# Beyond Static Estimates: Dynamic Simulation of Fire-Evacuation Interaction in Historical Districts
## 2 Methods
### Fire Simulation and Parameters
#### 2.3.1 Fire Development Model  
Fire development can be divided into four stages: ignition to flashover, flashover to full development, full development to collapse, and collapse to extinction [45]. The corresponding temperature and heat release characteristics are displayed in Figure 3a. The maximum combustion temperature, T\({}_{\text{max}}\), is set at 800 \({}^{\circ}\)C, while the peak heat release rate, Q\({}_{\text{max}}\), is set at 40 MW [46]. The parameter \(\alpha\) represents the temperature rise coefficient, \(\alpha^{\prime}\) is the HRR (Heat Release Rate), increase coefficient, \(\beta\) is the cooling coefficient, and \(\beta^{\prime}\) is the HRR decrease coefficient. Following ignition, the fire can spread to adjacent buildings through direct flame contact, thermal radiation, and heat plumes. An adjacent building is considered ignited when the net heat flux \(q_{w}\) exceeds 18 kW/m\({}^{2}\) and can be calculated as follows [46]:  
\[\left\{ \begin{aligned} q_{w}=q_{w,ij}-q_{w,ii}\\ q_{w,ij}=\varepsilon_{w}\big{[}(1-\sum\
olimits\theta_{R})\sigma T_{i }^{4}+\sum\
olimits\varphi_{R}q_{R}\big{]}+h_{w}\\ q_{w,ji}=\varepsilon_{w}\,\sigma T_{w}^{4}+h_{w}\,T_{w}\end{aligned} \right. \tag{1}\]  
Figure 3: Fire spread diagram: (**a**) Simplified curves of building temperature (left) and HRR (right), (**b**) Schematic diagram of heat conduction, (**c**) diagram of combined plumes. where \(i\) represents the exterior; \(j\) represents the interior; \(q_{w}\) is the heat flux at the wall surface; \(q_{w,ij}\) is the heat flux from exterior to interior; and \(q_{w,ji}\) represents the heat flux from interior to exterior. \(\varepsilon_{w}\) is the emissivity of the wall, typically set to 0.9 for timber walls; \(\varphi_{R}\) represents the configuration factor of external heat sources; \(q_{R}\) is the heat radiation power; and \(h_{w}\) is the convective heat transfer coefficient of the wall, calculated using the Dittus-Boelter correlation; \(T_{w}\) represents the wall surface temperature; \(T_{i}\) is the exterior temperature, determined using the plume model; and \(\sigma\) is the Stefan-Boltzmann constant. Thermal radiation \(q_{R}\) is calculated by relating the radiative heat flux to the diffusion of hot gases and flames through openings as follows [46]:  
\[ \begin{cases}q_{R}=\frac{1}{\varphi}\frac{q_{D}A_{D}}{A_{D}+A_{W}}=\frac{k}{ \varphi}q_{D}\\ q_{D}=\sigma T^{4}\end{cases}  \tag{2}\]  
where \(\varphi\) is the attenuation factor (typically set to 0.8), \(k\) represents the opening ratio of the wall; \(T\) is the average indoor temperature of the burning building; \(A_{D}\) is the area of the window, \(A_{w}\) is the area of the exterior wall (Figure 3b). The calculation of heat plumes follows the model proposed by [PERSON] and [PERSON] and is illustrated in Figure 3c. The mathematical models from [PERSON] and [PERSON] [46] are directly implemented in this study without modification, as these models calculate heat transfer in fire propagation, which we apply in NetLogo for real-time heat calculation of each patch to determine ignition status based on the calculated heat values. The temperature variation along the centerline of the heat plume is described by [46]  
\[\Delta T_{0}= \begin{cases}900\ \left(z/Q_{C}^{2/5}<0.08\right)\\ 60\left(z/Q_{C}^{2/5}\right)^{-1}\left(0.08\leq\ z/Q_{C}^{2/5}<0.2\right)\\ 24\left(z/Q_{C}^{2/5}\right)^{-5/3}\left(0.2\leq z/Q_{C}^{2/5}\right)\end{cases}  \tag{3}\]  
where \(z\) is the distance from a point on the heat plume to the building, and \(Q_{C}\) is the HRR of the burning building. The temperature rise \(\Delta T(r)\) for the surrounding environment is calculated as  
\[ \begin{cases}\Delta T(r)/\Delta T_{0}=exp\left[-\beta(r/l_{1})^{2}\right]\\ tan\theta=0.1\left[U_{\infty}/\left(Q^{\prime}g/\rho_{\infty}C_{p}T_{\infty} \right)^{1/3}\right]^{-3/4}\\ Q^{\prime}=Q_{c}/\sqrt{A_{Bfloor}}\end{cases}