## 2 Methods
### Dynamic Fire Thermal Damage Assessment Model  
High-temperature flames continuously emit thermal radiation to their surroundings. The radiative heat flux, defined as the amount of thermal energy transferred per unit area per unit time, is typically expressed in watts per square meter (W/m\({}^{2}\)). The heat flux \(q\)(x,t) received by a target at a specific position and time is a function of the fireball's surface radiative energy \(E(t)\), the geometric view factor between the fireball and the target \(F(x,t)\), and the atmospheric transmittance \(\tau_{a}(x,t)\). Based on these parameters, a dynamic radiative heat transfer model of the fire can be formulated as:  
\[q(x,t)=E(t)F(x,t)\tau_{a}(x,t) \tag{13}\]  
Thermal dose characterizes the cumulative thermal damage resulting from the combined effects of temperature and exposure duration. It reflects the overall impact of thermal energy on biological tissues or material surfaces. To evaluate the destructive effects on exposed targets, a thermal dose criterion is typically employed. This is achieved by integrating incident radiative heat flux \(q\)(x,t) over the duration of fire event \(t\), thereby establishing a time-dependent model of the fire-induced radiative thermal dose \(Q\):  
\[Q=\int_{0}^{t}E(t)F(x,t)\;\tau_{a}(x,t)dt \tag{14}\]This model captures the time-varying characteristics of thermal radiation during a fire event and provides a foundation for assessing dynamic thermal effects on surrounding targets. A multispectral camera responds to the incident radiative energy within its operational spectral range, and this radiative energy is inherently determined by the temperature of the emitting source. By integrating the Planck radiation function over the full spectral range, the total radiative energy emitted by a flame at a given temperature can be calculated [23]. \[E=\int_{0}^{\infty}\varepsilon C_{1}\lambda^{-5}\big{(}e^{C_{2}/(\lambda T)}-1 \big{)}^{-1}d\lambda=\varepsilon\sigma T^{4} \tag{15}\]  
where \(E\) is the total radiative energy emitted per unit area, \(T\) is the absolute temperature of the source, and \(\sigma=5.67\times 10^{-8}(\mathrm{W\cdot m^{-2}\cdot K^{-4}})\) is the Stefan-Boltzmann constant. The geometric view factor \(F(x,t)\) represents the ratio of radiative energy received per unit area on the target to the total radiative energy emitted by the flame source. It characterizes the spatial relationship between the radiation-emitting fireball and the exposed surface, accounting for both distance and orientation effects. A schematic illustration of the geometric configuration and corresponding view factor during flame radiation is shown in Figure 2.  
The target is at a certain orientation on the ground and the vertical distance from the center of the flame is X m. D is the flame diameter and H is the height of the flame center. The geometric view factor at different distances from the target are:  
\[F(x,t)=\frac{\mathrm{D}(t)^{2}}{4(\mathrm{H(t)}^{2}+\mathrm{X}^{2})} \tag{16}\]  
D(t) is the flame diameter as a function of time, H(t) is the height of the flame center as a function of time, X is the horizontal vertical distance from the target to the fireball (m), and \(t\) is the time (s). Atmospheric transmittance is a physical quantity that can fluctuate significantly. In actual temperature distribution testing, it is essential to accurately measure local air humidity and temperature. By applying a calculation formula that aligns with the test conditions, one can derive the atmospheric transmittance. This paper utilizes the temperature calibration target [24] from previous research to obtain the atmospheric transmittance in the field:  
\[\tau_{a}=\frac{G_{H}-G_{L}}{\alpha(L_{H}-L_{L})} \tag{17}\]  
Figure 2: Geometric view factor of the flame.