## 3 Results and Discussion
### Adjusted Model  
The humidity-adjusted annual heat gain/loss, Q\({}_{\text{eff}}\), in Equation (3) is adjusted by introducing the humidity as a relevant parameter in the following equations.  
\[\text{Q}_{\text{eff (year,c)}}=86,400\times U\times CDD\times(1-0.6\ AH\times\frac{( \text{X}_{opt}-\text{X}_{\text{ins}})}{\text{X}_{opt}}) \tag{22}\]  
\[\text{Q}_{\text{eff(year,h)}}=86,400\times U\times HDD\times(1-0.6\ AH\times \frac{(\text{X}_{opt}-\text{X}_{\text{ins}})}{\text{X}_{opt}}) \tag{23}\]  
where \(AH\) is the average humidity, its values are obtained from Table 5. Although based on sensible heat transfer, the adjusted equations for cooling and heating (Equations (22) and (23)) have been adjusted to account for latent heat effects. This was achieved by integrating the humidity factor through a performance reduction term based on deviation from the optimum insulation thickness (\(X_{opt}\)). Humidity can influence the thermal conductivity of insulation materials, particularly in regions with high moisture levels, such as coastal or humid climates. This adjustment aligns with previous findings in building energy studies that emphasize the influence of moisture on insulation properties, particularly for fibrous materials such as rock wool and polystyrene-based products. Studies such as [PERSON] (2012) [3] and [PERSON] et al. (2022) [12] highlight that thermal conductivity increases with moisture content, leading to higher heat transfer through the building envelope. The average annual humidity (AH) was introduced as a correction factor to represent moisture-dependent variations in insulation performance. The term \(\frac{(\text{X}_{opt}-\text{X}_{\text{ins}})}{\text{X}_{opt}}\) reflects the relative deviation from the optimum insulation thickness, and it is multiplied by the average humidity to model its impact on the insulation's thermal efficiency. This approach assumes that insulation performance degrades proportionally with both increasing humidity and under-insulation. The correction coefficient (0.6) is an empirically derived factor representing the approximate sensitivity of thermal conductivity to average humidity levels for typical insulation materials used in the region. The adjusted heat gain/loss equations for cooling and heating, designated as Q\({}_{\text{eff (year,c)}}\) and Q\({}_{\text{eff (year,h)}}\), respectively, explicitly reflect the combined effect of temperature and humidity.  
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline  & **Jeddah** **(\%)** & **Riyadh** **(\%)** & **Dhahran** **(\%)** & **Guraiat** **(\%)** & **Khamis M.** **(\%)** \\ \hline January & 66 & 45 & 68 & 65 & 67 \\ \hline February & 67 & 34 & 63 & 56 & 60 \\ \hline March & 61 & 26 & 48 & 43 & 55 \\ \hline April & 57 & 24 & 45 & 37 & 53 \\ \hline May & 53 & 17 & 33 & 31 & 45 \\ \hline June & 55 & 21 & 29 & 30 & 36 \\ \hline July & 50 & 15 &  
Figure 12: Comparison of calculated, simulated, and modeled annual heat gain in Dhahran for different insulation materials: (**a**) molded polystyrene; (**b**) polyurethane; (**c**) rock wool.  
Figure 13: Comparison of calculated, simulated, and modeled annual heat gain in Jeddah for different insulation materials: (**a**) molded polystyrene; (**b**) polyurethane; (**c**) rock wool.