# Thermal Analysis of Radiation Heat Transfer of Improved Fractal Solar Collectors
## 2 Design and Analysis
### Discrete Ordinates Radiation Model  
The Discrete Ordinates Radiation Model is a powerful and widely applicable model for simulating radiative heat transfer and can work over a wide range of optical thicknesses, including optically thin and thick media. Another key advantage is that it can also handle anisotropic scattering and non-gray radiation. The radiative transfer equation (RTE) is solved for a finite number of discrete solid angles, each associated with a fixed vector direction in the global Cartesian system. It can accurately model the directional nature of radiation and the wavelength dependence of radiative properties, which is desirable for many engineering applications. It can also consider the effects of participating media, such as gases and particles, on radiative heat transfer. The model is based on the discretization of the RTE into a set of interconnected equations for a finite number of discrete solid angles. The accuracy of the DO model depends on the number of discrete ordinates used; higher numbers generally provide better accuracy but increase computational cost. However, although it generally requires more computational power than some other radiation models, it offers superior accuracy and flexibility in a wide range of applications [44; 45]. In an absorbing, emitting, and scattering medium, the \(\overset{\rightarrow}{r}\) position in the \(\overset{\rightarrow}{s}\) direction is given in Equation (8). In the equation, \(\sigma\) represents the Stefan-Boltzmann constant (\(5.669\times 10^{-8}\) W/m\({}^{2}\) K\({}^{4}\)). Here, \(T\) is local temperature, \(\Phi\) is phase function, \(a\) is absorption  
Figure 3: Dimetric and top section views of fractal solar collector and parabolic dish models for (**a**) 3-row, (**b**) 4-row, and (**c**) 5-row. coefficient, \(\Omega^{\prime}\) is solid angle, \(\overset{\rightarrow}{r}\) position vector, \(\overset{\rightarrow}{s}\) is direction vector, \(\sigma_{s}\) is scattering coefficient, \(\overset{\rightarrow}{s}\) is scattering direction vector, \(s\) is path length, and \((a+\sigma_{s})\) is optical thickness or opacity of the medium. \[\frac{dl}{ds}\left(\overset{\rightarrow}{r},\overset{\rightarrow}{s}\right)+ (a+\sigma_{s})I\left(\overset{\rightarrow}{r},\overset{\rightarrow}{s} \right)=an^{2}\frac{\sigma T^{4}}{\tau}+\frac{\sigma_{s}}{4\pi}\int_{0}^{4\pi} I\left(c,\overset{\rightarrow}{s}\right)\Phi\left(\overset{\rightarrow}{s}, \overset{\rightarrow}{s}\right)d\Omega^{\prime} \tag{8}\]  
The refractive index \(n\) is an important parameter when considering radiation in semitransparent media. \(I\) defines the radiation intensity, which depends on the \(\overset{\rightarrow}{r}\) position and \(\overset{\rightarrow}{s}\) direction. This index is a fundamental factor affecting the speed and direction of light propagation in the medium. Figure 4 visualizes the process of heat transfer by radiation in detail. In this representation, the radiation intensity, expressed by the symbol \(I\), attracts attention with its spatial and directional dependence. In particular, the dependence of \(I\) on the \(\overset{\rightarrow}{r}\) position and \(\overset{\rightarrow}{s}\) direction shows how the radiation propagates and changes in three-dimensional space. This dependency contributes to more accurate modeling of heat transfer in complex geometries and heterogeneous environments by examining how radiation behaves at different points and in different directions. The solution technique used in the DO radiation model is similar to the methods applied in the fields of fluid dynamics and energy transfer. The distinctive feature of this model is that it evaluates the RTE in the \(\overset{\rightarrow}{s}\) direction as a field equation and does not use the ray tracing technique. Instead, RTE uses a limited number of discrete solid angles while solving Equation (9). Each solid angle is matched to a specific \(\overset{\rightarrow}{s}\) vector direction in the Cartesian coordinate system (x, y, z).