## 2 Numerical Model
### Validation of Methodology
The maximum temperature of the gas depends on its maximum value on the roof (\(\Delta T_{max}\)), which in turn was calculated by defining two different areas, depending on the dimensionless ventilation speed \(V^{\prime}\):  
\[\Delta T_{max} =\left\{ \begin{matrix}17.5\frac{\dot{Q}^{2/3}}{h_{0}^{3/3}},&V \leq 0.19\\ \frac{\dot{Q}}{u_{0}^{1/3}h_{0}^{3/3}},&V^{\prime}>0.19\end{matrix} \right. \tag{11}\]  
where \(\dot{Q}\) is the total heat release rate (kW), \(H_{ef}\) is the effective height of the tunnel (the height from the fire source to the ceiling) (m), \(u_{0}\) is the ventilation velocity, \(b_{f0}\) is the radius of the fire source (m), and \(V^{\prime}\) is defined as:  
\[V^{\prime}=\frac{u_{0}}{w^{*}} \tag{12}\]  
where \(w^{*}\) is the characteristic velocity of the plume and is expressed by the following equation:  
\[w^{*}=\left(\frac{g\dot{Q}}{b_{f0}\rho_{0}C_{p}T_{0}}\right)^{1/3} \tag{13}\]  
where \(g\) is the gravitational acceleration, (m/s), \(b_{f0}\) is the radius of the fire source (m), \(\rho_{0}\) is the density of air (kg/m\({}^{3}\)), \(C_{p}\) is the specific heat capacity (kJ/kgK), and \(T_{0}\) is the ambient temperature (K). The values calculated with the empirical model were compared to the numerical results for Scenario C, measured with thermocouples at the tunnel ceiling (z = 7 m) above and downwind the fire source (where x = 0 is directly above the fire source). A large difference in temperature values is shown in Figure 4. The figure shows the good agreement of the curves at 0 m during the initial stages of the fire. From this point on, the temperature decreases and follows a scattering trend. There is a consistent difference of around 400 \({}^{\circ}\)C between the empirical and the actual temperature values for every location, except for above the fire pool. This difference leads to the assumption that, since the flame length increases up to the point that it is equal to and greater than the effective height of the tunnel and starts crawling horizontally to the ceiling, the smoke layer surrounds the fire source. The results indicate that there was not enough oxygen for the fire to grow further, so the fire became ventilation controlled. Other possible reasons for these differences might be that there are limitations to the validity of parameters such as the geometry of the tunnel and the fire area (the FDS fire source has different dimensions than those used in the experiments). Additionally, for the validation of temperature, the results were compared with the hydrocarbon (HC) fire curve and modified hydrocarbon fire curve (HCM) used for fire resistance testing in structural designs and the RABT-ZTV train fire curve. The HC fire curve is used when there is a risk of liquid fires, e.g., in industrial plants (chemicals), on ships, pontoons, oil platforms, road, and railway tunnels, as well as for petrol and diesel pool fires. The HCM fire curve is derived from the HC curve and adjusted for French regulations. Both HC and HCM curves showed an abrupt temperature gradient in the first minutes, and their difference is the maximum temperature of 1100 and 1300 \({}^{\circ}\)C, respectively. The guideline for the equipment and operation of road tunnels (Richtlinie  
Figure 4: Temperatures at the tunnel ceiling (z = 7 m) above (x = 0 m) and downwind the fire source (x = 10, 30, 50 and 100 m) for Scenario C, calculated using the empirical model and the actual temperatures from the FDS output. fur die Ausstattung und den Betrieb von Strasentunnenl) (RABT) defines the so-called RABT tunnel fire. This is often used for the fire protection design of tunnel structures. The RABT tunnel fire curve is an artificial fire curve and is intended to cover the maximum temperatures resulting from tunnel fires [19]. The curve can be used in tunnel design and testing. The HC and HCM fire curve equations are shown below in Equation (14) and Equation (15), respectively.