## 2 Description  
The key elements in a network mathematical model of air distribution in a railway tunnel are: the topology of the tunnel [14; 15; 16], portals, the geometry and aerodynamic parameters of the tunnel, adit, ventilation shafts, crossings and other elements of the tunnel ventilation. Natural draft is modeled using fictitious pressure sources. The piston effect is taken into account in the model, including the fictitious pressure sources, to describe pressure changes ahead of and behind the train, resistance in the clearance space between the tunnel walls and the train, as well as dissipation resistance in front of the train. The dissipation resistance [13; 17] is used to model a reduction (deceleration) of the air flow rate in front of the train. Owing to this, it is possible to solve the problems concerning air distribution under the piston effect in tunnels with simple models of static air distribution. The piston effect in cases of static air distribution is modeled using the method of Krasyuk and Lugin [13]. Different pressures ahead of and behind the train are represented by two fictitious pressure sources--constant pressure fans (Figure 1). This allows for the adequate description of air pressure at the front face of the loco by fan 2 and the rarefaction behind the back face of the train by fan 1. There is also draught loss 3 matching air resistance in the tunnel clearance, and diffusion resistance 4. Figure 2 demonstrates the analytic model of integrating the piston effect model in the network model of the tunnel ventilation system. The natural draught (ND)--fan 13 in Figure 2--is modeled using the method with fictitious pressure sources [18]. Based on the height differences \(\Delta h\) of the portals and the temperature of the atmosphere, the pressures of the fictitious sources of ND are calculated using the basic law of hydrostatics. The weather conditions are assumed as per the areas of the extra-long Baikal-Amur Mainline tunnels: Severoumysky Tunnel--settlement Bagdarin (Buryatia); Kuznetsovsky Tunnel--settlement Ayan (Khabarovsk Territory); Baikal Tunnel--town Kirensk (Irkutsk Region) [19]. For cold and warm seasons (CS and WS), the monthly mean temperatures of the coldest month (January) and hottest month (July) are assumed, respectively. The pressures of the fictitious sources of ND are compiled in Table 1.  
The modeling of air flow governed by ND in a tunnel 6 km long, with a freight train 1000 m long inside it, shows that the air flow rate induced by the maximum ND of 354 Pa \begin{table}
\begin{tabular}{l l l l l l l} \hline \multicolumn{2}{c}{\(\Delta h\), \(\mathbf{m}\)} & \multicolumn{1}{c}{**5**} & \multicolumn{1}{c}{**10**} & \multicolumn{1}{c}{**15**} & \multicolumn{1}{c}{**20**} & \multicolumn{1}{c}{**25**} \\ \hline \multirow{2}{*}{Bagdarin} & WS & 60 & 120 & 179 & 240 & 300 \\  & CS & 71 & 141 & 213 & 283 & 354 \\ \hline \multirow{2}{*}{Ayan} & WS & 61 & 121 & 181 & 242 & 302 \\  & CS & 69 & 136 & 205 & 274 & 341 \\ \hline \multirow{2}{*}{Kirensk} & WS & 60 & 119 & 178 & 238 & 297 \\  & CS & 71 & 141 & 211 & 281 & 352 \\ \hline \end{tabular}
\end{table} 
Table 1: Pressures of fictitious sources of ND, Pa.  
Figure 1: Structure of network model of piston effect: 1—fan simulating negative pressure (air rarefaction) behind the tail car (\(Q^{b}\)—capacity and \(P^{b}_{SV}\)—static pressure of the fan); 2—fan simulating overpressure (air pressure increase) ahead of the loco’s front face (\(Q^{f}\)—capacity and \(P^{f}_{SV}\)—static pressure of the fan); 3—draught loss in the clearance between the train and tunnel walls, \(R_{Z}\); 4—diffusion resistance (air dispersion drag) \(R_{d}\); 5—train; \(V_{t}\)—train velocity; arrows show directions of air flow. Figure 2: Branches 1, 8, 10, 12—connections to atmosphere; 2 and 7—inlet and outlet portals; 3 and 6—tunnel; 4—\(R_{Z}\); 5—\(R_{d}\); 9—fan simulating air rarefaction behind tail car; 11—fan simulating increased of air pressure ahead of train; 13—fan simulating the natural drought. is not higher than 12.8 m\({}^{3}\)/s.