_2O$ gas species, followed with $600m^2/mol$ and $60m^2/mol$ for $C_{max}$ are respectively recommended. In two-dimensional geometries, variations in gas temperature across the $(x,y)$ plane induce corresponding changes in absorption and weighting coefficients. The geometry considered contains either a single gas $CO_2/N_2$, $H_2O/N_2$ or a mixture $CO_2/H_2O/N_2$, where $N_2$ represents a spectral window. In case of gas mixtures $CO_2/H_2O/N_2$ burning in the medium, the corresponding absorption and weighted coefficients are given by $k_g^{mix} = k_g^{CO_2} + k_g^{H_2O}$ and $a_g^{mix} = a_g^{CO_2} \cdot a_g^{H_2O}$. So, Planck-mean absorption coefficients provide an accurate description of radiative heat loss and total emission from the participating gas [2,34]. Once the discrete gray gas parameters ($k_g, a_g$) are computed, the Planck-mean absorption coefficients can be determined.

## 2.4. Planck-mean absorption coefficients

The Planck-mean absorption coefficients are obtained from line-by-line calculations using high-accuracy databases HITEMP [6] and HITRAN [7]. In the current SLW correlated method, the Planck-mean absorption coefficients are scaled using the high-resolution HITEMP-2010 database [6], according to Ref. [2] and given in Eq. (6):

$$\kappa_p = \frac{\int_0^\infty k_\eta I_{b\eta} d\eta}{\int_0^\infty I_{b\eta} d\eta} = \frac{1}{I_b} \sum_i I_{b\eta_i} S_i. \quad (6)$$

The total blackbody intensity is given by $I_b = \frac{\sigma_B T^4}{\pi}$, $I_{b\eta}$ and $k_\eta$ denote the spectral blackbody intensity and spectral absorption coefficients, respectively. The line-integrated coefficient, $S_i = \int_0^\infty k_{\eta_i} d\eta$ also depends on pressure and temperature. Using the HITEMP-2010 database, $\kappa_p$ represents the smoothly varying contribution of each spectral line, with each gray gas g corresponding to the contribution of its individual spectral lines. The Planck-mean absorption coefficient is the product of the absorption coefficients $k_g$ and weighting coefficients $a_g$ for each gray gas in the spectrum [36], such that $\kappa_p = \sum_{g=0}^{N_g} a_g k_g$, and $\kappa_p^{mix} =$
$\sum_{g=0}^{N_g} a_g^{mix} k_g^{mix}$ in case of gas mixture.

### 3. Governing equations

#### 3.1. Radiative heat transfer

The semi-transparent non-gray gases inside the combustion chamber absorb and emit radiation but do not scatter it and are enclosed by black surfaces at the boundaries $\partial \mathcal{S}_e$. The entire spectral range $\eta \in [0, +\infty[$ of the absorbing gases is therefore divided into a finite number of discrete gray gases $N_g$ each with a spectral width $\Delta g$. To optimise computational resources while maintaining good accuracy, the radiative transfer equation (RTE) is solved using the Planck-mean absorption coefficients $\kappa_p$, as given in Eq. (7):

$$ \sin\theta \left[ \cos\varphi \frac{\partial I(x, y, \theta, \varphi)}{\partial x} + \sin\varphi \frac{\partial I(x, y, \theta, \varphi)}{\partial y} \right] = -\kappa_p [I(x, y, \theta, \varphi) - I_b[T(x, y)]], \quad (7) $$

subjected to the boundary conditions $I(x, y, \theta, \varphi) = I_b[T_w]$, for $(x, y) \in \partial \mathcal{S}_e$, $(\theta, \varphi) \in \Theta^\Omega$,

where $I_b = \frac{\sigma_B T^4}{\pi}$ is the blackbody intensity, $T_w$ denotes the wall temperature, $T$ is the gas temperature within the medium, and $(\theta, \varphi)$ are the zenith and azimuth angles of the unit radiation propagation vector $\vec{\Omega}$.

##### 3.1.1. Radiative integral solutions

The exact integral solution of Eq. (7), corresponding to all non-isothermal discrete gray gases $g = 1, 2, \dots, N_g$, is given by Eq. (8):

$$ I(x, y, \theta, \varphi) = I_w e^{-\kappa_p \delta(x, y, \theta, \varphi)} + \kappa_p \int_0^{\delta(x, y, \theta, \varphi)} I_b[T(x', y')] e^{-\kappa_p \delta} d\delta, \quad (8) $$

where $(x', y') \in [0, x] \times [0, y]$, $\delta$ and $\delta'$ denote respectively the curvilinear abscissa of ray pathways in the medium. The term $I_w$ represents the unidirectional radiation leaving each wall of the enclosure and is defined as $I_w = I_b[T_w]$, $(x, y) \in \partial \mathcal{S}_e$.

The incident radiation within the participating medium is obtained from Eq. (9):

$$ G(x, y) = 2 \int_{\theta=0}^{\theta=\frac{\pi}{2}} \int_{\varphi=0}^{\varphi=2\pi} I(x, y, \theta, \varphi) \sin\theta d\theta d\varphi. \quad (9) $$

According to the ray-tracing method, the exact semi-analytical contribution to the incident radiation originating from the East wall of the enclosure containing the non-gray gases is given by Eq. (10):

$$ G_{\Gamma_E}(x_i, y_j) = \frac{2\sigma_B T_E^4}{\pi} \left[ \varphi_{E_1}^{ij} \sum_{l=1}^{N_\varphi} \omega_l K_i^2 \left( \frac{\kappa_p (L-x_l)}{\cos\varphi_{l_1}} \right) + \varphi_{E_2}^{ij} \right. \\ \left. \times \sum_{l=1}^{N_\varphi} \omega_l K_i^2 \left( \frac{\kappa_p (L-x_l)}{\cos\varphi_{l_2}} \right) \right] + \frac{2\sigma_B}{\pi} (L-x_l) \varphi_{E_1}^{ij} \sum_{l=1}^{N_\varphi} \omega_l K_i^2 \left( \frac{\kappa_p (L-x_l)}{\cos\varphi_{l_1}} \right) + \frac{2\sigma_B}{\pi} (L-x_l) \varphi_{E_2}^{ij} \\ \times \sum_{m=1}^{M} \omega_m \omega_l T^4 (x'_i, y'_j) K_i^1 \left( \frac{\kappa_p (L-x_l)\delta_m}{\cos\varphi_{l_2}} \right) + \sum_{m=1}^{M} \omega_m \omega_l T^4 (x'_i, y'_j) K_i^1 \left( \frac{\kappa_p (L-x_l)\delta_m}{\cos\varphi_{l_2}} \right), \quad (10) $$

where, $\varphi_{E_1}^{ij} = \tan^{-1}\left\{\frac{H-y_1}{L-x_1}\right\}$, $\varphi_{E_2}^{ij} = \tan^{-1}\left\{\frac{y_1}{L-x_1}\right\}$, $\varphi_{l_1} = \delta_l \times \varphi_{E_1}^{ij}$, $\varphi_{l_2} = \delta_l \times \varphi_{E_2}^{ij}$. The specific functions $K_1$ and $K_2$, are respectively the Bickley Naylor functions of first and second order [31], and are defined as:

$$ K_i(u) = \int_{\theta=0}^{\theta=\frac{\pi}{2}} e^{-\frac{u}{\sin\theta}} (\sin\theta)^{n-1} d\theta, n \in \mathbb{N}, u \in \mathbb{R}^+, \quad (11) $$

with, $K_{2n}(0) = \frac{2^{2(n-1)(n-1)!}}{(2n-1)!}$ and $K_{2n+1}(0) = \frac{\pi(2n-1)!}{2^{2n}n!}$. For, $u \gg 1$, Eq. (11) can be expanded into asymptotic series defined by Eq. (12):

$$ K_{i_n}(u) \approx \sqrt{\frac{\pi}{2u}} e^{-u} \left\{ 1 - \frac{(4n+1)}{8u} + \frac{3(6n^2 + 24n + 3)}{2!(8u)^2} - \dots \right\}, n \in \mathbb{N}. \quad (12) $$

Finally, an exact semi-analytical solution for the incident radiation in the medium, as defined in Eq. (9), is obtained by applying the methodology leading to Eq. (10) to the West, North, and South walls. The resulting expression is given by Eq. (13):

$$ G(x_i, y_j) = \sum_{\Gamma=E, \Gamma_W, \Gamma_N, \Gamma_S} G_{\Gamma}(x_i, y_j), \quad \forall (x_i, y_j) \in \{\mathcal{S}\} \quad (13) $$

Once the unidirectional radiative intensity is known, the radiative heat flux can be obtained using Eq. (14) below:

$$ q^r(x, y) = 2 \int_{\theta=0}^{\theta=\frac{\pi}{2}} \int_{\varphi=0}^{\varphi=2\pi} I(x, y, \theta, \varphi) \sin^2\theta d\theta d\varphi, \quad (14) $$

The radiative heat flux is evaluated at the East, West, North, and South walls using Eq. (15):

$$ q^r(x_i, y_j) = \sum_{\Gamma=E, \Gamma_W, \Gamma_N, \Gamma_S} q_{\Gamma}^r(x_i, y_j), \quad \forall (x_i, y_j) \in \{\mathcal{S}\} \quad (15) $$

The radiative source $S^r = -\vec{\nabla} \cdot \vec{q}^r(x, y)$ represents the decrease in radiation due to flux divergence and is expressed in terms of Planck-mean absorption coefficients, as given by Eq. (16):

$$ S^r(x_i, y_j) = \kappa_p [G(x_i, y_j) - 4\pi I_b [T(x_i, y_j)]], \quad \forall (x_i, y_j) \in \{\mathcal{S}\}. \quad (16) $$

The respective expressions of dimensionless thermal quantities, such as $G^r$, $T^r$ and $S^r$ are given in Appendix. B.

### 3.2. Conduction heat transfer

The conductive behavior of the gases is governed by Fourier's law, as shown in Eq. (17):

$$ \vec{q}^c = - \kappa_c \vec{\nabla} T(x, y), \quad (17) $$

where, $\kappa_c$ represents the thermal conductivity of the participating gas and is calculated at a given temperature $T$ using the Sutherland relation from Ref. [30], as given in Eq. (18):

$$ \kappa_c(T) = \frac{273 + S_u}{T + S_u} \kappa_{c273} \left( \frac{T}{273} \right)^{3/2}, \quad (18) $$

where $S_u$ is the Sutherland constant, which depends on the nature of the gas involved, and $\kappa_{c273}$ is the thermal conductivity of the gas at $T = 273$K. From analysis proposed in Ref. [37] and the tabulated values of thermal conductivities recommended by Hubert et al. [38] for pressures up to 200 MPa and temperatures up to 1100 K, the value of the Sutherland constant for carbon dioxide was computed as $(S_u)_{CO_2} \approx 1073.65$ K. Similarly, using reference data for water vapor at temperatures up to 1173 K and at standard pressure, as proposed in Ref. [39], the corresponding Sutherland constant was calculated as $(S_u)_{H_2O} \approx 448.56$ K.

### 3.3. Combined conduction-radiation formulation

The steady-state energy equation shows how heat transfer by radiation is dissipated due to the conductive effects of gases and walls. Therefore, the divergence of the sum of the conductive and radiative heat fluxes is zero, as expressed in Eq. (19):

$$ \frac{\partial}{\partial x} \left( \kappa_c \frac{\partial T}{\partial x} \right) + \frac{\partial}{\partial y} \left( \kappa_c \frac{\partial T}{\partial y} \right) - \vec{\nabla} \cdot \vec{q}^r(x, y) = 0. \quad (19) $$
The conjugate problem is solved across a set of discrete gray gases using the Planck-mean absorption coefficients. The radiative source $S^r(x_i, y_j)$, derived from Eq. (16), is combined with the energy equation in Eq. (19) to iteratively obtain the temperature distribution $T(x_i, y_j)$ of the participating gases, as given in Eq. (20) below:

$$ egin{cases} 
\kappa_c \Delta T(x_i, y_j) = \kappa_p [4\pi I_b [T(x_i, y_j)] - G(x_i, y_j)] \\ 
T_w(x_i, y_j) = b_0(x_i, y_j), \\ 
\kappa_p = \sum_{g=0}^{N_g} a_g k_g 
\end{cases} 
, \quad (x_i, y_j) \in \mathcal{D}_i. \tag{20} $$

Here, $b_0(x, y)$ is the prescribed wall temperature according to Dirichlet boundary conditions. Due to the fourth-power dependence of the Planck source $I_b$, on temperature, the solutions may diverge during computation. Therefore, a dimensionless form of Eq. (20) is proposed, as given in Eq. (21) below:

$$ egin{cases} 
\Delta T^*(x^*, y^*) = \beta_g [T^{*4}(x^*, y^*) - G^*(x^*, y^*)] \\ 
T_w^*(x^*, y^*) = c_0(x^*, y^*) \\ 
\kappa_p = \sum_{g=0}^{N_g} a_g k_g 
\end{cases} 
, \tag{21} $$

here the constant, $\beta_g = \kappa_p^2 L^2 / N_{pl}$ and the conduction-radiation parameter $N_{pl} = k_c k_p / 4 \sigma_B T_{hot}^3$, describe which mode conduction or radiation predominates as both heat transfer mechanisms propagate through the medium. $c_0$ represents the dimensionless temperature at the walls. Other useful dimensionless parameters are defined in Appendix B.

The Crank-Nicolson method [40] is applied under the steady-state assumption, using an implicit finite-difference discretization scheme on the left-hand side of Eq. (21) due to its accuracy and suitability for diffusion problems with Dirichlet boundary conditions. The $(n+1)^{th}$ iteration of the dimensionless temperature distribution can be numerically computed using a constant coefficient for stability, $F_N^* = 1/4$, because of the steady-state operation, following Eq. (22) below:

$$ (T^*)^{n+1}_{ij} = F_N^* [(T^*)^n_{i-1,j} + (T^*)^n_{i+1,j} + (T^*)^n_{i,j-1} + (T^*)^n_{i,j+1}] + \beta_g F_N^* (\Delta x^*)^2 [(G^*)^n_{ij} - (T^{*4})^n_{ij}] \tag{22} $$

The geometric dimensions are chosen such that $L = 2H$. By analogy, a dimensionless mesh grid is selected with $N_x = 2N_y$, and a truncation error of order $0[(\Delta x^*)^2 \times (\Delta y^*)^2]$, as shown in [29], is considered when solving the transient problem. Once $T^*$ is computed, the corresponding dimensionless total heat flux vector for all participating gases in the medium is obtained from Eq. (23):

$$ \bar{q}^{t*}(x_i, y_j) = -\frac{N_{pl}}{k_p} \bar{\nabla} T^*(x_i, y_j) + \bar{q}^{*s}(x_i, y_j) \tag{23} $$

Here, $\bar{q}^{*s}(x_i, y_j) = -\frac{N_{pl}}{k_p} \bar{\nabla} T^*(x_i, y_j)$, represents the dimensionless conductive heat flux. The numerical values of these thermal quantities, discussed above, are physically interpreted and analysed in the next section.

# 4. Computation, validation and numerical results

The aim of this subsection is to present the computational methodology, and the validation of results obtained using the SLW correlated model alongside the conjugate conductive-radiative governing equations. The chemical species considered as participating gases within the geometry are CO$_2$/N$_2$, H$_2$O/N$_2$ and CO$_2$/H$_2$O/N$_2$ are considered as

participating gases inside the geometry. The profiles of temperature distributions and total heat flux are highlighted. The computations were performed on a desktop computer with an Intel(R) Core (TM) i7-6700HQ CPU@ 2.60GHz 2.59 GHz, using FORTRAN95 software.

## 4.1. Computation

The set of algorithms implemented in this study follows one or more computational steps, summarized below:

a. Initialize temperatures $T_b = T^{loc}$ and $T_g = T^{ref}$, the total number of discrete gray gases $N_g$, and the gas composition $Y_s$ from Eq. (3). Then, compute the values of the supplemental absorption cross sections $\tilde{C}_g$.

b. Combine $\tilde{C}_g$ from the SLW-correlated model using the ALBDF set in Eq. (2) and the Pearson correlation coefficients [17,18]. Then, compute the discrete absorption coefficients $k_g$, their corresponding weights $a_g$, and deduce the Planck-mean absorption coefficient $\kappa_p = \sum_{g=0}^{N_g} a_g k_g$ from Eq. (6).

c. Using the data of $\kappa_p$, solve the RTE from Eq. (7) and iteratively compute the dimensionless quantities, including the incident radiation $G^*$ from Eq. (13) and the radiative source $S^r$ from Eq. (16), at radiative equilibrium.

d. Compute thermal conductivity $\kappa_c$ at $T_g$ using Eq. (18).

e. Use the energy equation from Eqs. (19–22), along with the data for $\kappa_p$, $G^*$ and $S^r$, compute the dimensionless conduction-radiation temperature $T^*$ and the total heat flux $q^*$.

f. Perform the previous steps (d-e) until convergence, following a criterion $Max \left[ \frac{(T^*)^{n+1}_{ij} - (T^*)^n_{ij}}{(T^*)^n_{ij}} \right] \le 10^{-5}$.

The number of discrete gray gases used to compute the Planck-mean coefficients with the SLW-correlated model, the quadrature angles employed in the RTE solver, the number of cells used for meshing the geometry, and the conduction-radiation parameters in the energy equation all have a significant effect on the CPU time required for the iterative scheme and on the resulting thermal quantities. In contrast, computing the dimensionless quantities $S^r$, $G^*$, and $T^*$ with $\kappa_p$ requires relatively little computational time and energy. Furthermore, the ray-tracing method employed is highly dependent on the ray pathways $\delta(x, y, \theta, \varphi)$, which are determined by the directional angles $(\theta, \varphi)$, of the solid angle $\Omega$ and the mesh resolution of the geometry $N_x \times N_y$. Consequently, there is no predefined sample size or number of rays that can be selected prior to the simulation. Instead, as the geometric mesh is refined, the number of rays generated from the angular quadrature $N_\theta \times N_\varphi$ increases, leading to improved solution accuracy.

## 4.2. Validation

### 4.2.1. SLW correlated model

To validate the SLW-correlated model coupled with the RTE solver using the ray-tracing method, the radiative source at the centerline of the enclosure is compared with the results of Test Case 5 studied by Chu et al. [41], using the LBL and SNBCK methods.

Fig. 2-(a) describes the numerical simulation of the prescribed non-isothermal temperature under inhomogeneous conditions used to compute the results, for an enclosure with $L \times H = 1m \times 0.5m$. A careful examination of the radiative source profile shown in Fig. 2-(b) indicates that the results predicted using the SLW-correlated model with the ray-tracing method are in good agreement with the LBL and SNBCK results reported in Ref. [41]. In contrast, the radiative source computed using the SNB-CK model exhibits slight deviations in the central regions of the rectangular geometry.
<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Validation of SLW correlated model with 10%CO₂ + 20%H₂O, N_g = 20. (a) Dimensionless non-isothermal temperature distribution T/T₀ (b) radiative source along a centerline following y-direction.</figcaption></figure>

<figure><img src="image_5.png" /></figure>

### 4.2.2. Conduction-radiation problem

To validate the conjugate conduction–radiation solver developed in this study, the conduction–radiation temperature profile $T^*(0.5, y^*)$, with $y^* = y/L$ along the vertical centerline of the rectangular geometry shown in Fig. 3, is compared with the results reported by Yuen and Takara [42]. The case study considers a section of $L \times H = 1m \times 1m$, an optical thickness of $k_p = 0.1m^{-1}$, and conduction-radiation parameters $N_{pl} \in \{0.001, 1\}$, used for numerical computations. The medium absorbs and emits radiation but does not scatter it, with black walls ($\epsilon = 1$). The bottom wall is maintained at a hot temperature ($T_s = T_{hot}$), while the initial temperature of the medium and the other walls is cold ($T_0 = T_{cold} = T_{hot}/2$).

The temperature profile shown in Fig. 3 agrees well with that reported by Yuen and Takara [42], which was computed using the diffusion approximation for the RTE and the finite difference method for the energy equation. A discrete angular quadrature of $M_\phi \times N_\theta = 6 \times 6$ and a mesh grid of $N_x \times N_y = 41 \times 41$ for the energy equation were used in the present computations. It is noted that conduction dominates when $N_{pl} = 1$, whereas radiative transfer becomes significant when $N_{pl} = 0.001$.

## 4.3. Numerical results

The aim of this sub-section is first to present the thermal properties of

<figure><img src="image_6.png" /><figcaption>**Fig. 3.** Dimensionless conduction-radiation temperature profile $T^*(0.5, y^*)$ along the vertical centerline of the rectangular geometry.</figcaption></figure>

the gases (Planck-mean absorption coefficients and thermal conductivity) and then to analyze the effects of conduction-radiation solvers, gas temperature, number of gray gases, and gas mixture ratios on the temperature and heat fluxes in the medium. According to Fig. 1(a-b), the participating gases enter the rectangular geometry $L \times H = 1m \times 0.5m$ through the West wall ($\Gamma_W$), with $T_W = T_{hot} = T_g$. The other walls and the initial temperature in the medium are cold ($T_E = T_N = T_S = T_{cold}$, $T_0 = T_{cold} = T_{hot}/2$). The gases absorb and emit radiation but do not scatter it, conduct heat with a constant thermal conductivity computed at $T_g$, and all four walls of the geometry are black ($\epsilon = 1$). A total of $N_g = 20$ gray gases were used, with a reference temperature $T^{ref} = T_b = T_g$, where $T_g \in \{500K, 1000K, 3000K\}$, to compute the Planck-mean absorption coefficients $\kappa_p$. The shapes of absorption spectral bands corresponding to the 10%CO₂ and 20%H₂O species are presented in Ref. [43], for a better understanding. Additionally, boundary conditions considered for numerical simulations are summarized in the Table 1.

### 4.3.1. Thermal properties of gases

The thermal conductivity $\kappa_c$, and the dimensionless conduction-radiation $N_{pl}$ of the participating gases, 10%CO₂/N₂, 20%H₂O/N₂ and their mixture 10%CO₂/N₂ + 20%H₂O/N₂ at a constant pressure of $P = 1$ atm and gas temperatures $T_g \in \{500K, 1000K, 3000K\}$ were computed and summarized in Table 2. Controlling the thermal conductivity of semi-transparent gases under varying environmental conditions of pressure and temperature remains challenging. However, knowledge of these values provides important data and insight into thermal transfer through participating gases. Shieh et al. [37] demonstrated that the thermal conductivity of carbon dioxide at high pressures and variable temperatures is significantly lower than that of water vapor. As shown in Table 2, for both low and high gas temperatures, the corresponding thermal conductivity remains very low. The range of the dimensionless conduction-radiation parameter further confirms this behavior; specifically, the largest value of $N_{pl}$ is approximately $N_{pl} = 10^{-2}$, indicating the predominance of radiative heat transfer.

### 4.3.2. Effects of conductive-radiative solver

Discrete angles, mesh grids, and CPU time are the key parameters used to assess the impact of the conductive-radiative solver on temperature distribution.

* Discrete angles

<table><caption>**Table 1**
Boundary conditions for numerical simulations.</caption><thead><tr><th>∂T<sub>e</sub></th><th>Γ<sub>W</sub></th><th>Γ<sub>E</sub></th><th>Γ<sub>N</sub></th><th>Γ<sub>S</sub></th></tr></thead><tbody><tr><td>I<sub>w</sub>/σ<sub>B</sub>T<sup>4</sup><sub>hot</sub></td><td>1/π</td><td>1/2π</td><td>1/2π</td><td>1/2π</td></tr></tbody></table>
<caption>**Table 2**</caption>

Thermal conductivity $κ_c$ and dimensionless conduction-radiation parameter $N_{pl}$ of carbon dioxide, computed for the set of temperatures $T_g = \{500K, 1000K, 3000K\}$.

<table><thead><tr><th rowspan="2">T<sub>g</sub>(K)</th><th rowspan="2">N<sub>g</sub></th><th colspan="2">10%CO<sub>2</sub>/N<sub>2</sub></th><th colspan="2">20%H<sub>2</sub>O/N<sub>2</sub></th><th colspan="2">10%CO<sub>2</sub>/N<sub>2</sub> + 20%H<sub>2</sub>O/N<sub>2</sub></th></tr><tr><th>N<sub>pl</sub></th><th>κ<sub>c</sub> (W.m<sup>-1</sup>.K<sup>-1</sup>)</th><th>N<sub>pl</sub></th><th>κ<sub>c</sub> (W.m<sup>-1</sup>.K<sup>-1</sup>)</th><th>N<sub>pl</sub></th><th>κ<sub>c</sub> (W.m<sup>-1</sup>.K<sup>-1</sup>)</th></tr></thead><tbody><tr><td rowspan="3">500</td><td>5</td><td>6.7 × 10<sup>-3</sup></td><td rowspan="3">0.033</td><td>7.6 × 10<sup>-3</sup></td><td rowspan="3">0.044</td><td>1.51 × 10<sup>-2</sup></td><td rowspan="3">0.040</td></tr><tr><td>10</td><td>5.4 × 10<sup>-3</sup></td><td>5.8 × 10<sup>-3</sup></td><td>1.19 × 10<sup>-2</sup></td></tr><tr><td>20</td><td>5.1 × 10<sup>-3</sup></td><td>5.4 × 10<sup>-3</sup></td><td>1.12 × 10<sup>-2</sup></td></tr><tr><td rowspan="3">1000</td><td>5</td><td>2.1 × 10<sup>-3</sup></td><td rowspan="3">0.070</td><td>5.3 × 10<sup>-4</sup></td><td rowspan="3">0.082</td><td>2.86 × 10<sup>-3</sup></td><td rowspan="3">0.078</td></tr><tr><td>10</td><td>1.8 × 10<sup>-3</sup></td><td>4.1 × 10<sup>-4</sup></td><td>2.48 × 10<sup>-3</sup></td></tr><tr><td>20</td><td>1.8 × 10<sup>-3</sup></td><td>3.8 × 10<sup>-4</sup></td><td>2.39 × 10<sup>-3</sup></td></tr><tr><td rowspan="3">3000</td><td>5</td><td>2.8 × 10<sup>-5</sup></td><td rowspan="3">0.187</td><td>2.9 × 10<sup>-6</sup></td><td rowspan="3">0.180</td><td>3.10 × 10<sup>-5</sup></td><td rowspan="3">0.182</td></tr><tr><td>10</td><td>2.7 × 10<sup>-5</sup></td><td>2.2 × 10<sup>-6</sup></td><td>2.92 × 10<sup>-5</sup></td></tr><tr><td>20</td><td>2.7 × 10<sup>-5</sup></td><td>2.1 × 10<sup>-6</sup></td><td>2.87 × 10<sup>-5</sup></td></tr></tbody></table>

**Table 3** highlights the respective values of $T^*$ at the center coordinates of the rectangular geometry for different solid angle quadratures $M_φ × N_θ$. The minimum value, $T^* = 0.52514$ occurs for $10\%CO_2/20\%H_2O/N_2$, with $M_φ × N_θ = 4 × 4$, while the maximum, $T^* = 0.61796$, is observed for $20\%H_2O/N_2$ with $M_φ × N_θ = 6 × 6$. The calculated $T^*$ values are similar across different gas temperatures $T_g \in \{500K, 1000K, 3000K\}$, but they increase as the quadrature resolution increases.

* Mesh grid

**Table 4** summarises the values of $T^*$ at the center-point coordinates of the rectangular geometry for different mesh grid resolutions. The $T^*$ results are reported with five-digit precision and become more representative as the mesh is refined. Regardless of the choice of $N_x × N_y$, the values of $T^*$ increases as well as $T_g$ increases, indicating that the participating medium transitions from optically thick to optically thin behavior with respect to thermal radiation. Moreover, for any $N_x × N_y$ and $T_g$, the value of $T^*$ at $(x^*, y^*) = (0.5, 0.25)$ is consistently higher when the thermal properties are computed using a $20\%H_2O/N_2$.

* CPU time

The **Table 5** summarises the CPU times recorded during the computation of $T^*$, for different solid angle $M_φ × N_θ$. For the same quadrature and gas temperature $T_g$, the CPU times are higher for the $10\%CO_2/N_2$ mixture than for $20\%H_2O/N_2$. The maximum CPU time recorded is 517.984 s, corresponding to $10\%CO_2/N_2$ at $T_g = 3000K$. In general, CPU time increases with increasing gas temperature $T_g$. Overall, the CPU times reported in **Table 5** indicate that the proposed semi-analytical method, combining ray-tracing techniques with gas kinetics

<caption>**Table 3**</caption>

Effect of angular's discrete direction on dimensional conduction-radiation temperature $T^*$ at $(x^*, y^*) = (0.5, 0.25)$, $N_x × N_y = 221 × 111$ for a rectangular geometry.

<table><caption>**Table 4**</caption><thead><tr><th colspan="5" style="text-align: center;">T<sup>*</sup> = T/T<sub>hot</sub></th></tr><tr><th></th><th style="text-align: center;">M<sub>φ</sub> ×<br>N<sub>θ</sub></th><th style="text-align: center;">T<sub>g</sub> =<br>500K</th><th style="text-align: center;">T<sub>g</sub> =<br>1000K</th><th style="text-align: center;">T<sub>g</sub> =<br>3000K</th></tr></thead><tbody><tr><td rowspan="3">10%CO<sub>2</sub>/N<sub>2</sub></td><td style="text-align: center;">4 × 4</td><td style="text-align: center;">0.55560</td><td style="text-align: center;">0.53855</td><td style="text-align: center;">0.60224</td></tr><tr><td style="text-align: center;">5 × 5</td><td style="text-align: center;">0.55684</td><td style="text-align: center;">0.54546</td><td style="text-align: center;">0.60220</td></tr><tr><td style="text-align: center;">6 × 6</td><td style="text-align: center;">0.55719</td><td style="text-align: center;">0.54654</td><td style="text-align: center;">0.60221</td></tr><tr><td rowspan="3">20%H<sub>2</sub>O/N<sub>2</sub></td><td style="text-align: center;">4 × 4</td><td style="text-align: center;">0.56514</td><td style="text-align: center;">0.59906</td><td style="text-align: center;">0.61794</td></tr><tr><td style="text-align: center;">5 × 5</td><td style="text-align: center;">0.56575</td><td style="text-align: center;">0.59902</td><td style="text-align: center;">0.61795</td></tr><tr><td style="text-align: center;">6 × 6</td><td style="text-align: center;">0.56589</td><td style="text-align: center;">0.59902</td><td style="text-align: center;">0.61796</td></tr><tr><td rowspan="3">10%CO<sub>2</sub>/N<sub>2</sub> +<br>20%H<sub>2</sub>O/N<sub>2</sub></td><td style="text-align: center;">4 × 4</td><td style="text-align: center;">0.52514</td><td style="text-align: center;">0.53311</td><td style="text-align: center;">0.60100</td></tr><tr><td style="text-align: center;">5 × 5</td><td style="text-align: center;">0.53431</td><td style="text-align: center;">0.53874</td><td style="text-align: center;">0.60096</td></tr><tr><td style="text-align: center;">6 × 6</td><td style="text-align: center;">0.53768</td><td style="text-align: center;">0.54076</td><td style="text-align: center;">0.60098</td></tr></tbody></table>

Effect of mesh grid on dimensionless conduction-radiation temperature $T^*$ at $(x^*, y^*) = (0.5, 0.25)$, $M_φ × N_θ = 6 × 6$ for a rectangular geometry.

<table><caption>**Table 5**</caption><thead><tr><th colspan="5" style="text-align: center;">T<sup>*</sup> = T/T<sub>hot</sub></th></tr><tr><th></th><th style="text-align: center;">N<sub>x</sub> × N<sub>y</sub></th><th style="text-align: center;">T<sub>g</sub> =<br>500K</th><th style="text-align: center;">T<sub>g</sub> =<br>1000K</th><th style="text-align: center;">T<sub>g</sub> =<br>3000K</th></tr></thead><tbody><tr><td rowspan="4">10%CO<sub>2</sub>/N<sub>2</sub></td><td style="text-align: center;">121 × 61</td><td style="text-align: center;">0.55710</td><td style="text-align: center;">0.54653</td><td style="text-align: center;">0.60209</td></tr><tr><td style="text-align: center;">201 ×<br>101</td><td style="text-align: center;">0.55718</td><td style="text-align: center;">0.54656</td><td style="text-align: center;">0.60220</td></tr><tr><td style="text-align: center;">501 ×<br>251</td><td style="text-align: center;">0.55735</td><td style="text-align: center;">0.54675</td><td style="text-align: center;">0.60224</td></tr><tr><td style="text-align: center;">121 × 61</td><td style="text-align: center;">0.56578</td><td style="text-align: center;">0.59894</td><td style="text-align: center;">0.61795</td></tr><tr><td rowspan="4">20%H<sub>2</sub>O/N<sub>2</sub></td><td style="text-align: center;">201 ×<br>101</td><td style="text-align: center;">0.56587</td><td style="text-align: center;">0.59900</td><td style="text-align: center;">0.61795</td></tr><tr><td style="text-align: center;">501 ×<br>251</td><td style="text-align: center;">0.56601</td><td style="text-align: center;">0.59905</td><td style="text-align: center;">0.61796</td></tr><tr><td style="text-align: center;">121 × 61</td><td style="text-align: center;">0.53817</td><td style="text-align: center;">0.54094</td><td style="text-align: center;">0.60094</td></tr><tr><td style="text-align: center;">201 ×<br>101</td><td style="text-align: center;">0.53773</td><td style="text-align: center;">0.54080</td><td style="text-align: center;">0.60097</td></tr><tr><td rowspan="4">10%CO<sub>2</sub>/N<sub>2</sub> +<br>20%H<sub>2</sub>O/N<sub>2</sub></td><td style="text-align: center;">501 ×<br>251</td><td style="text-align: center;">0.53792</td><td style="text-align: center;">0.54099</td><td style="text-align: center;">0.60101</td></tr></tbody></table>

Computational time of dimensionless conduction-radiation temperature $T^*$, with Planck-mean absorption coefficients $κ_p$, $N_x × N_y = 201 × 101$ for a rectangular geometry.

<table><thead><tr><th colspan="5" style="text-align: center;">CPU time (s) for T<sup>*</sup> = T/T<sub>hot</sub></th></tr><tr><th></th><th style="text-align: center;">M<sub>φ</sub> ×<br>N<sub>θ</sub></th><th style="text-align: center;">T<sub>g</sub> =<br>500K</th><th style="text-align: center;">T<sub>g</sub> =<br>1000K</th><th style="text-align: center;">T<sub>g</sub> =<br>3000K</th></tr></thead><tbody><tr><td rowspan="3">10%CO<sub>2</sub>/N<sub>2</sub></td><td style="text-align: center;">4 × 4</td><td style="text-align: center;">291.920</td><td style="text-align: center;">304.921</td><td style="text-align: center;">315.420</td></tr><tr><td style="text-align: center;">5 × 5</td><td style="text-align: center;">264.826</td><td style="text-align: center;">267.187</td><td style="text-align: center;">270.733</td></tr><tr><td style="text-align: center;">6 × 6</td><td style="text-align: center;">352.765</td><td style="text-align: center;">348.233</td><td style="text-align: center;">517.984</td></tr><tr><td rowspan="3">20%H<sub>2</sub>O/N<sub>2</sub></td><td style="text-align: center;">4 × 4</td><td style="text-align: center;">245.328</td><td style="text-align: center;">298.733</td><td style="text-align: center;">309.327</td></tr><tr><td style="text-align: center;">5 × 5</td><td style="text-align: center;">262.889</td><td style="text-align: center;">264.140</td><td style="text-align: center;">263.500</td></tr><tr><td style="text-align: center;">6 × 6</td><td style="text-align: center;">350.718</td><td style="text-align: center;">342.577</td><td style="text-align: center;">416.061</td></tr><tr><td rowspan="3">10%CO<sub>2</sub>/N<sub>2</sub> +<br>20%H<sub>2</sub>O/N<sub>2</sub></td><td style="text-align: center;">4 × 4</td><td style="text-align: center;">370.625</td><td style="text-align: center;">342.218</td><td style="text-align: center;">444.608</td></tr><tr><td style="text-align: center;">5 × 5</td><td style="text-align: center;">264.358</td><td style="text-align: center;">272.000</td><td style="text-align: center;">262.515</td></tr><tr><td style="text-align: center;">6 × 6</td><td style="text-align: center;">339.405</td><td style="text-align: center;">339.248</td><td style="text-align: center;">344.983</td></tr></tbody></table>

and the energy equation, requires fewer computational resources and achieves reduced computation times.

### 4.3.3. Effect of gase's temperature

The gas temperature directly affects the values of absorption coefficients and thermal conductivity. It significantly modifies the temperature profile at radiative equilibrium as well as the conduction-radiation heat flux within the medium.

* Dimensionless conduction-radiation temperature

The gas temperatures $T_g = \{500K, 1000K, 3000K\}$, listed in **Table 2**
are used as input values, (with $T_g = T_{hot}$) to compute the thermal properties, including absorption coefficients and thermal conductivity. These properties are then incorporated into the iterative subroutine to evaluate the dimensionless conduction-radiation temperature in the combustion chamber. Therefore, due to the different behavior of thermal properties in conjugate heat transfer, their effects on the temperature profiles along the respective centerlines of the two-dimensional geometry are analysed both quantitatively and qualitatively. In contrast, in Ref. [23], the thermal conductivity is evaluated using an average temperature of the hot and cold walls. The Figs. 4(a-b) highlight the plots of $T^*$ for $T_g = 500K$. In Fig. 4-(a), the $T^*$ profiles in the participating medium decreases along $x^* \in [0, 0.975]$, and then increase along $x^* \in [0.975, 1]$. The curves corresponding to the different gas species remain very close to each other due to radiative effects, resulting in smooth variations of the conjugate heat transfer. In Fig. 4-(b), the $T^*$ follow symmetric parabolic curves, with maximum values predicted by $20\%H_2O/N_2$ mixture.

A simultaneous smooth decrease and increase of $T^*$ for all gas species over $y^* \in [0, 0.022]$ and $y^* \in [0.977, 1]$ is observed, yielding symmetric profiles of identical magnitude. This behavior of $T^*$, is attributed to polarisation effects within the medium, which are directly related to its thermal characteristics and the boundary conditions at the walls.

The Figs. 5(a-b) present the distributions of $T^*$ for $T_g = 1000K$. In Fig. 5-(a), the $T^*$ profile for the $20\%H_2O/N_2$ mixture decreases monotonically along $x^* \in [0, 1]$, whereas the profiles for $10\%CO_2/N_2$ and the mixture $10\%CO_2/20\%H_2O/N_2$ mixture initially decrease over $x^* \in [0, 0.975]$, and then increase over $x^* \in [0.975, 1]$. In addition, a growing difference in the magnitude of $T^*$ is observed over $x^* \in [0.22, 1]$, with higher values associated with the $20\%H_2O/N_2$ mixture compared to those of $10\%CO_2/N_2$ and the mixed composition. In Fig. 5-(b), the difference in the magnitude of $T^*$ for the $20\%H_2O/N_2$ mixture becomes more pronounced over the entire $x^* \in [0, 1]$ domain.

For $T_g = 3000K$, the plots are displayed in Fig. 6(a-b), where the curve related to $20\%H_2O/N_2$ decreases from $T^* \in [1, 0.5]$ for $x^* \in [0, 0.5]$. There is also a significant difference in magnitude in $T^*$ profiles along $x^* \in [0.022, 0.977]$, for $20\%H_2O/N_2$ along the vertical centerline.

Finally, the shapes of the curves describing the dimensionless temperature $T^*$, computed for the $10\%CO_2/N_2$ case, are considerably more sensitive to variations in gas temperature than those obtained for the $20\%H_2O/N_2$.

### Dimensionless conductive, radiative and total heat fluxes

The Figs. 7(a-b), Figs. 8(a-b), Figs. 9(a-b) respectively present the dimensionless conductive heat flux $q^{c*}$, radiative heat flux $q^{r*}$, and total heat flux $q^{t*}$ along the East wall ($\Gamma_E$) of the rectangular geometry. The thermal properties of the gas species are computed for gas temperatures $T_g \in \{500K, 1000K, 3000K\}$. The resulting profiles exhibit parabolic shapes and show slight discontinuities near $y^* = 0$ and $y^* = 0.5$, which are attributed to the weak contribution of hot rays reaching the East wall ($\Gamma_E$). In the participating medium, heat propagates with conduction and radiation contributing to opposite directions, so the difference between the two heat fluxes increases with rising $T_g$. Fig. 7(a-b) compare the heat flux profiles for $10\%CO_2/N_2$ and $20\%H_2O/N_2$ non-gray gases at $T_g = 500K$. The magnitudes of $q^{c*}$, $q^{r*}$ and $q^{t*}$ computed with the thermal properties of $20\%H_2O/N_2$ are higher than those for $10\%CO_2/N_2$. Along the West wall, the magnitude of $q^{c*}$ is comparable to $q^{r*}$ despite the relatively low temperature, indicating that $q^{t*}$ is primarily dominated by thermal radiation.

In Figs. 8(a-b), $q^{t*}$ increases due to the higher gas temperature $T_g = 1000K$, while $q^{c*}$ continues to decrease. For $20\%H_2O/N_2$, $q^{t*}$ closely follows $q^{r*}$ compared to $10\%CO_2/N_2$. Furthermore, the $q^{r*}$ profile in Fig. 8-(b) exhibits more discontinuities for $10\%CO_2/N_2$ than for $20\%H_2O/N_2$ in Fig. 8-(a). In Figs. 9(a-b), although the conductive contribution $q^{c*}$ is of the order of $10^{-5}$, the total heat flux is essentially governed by radiative heat transfer due to the high gas temperature, $T_g = 3000K$.

### 4.3.4. Effect of the number of discrete gray gases

The present SLW correlated model provides good accuracy in predicting thermal quantities, with its performance controlled by the number of discrete gray gases $N_g$. Indeed, increasing the number of gray gases, such as $N_g = 20$, as used in Refs. [17, 18], leads to higher accuracy, although it is accompanied by an increase in CPU time.

#### Dimensionless conduction-radiation temperature

The Figs. 10(a-b) present the conduction-radiation absolute error $|T_{N_g=20}^* - T_{N_g=05.10.15}^*| \times 100$ for gas mixtures containing $10\%CO_2$ and $20\%H_2O$, with the remaining 80% and 90% composed of $N_2$, respectively. The results are computed using different numbers of gray gases ($N_g = 05, 10, 15$) along the vertical centerline of the rectangular geometry. Taking $N_g = 20$ as the benchmark solution, the absolute error decreases with increasing $N_g$, while noticeable inaccuracies are observed

<figure><img src="image_7.png" /><figcaption>Fig. 4. Dimensionless conduction-radiation temperature $T^*$ with Planck-mean absorption coefficients along the centerlines of the combustion chamber for $10\%CO_2/N_2$, at $T_g = 500K$: (a) $y = H_x/2$, (b) $x = H_x/2$.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Fig. 5. Dimensionless conduction-radiation temperature $T^*$ with Planck-mean absorption coefficients along the centerlines of the combustion chamber for 10%CO$_2$/N$_2$, at $T_g = 1000K$: (a) $y = H_y/2$, (b) $x = H_x/2$.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 6. Dimensionless conduction-radiation temperature $T^*$ with Planck-mean absorption coefficients along the centerlines of the combustion chamber for 10%CO$_2$/N$_2$, at $T_g = 3000K$: (a) $y = H_y/2$, (b) $x = H_x/2$.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 7. Dimensionless conductive $q^*$, radiative $q^{*\text{r}}$ and total heat flux $q^{*\text{t}}$ profiles with Planck-mean absorption coefficients along the East wall ($\Gamma_E$) of the combustion chamber at $T_g = 500K$: (a) 10%CO$_2$/N$_2$ (b) 20%H$_2$O/N$_2$.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Fig. 8. Dimensionless conductive $q^*$, radiative $q^*$ and total heat flux $q^{*\tau}$ profiles with Planck-mean absorption coefficients along the East wall ($\Gamma_E$) of the combustion chamber at $T_g = 1000$K: (a) $10\%CO_2/N_2$, (b) $20\%H_2O/N_2$.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 9. Dimensionless conductive $q^*$, radiative $q^*$ and total heat flux $q^{*\tau}$ profiles with Planck-mean absorption coefficients along the East wall ($\Gamma_E$) of the combustion chamber at $T_g = 3000$K: (a) $10\%CO_2/N_2$, (b) $20\%H_2O/N_2$.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 10. Absolute error of the dimensionless conduction-radiation temperature along a vertical centerline of the combustion chamber computed with different gray gas number at $= 1000$K: (a) $10\%CO_2$, (b) $20\%H_2O$.</figcaption></figure>
for small values of $N_g$, particularly for $N_g = 05$.

* Dimensionless total heat flux

Figs.11(a-b) display the absolute error of the dimensionless total heat flux, $|q_{N_g=20}^* - q_{N_g=05,10,15}^*| \times 100$, along the East wall ($\Gamma_E$), for gas mixtures containing 10%CO₂ and 20%H₂O with $N_g = 05, 10, 15$. The total heat flux prediction appears less sensitive to the number of gray gases for the 10%CO₂ mixture than for 20%H₂O. Moreover, the contribution of radiative heat flux errors to the absolute error of $q^*$ is smaller than that of conductive heat flux, as it is directly related to the conjugate temperature gradient.

### 4.3.5. Effect of different gas species

Heat transfer does not behave identically when interacting with different gas species due to their distinct chemical properties and absorption spectral bands. The case of pure carbon dioxide and pure water vapor is analysed in Figs.12(a-b), where the dimensionless conduction-radiation temperature along the vertical centerline and the total heat flux along the East wall are examined. In Fig.12-(a), the temperature distribution associated to pure water vapor appears more spread than that of pure carbon dioxide.

The absolute error $|T_{pure\ CO_2}^* - T_{pure\ H_2O}^*|$ shown in Fig.12-(a), oscillates around 0.2. In Fig. 12-(b), the total heat flux corresponding to pure water vapor appears less spread than that of pure carbon dioxide due to the conductive heat flux contribution. Consequently, the absolute error $|q_{pure\ CO_2}^* - q_{pure\ H_2O}^*|$ attains an average value of approximately 0.009.

### 4.3.6. Effect of gas mixture ratios

Figs.13(a-b) present the dimensionless conduction–radiation temperature and total heat flux profiles along the vertical centerline and the West wall of the rectangular geometry for $T_g = 1000K$.

Three mole-fraction cases, 20%CO₂ + 40%H₂O, 30%CO₂ + 30%H₂O and 40%CO₂ + 20%H₂O are investigated, with the remaining fraction assigned to nitrogen, treated as a non-participating gas. When the mole fraction of carbon dioxide is lower than that of water vapor, as in the 20%CO₂ + 40%H₂O case, the temperature and total heat flux distributions are less spread than in the 40%CO₂ + 20%H₂O case. The temperature and total heat flux profiles obtained for the 30%CO₂ + 30%H₂O mixture is like those corresponding to the 40%CO₂ + 20%H₂O case. These differences in the shapes of thermal quantities arise from variations in chemical properties and spectral absorption bands.

Finally, it is clearly demonstrated that various physical parameters

significantly influence the behavior of the thermal quantities examined in this work. Among them, gas temperature remains the principal parameter, as it directly affects the thermal properties of the gas and the boundary conditions of the conduction-radiation solver.

## 4.4. Numerical simulations

The aim of this subsection is to present suitable numerical simulations of the conjugate conduction-radiation temperature and total heat flux distributions, within the rectangular combustion chamber of dimensions $L \times H$, with $L = 2H$. Computations are performed using different values of constant thermal conductivity $\kappa_c$, assuming black walls (unit emissivity), and radiative properties defined as $\kappa_p = \sum_{g=0}^{N_g} a_g k_g$. Gas compositions of $Y_{CO_2} = 0.1$ and $Y_{H_2O} = 0.2$, are considered to carry out the proposed numerical simulations. A dimensionless conduction-radiation parameter, $N_{pl} = \kappa_c \kappa_p / (4 \sigma_B T_{hot}^3)$, is introduced to illustrate whether conduction or radiation heat transfer predominates within the participating medium. The West wall ($\Gamma_W$) is maintained at a constant hot temperature, $T_g = T_b = \{500K, 1000K, 3000K\}$, while the remaining walls are treated as cold, with, $T_N = T_S = T_E = T_{hot} / 2$. The initial temperature of the medium is also set to $T_0 = T_{hot} / 2$. Then, $k_c$ is calculated for the set of temperatures, $T_g \in \{500K, 1000K, 3000K\}$, representing low, intermediate, and high temperature ranges, respectively, to characterise the conductive heat transfer from the West wall toward the remaining walls. A structured mesh grid of $N_x \times N_y = 2001 \times 1001$ is employed to ensure adequate resolution and accurate representation of thermal propagation within the domain.

### 4.4.1. Dimensionless conduction-radiation temperature distribution

The numerical simulations presented in Figs.14(a-c) to Figs.16(a-c), illustrate the evolution of the dimensionless temperature distribution $T^*$ resulting from conjugate conduction-radiation heat transfer within the combustion chamber.

Hot gases enter the 2D geometry at the same temperature as the West wall ($\Gamma_W$), and then propagate through the medium. The operating temperature within the combustion chamber is reached after several iterations, once equilibrium is established. The 2D simulations show a decrease in the dimensionless temperature $T^*$ from the inlet ($\Gamma_W$) to the outlet ($\Gamma_E$) wall, because of the absorption of heat within the participating medium.

Figs. 14(a-c) present the dimensionless temperature distribution $T^*$ for 10%CO₂/N₂, 20%H₂O/N₂ and 10%CO₂/20%H₂O/N₂ respectively, at $T_g = 500K$. The magnitude of $T^* \in [0.4, 1]$ for the individual gases,

<figure><img src="image_14.png" /><figcaption>Fig. 11. Absolute error of the dimensionless total heat flux along the East wall of the combustion chamber computed with different gray gas number at $T = 1000K$: (a) 10%CO₂, (b) 20%H₂O.</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>**Fig. 12.** (a-b). Effect of different gas species, pure CO₂ and H₂O, T_g = 1000K: (a) dimensionless conduction-radiation temperature along the vertical centerline, (b) dimensionless total heat flux along the East wall.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>**Fig. 13.** (a-b). Effect of different gas mixture ratios, 20%CO₂ + 40%H₂O, 30%CO₂ + 30%H₂O and, 40%CO₂ + 20%H₂O: (a) dimensionless conduction-radiation temperature, (b) dimensionless total heat flux.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>**Fig. 14.** Dimensionless temperature distribution T* in the combustion chamber computed at T_g = 500K: (a) 10%CO₂/N₂, (b) 20%H₂O/N₂, (c) 10%CO₂/N₂ + 20%H₂O/N₂.</figcaption></figure>

while it spans into $T^* \in [0.3, 1]$ for the gas mixture.

Differences in the dimensionless conduction-radiation parameter $N_{pl}$ for each gas species significantly influence the temperature behavior $N_{pl} = 5.19 \times 10^{-3}$ corresponding to 10%CO₂/N₂, the medium appears more resistant to thermal propagation compared to 20%H₂O/N₂ with $N_{pl} = 5.47 \times 10^{-3}$. The 10%CO₂/20%H₂O/N₂ case, with $N_{pl} = 1.12 \times 10^{-2}$, represents a weighted contribution from both gas species. At

equilibrium, the hottest region occurs near the inlet, while the coldest points are located at the corners $\Gamma_N \cap \Gamma_E$ and $\Gamma_S \cap \Gamma_E$. The conductive heat transfer is particularly effective within the sub-domain $\{\mathcal{S}_{hot}\} = \{\forall(x,y) \in [0, 0.2] \times [0, 0.5]\}$. Figs. 15(a-c) present the dimensionless temperature distribution $T^*$ for 10%CO₂/N₂, 20%H₂O/N₂ and 10%CO₂/20%H₂O/N₂ non-gray gases at $T_g = 1000K$. The temperature propagates from the hot region near the West wall ($\Gamma_W$) toward the East
<figure><img src="image_18.png" /><figcaption>**Fig. 15.** Dimensionless temperature distribution $T^*$ in the combustion chamber computed at $T_g = 1000K$: (a) $10\%CO_2/N_2$, (b) $20\%H_2O/N_2$, (c) $10\%CO_2/N_2 + 20\%H_2O/N_2$.</figcaption></figure>

wall ($\Gamma_E$), exhibiting a pattern similar to that observed for $T_g = 500K$.

One observes a large eccentricity and significant depth of heat penetration, represented by successive concentric curves, compared with those obtained when $T_g = 500K$. Although the thermal conductivity of gases is higher, it remains insufficient to substantially affect the coupled thermal transfer; therefore, thermal radiation continues to dominate the temperature propagation. When $N_{pl} = 1.81 \times 10^{-3}$, two symmetric cold regions appear near the corners at $\Gamma_N \cap \Gamma_E$ and $\Gamma_S \cap \Gamma_E$, which doesn't appear in Fig. 15-(b) for water vapor, due to differences in absorption spectral bands.

For $N_{pl} = 3.85 \times 10^{-4}$, as shown in Fig. 15-(b), the shape of concentric curves forming the map of hot points become less eccentric and shallower compare with the case observed in Fig. 15-(a). Due to the non-neglectable value of the thermal conductivity, $k_c = 0.082 \text{ W.m}^{-1}.\text{K}^{-1}$, and the very small value of $N_{pl}$, the conduction effect is less pronounced near the North ($\Gamma_N$) and South ($\Gamma_S$) walls. For $N_{pl} = 2.39 \times 10^{-3}$, thermal characteristics of carbon dioxide and water vapor are combined. The resulting thermal conductivity is a weighted sum of the contributions from each gas species. Interestingly, the contribution of carbon dioxide to heat transfer predominates despite its lower chemical ratio ($Y_{CO_2} = 0.1$), compared with water vapor ($Y_{H_2O} = 0.2$). Physically, this occurs because, at the same temperature ($T_g = 500K$), the Planck-mean absorption coefficient of $10\%CO_2/N_2$ ($\kappa_p = 4.461 \text{ m}^{-1}$) is greater than that of $20\%H_2O/N_2$ ($\kappa_p = 3.530 \text{ m}^{-1}$).

Figs. 16(a-c) respectively represent the temperature distribution $T^*$ of $10\%CO_2/N_2$, $20\%H_2O/N_2$ and $10\%CO_2/20\%H_2O/N_2$ non-gray gases displayed for $T_g = 3000K$. The participating medium is optically thin due to the low values of the absorption coefficients. Although the corresponding values of the conduction-radiation parameter $N_{pl}$ remain very small, on the order of $10^{-5}$ and $10^{-6}$ order, the associated thermal conductivities are non-negligible. For $N_{pl} = 2.73 \times 10^{-5}$, thermal conduction governs heat propagation in the sub-domain $\{\mathcal{F}_{hot}\} = \{\forall (x,y) \in [0, 0.2] \times [0, 0.5]\}$ due to wall effects; subsequently, thermal radiation occurs toward the east wall ($\Gamma_E$). This case illustrates that both

conduction and radiation dominate within specific sub-domains of the geometry. For $N_{pl} = 2.11 \times 10^{-6}$, thermal radiation governs heat propagation of water vapor throughout the entire domain, and thermal waves appear to propagate freely within the participating medium. The temperature magnitude near the cold walls ($\Gamma_E, \Gamma_N, \Gamma_S$) remains almost unchanged, even at the equilibrium state. For $N_{pl} = 2.87 \times 10^{-5}$, this case represents one of the most significant study scenarios, in which both conduction and radiation interact strongly during the combustion of the $10\%CO_2/20\%H_2O/N_2$ gas mixture. Even when gas properties are evaluated at high temperatures such as $T_g = 3000K$, the contribution of carbon dioxide remains more significant in the numerical simulations of the temperature distribution.

### 4.4.2. Dimensionless total heat flux

The numerical simulations presented in Figs. 17(a-c) to Figs. 19(a-c), highlight the distribution of the total heat flux within the geometry. The behavior of the energy transfer $q^{\varepsilon}$ depends on the conductive heat transfer $q^{c\varepsilon}$, which itself is governed by the conduction-radiation temperature $T^*$, as discussed in sub-section 4.4.1. The numerical results corresponding to a gas inlet temperature of $T_g = 500K$ are shown in Fig. 17(a-c). The different values of the conduction-radiation parameter arise from variations in Planck-mean absorption coefficients and thermal conductivities of the gas species. Although the absorption behavior of gases differs across their spectral bands, the medium remains optically thick at lower gas temperatures; consequently, the total heat flux $q^{c\varepsilon}$ exhibits limited spatial spreading within the medium.

In Fig. 18(a-c), the gas temperature is set to $T_g = 1000K$; as a result, conductive heat transfer decreases, giving rise to radiative heat transfer. The Figs. 19(a-b) highlight the numerical simulations of the dimensionless total heat flux for $T_g = 3000K$. In this case, the contribution of conductive heat transfer can be neglected in comparison with the radiative mode.

In conclusion, the conjugate conduction-radiation heat transfer in the presence of non-gray gases, computed using Planck-mean absorption coefficients, represents an effective approach for predicting combustion

<figure><img src="image_19.png" /><figcaption>**Fig. 16.** Dimensionless temperature distribution $T^*$ in the combustion chamber computed at $T_g = 3000K$: (a) $10\%CO_2/N_2$, (b) $20\%H_2O/N_2$, (c) $10\%CO_2/N_2 + 20\%H_2O/N_2$.</figcaption></figure>
<figure><img src="image_20.png" /><figcaption>Fig. 17. (a-c). Dimensionless total heat flux $q^{t*}$ in the combustion chamber computed for $T_g = 500$K: (a) $10\%CO_2/N_2$, (b) $20\%H_2O/N_2$, (c) $10\%CO_2/N_2 + 20\%H_2O/N_2$.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>Fig. 18. (a-b). Dimensionless total heat flux $q^{t*}$ in the combustion chamber computed for $T_g = 1000$K: (a) $10\%CO_2/N_2$, (b) $20\%H_2O/N_2$, (c) $10\%CO_2/N_2 + 20\%H_2O/N_2$.</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>Fig. 19. (a-b). Dimensionless total heat flux $q^{t*}$ in the combustion chamber computed for $T_g = 3000$K: (a) $10\%CO_2/N_2$, (b) $20\%H_2O/N_2$, (c) $10\%CO_2/N_2 + 20\%H_2O/N_2$.</figcaption></figure>

chamber behavior. The current methods become difficult to implement when the section of the burner has an irregular geometry. Nevertheless, this method provides greater accuracy when gas temperatures reach higher values ($T_g \ge 1000$K).

### 4.5. Three-dimensional implementation strategy

In most real-world engineering scenarios involving combustion processes, these phenomena occur in burners with three-dimensional geometries. Nevertheless, the coupled conductive-radiative problem can be modeled in the $(x,y)$ plane, by assuming invariance along the z-axis. Under this assumption, the resulting radiative quantities provide a realistic representation of heat transfer within the burner for a rectangular cross-section. Mathematically, the solid angle $\Omega(\theta, \varphi)$, which generally has three components, reduces to two because $\cos\theta = 0$. In contrast, when modeling the conjugate thermal problem in a fully three-dimensional furnace with a finite z-direction, the solid angle $\Omega(\theta, \varphi)$, the ray path length $\delta(x, y, z, \theta, \varphi)$ and the temperature distribution $T(x', y', z')$ associated with internal emission all vary. Consequently, considering a full 3D domain ($\cos\theta \neq 0$) significantly increases the computational cost

of the numerical simulations. Therefore, the number of rays increases within the volume defined by the solid angle $\Omega(\theta, \varphi)$, increasing the complexity of solving the radiative transfer equation.

## 5. Conclusion and remarks

This study investigated a combined conduction-radiation heat transfer problem involving non-gray gas kinetics, using a correlated SLW model applied to a two-dimensional combustion chamber. Carbon dioxide ($10\%CO_2/N_2$), water vapor ($20\%H_2O/N_2$), and their mixture were considered as participating gases, with $N_2$ essentially acting as a spectral window. Planck-mean absorption coefficients and thermal conductivities of the participating gases were calculated and tabulated for a set of temperatures ($T_g \in \{500K, 1000K, 3000K\}$). A semi-analytical method combined with a ray-tracing technique was applied to solve the radiative transfer equation. Accordingly, radiative integral solutions were accurately evaluated using Gaussian quadrature together with first- and second-order specific Bickley-Naylor functions. The energy equation was solved under steady-state conditions using the finite difference method with a Crank-Nicolson scheme. The influence of gas tempera-
ture, number of discrete gray gases, mixture ratios, and gas species was then examined. The results indicate that both the accuracy and computational efficiency of the Planck-mean absorption coefficient are strongly governed by the number of discrete gray gases and the gas temperature. The computed conduction-radiation temperature and total heat flux distributions demonstrate satisfactory computational efficiency and reliable accuracy. Differences in the absorption spectral bands of the gas species resulted in an average absolute error of 20% in the temperature distribution and 1% in the total heat flux in the rectangular geometry at radiative equilibrium. Further research work would incorporate diffuse and reflective wall hypotheses and be coupled with reactive flow governing equations.

### CRediT authorship contribution statement

Julien Sorel Djeumegni: Writing – original draft, Validation,

Methodology, Conceptualization. **Vital Le Dez:** Validation, Supervision, Formal analysis. **Myriam Lazard:** Supervision, Resources.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgment

This work was partially funded by the French "investments for the future" program managed by the National Agency for Research (ANR) under contract ANR-10-LABX-22-01 (Labex SOLSTICE), supported by CNRS-Institut Pprime-University of Poitiers-ENSMA, UPR 3346 and University of Nantes, ONIRIS, CNRS, GEPEA, UMR-6144.

## Appendix A. Annex 1

* $H_2O$

$$ 
\begin{aligned}
F_w(C, T_g, T_b, Y_w) &= \frac{1}{2} \tanh \left[ P_w(T_g, T_b, \xi - \xi_p) \right] + \frac{1}{2} \\
P_w(T_g, T_b, \xi - \xi_p) &= \sum_{l=0}^{3} \sum_{m=0}^{3} \sum_{n=0}^{3} b_{lmn} \left( \frac{T_g}{2500} \right)^n \left( \frac{T_b}{2500} \right)^m (\xi - \xi_p)^l \\
\xi_p &= \sum_{l=0}^{3} \sum_{m=0}^{3} \sum_{n=0}^{3} u_{lmn} \left( \frac{T_g T_b}{2500^2} \right)^n \xi^m \psi_w^{l+1} \\
\psi_w &= \frac{1}{10} \ln(100p_e) \\
p_e &= (1 + 8.17Y_w)p \\
\xi &= \ln(C), \quad 1.10^{-4} \le C \le 60 \quad \left[ \frac{m^2}{mol} \right]
\end{aligned}
$$

* $CO_2$

$$ 
\begin{aligned}
(C, T_g, T_b) &= \frac{1}{2} \tanh \left[ P_c(T_g, T_b, \xi - \xi_p) \right] + \frac{1}{2} \\
P_c(T_g, T_b, \xi) &= \sum_{l=0}^{3} \sum_{m=0}^{3} \sum_{n=0}^{3} d_{lmn} \left( \frac{T_g}{2500} \right)^n \left( \frac{T_b}{2500} \right)^m (\xi - \xi_p)^l \\
\xi_p &= \sum_{l=0}^{3} \sum_{m=0}^{3} \sum_{n=0}^{3} v_{lmn} \left( \frac{T_g T_b}{2500^2} \right)^n \xi^m \psi_c^{l+1} \\
\psi_c &= \frac{1}{10} \ln(100p) \\
\xi &= \ln(C), \quad 1.10^{-4} \le C \le 600 \quad \left[ \frac{m^2}{mol} \right]
\end{aligned}
$$

Correlation coefficients $b_{lmn}$, $c_{lmn}$ and $d_{lmn}$ are computed at atmospheric pressure, whereas $u_{lmn}$, $v_{lmn}$ and $w_{lmn}$ at variable pressure conditions and available following the link http://albdf.byu.edu

## Appendix B. Annex 2

Space coordinates: $x^* = x/L$, $y^* = y/L$.

Temperatures: $T^* = T_{hot}^4$, $T_0^* = T_{hot}^4$.

Radiative quantities: $G^* = \frac{G}{4\sigma_B T_{hot}^4}$; $q^* = \frac{q'}{4\sigma_B T_{hot}^4}$ and $S^* = \frac{S}{4\sigma_B T_{hot}^4}$.

Total heat flux: $q_x^* = \frac{q_x}{4\sigma_B T_{hot}^4}$; $q_y^* = \frac{q_y}{4\sigma_B T_{hot}^4}$.

Planck number: $N_{pl} = \frac{k_c k_p}{4\sigma_B T_{hot}^3}$.

### Data availability

The data that has been used is confidential.

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