## 4 Conclusions  
In this work, three distinct relative roughness pitches (P/e = 7.14; 10.71; 14.29) and three distinct roughness sections (square, semi-circle, and triangle) are used for the numerical analysis of the solar air heater, with Reynolds numbers ranging from 4000 to 18,000. Numerical data obtained using the computational fluid dynamic technique and the innovative HFO-1 swarm optimization technique are used to create two novel correlation models for Nusselt number prediction. HFO-1 is for the first time used in the literature to optimize the parameters (coefficients) of the correlation models. Furthermore, two different optimizations using a non-swarm variant (GRG) and a swarm variant (HFO-1) are comparatively studied. In the semicircular section, the Nusselt number reached its maximum value of 87.220, P/e = 7.14, and the Reynolds number reached 18,000. The Nusselt number increased due to the decrease in P/e value. The Nusselt number is analyzed based on the kind of geometry; its minimal value is found in the square section, and its maximum value is observed in the semi-circle section. In the study, two different equations (M1 and M2) were proposed to obtain the Nusselt number. The Nusselt number was obtained using HFO-1 with an error of 2.15% based on M1 and with an error of 0.85% based on M2. Moreover,  
Figure 6: Nusselt number prediction using GRG and HFO-1 algorithm based on M2 ((a): square, (b): semicircular, (c): triangular). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Model** & **Geometry** & **MAE** & **MAPE** & **MSE** & **RMSE** & **R\({}^{2}\)** \\ \hline \multirow{4}{*}{M1} & Square & 33.949 & 99.298 & 60.735 & 37.338 & 0.030 \\  & Semicircular & 43.606 & 99.410 & 66.418 & 42.051 & 0.000 \\  & Triangle & 36.720 & 99.316 & 64.002 & 40.000 & 0.051 \\  & Mix type & 46.260 & 99.440 & 69.957 & 45.188 & 0.041 \\ \hline \multirow{4}{*}{M2} & Square & 55.723 & 99.597 & 80.177 & 55.476 & 0.767 \\  & Semicircular & 56.227 & 99.608 & 80.638 & 55.996 & 0.574 \\ \cline{1-1}  & Triangle & 40.835 & 99.432 & 68.042 & 43.471 & 0.748 \\ \cline{1-1}  & Mix type & 23.384 & 99.275 & 88.090 & 65.489 & 0.550 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Performance improvement of HFO-1 over GRG for models M1 and M2 (%). the achievement of HFO-1 with the proposed correlation model is 50% better than the achievement of the traditional model on average. Approximately 99.5% of the variation in the Nusselt number is explained by the equation established using the coefficients found by the HFO-1 algorithm. The results show that the proposed correlation model is very successful in predicting the Nusselt number for the assumed roughness parameters of the absorber plate. The proposed correlation model has an increased complexity, causing the HFO-1 to be better than the traditional optimization tool. As the model complexity increases, a more sophisticated optimization tool is required. However, increasing the complexity of the model may disrupt its generalization ability for the other types of absorber plates. Thus, for future studies, the proposed model is recommended to be studied with different types of absorber plates. Also, the proposed model is recommended to be tested for the prediction of friction factors. This study highlights the importance of using swarm intelligence in predicting the Nusselt number with novel correlation models. Conceptualization, A.K., U.E. and Z.Y. ; methodology, A.K., U.E., Z.Y. and F.K. ; software, A.K., U.E., Z.Y. and F.K. ; validation, A.K. ; formal analysis, A.K., U.E. and Z.Y. ; investigation, A.K. ; resources, A.K. ; data curation, A.K., U.E. and Z.Y. ; writing--original draft preparation, A.K., U.E., Z.Y. and F.K. ; writing--review and editing, A.K., U.E., Z.Y. and F.K. ; visualization, A.K. and U.E. ; supervision, A.K., U.E., Z.Y. and F.K. ; project administration, A.K., U.E., Z.Y. and F.K. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable.