## 3 Results and Discussion
### Optimal Insulation Thickness Results  
Equations (10)-(19) are adjusted based on the suggested correction factor introduced in Equations (22) and (23) to better improved total costs for cooling energy and insulation material. As the insulation thickness (\(X_{ins}\)) increases, the annual cost of cooling energy (\(C_{ac}\)) decreases. The total cost (\(C_{tc}\)) includes both the cost of the electricity and the cost of the insulation material. Furthermore, the insulation cost rises linearly with the thickness of the insulation. The optimum insulation thickness is estimated by determining the thickness corresponding to the lowest overall cost. Figures 15-19 show the change in the insulation cost (\(C_{ins}\)), annual cooling energy (\(C_{ac}\)), and the total cost (\(C_{tc}\)) in the selected five cities for the three insulation materials. The results show that \(C_{tc}\) decreases as \(X_{ins}\) increases, but the decrease is abrupt initially and then becomes gradual as \(X_{ins}\) increases. As a result, at low \(X_{ins}\), a significant percentage fall in (\(C_{tc}\)) can be achieved, and this percentage drop diminishes as \(X_{ins}\) increases. Since the cost of insulation material increases linearly with the thickness while the cost of energy consumption is proportional to annual heat transmission, it is expected that the overall cost initially decreases as \(X_{ins}\) increases, then rises as \(X_{ins}\) increases more. This behavior indicates the existence of a minimum overall cost that corresponds to an optimal insulation thickness. Based on the insulations' thermal conductivity values, polyurethane has the smallest \(Q_{year}\) for a given \(X_{ins}\), followed by polystyrene and finally rock wool. The same trends can be seen in [33] for Riyadh and Jeddah. As shown, all curves yield minimum values reflecting minimum total costs (\(C_{t,min}\)) for each insulation material, which correspond to optimum insulation thicknesses (\(X_{opt}\)). The optimum thicknesses, corresponding payback periods, and minimum total cost values for all types of insulations and climate conditions are summarized in Table 6. Polystyrene has the lowest \(C_{t,min}\), rock wool, polyurethane with the highest \(C_{t,min}\), and least \(X_{opt}\). Polystyrene  
Figure 14: Comparison of calculated, simulated, and modeled annual heat gain in Khamis M. for different insulation materials: (**a**) molded polystyrene; (**b**) polyurethane; (**c**) rock wool. is the most economical insulation, according to \(C_{t,min}\). These results are consistent with other work [33]. Figure 16: Relationship between insulation cost, annual cooling energy, and total cost in Guraiat for different insulation materials: (**a**) molded polystyrene; (**b**) polyurethane; (**c**) rock wool. Figure 15: Relationship between insulation cost, annual cooling energy, and total cost in Riyadh for different insulation materials: (**a**) molded polystyrene; (**b**) polyurethane; (**c**) rock wool. Figure 17: Relationship between insulation cost, annual cooling energy, and total cost in Dhahran for different insulation materials: (**a**) molded polystyrene; (**b**) polyurethane; (**c**) rock wool. Figure 18: Relationship between insulation cost, annual cooling energy, and total cost in Jeddah for different insulation materials: (**a**) molded polystyrene; (**b**) polyurethane; (**c**) rock wool. It is worth noting that the total cost includes the cost of insulation material as well as the cost of energy used due to cooling and heating loads. Associated costs, such as