## 3 Results and Discussion
### Optimal Insulation Thickness Results
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**City** & **Insulation** & \(\mathbf{C_{t,min}}\) **(SAR/m2)** & \(\mathbf{PP}\) **(Year)** & \(\mathbf{X_{opt}}\) **(cm)** & \(\mathbf{R_{opt}}\) **(m2.K/W)** & \(\mathbf{U_{opt}}\) **(W/m2.K)** \\ \hline \multirow{3}{*}{Khamis M. (CDD = 3390 \({}^{\circ}\)C-day)} & MP & 34.89 & 2.06 & 4.96 & 1.46 & 0.69 \\ \cline{2-6}  & PB & 44.12 & 2.74 & 2.34 & 0.98 & 1.03 \\ \cline{2-6}  & RW & 39.69 & 2.4 & 4.96 & 1.18 & 0.85 \\ \cline{2-6}  & MP & 35.95 & 2.01 & 5.14 & 1.51 & 0.66 \\ \cline{2-6}  & PB & 45.52 & 2.65 & 2.43 & 1.01 & 0.99 \\ \cline{2-6}  & RW & 40.92 & 2.34 & 5.15 & 1.23 & 0.82 \\ \hline \multirow{3}{*}{Riyadh (CDD = 5688 \({}^{\circ}\)C-day)} & MP & 46.72 & 1.6 & 6.94 & 2.04 & 0.49 \\ \cline{2-6}  & PB & 59.82 & 2.12 & 3.39 & 1.41 & 0.71 \\ \cline{2-6}  & RW & 53.47 & 2.86 & 7.05 & 1.68 & 0.60 \\ \cline{2-6}  & MP & 47.91 & 1.63 & 7.13 & 2.10 & 0.48 \\ \cline{2-6}  & PB & 61.41 & 2.07 & 3.5 & 1.46 & 0.69 \\ \cline{2-6}  & RW & 54.86 & 1.8 & 7.2 & 1.71 & 0.58 \\ \hline \multirow{3}{*}{Jeddah. (CDD = 6587 \({}^{\circ}\)C-day)} & MP & 50.66 & 1.48 & 7.59 & 2.23 & 0.45 \\ \cline{2-6}  & PB & 65.07 & 1.97 & 3.74 & 1.56 & 0.64 \\ \cline{1-1} \cline{2-6}  & RW & 58.07 & 1.73 & 7.75 & 1.85 & 0.54 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Overview of findings for molded polystyrene (MP), polyurethane board (PB), and rock wool (RW) across five different climate conditions. Figure 19: Relationship between insulation cost, annual cooling energy, and total cost in Khamis M. for different insulation materials: (**a**) molded polystyrene; (**b**) polyurethane; (**c**) rock wool. installation expenses, are not considered. These charges can be applied to the net cost without affecting \(X_{opt}\). Furthermore, the results presented in Table 6 illustrate the impact of climatic conditions on the optimal R-value and U-value for various insulation materials. As expected, cities with higher Cooling Degree Days (CDDs), such as Jeddah and Dhahran, exhibit higher optimal R-values and lower U-values, indicating the necessity for thicker insulation to mitigate cooling loads effectively. For instance, in Jeddah, the highest cooling demand location molded polystyrene (MP) achieves an R-value of 2.23 m\({}^{2}\)K/W with a corresponding U-value of 0.45 W/m\({}^{2}\)K, which is below the SBC 601 and SBC 602 maximum U-value requirements [40]. Conversely, locations with lower cooling demands, such as Khamis Mushait and Guriat, require thinner insulation, resulting in relatively higher U-values. The results emphasize that a uniform insulation requirement, as prescribed by SBC, does not account for climatic variations and may lead to suboptimal energy savings. Our findings demonstrate the necessity of climate-responsive insulation strategies to enhance energy efficiency while considering economic feasibility  
It should be stated that the payback period values are not mere values since inflation and the cost of money are considered. The payback period was determined using the same analysis, representing the number of years it takes for the initial insulation cost to be offset by energy savings. The payback period increases with X\({}_{\text{ins}}\) almost linearly. For Dhahran, Polystyrene and rock wool yield similar results, but polyurethane has a much more extended payback period due to its higher price. The molded polystyrene yields the shortest payback period in all cities due to its low cost and relatively low conductivity, as shown in Table 6. The noticed short payback period in all cities and insulation materials significantly promotes insulation use. Riyadh, Dhahran, and Jeddah have close results because the climate in the three cities is cooling-dominated with relatively similar CDD. On the other hand, Guriat and Khamis M. yield almost the same payback period for each insulation material. Payback period values are similar to other studies [41].