# Experimental Study of the Air Demand of a Spillway Tunnel with Multiple Air Vents
## 4 Calculation Equation for Air Demand  
### Solution Procedure for Air Demand  
The above model tests reveal that hydraulic factors, such as water velocity and flow depth, and structural factors, including the air vent area and the location of air vents, significantly affect the air demand of spillway tunnels with multiple air vents. This section proposes a novel approach to calculate air discharge through air vents on the basis of energy, momentum, and force balance equations. Notably, free-surface flow is the desired flow pattern and spray flow is affected by scale and model effects. Therefore, the method was limited to free-surface flow. The air discharge value in the bubble suspension layer is approximately 10% in the air vent for free-surface flow [17]. The volume of droplets in the droplet jump layer can compensate for the volume of air in the bubble suspension layer. Thus, the air discharge in the bubble suspension layer is neglected in the derivation. In this study, the volume of air can be divided into two main categories, as shown in Figure 11: the volume of air between section \(0_{i}-0_{i}\), away from each air vent inlet, and sections \(1_{i}-1_{i}\) and \(2_{i}-2_{i}\) serves as the control body \(T_{1,i+1}\), described in magenta; while the volume of air between sections \(2_{i}-2_{i}\) and \(1_{i+1}-1_{i+1}\) serves as the control body \(T_{2,i}\), described in red. The energy equation for the control body \(T_{1,i}\) from section \(0_{i}-0_{i}\) to \(2_{i}-2_{i}\) is as follows:  
\[Z_{0,i}+\frac{p_{0,i}}{\rho_{\textit{a}}g}+\frac{U_{0,i}^{2}}{2g}=Z_{2,i}+\frac {p_{2,i}}{\rho_{\textit{a}}g}+\frac{U_{A,i}^{2}}{2g}+\left(\lambda_{i}\frac{l_ {a,i}}{4R_{i}}+(\Sigma\xi)_{i}\right)\frac{U_{a,i}^{2}}{2g} \tag{1}\]  
where \(Z_{0,i}\) and \(Z_{2,i}\) are the \(z\)-direction positions of the midpoints of sections \(0_{i}-0_{i}\) and \(2_{i}-2_{i}\), respectively. The corresponding negative pressures are \(p_{0,i}\) and \(p_{2,i}\), and the average air velocities are \(U_{0,i}\) and \(U_{A,i}\). Furthermore, \(U_{a,i}\), \(l_{a,i}\), \(l_{a,i}\), \(B_{a,i}\), \(R_{i}=\frac{B_{a,i}l_{a,i}}{2(B_{a,i}+l_{a,i})}\), \(\lambda_{i}\), and \((\Sigma\xi)_{i}\) represent the average air velocity, length, height, width, hydraulic diameter, resistance coefficient, and the local loss coefficients of the \(i\)th air vent, respectively. The pressure \(p_{0,i}\) away from the inlet approaches atmospheric pressure, and the height difference between sections \(0_{i}-0_{i}\) and \(2_{i}-2_{i}\) is much smaller than the negative pressure head \(\Delta H_{2,i}=\frac{p_{2,i}}{\rho_{\textit{a}}g}\), thus the height term is negligible in the derivation. \(U_{A,i}\) can be  
Figure 11: Schematic diagram of air–water stratified flow. calculated in Equation (2) by the continuity equation, and substituting \(\delta_{i}=\lambda_{i}\frac{l_{i,i}}{4R_{i}}+(\sum\xi)_{i}\) into Equation (1) yields Equation (3) for pressure head \(\Delta H_{2,i}\). \[U_{A,i}A_{a,i}=A_{a,i-1}U_{A,i-1}+a_{i}U_{a,i} \tag{2}\]  
\[\Delta H_{2,i}=-(\frac{U_{A,i}^{2}}{2g}+\delta_{i}\frac{U_{a,i}^{2}}{2g}) \tag{3}\]  
where \(A_{ai}\) is the residual tunnel cross-sectional area above the water surface between sections \(2_{i}-2_{i}\) and \(1_{i+1}-1_{i+1}\), and \(a_{i}\) is the cross-sectional area of the \(i\)th air vent; the first vent is just downstream of the gate leading to \(U_{A,0}=0\). The control body \(T_{2,i}\) in Figure 11 is subjected to drag force \(f_{2w,i}\) at the wall, drag force \(f_{2w,i}\) at the air-water interface, and negative air pressure (\(p_{1,i+1}A_{a,i}\), \(p_{2,i}A_{a,i}\)) at sections \(1_{i+1}-1_{i+1}\) and \(2_{i}-2_{i}\). The drag force \(f_{2w,i}\) can be expressed as \(f_{2w,i}=\tau_{w,i}L_{i}B_{i}\), where \(L_{i}B_{i}\) is the area of the air-water interface between sections \(2_{i}-2_{i}\) and \(1_{i+1}-1_{i+1}\).