# Thermal Analysis of Radiation Heat Transfer of Improved Fractal Solar Collectors
## 2 Design and Analysis
### Experimental Setup  
Experimental studies were carried out in Kazakhstan/Turkestan (68.23\({}^{\circ}\) latitude and 43.3051\({}^{\circ}\) longitude) under the climatic conditions of 1 March 2023, using a 3-row toroidal fractal structure. The experiments were conducted under environmental conditions with an average wind speed of 2.5 m/s, 45% relative humidity, and 2/8 cloud cover. The wind speed increases heat losses through convection on the collector's outer surface, contributing to the discrepancy between experimental results and numerical analyses. An increase in relative humidity enhances the air's heat retention capacity and influences heat losses from the collector to the environment. Changes in cloud cover affect the amount of solar radiation and, consequently, the energy absorbed by the collector. The collector with a 3-row toroidal fractal array used in the experiments (shown in Figures 1 and 2) was designed with an absorber structure with a highly selective coating [41]. In the design phase, the formula for the surface area of the absorber depending on the radius was calculated using Equation (1). In Equations (1)-(5), \(R\) is the distance from the center of the circle to the axis of rotation of the absorber; \(r\) is the radius of the absorber; \(c\) is the inner diameter; and \(b\) is the outer circle. As seen from Equation (1), the system capacity can be changed depending on the number of fractal arrays according to the circle dimensions of the toroidal structure [40]. \[A=4\pi r^{2}Rr \tag{1}\]  
\[b=2\pi(R-r) \tag{2}\]  
\[c=2\pi(R+r) \tag{3}\]  
\[R=(b+c)/4\pi \tag{4}\]  
\[r=(b-c)/4\pi \tag{5}\]After calculating the radii along the circles of the absorber, the surface area (\(A\)) and volume (\(V\)) of the absorber are calculated using Equations (6) and (7), respectively, by substituting the expressions obtained for the radii in the standard formulas. \[A=4\pi r^{2}Rr=\left(b^{2}-c^{2}\right)/4 \tag{6}\]  
\[V=2\pi^{2}Rr^{2}=(b-c)\left(b^{2}-c^{2}\right)/32\pi \tag{7}\]  
The water transferred to the system is heated by the sun rays while passing through the first toroidal fractal. In this case, the heat losses of the first fractal increase while reaching the second fractal by convection and providing an additional thermal energy source. Thus, the heat losses of the first and second toroids serve as an additional thermal energy source for the third toroidal. The heating water is removed from the system by rising from the toroidal fractal pipes, so the water temperature in each fractal absorber is different. Meanwhile, the parabolic dish concentrates the sun rays falling on the surface area to the focal points on the fractal, causing thermal losses to decrease. In this way, the thermal energy obtained is transferred from the focusing region to the working fluid and sent to the thermodynamic circulation. The speed of the water entering the toroidal pipe is 0.3 m/s at 20 \({}^{\circ}\)C. The pipe used in the experimental setup has an outer diameter of 10 mm and is coated with a 1.6 mm thick galvanized layer, which is suitable for pipes of this size. The material of the pipe is ST37.2, an unalloyed structural steel known for its weldability and corrosion resistance. In the experiment, the temperature data of the water entering and leaving the collector were transmitted to the datas logger for each minute with a K-type temperature sensor (Testo 176 T4, Testo 175 T1, Testo 454, and Testo 350) (Testo SE, Titisee-Neustadt, Germany), and the data were recorded for 60 min. Statistical analyses of the experimental studies are  
Figure 1: 3-row toroidal fractal array [40]. Figure 2: Toroidal collector with 3-row fractal structure (**a**) front surface (**b**) back surface. presented in Table 1. The Pearson correlation coefficients between three measurement sets (test, retest 1, and retest 2) were calculated to assess measurement reliability. The correlation analysis results, shown in Table 1, demonstrate the statistical relationships between repeated measurements [42].