## 4 Results and Discussion
### Heat Transfer Analysis  
In case of a fire, it is necessary to limit the outer surface temperature of the refuse chute to a certain level. In this way, the impact of a fire in the refuse chute will be contained inside, without affecting the surrounding environment. In this design, the outer surface temperature was limited to 65 \({}^{\circ}\)C for potential human contact during a fire. To achieve this, insulation was applied around the refuse chute using rock wool with a service temperature of 650 \({}^{\circ}\)C, according to the data sheet documents. To ensure that the inner surface temperature of the waste chute is 650 \({}^{\circ}\)C and the outer surface temperature is 65 \({}^{\circ}\)C during a fire, the required thickness of rock wool was calculated using [PERSON]'s law of heat conduction. The insulation is shown in Figure 13.  
The calculation of the required insulation thickness is based on [PERSON]'s law of heat conduction, which states that the rate of heat transfer through a material is proportional to the negative slope of the temperature and the area over which the heat is transferred. Mathematically, [PERSON]'s law is basically obtained as shown in Equation (4). \[Q=-k\frac{dT}{dX} \tag{4}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Area of Use** & **Evaluation** & **Waste Volume per Week** \\ \hline House & Per house/floor & 0.3 m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Waste volume per week (\(V_{w}\)) [20; 42]. Here, \(Q\) represents the heat transfer rate, \(k\) represents the thermal conductivity of the material, and \(\frac{dT}{dX}\) represents the temperature gradient. The heat conduction Equations (5)-(9), based on Equation (4), considering the thickness of the inner stainless-steel sheet as \(\mathrm{d_{si}}\), the thermal conductivity of the inner stainless-steel sheet as \(\mathrm{d_{r}}\), the thickness of the rock wool as \(\mathrm{d_{r}}\), the thermal conductivity of the rock wool as \(\mathrm{d_{r}}\), the thickness of the outer stainless-steel sheet as \(\mathrm{d_{so}}\), the thermal conductivity of the outer stainless-steel sheet as \(\mathrm{\lambda_{o}}\), and the surface temperatures as \(T_{4}\), \(T_{3}\), \(T_{2}\), and \(T_{1}\), are expressed as follows [48]. \[q_{1}=k_{1}\cdot\Delta T_{1},\ \Delta T_{1}=T_{4}-T_{1} \tag{5}\]  
\[\frac{1}{k_{1}}=\frac{1}{\lambda_{i}}+\frac{1}{\lambda_{r}}+\frac{1}{\lambda_{ o}} \tag{6}\]  
\[q_{1}=q_{2} \tag{7}\]  
\[q_{2}=k_{2}\cdot\Delta T_{2},\Delta T_{2}=T_{2}-T_{1} \tag{8}\]  
\[\frac{1}{k_{2}}=\frac{1}{\lambda_{o}} \tag{9}\]  
The thickness of the inner stainless steel sheet (\(\mathrm{d_{si}}\)) was 1.5 mm, and the thermal conductivity of the inner stainless steel sheet (\(\mathrm{\lambda_{i}}\)) was 17 W/m\(\cdot\)K [49]. The thermal conductivity of the rock wool (\(\mathrm{\lambda_{r}}\)) was 0.034 W/m\(\cdot\)K [50]. The thickness of the outer stainless steel sheet (\(\mathrm{d_{so}}\)) was 0.5 mm, and the thermal conductivity of the outer stainless steel sheet (\(\mathrm{\lambda_{o}}\)) was 17 W/m\(\cdot\)K [49]. With surface temperatures \(T_{4}=650\ ^{\circ}\)C, \(T_{2}=70\ ^{\circ}\)C, and \(T_{1}=65\ ^{\circ}\)C, the thickness of the rock wool (\(\mathrm{d_{r}}\)) was determined as 79.27 mm \(\equiv 80\) mm. Thus, a design with an insulation thickness of 80 mm was implemented for fire safety. A study report conducted in Singapore [51] indicated that 39.8% of residential fires originated from refuse chutes. This significantly high rate highlights the importance of maintaining and cleaning refuse chutes. During a fire, it is essential to provide clear alerts to residents in the building. To fully extinguish the fire and prevent any new flare-ups, all floors must be checked repeatedly to ensure the flame is completely out. As seen, even when extinguished, the psychological impacts of the fire cannot be quickly mitigated. Therefore, preventive measures must be taken before a fire occurs rather than after it ignites.