## 3 Numerical Simulations
### Heat Transfer Model  
Radiative heat transfer occurred between the radiant panel and the coating surfaces. The coatings were very thin, and the thickness was low compared to the area of the coatings. Moreover, the radiative effect of the air on the coatings was weak. Therefore, the following assumptions and simplifications concerning the heat transfer process were made:  
1. The film was isotropic and homogeneous, the coating temperature was uniform and consistent before curing, and the film thickness did not change during curing;
2. The density, thermal conductivity, specific heat capacity, and other physical properties of the coating and substrate were constant and did not change with temperature;
3. The thickness of the coating was set to \(e_{p}\), and heat transfer only occurred in the \(Z\) direction;
4. Assuming that the heat transfer process was uniform, the heat transfer model in the substrate was simplified as one-dimensional heat transfer along the \(Z\) direction;
5. The radiative effect of air was not considered. Due to the complexity of radiative heat transfer and the convenience of computational models for establishing stable heat transfer simulations, the model was simplified with the following assumptions:  
1. Infrared rays have a specific penetration capability, and objects can be simultaneously heated within this penetration distance. Thus, the infrared heating rate of the coating was higher than that of hot-air heating. The thickness of the coating (100 \(\upmu\)m) was within this infrared penetration zone. Therefore, it was assumed that the temperature at every point in the coating simultaneously increased. Based on this analysis, the heat transfer process was simplified by equating the infrared radiation heating of the coating with the heating of the coating via an internal heat source;
2. The coating was very thin, so it was assumed that no attenuation of infrared radiation occurred along the thickness direction. Therefore, the intensity of the equivalent internal heat source within the coating was equal across all points in the coating. Figure 16 shows the relevant heat transfer equations for the powder coating and the metal substrate. Due to the low coating thickness, the metal substrate was relatively thin and had good thermal conductivity. The coating was measured multiple times with a thermocouple during the experimental test process. The heat balance of the powder coating and metal substrate system is shown below [13; 29]:  
\[\frac{dT}{dt}\Big{(}C_{p_{p}}\rho_{p}e_{p}+C_{p_{i}}\rho_{s}e_{s}\Big{)}= \overline{Q}_{abs}-\overline{Q}_{emitted}-\overline{Q}_{convection}-\overline {Q}_{reaction} \tag{1}\]  
\[\overline{Q}_{abs}Ap=\eta_{tot}Q_{chem} \tag{2}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material** & **Length (mm)** & **Width (mm)** & **Thickness (mm)** \\ \hline Coating & 150 & 70 & 0.1 \\ Substrate & 150 & 70 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Dimensions of the coating and substrate. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Material** & **Thermal** & **Specific Heat** & & & \\
**Conductivity** & **Capacity** & **Rg/m\({}^{3}\)** & & \\
**W/m\({}^{\circ}\)C** & **J**/kg\({}^{\circ}\)C & & & \\ \hline Coating & 0.1 & 1500 & 1590 & 0.91 \\ Substrate & 44.5 & 460 & 8237 & 0.92 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Thermophysical properties of the coating and substrate. The heat released by the curing chemical reaction was negligible compared to the other heat flux values. Thus, it was ignored in the calculation [13].