## 2 Mathematical Model
### Economic Assessment and \(Co_{2}\) Emission Calculations  
When calculating the total cost of such an investment, the heating-related fuel cost during the winter months, the cooling-related electricity cost during the summer months, and the insulation cost, depending on the applied thickness, are taken into account. Additionally, to obtain more realistic results, the cost of wall material and plaster, aside from the insulation material, are also considered (see Equation (14)). The life-cycle cost (LCC) analysis method, which is frequently used in the literature, has been employed in economic analysis. In this method, a term defined as present worth factor (\(PVF\)) is used to discount future annual energy costs to present value. The \(PVF\) is calculated based on the lifespan of the investment (\(N\)), the inflation rate (\(g\)), and the interest rate (\(i\)) as shown in Equation (11) [33]. \[PVF=\frac{\left(\frac{1+i}{1+\delta}\right)^{N}-1}{\left(\frac{i-g}{1+\delta} \right)\left(\frac{1+i}{1+\delta}\right)^{N}} \tag{11}\]  
In this study, the inflation rate and interest rate are chosen as 17% and 20%, respectively. These financial parameters are selected based on published data from the Turkish Statistical Institute and the Central Bank of Turkiye [42; 43]. The expense associated with applying insulation can be determined by the product of the cost of the insulation material (\(C_{ins}\)) and the thickness of the insulation (\(x\)). Similarly, the total cost of plaster is calculated by multiplying its unit price (\(C_{plast}\)) by its thickness, and the total cost of the wall material is determined by multiplying its unit price (\(C_{null}\)) by  
Figure 1: Wall Components of the Base Case Scenario. its thickness. The installation costs, denoted as \(C_{inst}\), are considered to be 9 USD/m\({}^{2}\). In addition to these items, the total cost also includes the energy expenses for heating (\(C_{Hent}\)) and cooling (\(C_{Coal}\)), as shown in Equations (12) and (13) [33]. The annual total cost (\(C_{T,A}\)) can be determined using Equation (14). \[C_{Hent}=\frac{86,400 HDDC_{f}PWF}{\left(\frac{1}{h_{i}}+\frac{1}{h_{o}}+\frac{x_ {inst}}{k_{plast}}+\frac{x_{null}}{k_{null}}+\frac{x_{ins}}{k_{ins}}\right)H_{u}\eta} \tag{12}\]  
\[C_{Coal}=\frac{86,400 CDD_{e}PWF}{\left(\frac{1}{h_{i}}+\frac{1}{h_{o}}+\frac{x_ {inst}}{k_{plast}}+\frac{x_{null}}{k_{null}}+\frac{x_{ins}}{k_{ins}}\right)COP} \tag{13}\]  
\[C_{T}=C_{ins}x_{ins}+C_{plast}x_{plast}+C_{null}x_{null}+C_{inst}+C_{Heat}+C_{ Cool} \tag{14}\]  
Within these equations, \(C_{f}\) indicates the cost of fuel, while \(C_{e}\) denotes the cost of electricity. The lower heating value of the heating source is represented by \(H_{u}\), and the thermal conductivity of all wall components is symbolized by \(k\). Tables 1 and 2 present the relevant values  
Another purpose of applying insulation in this study is to reduce carbon dioxide emissions released into the atmosphere. In this context, emissions are divided into two categories: operational and embodied emissions. Operational emissions refer to the CO\({}_{2}\) released into the environment from the fuel consumed during heating and cooling. Embodied emissions are related to the CO\({}_{2}\) released during processes such as production, transportation, and installation of wall materials. This study evaluates the environmental impact of insulation applications by analyzing both operational and embodied emissions. This approach ensures that the results are more comprehensive and realistic. From Equations (15)-(19), allow for the calculation of annual CO\({}_{2}\) emissions for heating (\(M_{\text{CO}_{2},heat}\)), cooling (\(M_{\text{CO}_{2},cool}\)), insulation materials (\(M_{\text{CO}_{2},ins}\)), wall materials (\(M_{\text{CO}_{2},null}\)), and plaster (\(M_{\text{CO}_{2},plast}\)). Each emission source can be individually determined using these equations. Total amount of CO\({}_{2}\) released to the atmosphere from the building used throughout the year is presented in Equation (20).