## 3 Results and Discussion
### Optimum Insulation Thicknesses for Different Directions and Seasonal Usages
Conversely, in scenarios where a building is exclusively heated or cooled, the impact of wall direction on the required insulation thickness becomes more pronounced. Calculations reveal a notable 0.50 cm disparity in insulation thicknesses for heating purposes between the wall demanding the thickest insulation (5.02 cm) and the one necessitating the thinnest (4.52 cm). Moreover, calculations show a noticeable difference of 0.53 cm in how thick the insulation needs to be for cooling. This distinction exists between the wall needing the thickest insulation, at 2.09 cm, and the one needing a thinner layer, at 1.56 cm. In addition, correlations providing the total cost based on the applied insulation thickness for each direction and various building usage scenarios have been determined and presented in Table 3.  
Figure 8: Typical variation of costs with changing insulation thicknesses (for the north-facing wall). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & N & NE & E & SE & S & SW & W & NW \\ \hline Only Heated, x\({}_{\text{opt,H}}\) (cm) & 5.02 & 5.01 & 4.88 & 4.66 & 4.52 & 4.66 & 4.88 & 4.75 \\ Only Cooled, x\({}_{\text{opt,C}}\) (cm) & 1.56 & 1.58 & 1.72 & 1.95 & 2.09 & 1.95 & 1.72 & 1.80 \\ Both Heated and Cooled, x\({}_{\text{opt,A}}\) (cm) & 6.08 & 6.07 & 6.02 & 5.95 & 5.92 & 5.95 & 6.02 & 5.96 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Optimum insulation thicknesses for different directions and seasonal usages. \begin{table}
\begin{tabular}{c c} \hline \hline
**Only Heated, x\({}_{\text{opt,H}}\) (cm)** \\ \hline N & Cost = \(26.43997-622.0711^{\text{x}}\) + \(11259.94\times^{2}\) - \(79956.95^{\text{x}}\)\({}^{3}\) + \(208548.9\times^{4}\) \\ NE & Cost = \(26.35419-619.6636^{\text{x}}\)\({}^{1}\) + \(11223.41^{\text{x}}\)\({}^{2}\) - \(79697.54\times^{3}\) + \(207872.3^{\text{x}}\)\({}^{4}\) \\ E & Cost = \(25.41728-593.3679^{\text{x}}\)\({}^{1}\) + \(10824.41^{\text{x}}\)\({}^{2}\) - \(76864.23^{\text{x}}\)\({}^{3}\) + \(200482.3^{\text{x}}\)\({}^{4}\) \\ SE & Cost = \(23.96003-552.4683^{\text{x}}\)\({}^{1}\) + \(10203.81^{\text{x}}\)\({}^{2}\) - \(72457.36^{\text{x}}\)\({}^{3}\) + \(188988.0^{\text{x}}\)\({}^{4}\) \\ S & Cost = \(23.07432-527.6099^{\text{x}}\)\({}^{1}\) + \(9826.619^{\text{x}}\)\({}^{2}\) - \(69778.91^{\text{x}}\)\({}^{3}\)\({}^{1}\) + \(182001.9^{\text{x}}\)\({}^{4}\) \\ SW & Cost = \(23.96003-552.4683^{\text{x}}\)\({}^{1}\) + \(10203.81^{\text{x}}\)\({}^{2}\) - \(72457.36^{\text{x}}\)\({}^{3}\)\({}^{1}\) + \(188988.0^{\text{x}}\)\({}^{4}\) \\ W & Cost = \(25.41728-593.3679^{\text{x}}\)\({}^{1}\) + \(10824.41^{\text{x}}\)\({}^{2}\) - \(76864.23^{\text{x}}\)\({}^{3}\)\({}^{2}\) + \(200482.3^{\text{x}}\)\({}^{4}\) \\ NW & Cost = \(24.58018-569.8737^{\text{x}}\)\({}^{1}\) + \(10467.92^{\text{x}}\)\({}^{2}\) - \(74332.77^{\text{x}}\)\({}^{3}\) + \(193879.6^{\text{x}}\)\({}^{4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Correlations for different directions and seasonal usages. Figure 9: Heating (**a**), cooling (**b**), and annual (**c**) optimum insulation thickness variations for various directions.