## 3 Results and Discussion  
Two optimization methods, HFO-1 and GRG, are applied to SAH optimization with the algorithm parameters given in Table 1. The boundaries of the search space may vary depending on the problem. Here, different lower and upper bounds were used, and optimum parameters were found to be in the range \([-2,\,4]\). The HFO-1 optimization algorithm has been computed in MATLAB R2023a, whereas GRG optimization has been computed in Excel. In this study, a thermo-hydraulic performance analysis and numerical study using the CFD method of heat transfer was conducted on a solar air heater with various rib types. An intentionally roughened solar air heater with three distinct relative roughness pitches (P/e) on the absorber plate and three different rib patterns (square, semicircular, and triangular) was examined. Details of CFD studies are given in the previous study [9]. Nusselt numbers obtained as a result of the CFD analysis were compared with empirical results obtained from regular channel solar heaters under equivalent operating conditions. The Nusselt numbers were observed to be consistent with the Dittus-Boelter correlation (MAPE = 0.093). Figure 3 illustrates how the Nusselt number varies with the Reynolds number on the smooth duct. \[Nu_{s}=0.022\;Re^{0.8}\;Pr^{0.4} \tag{14}\]  
Regarding the Nusselt number for corrugated surfaces, the absorber plate contains three different kinds of ribs with three distinct relative roughness pitches (P/e) intentionally roughened to enhance heat transfer. The addition of ribs to the hot absorber plate significantly enhances the heat transfer of solar air heaters. This is because the ribs increase the surface area available for convection, which in turn increases convection heat transmission while decreasing conduction heat transfer. The semicircular section displays the Nusselt number's greatest value, while the square section displays its minimum value when the Nusselt number is analyzed based on the geometry type. Figure 4 shows the relationship between the Reynolds number and the Nusselt number for absorber plates that are smooth and corrugated and have different P/e. The Nusselt number for all corrugated surfaces peaked at P/e = 7.14 and a Reynolds number of 18,000 [9]. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Optimization** & **Population** & **Search Range** & **Algorithm Dependent Parameters** \\
**Algorithm** & **Size** & & \\ \hline HFO-1 & 30 & \([-2,\,4]\) & mixing factor: 8, qo: 2, \\  & & maximum iteration: 15,000 \\ GRG & 100 & \([-2,\,4]\) & precision: \(10^{-4}\), mutation rate: 0.075 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter settings of the optimization algorithms. Figure 3: Nusselt number variation with Reynolds number [9]. The equations of the proposed correlation models regarding the Nusselt number for corrugated surfaces are given in Table 2, where \(C_{0}\), \(C_{1}\), \(C_{2}\), \(C_{3}\), \(C_{4}\), and \(C_{5}\) are the tuning parameters that are to be optimized by the optimization algorithms. The correlations were established using numerical data and optimization algorithms (HFO-1 and GRG). Equation (15) shows that the Nusselt number can be expressed as a function of Re and P/e [10]. \[Nu_{r}=f(Re,P/e) \tag{15}\]  
Coefficients of the models were determined, for the equations given in Table 2, by separately optimizing them for three different types of geometry data. In addition, the equations were also optimized for all data, regardless of geometry, indicated by the mix type geometry in Tables 3 and 4. For the proposed models, the coefficients of the equations were found using both the GRG optimization and the HFO-1 algorithm. Then, the experimental results are provided in terms of various evaluation metrics, given in Tables 3 and 4, demonstrating both the optimized coefficients and the achievement of the proposed models in predicting the Nusselt numbers. According to the results, the Nusselt numbers are predicted to be close to the simulation values, acquired using CFD, for all geometries.