# Effects of the Ventilation System by Using Jet Fans during a Fire in Road Tunnels
## 2 Effects of the Ventilation System in a One-Way Road Tunnel during a Fire Using Analytical Methods
The pressure loss due to the friction of the air along the tunnel walls is given by the following equation:  
\[v_{c}=K_{Fr}\cdot K_{g}\cdot\sqrt[3]{\frac{g\cdot H\cdot HRR}{\rho\cdot C_{p} \cdot A\cdot T_{f}}} \tag{4}\]4  
where:  
\(\lambda\) is the friction coefficient of the tunnel wall, also known as the Moody coefficient (set equal to 0.015-0.020 for concrete walls);  
\(D_{hyd}\) is the hydraulic diameter of the tunnel cross-section, equal to 4 times the ratio between the area and the perimeter of the cross-section;  
\(L_{tot}\) is the total length of the tunnel;  
\(L_{fire}\) is the distance of the fire from the tunnel entrance. Finally, a fire induces a concentrated pressure loss \(\Delta p_{fire}\), equal to about 60-100 Pa for medium-high thermal power fires, due to the air and fumes turbulence and accelerations. It is possible to use the following equation to determine an estimate of \(\Delta p_{fire}\) (in Pa) as a function of the fire heat output \(HRR\) (in Watts):  
\[v_{c}=K_{Fr}\cdot K_{g}\cdot\sqrt[3]{\frac{g\cdot H\cdot HRR}{\rho\cdot C_{p} \cdot A\cdot T_{f}}} \tag{4}\]5  
The air velocity \(v\) in the upstream section of the fire is then obtained by equating the sum of the pressure increases due to the jet fans and the pressure losses detected along the tunnel during a fire (the pressure loss due to the back-layering phenomenon is neglected due to the fact that it is generally lower than the other terms):  
\[v_{c}=K_{Fr}\cdot K_{g}\cdot\sqrt[3]{\frac{g\cdot H\cdot HRR}{\rho\cdot C_{p} \cdot A\cdot T_{f}}} \tag{4}\]6  
Considering as a precaution that the fans downstream of a medium-high thermal power fire can go out of service due to the high air temperatures and that the thermal power lost by convection and radiation on the walls of the tunnel is negligible due to the sudden increase in the temperature of the same walls, the air velocity v can be obtained from the following equation (Cardano solution):  
\[v_{c}=K_{Fr}\cdot K_{g}\cdot\sqrt[3]{\frac{g\cdot H\cdot HRR}{\rho\cdot C_{p} \cdot A\cdot T_{f}}} \tag{4}\]7  
where:  
\[v_{c}=K_{Fr}\cdot K_{g}\cdot\sqrt[3]{\frac{g\cdot H\cdot HRR}{\rho\cdot C_{p} \cdot A\cdot T_{f}}} \tag{4}\]8  
\[v_{c}=K_{Fr}\cdot K_{g}\cdot\sqrt[3]{\frac{g\cdot H\cdot HRR}{\rho\cdot C_{p} \cdot A\cdot T_{f}}} \tag{4}\]9  
\(m\) is the total number of jet fans active in the upstream zone of the tunnel;  
All quantities are expressed in the international system: m, kg, N, s, J, K, W.  
To analyze the evolution of the back-layering length over time from the start of the fire (\(t=0\)) it is necessary to adopt an iterative process (Figure 4), increasing the time t for predefined intervals and evaluating the main characteristics of the fire. The iterative process requires an initial air velocity \(v\) at time \(t=0\) (in the absence of fire), which can be obtained from the following simple equation:  
\[T_{f}=\frac{HRR}{\rho\cdot C_{p}\cdot A\cdot v}+T \tag{5}\]0The final result of the procedure is represented by the evolution over time of: the air temperature downstream of the fire, the upstream air velocity, and the back-layering length of the fumes towards the upstream entrance. If a fire occurs at a point where no active jet fans upstream of it are conceivable, it is appropriate to assume the absence of mechanical ventilation. In this case, the air velocity upstream of the fire must be cautiously set equal to 0. The back-layering distance can, however, be calculated using Equations (2) and (3) for the determination of the smoke flow rates produced by the fire, assuming that the fumes are divided equally in the two directions starting from the point of formation of the fire:  
\[T_{f}=\frac{HRR}{\rho\cdot C_{p}\cdot A\cdot v}+T \tag{5}\]1  
\[T_{f}=\frac{HRR}{\rho\cdot C_{p}\cdot A\cdot v}+T \tag{5}\]2  
where we used m for \(L_{bl}\), m\({}^{2}\) for \(A\), MW for \(HRR\), and min for time \(t\). In these cases, the length \(L_{bl}\) can rapidly increase over time, causing serious risks for people at a short distance from the fire.