## 5 Discussion  
Figure 5 shows the parameters of the fans that simulate the pressure front ahead of the train, as well as the rarefaction front behind the tail car. It is important to note that until the train is fully in the tunnel, the rarefaction front at the back of the tail car is not ventilated. For this reason, the model (Figure 1) in this time interval is free from fan 1 which simulates air rarefaction behind the tail car. In the case under study, such a model is valid for the time span 0 > t > 100.9 s. The draught loss in the tunnel clearance is a variable that is dependent on the length of the train entering the tunnel. In the exit section of the tunnel, when t > 278 s, the model is free from fan 2, which simulates the increased air pressure in front of the head car, and the draught loss in the tunnel clearance is also a variable. Thus, the modeling of air distribution caused by the piston effect in an extra-long railway tunnel has three stages: 1--entry of the train into a tunnel; 2--movement of the whole train within a tunnel; 3--exit of the train from a tunnel. Figure 5 indicates the trends of the pressure on the LF and TB for each of the listed stages. Let the pressure of the fan simulating the front surface of the loco be equal to the average value of the pressure trend in the fitting section. When the train enters a tunnel, P1f = 124 Pa; when the whole train moves within a tunnel, P2f = 60 Pa; when the train leaves a tunnel, P3f = 60 Pa. Hence, the pressure of the fan simulating the tail surface of the moving train inside the tunnel is P2b = 60 Pa, and is P3b = 40 Pa when the train leaves the tunnel. In Figure 6, we see how the diffusion resistance Rd (Figure 1) depends on the distance between the front face of the loco and the tunnel section where the air flow rate is required to be determined. By way of example, we analyze the variation in the air flow rate in section 2-2 when the loco approaches it (Figure 6). When the train is at the distance of 1500 m, the air flow rate in section 2-2 is less than 10 m\({}^{3}\)/s in the 3000 m long tunnel, while the average air flow velocity is under 0.2 m/s. Thus, this flow rate is assumed to be undisturbed. \begin{table}
\begin{tabular}{c c c} \hline
**Time Interval** & **3 km** & **6 km** \\ \hline I & \(Q_{\rm t}=1.05t-15.65\) & \(Q_{\rm t}=0.7t-10.4\) \\ \hline II & \(Q_{\rm t}=-0.0016t^{2}+0.883t+22\) & \(Q_{\rm t}=-0.00039t^{2}+0.36t+30.8\) \\ \hline III & \(Q_{\rm t}=-0.478t+284.5\) & \(Q_{\rm t}=-0.299t+280.43\) \\The diffusion resistance \(R_{d}\) is a function of the distance \(l\) between the front face of the train and the point at which the air velocity is to be determined. The air velocity is calculated so that the flow rate decreases linearly, and at the distance of 1500 m from the loco front, the air flow caused by the piston effect attenuates down to the velocity of an undisturbed flow. We relate this effect to the diffusion resistance \(R_{d}(S)\) and the distance from the front of the loco to the tunnel section of interest (Figure 7):  
\[R_{d}(S)=5\times 10^{-13}.S^{3}-9\times 10^{-10}.S^{2}+7\times 10^{-7}.S-8\times 1 0^{-5},k\upmu.\]  
In this fashion, the mathematical model of the air flow caused by the piston effect is given by the system of equations:  
\[\left\{ \begin{array}{l}P_{SV}^{f}=\!\!0.47919V_{t}^{2};\\ P_{SV}^{b}=-\!\!0.41748V_{t}^{2}.\end{array} \right.\quad\text{--for the 3 km long tunnel;}\]  
\[R_{d}(S)=4.6\times 10^{-13}.S^{3}-8.27\times 10^{-10}.S^{2}+7.93\times 10^{-7}.S-8.333\times 10^{-5}\]  
\[\left\{ \begin{array}{l}P_{SV}^{f}=\!\!0.5782V_{t}^{2};\\ P_{SV}^{b}=-\!\!0.47929V_{t}^{2}.\end{array}