# Research on Thermal and Heat Insulation Properties of Aerogel Heat-Insulating Reflective Coatings
## 4 Theoretical Heat Transfer Calculations
### Calculation of Radiant Heat Transfer  
The heat involved in a coating consists of four components [27; 28; 29], the thermal radiation \(q_{rad}\) emitted outward by the coating itself, the heat \(q_{sun}\) radiated by the sun to the specimen, the heat \(q_{air}\) released by the surrounding atmosphere to the surface of the sample, and the heat loss \(q_{loss}\) generated outward by the sample itself. Hence, the total power of the coating to radiate the heat insulation can be expressed as follows:  
\[q_{re}=q_{rad}-q_{sun}-q_{air}-q_{loss} \tag{3}\]  
\[q_{md}=\int\varepsilon(\lambda)I_{b}(\lambda,T_{s})\mathrm{d}\lambda \tag{4}\]  
\[q_{sun}=\int\varepsilon(\lambda)I_{\mathrm{sun}}(\lambda)\mathrm{d}\lambda \tag{5}\]  
\[q_{air}=\int\varepsilon(\lambda)I_{air}(\lambda)\mathrm{d}\lambda \tag{6}\]  
\[q_{loss}=H(T_{air}-T_{s}) \tag{7}\]  
where \(\varepsilon(\lambda)\) is the spectral emissivity of the specimen; \(I_{b}(\lambda,T_{s})\) is the blackbody radiative force at different wavelengths for a blackbody at a temperature of \(T_{s}\); \(I_{\mathrm{sun}}(\lambda)\) is the standard AM1.5 solar spectrum; \(I_{air}(\lambda)\) is the ambient radiative force, calculated from \(I_{air}(\lambda)=\varepsilon_{air}(\lambda)I_{b}(\lambda,T_{air})\); \(\varepsilon_{air}(\lambda)\) is the atmospheric transmittance, \(I_{b}(\lambda,T_{air})\) is the temperature for the \(T_{air}\) equivalent blackbody radiative force; H is the integrated radiative heat  
Figure 6: Enperature field distribution of the specimen during the stable heating stage: (**a**) temperature field distribution diagram under Condition 1; (**b**) temperature field distribution diagram under Condition 2.  
transfer coefficient of the coating surface; \(T_{air}\) and \(T_{s}\) are the ambient temperature around the specimen and the specimen surface temperature, respectively; the related calculation plots are shown in Figure 7.  
Figure 7a shows the absorptivity and emissivity of an ideal radiative cooling material in different wavelength bands. We want the coatings to have lower absorption in the solar radiation band (250-2500 nm) and higher emissivity in the atmospheric window (7-14 um). We tested the absorptivity and emissivity of the aerogel coating and white coating in the corresponding bands, as shown in Figure 7b,c. The test results show that the aerogel coatings have lower absorption than white coatings in 300-1500 nm wavelengths. However, the emissivity of the two coatings is very close in the 7-14 um wavelength. This shows that the aerogel coatings have better radiative cooling ability in the visible and near-infrared wavelengths. The total power, \(q_{sun}\), of solar radiation energy absorbed by aerogel coatings and ordinary facade coatings was calculated to be 102.7 W/m\({}^{2}\) and 206.2 W/m\({}^{2}\). It has been shown [29] that ideal radiatively cooled materials have high emissivity in the wavelength range of \(\geq\)4 um, and the leading radiant energy bands in the atmospheric window are 7-14 um. Therefore, we integrated only 7-14 um in calculating the \(q_{air}\) and figured that the aerogel coatings and ordinary facade coatings receive atmospheric radiant powers of 117.4 W/m\({}^{2}\) and 115.8 W/m\({}^{2}\). The intensities of atmospheric radiation received by the aerogel coating and the regular exterior coating were calculated to be 117.4 W/m\({}^{2}\) and 115.8 W/m\({}^{2}\). Similarly, because the integration band of \(q_{rad}\) is also 7-14 um, the calculation results of the aerogel coating and white exterior wall coating were 366.1 W/m\({}^{2}\) and 370.6 W/m\({}^{2}\), respectively. When the coating surface was at the same temperature as the ambient temperature (\(q_{loss}\) = 0 W/m\({}^{2}\)), the net radiant  
Figure 7: Plot of radiant power calculation. (**a**) Schematic diagram of radiant power calculation [27,30,31]; (**b**) standard solar spectrograms and coatings’ solar spectral absorption diagrams; (**c**) graph of atmospheric transmittance and infrared emissivity of coatings.