# Research on Thermal and Heat Insulation Properties of Aerogel Heat-Insulating Reflective Coatings
## 4 Theoretical Heat Transfer Calculations
### Calculation of Heat Transfer Barrier  
The equivalent thermal resistance value is currently an essential index for evaluating the thermal performance of building materials. According to [PERSON]'s law of heat conduction, the heat transfer power of a one-dimensional single-layer wall can be simplified as Equation (8), and the equivalent thermal resistance value can be used to calculate the heat transfer power within a multi-layer wall for composite walls composed of multi-layer materials. The thermal resistance value of single-layer materials is calculated by Equation (9), and the equivalent thermal value is the sum of the thermal resistance values of each layer. To be able to compare the heat transfer power of aerogel board and XPS board, it was assumed that the temperature difference between the two sides of the wall was 10 \({}^{\circ}\)C (\(T_{1}-T_{2}\) = 10 \({}^{\circ}\)C). The contribution of the coating primer and XPS board binder to the equivalent thermal resistance value was ignored in the calculations; the materials of each layer are tightly bonded to each other, and the temperature of the two fabrics on the bonding surface was the same. The rest of each calculation parameter is shown in the previous Table 1, and the calculation results are shown in Table 3.  
\[q=\frac{T_{1}-T_{2}}{R} \tag{8}\]  
\[R_{i}=\frac{\text{d}_{i}}{\lambda_{i}} \tag{9}\]  
\[R=\sum R_{i} \tag{10}\]  
where \(T_{1}\)  and \(T_{2}\) are the temperatures on both sides of the wall, inside and outside; \(R_{i}\) and \(R\) are the single-layer wall's thermal resistance value and the equivalent thermal resistance value of the multi-layer composite wall; and \(\text{d}_{i}\) and \(\lambda_{i}\)  are the thickness of each layer of the wall and the coefficient of thermal conductivity.  
Calculation results show that the aerogel coating has a significant radiative cooling ability compared with the white wall coating due to the aerogel material's high reflective properties. When the temperature difference between the two surfaces of the specimen is assumed to be fixed, the aerogel coating does not show its high reflective properties, and it relies on its internal porous structure to block heat transfer as traditional thermal insulation materials do. The thermal conductivity of our aerogel coating was higher than that of the XPS insulation board, and the thickness of the coating was so thin that the heat flow density through the aerogel board was 7.87 W/m\({}^{2}\) higher than that of the XPS board. Therefore, the temperature difference between the hot and cold ends of the XPS board should be significantly higher than that of the blank and aerogel boards, which is consistent with the results of our tests.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline  & & \multicolumn{2}{c}{\(R_{i}\) K/W} & & & & \\ \cline{2-6}  & **Plasterboards** & **Aerogel Coating** & **White Coating** & **XPS** & **K/W** & **W/m\({}^{2}\)** \\ \hline Blank board & 0.81 & / & 0.10 & / & 0.91 & 10.98 \\ Aerogel board & 0.81 & 0.24 & / & / & 1.05 & 9.55 \\ XPS board & 0.81 & / & / & 5.14 & 5.94 & 1.68 \\ \hline \end{tabular}
\end{table}
Table 3: Calculation of heat transfer power.