## 3 Results and Discussion
### Cost Saving Rates with Payback Period  
The total lifetime cost of walls with and without insulation across different orientations are illustrated in Figure 10. The results clearly show that insulating the walls significantly \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{_Study_} & \multirow{2}{*}{_City_} & \multicolumn{5}{c}{**Insulation Thickness (cm)**} \\  & & N & E & S & W \\ \hline Present Study & Bursa & 5.02 & 4.88 & 4.52 & 4.88 \\ [PERSON] et al. (2016) & Istanbul & 5.15 & 4.79 & 4.33 & 4.77 \\ \hline \hline \multicolumn{5}{c}{**Heating Degree Days (HDDs)**} \\  & & N & E & S & W \\ \hline Present Study & Bursa & 1466 & 1409 & 1327 & 1409 \\ [PERSON] et al. (2016) & Istanbul & 1719 & 1633 & 1535 & 1628 \\ \hline \multicolumn{5}{c}{**Incoming Solar Radiation (W/m2)**} \\  & & N & E & S & W \\ \hline Present Study & Bursa & 52 & 97 & 112 & 97 \\ [PERSON] et al. (2016) & Istanbul & 50 & 90 & 109 & 94 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparative analysis with another study. \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{2}{c}{**Only Coiled, \(\rm x_{opt,C}\) (cm)**} \\ \hline N & Cost = 7.828451 \(-\) 99.71533* \(+\) 3333.880*\(\times\)2 \(-\) 23673.97*\(\times\)3 \(+\) 61747.98*\(\times\)4 \\ NE & Cost = 7.877767 \(-\) 101.0994* \(+\) 3354.890*\(\times\)2 \(-\) 23823.10*\(\times\)3 \(+\) 62136.96*\(\times\)4 \\ E & Cost = 8.430012 \(-\) 116.5989* \(+\) 3590.074*\(\times\)2 \(-\) 25493.15*\(\times\)3 \(+\) 66492.88*\(\times\)4 \\ SE & Cost = 9.345131 \(-\) 142.2829*\(\times\) 3979.794*\(\times\)2 \(-\) 28260.55*\(\times\)3 \(+\) 7371.100*\(\times\)4 \\ S & Cost = 9.948581 \(-\) 159.2194*\(\times\) + 4236.784*\(\times\)2 \(-\) 30085.44*\(\times\)3 \(+\) 78470.79*\(\times\)4 \\ SW & Cost = 9.345131 \(-\) 142.2829*\(\times\) 3979.794*\(\times\)2 \(-\) 28260.55*\(\times\)3 \(+\) 73711.00*\(\times\)4 \\ W & Cost = 8.430012 \(-\) 116.5989*\(\times\)3 \(+\) 3590.074*\(\times\)2 \(-\) 25493.15*\(\times\)3 \(+\) 66492.88*\(\times\)4 \\ NW & Cost = 8.746919 \(-\) 125.4933*\(\times\) 3725.034*\(\times\)2 \(-\) 26451.50*\(\times\)3 \(+\) 68992.52*\(\times\)4 \\ \hline \hline \multicolumn{2}{c}{**Both Heated and Cooled, \(\rm x_{opt,A}\) (cm)**} \\ \hline N & Cost = 34.26842 \(-\) 841.7864*\(\times\) 14593.83*\(\times\)2 \(-\) 103630.9*\(\times\)3 \(+\) 270296.9*\(\times\)4 \\ NE & Cost = 34.23196 \(-\) 840.7630*\(\times\) 14578.30*\(\times\)2 \(-\) 103520.6*\(\times\)3 \(+\) 270009.3*\(\times\)4 \\ E & Cost = 33.84729 \(-\) 829.9668*\(\times\) 14414.48*\(\times\)2 \(-\) 102357.4*\(\times\)3 \(+\) 266975.1*\(\times\)4 \\ SE & Cost = 33.30516 \(-\) 814.7511*\(\times\) 14183.61*\(\times\)2 \(-\) 100717.9*\(\times\)3 \(+\) 262699.0*\(\times\)4 \\ S & Cost = 33.02290 \(-\) 806.8294*\(\times\) 14063.40*\(\times\)2 \(-\) 99864.35*\(\times\)3 \(+\) 260472.7*\(\times\)4 \\ SW & Cost = 33.30516 \(-\) 814.7511*\(\times\) 14183.61*\(\times\)2 \(-\) 100717.9*\(\times\)3 \(+\) 262699.0*\(\times\)4 \\ W & Cost = 33.84729 \(-\) 829.9668*\(\times\) 14414.48*\(\times\)2 \(-\) 102357.4*\(\times\)3 \(+\) 266975.1*\(\times\)4 \\ NW & Cost = 33.32710 \(-\) 815.3670*\(\times\) 14192.95*\(\times\)2 \(-\) 100784.3*\(\times\)3 \(+\) 262872.1*\(\times\)4 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: _Cont._reduces the total cost over the building's lifetime, regardless of directions. The north-facing wall, which experiences the lowest amount of solar radiation and consequently the highest heating demand, achieves the most substantial cost reduction of 38.9%. Conversely, the south-facing wall, which receives the highest solar radiation and benefits more from natural heating, shows the lowest cost reduction of 37.0%. Overall, the average cost reduction across all directions is approximately 38.0%, demonstrating the financial benefits of applying optimum insulation. Figure 10 highlights that while there are slight variations in cost savings depending on the wall orientation, the implementation of insulation leads to considerable long-term economic advantages for all orientations.