# Thermal Analysis of Radiation Heat Transfer of Improved Fractal Solar Collectors
## 2 Design and Analysis
### Discrete Ordinates Radiation Model
Thus, by solving a transport equation for each direction, the total number of solved equations is equal to the number of determined directions. Therefore, this method makes the solution of the problem more systematic and organized and offers the flexibility to balance the sensitivity and computational load of the model. Therefore, it significantly increases the adaptability of the DO model to various geometric structures and complex radiation scenarios, thus expanding the application area of the model. \[\
abla.\left(I_{\lambda}\left(\overset{\rightarrow}{r},\overset{\rightarrow}{s} \right)\overset{\rightarrow}{s}\right)+(a_{\lambda}+\sigma_{s})I_{\lambda} \left(\overset{\rightarrow}{r},\overset{\rightarrow}{s}\right)=a_{\lambda}n ^{2}I_{b\lambda}+\frac{\sigma_{s}}{4\pi}\int_{0}^{4\pi}I_{\lambda}\left( \overset{\rightarrow}{r},\overset{\rightarrow}{s}\right)\Phi\left(\overset{ \rightarrow}{s},\overset{\rightarrow}{s}\right)d\Omega^{\prime} \tag{9}\]  
Equation (10) is used to calculate the spectral intensity \(I_{\lambda}\left(\overset{\rightarrow}{r},\overset{\rightarrow}{s}\right)\) for RTE. Here, \(I_{b\lambda}\) represents the black body intensity given by the Planck function. In addition, \(a_{\lambda}\) is the spectral absorption coefficient, and \(\lambda\) is the wavelength. When calculating the scattering  
Figure 4: Process of radiative heat transfer. coefficient, it is assumed that \(n\) is independent of the wavelength and the scattering phase function (a \(\sigma\) T\({}^{4}/\pi\))\(ds\)  
\[\
abla_{\cdot}\Big{(}I_{\lambda}\Big{(}\overset{\rightarrow}{r},\overset{ \rightarrow}{s}\Big{)}\overset{\rightarrow}{s}\Big{)}+(a_{\lambda}+\sigma_{s} )I_{\lambda}\Big{(}\overset{\rightarrow}{r},\overset{\rightarrow}{s}\Big{)} =a_{\lambda}n^{2}\frac{\sigma T^{4}}{\pi}+\frac{\sigma_{s}}{4\pi}\int_{0}^{4 \pi}I_{\lambda}\Big{(}\overset{\rightarrow}{r},\overset{\rightarrow}{s} \Big{)}\Phi\Big{(}\overset{\rightarrow}{s}\cdot\overset{\rightarrow}{s} \Big{)}d\Omega^{\prime} \tag{10}\]  
\[[F(0\to n\lambda_{2}T)-F(n\lambda_{1}T)]n^{2}\frac{\sigma T^{4}}{\pi} \tag{11}\]  
The fraction of the radiant energy emitted by a black body, \(F(0\to n\lambda T)\), is given in Equation (11). Here, in a medium with a refractive index of \(n\), at a temperature of \(T\), the wavelength range is between \(\lambda_{1}\) and \(\lambda_{2}\), which are the wavelength limits of the band. The total intensity \(I\Big{(}\overset{\rightarrow}{r},\overset{\rightarrow}{s}\Big{)}\) in the \(\overset{\rightarrow}{s}\) direction at the \(\overset{\rightarrow}{r}\) position is calculated using Equation (12). \[I\Big{(}\overset{\rightarrow}{r},\overset{\rightarrow}{s}\Big{)}=\sum_{k}I_{ \lambda_{k}}\Big{(}\overset{\rightarrow}{r},\overset{\rightarrow}{s}\Big{)} \Delta\lambda_{k} \tag{12}\]  
The energy matching of the DO model is shown in Equation (13). where \(\beta_{i}^{T}=16\kappa\sigma T_{i}^{*3}\Delta V_{i}\), \(S_{i}^{T}=12\kappa\sigma T_{i}^{*4}\Delta V_{i}\), \(\kappa\) is the absorption coefficient, and \(\Delta V_{i}\) is the control volume. The coefficient \(\mu_{ij}^{T}\) and the source term \(S_{i}^{J}\) are due to the discretization of the convection and diffusion terms and the non-radiative source terms. \[\sum_{j=1}^{N}\mu_{ij}^{T}T_{j}-\beta_{i}^{T}T_{i}-\alpha_{i}^{T}\sum_{k=1}^{L }k_{i}^{k}\omega_{k}-S_{i}^{T}S_{i}^{h} \tag{13}\]  
As seen in Figure 5, a schematic representation of the radiation on an opaque wall is presented. The DO model allows opaque walls to be modeled under various conditions. The model can consider walls in the following two different scenarios: first, walls located inside a space and surrounded by liquid or solid regions on both sides. and second, walls located outside the space and in contact with a liquid or solid region on only one side. Another important feature of the DO model is that it can evaluate the properties of walls in different ways in radiation calculations.