## 3 Methodology  
### The Physico-Mathematical Modeling of Heat Transfer  
The physical model describing natural convection flow combined with radiation within a cavity encompasses the transport equations for mass, momentum, energy [43], and radiative transfer [44] (Equations (1)-(6)), which are solved numerically via the finite volume method [45]. The simulations are conducted for a three-dimensional, steady, and turbulent flow regime. The thermo-physical properties of the fluid are considered temperature-dependent, as the Boussinesq approximation is not applicable within the temperature range investigated [46].  
The governing equations are as follows:  
Continuity equation:  
\[\frac{\partial\rho u}{\partial x}+\frac{\partial\rho v}{\partial y}+\frac{ \partial\rho w}{\partial z}=0 \tag{1}\]  
Momentum equations:  
\[\begin{array}{l}\Pi\frac{\partial\rho\overline{u}}{\partial x}+\
abla\frac {\partial\rho\overline{u}}{\partial\underline{v}}+\overline{w}\frac{\partial \rho\overline{u}}{\partial\underline{c}}=-\frac{\partial\overline{P}^{\prime} }{\partial x}+\frac{\partial}{\partial x}\left(\mu\frac{\partial\overline{u} }{\partial x}\right)+\  
The unit thermal flux \(q_{s}\) transferred at the contact surface can be expressed by [PERSON]'s law, as a function of the thermal conductivity \(\lambda_{f}\) and the temperature gradient at the surface (\(y=0\)).  
\[q_{s}(x)=-\lambda_{f}\frac{\delta t(x,y)}{\delta y}\quad\left[\text{W}/\text{m} ^{2}\right] \tag{8}\]  
Beyond the conductive boundary layer adjacent to the surface, heat transfer within the fluid occurs through both conduction and predominantly by convective transport due to the fluid's bulk motion. Therefore, the local convective heat transfer coefficient, \(\alpha\), can be expressed as:  
\[\alpha(x)=\frac{q_{s}(x)}{t_{s}(x)-t_{f}}=-\lambda_{f}\frac{\delta t(x,y)}{ \delta y}*\frac{1}{t_{s}(x)-t_{f}}\quad\left[\text{W}/\text{m}^{2}\right] \tag{9}\]  
One of the earliest efforts addressing the laminar case through an integral approach for \(Pr\) > 0,6, proposing that the ratio of thermal to velocity boundary layer thickness, \(\Delta\), depends solely on the Prandtl number, was presented by [47]. The derived results demonstrated strong agreement with both experimental findings [48] and numerical solutions obtained via the similarity method [49]. Given the lack of literature on the simultaneous development of viscous and thermal layers, the current theory is formulated under the simplifying assumption that these laminar outcomes can be reasonably extended to turbulent flow conditions as follows:  
\[\Delta(Pr)=\frac{\partial_{t(x)}}{\partial_{(x)}} \tag{10}\]  
The values of the \(\Delta\) ratio for different Prandtl numbers cover the physical situation where \(\partial_{(x)}\) > \(\partial_{t(x)}\), which correspond to common fluids such as \(Pr\) > 0.636. Therefore, using assumption (10), the integral forms of the energy and motion equations in the boundary layer are as described in [50].  
The local Nusselt number \(Nu_{x}\) is defined by  
\[Nu_{x}=\frac{g_{w}x}{\theta_{w}\lambda} \tag{11}\]  
Introducing \(\Pi_{\Delta}\) as a function of the \(\Delta\) boundary layer ratio  
\[\Pi_{\Delta}=\Delta^{\frac{8}{9}}\left(\frac{7}{72}-\frac{7}{60}\Delta+\frac{ 21}{253}\Delta^{2}-\frac{14}{435}\Delta^{3}+\frac{7}{1332}\Delta^{4}\right) \tag{12}\]  
both the dynamical boundary layer and the Nusselt number result from the Prandtl number (\(\Delta\) is only \(Pr\) dependent) and are expressed by  
\[\frac{\delta}{x}=0.0448(Ra_{x}^{*})^{-\frac{1}{14}}\left[\frac{Pr^{-\frac{16} {3}}}{\Delta(\Pi_{\Delta})^{10}}\left(1+\frac{0.0823}{(\Pi_{\Delta})Pr^{\frac {10}{15}}}\right)\right]^{\frac{1}{14}} \tag{13}\]  
\[Nu_{x}=0.0631(Ra_{x}^{*})^{\frac{2}{9}}\left[\frac{\Pi_{\Delta}^{\frac{1}{2}} Pr^{\frac{1}{2}}}{\Delta}\left(1+\frac{0.0823}{(\Pi_{\Delta})Pr^{\frac{10}{15}}} \right)\right]^{-\frac{2}{7}} \tag{14}\]  
\(Ra_{x}^{*}\) is the local modified Rayleigh number defined by  
\[Ra_{x}^{*}=\frac{g\beta\varphi_{u}x^{4}}{\lambda v^{2}}Pr. \tag{15}\]