# Aerodynamic Analysis of the Opening Hood Structures at Exits of High-Speed Railway Tunnels
## 2 Theoretical Analysis of Key Parameters for Exit Hoods  
When a train enters a tunnel at high speed, the confinement of the tunnel walls causes the air ahead to generate compression waves. These compression waves propagate forward within the tunnel at the speed of sound. Upon reaching the tunnel exit, they generate MPWs that radiate into external space. However, when an opening hood is placed at the exit, the compression waves passing through the opening area release a portion of their pressure in advance. This physical phenomenon is analogous to gas leakage from a sealed container. Hence, the thermodynamic model of the exit hood can be simplified to that of gas leakage in a rigid, sealed container. Moreover, the controlled volume method is utilized to analyze the key factors influencing pressure changes within the air area. The simplified thermodynamic system of the tunnel exit hood is shown in Figure 1.  
Taking \(V_{1}\), \(V_{2}\), and \(V_{3}\) as control volumes from Figure 1, the analysis process follows the following assumptions:  
1. The air within the control volumes is treated as an ideal gas, and the total energy change of the system is attributed to thermodynamic energy. 2. The tunnel walls act as adiabatic boundaries, and the system does not perform mechanical work on the external environment. 3. The specific enthalpy entering and exiting through the opening interface is equal to the average specific enthalpy of the air within the control volume at that moment. 4. The change in mass within the control volume during differential processes is entirely caused by variations in the mass of the air. The energy equation [17] (pp. 46-49) for the opening hood structure system is given by Equation (1). Substituting \(h=pv+u\) into Equation (1) and simplifying the calculation in conjunction with the assumptions yields Equation (2) [18]:  
\[\delta Q=\mathrm{d}E_{CV}+h_{out}\delta m_{out}-h_{in}\delta m_{in}+\delta W_{i} \tag{1}\]  
\[m\mathrm{d}u=pv\mathrm{d}m \tag{2}\]  
where \(\delta Q\) is the energy received by the system from the external environment, \(\mathrm{d}E_{CV}\) is the change in total energy of the system, \(h_{in}\), \(h_{out}\) is the enthalpy of the infinitesimal working substance flowing into and out of the system, \(\delta m_{in}\), \(\delta m_{out}\) is the mass of the infinitesimal working substance flowing into and out of the system, \(\delta W_{i}\) is the mechanical work done by the system on the external environment, \(h\) is the average specific enthalpy of the air within the control volume, \(p\) is the transient average pressure within the control volume,  
Figure 1: Schematic diagram of control volume method analysis of tunnel exit hood. is the specific volume, \(u\) is the specific thermodynamic energy, \(m\) is the total mass of the air within the control volume, and d\(m\) is the net mass of air flowing in and out of the control volume through its interfaces. When an ideal gas within the control volume is in equilibrium, Equations (3)-(5) can be derived from the Clapeyron equation. Equation (5) is the differential form of the Clapeyron equation:  
\[pv=R_{g}T \tag{3}\]  
\[pV=mR_{g}T \tag{4}\]  
\[\frac{\mathrm{d}p}{p}+\frac{\mathrm{d}V}{V}=\frac{\mathrm{d}m}{m}+\frac{ \mathrm{d}T}{T} \tag{5}\]  
where \(R_{g}\) is the gas constant, \(T\) is temperature, and \(V=V_{1}+V_{2}+V_{3}\) is the volume of the entire system. The equation for the specific heat capacity of an ideal gas [17] is given by Equation (6). By substituting Equations (3) and (6) into Equation (2), we can derive Equation (7). In Equation (7), since the control volume of the entire system remains constant, combining the Clapeyron equation with Equation (7) leads to the average pressure change caused by mass variation within the exit hood system, as shown in Equation (8). Due to the influence of the thermal motion of air molecules, it can be estimated in the subsequent analysis that each point on the opening cross-section of the hood structure has the same flow velocity.