## 1 Introduction
### Energy Modeling  
Energy modeling was conducted on a simple structure to verify the results in Section 2.3 and potentially substantiate the equation in the previous section. A three-meter by three-meter cube structure was modeled with a wall section typical of Figure 4. The model was intentionally simplified to enable a focused analysis of wall performance while minimizing other influences. The structure was modeled as a stand-alone structure with all surfaces exposed to the outside. Furthermore, a simple HVAC system was defined with a COP of three which was set at a base temperature of 18.3 \({}^{\circ}\)C to align the simulation with the CDD/HDD calculation and the base temperature was kept the same as defined for the calculation. Additional building energy parameters were incorporated as defined in previous research for the location [9], including parameters for thermal zoning, occupancy schedules, internal loads, and Domestic Hot Water (DHW) consumption.  
The energy simulations were carried out using DesignBuilder (Version 5.5.2), a widely used building performance simulation tool that operates on the EnergyPlus engine. EnergyPlus is a validated and commonly accepted simulation engine known for its accuracy in dynamic thermal modeling. Simulations were conducted for five distinct climatic regions in Saudi Arabia (Figure 3), using corresponding weather files provided by EnergyPlus.  
Firstly, the model was calibrated by comparing its simulated EUI with values reported in established energy studies for Dhahran City. The baseline simulation was conducted using EnergyPlus, incorporating local climate data, building characteristics, and operational parameters. The resulting EUI was then benchmarked against peer-reviewed studies to assess the model's accuracy [6; 9]. The simulated EUI of 141.4 kWh/m\({}^{2}\)/year exhibited a deviation of less than 5% from reported values, which falls within acceptable calibration tolerances. To ensure alignment with empirical data, the model was iteratively refined by adjusting key parameters, including insulation properties, envelope characteristics, occupancy schedules, and internal heat gains, until the deviation fell within the acceptable threshold. The model was most responsive to variations in insulation thickness, highlighting its significant impact on energy performance.  
Next, 15 simulations are performed for each location, with five for each insulation type. Simulations were run with the wall section varying based on the insulation type and  
Figure 4: External wall structure [39].  
thickness (Tables 3 and 4). The insulation thickness varied, with two points above the \(X_{opt}\) and two below. To ensure that the simulation results provide only the energy consumption due to solar heat gain through external walls for validation and comparison with the equation results, all other energy-consuming parameters were eliminated, including the occupancy, lighting system, etc. In addition, the heat gain by the roof was also eliminated by setting the insulation R-value of the roof to the maximum. Then, the total net energy was divided by the total surface area of the walls to obtain conduction gain per square meter of wall area. The chosen approach ensures that the impact of only wall insulation materials can be thoroughly analyzed without additional variables influencing the results. This simplification of the energy model is necessary to validate and modify the calculation. However, we acknowledge that in a whole-building energy assessment, accounting for all heat loss mechanisms would be necessary.