## 2 Materials and Methods  
### Numerical Analyses  
The thermo-hydraulic performance was analyzed with the CFD method between 4000 and 18,000 Reynolds numbers of a solar air heater with various kinds of ribs. The study focused on an artificially roughened solar air heater that had three different types of ribs (square, semi-circle, and triangle) on the absorber plate with three various relative roughness pitches (P/e = 7.14; 10.71; 14.29). Also, the constant value of e/D = 0.042 was used to determine the relative roughness height. According to [PERSON] et al. (2018), solar air heating systems with Reynolds numbers between 3800 and 18,000 performed better thermo-hydraulically. Figure 2 displays the schematic view of the solar air heater. A plate made of aluminum is connected to the upper test wall of the air heater, and a continuous heat flux of 1000 W.m\({}^{-2}\) is applied to the test wall. The other walls are adiabatic, and heat losses due to environmental interactions are neglected. It is presumed that the air enters the duct at an ambient temperature (300 K) and at a constant velocity. The air inlet velocities  
Figure 1: Main stages of the study. were determined using Equation (1) based on the Reynolds number. At the exit of the duct, a pressure outlet condition is applied to maintain a constant atmospheric pressure of 1.013 \(\times\) 105 Pa. Adiabatic boundary and no-slip wall conditions were applied across the walls. The entrance, test, and exit parts make up the three components of the geometric model. Equations (2) and (3) are used to determine the minimum values for the entrance and exit lengths by ASHRAE Standard 93 GA30329 [39]. \[v=\frac{Re\mu}{\rho D_{h}} \tag{1}\]  
\[L_{1}=5\sqrt{HW} \tag{2}\]  
\[L_{3}=2.5\sqrt{HW} \tag{3}\]  
\[D_{h}=\frac{4A}{P_{w}} \tag{4}\]  
ANSYS software was used to generate the mesh structure, one of the key elements of the numerical analysis. The efficiency of the system depends on mesh structure optimization. In order to ascertain the ideal cell number with increasing nodes, the studies were sequentially completed. The ideal number of cells was found to be in the scenario where the friction factor is less than 2% and the alteration in the Nusselt number is less than 1%. Beyond the optimal number, adding nodes will increase computational effort without improving the correctness of the result. An investigation of the mesh form was conducted at P/e = 7.14 and Re = 16,000. The analysis yielded an optimal cell size of 0.22 mm. Heat transfer and forced turbulent fluid flow are defined by the governing equations (continuity, momentum, and energy). Equations (5)-(8) provide these equations. \[\frac{\partial}{\partial x_{i}}(\rho u_{i})=0 \tag{5}\]  
\[\frac{\partial}{\partial x_{i}}\big{(}\rho u_{i}u_{j}\big{)}=-\frac{\partial p }{\partial x_{i}}+\frac{\partial}{\partial x_{j}}\bigg{[}\mu\left(\frac{ \partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right) \bigg{]}+\frac{\partial}{\partial x_{j}}\big{(}-\rho u_{i}u_{j}\big{)} \tag{6}\]  
\[\frac{\partial}{\partial x_{i}}(\rho u_{i}T)=-\frac{\partial}{\partial x_{i}} \bigg{(}(\tau+\tau_{t})\frac{\partial T}{\partial x_{j}}\bigg{)} \tag{7}\]  
where \(\tau\) and \(\tau_{t}\) are molecular thermal diffusivity and turbulent thermal diffusivity, respectively, expressed as follows:  
\[\tau=\frac{\mu}{Pr}\tau_{t}=\frac{\mu_{t}}{Pr_{t}} \tag{8}\]  
Figure 2: Schematic view of solar air heater. The Renormalization-group (RNG) k-\(\epsilon\) is a frequently utilized turbulent flow in the channel that was simulated using a turbulence model. In their study, [PERSON] and [PERSON] thoroughly investigated several turbulence models that can be applied to the numerical analysis of solar air heaters. The RNG k-\(\epsilon\) model is the frequently chosen model for solar air heaters [40]. Furthermore, numerous studies have shown that this model produces correct results [41; 42; 43]. The analysis employed a second-degree discretization technique. All dependent variables were found to have a convergence threshold of \(10^{-3}\).