# Energy-Saving Network Ventilation Technology of Extra-Long Tunnel in Climate Separation Zone
## 2 Materials and Methods
### Calculation Methods of the Natural Wind Velocity in the Tunnel
#### 2.1.2 Calculation Methods of the Natural Wind Velocity in the Tunnel with Inclined Shafts  
Calculation Method of Ultra-Static Pressure Difference and Dynamic Pressure in the Tunnels with Shafts  
As shown in Figure 2, point 3 at the shaft is set as the calculation reference point of ultra-static pressure. The ultra-static pressure difference relative to point 1 at the lower tunnel entrance, point 2 at the higher tunnel entrance and point 3 at the inclined shaft exit are \(\Delta P_{u1}\), \(\Delta P_{u2}\) and \(\Delta P_{u3}\), respectively (\(\Delta P_{u3}=0\)). With the ultra-static pressure difference, the air in the tunnel and the shaft starts to flow  
Figure 1: Calculation method of natural wind in the tunnel without Shafts. and finally becomes steady. At this time, the ultra-static pressure differences in the tunnel will be stable values. When the airflow is stable, \(\Delta P_{u4}\)' stands for the sum of the ultra-static pressure \(\Delta P_{u4}\) and dynamic pressure \(h_{v}\). Without the pressure losses of conflux and split flow taken into account, the value of \(\Delta P_{u4}\)' at the cross sections a-a', b-b' and c-c' will be equal. Then, the pressure difference of the sum of \(\Delta P_{u4}\) and \(h_{v}\) at the two tunnel ends and the inclined shaft exit with respect to the inclined shaft bottom are obtained (hereinafter referred to as differential pressure). When the airflow \(Q_{i}\) (\(i=1,2,3\)) flows in Node 4, the value of \(Q_{i}\) is positive, otherwise, it is negative. The laws of airflow resistance are shown as Equations (7)-(9). \[\Delta P_{u1}\text{'}-\Delta P_{u4}\text{'}=(R_{1}+R_{1}\text{'})\cdot Q_{1} \cdot|Q_{1}| \tag{7}\]  
\[\Delta P_{u2}\text{'}-\Delta P_{u4}\text{'}=(R_{2}+R_{2}\text{'})\cdot Q_{2} \cdot|Q_{2}| \tag{8}\]  
\[\Delta P_{u3}\text{'}-\Delta P_{u4}\text{'}=(R_{3}+R_{3}\text{'})\cdot Q_{3} \cdot|Q_{3}| \tag{9}\]  
where \(R_{i}\) and \(R_{i}\text{'}\) (\(i=1,2,3\)) is the frictional resistance and the local resistance of the tunnel, respectively. For Node 1, in the ventilation network, when the air density changes little, the air quantity flowing in Node \(i\) is equal to that flowing out of Node \(i\), i.e., the algebraic sum at any node is zero, as shown in Equations (10) and (11). \[Q_{1}+Q_{2}+Q_{3}=0 \tag{10}\]  
\[Q_{i}=S_{i}\cdot v_{i}\{i=1,2,3\} \tag{11}\]  
As a middle node in the ventilation system, the air flows in and out. When the air flows from Node \(i\) into Node 4, \(\Delta P_{u4}\text{'}<\Delta P_{ui}\). The calculation steps are listed as follows:  
1. Suppose \(\Delta P_{u4}\text{'}=\min(\Delta P_{u1}\text{'},\Delta P_{u2}\text{'}, \Delta P_{u3}\text{'})\). 2. Insert \(\Delta P_{u4}\text{'}\) into Equations (7)-(9), then get the air quantities \(Q_{1}\), \(Q_{2}\) and \(Q_{3}\). 3. Calculate the sum of \(Q_{1}\), \(Q_{2}\) and \(Q_{3}\), if \(Q_{1}+Q_{2}+Q_{3}<\varepsilon\), the calculation is over. \(\varepsilon\) is the control accuracy and its value is very small, such as 0.1. The calculated \(Q_{1}\), \(Q_{2}\) and \(Q_{3}\) are the solutions of the calculation. Otherwise, conduct Step 4. 4. Suppose \(\Delta P_{u4}\text{'}=\Delta P_{u4}\text{'}+0.01\), repeat Steps 2 and 3. The value of \(\Delta P_{u4}\text{'}\) will not be larger than all of \(\Delta P_{u1}\text{'}\), \(\Delta P_{u2}\text{'}\) and \(\Delta P_{u3}\text{'}\), so another end condition of the calculation is \(\Delta P_{u4}\text{'}=\max(\Delta P_{u1}\text{'},\Delta P_{u2}\text{'}, \Delta P_{u3}\text{'})\). When \(\Delta P_{u4}\text{'}=\max(\Delta P_{u1}\text{'},\Delta P_{u2}\text{'}, \Delta P_{u3}\text{'})\), the calculation will end with no results. This calculation can be used for the air volume of each section in the tunnel. However, when it involves a cyclic process, the calculation method is no longer appropriate. Figure 2: Calculation method of natural wind in the tunnel with shafts.