# Experimental Study of the Air Demand of a Spillway Tunnel with Multiple Air Vents
## 4 Calculation Equation for Air Demand
### Comparison of the Prototype and Model Test Dates  
This section details the practical implementation of the proposed air demand calculation equation. The validation efforts faced inherent challenges arising from two primary limitations: (1) The spillway tunnel's multi-vent configuration, and (2) the method's sensitivity to precise flow characteristics. These constraints, combined with the methodology's restriction to free-surface flow conditions, complicated parameter identification. To address these challenges, the required parameters were taken from two literature datasets (catalog in Table 4): Nuozhadu Dam [35] and Jinping-I Dam [36]. The calculation equation was further applied to four model test cases (cases (1)-(4)), with the computational results illustrated in Figure 15.  
The model test Froude numbers (Fr \(<\) 19) align with prototype observation ranges (Fr \(<\) 8), where air entrainment and scale effects are negligible in free-surface flow [1; 37; 38]. This alignment validates the Section 4.2 model for comparing observed prototype air demand (\(Q_{a,\text{obs}}\)) with predictions, as shown in Figure 16. For prototype data from the Nuozhadu and Jinping-I projects, the predictions closely follow the 45\({}^{\circ}\) equivalence line, exhibiting a mean absolute percentage error (MAPE) of 19.95% and \(R^{2}\) = 0.980, with all data trends captured within a \(\pm\)25% deviation range. The model experiments (cases (1)-(4)) demonstrate even stronger agreement, yielding an MAPE of 4.38% and \(R^{2}\) = 0.993, with deviations confined to \(\pm\)15%. These results confirm robust consistency between the spillway tunnel air demand predictions and both the prototype and experimental measurements. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Project**} & \multicolumn{3}{c}{**Spillway Tunnel**} & \multicolumn{3}{c}{**Air Vent Area (m\({}^{2}\))**} & \multicolumn{6}{c}{**\(U_{a}\) (m/s)**} \\ \cline{2-13}  & \(B\) **(m)** & \(h\) **(m)** & \(L\) **(m)** & \(a_{1}\) & \(a_{2}\) & \(a_{3}\) & \(a_{4}\) & \multirow{2}{*}{\(H_{w}\) **(m)**} & \multirow{2}{*}{\(U_{w}\) **(m)**} & \multirow{2}{*}{\(U_{w}\) **(m)**} & \multirow{2}{*}{\(\#1\)} & \multirow{2}{*}{\(\#2\)} & \multirow{2}{*}{\(\#3\)} & \multirow{2}{*}{\(\#4\)} \\ \cline{2-13} Nuozhadu & \multirow{2}{*}{12} & \multirow{2}{*}{16} & \multirow{2}{*}{497.8} & \multirow{2}{*}{31} & \multirow{2}{*}{8} & \multirow{2}{*}{9.6} & \multirow{2}{*}{12} & 1.7 & 35.6 & 44.7 & 10.8 & 18.8 & 10.8 \\ Left & & & & & & & & 9.0 & 34.8 & 98.8 & 40.6 & 21.7 & 32.5 \\ \hline \multirow{3}{*}{\
\begin{tabular}{} \end{tabular} } & \multirow{3}{*}{12} & \multirow{3}{*}{16.5} & \multirow{3}{*}{382.7} & \multirow{3}{*}{36} & \multirow{3}{*}{9.42} & \multirow{3}{*}{9.42} & \multirow{3}{*}{9.42} & 3.4 & 36.5 & 91.4 & 79.0 & 94.5 & 10.8 \\  & & & & & & & 5.1 & 36.4 & 94.5 & 92.0 & 101.7 & 15.3 \\  & & & & & & & 6.8 & 36.2 & 119.3 & 105.1 & 121.7 & 15.3 \\  & & & & & & & 8.5 & 36.1 & 122.2 & 113.7 & 131.5 & 15.3 \\ \hline \multirow{3}{*}{
\begin{tabular}{} \end{tabular} } & \multirow{3}{*}{13} & \multirow{3}{*}{17} & \multirow{3}{*}{1407} & \multirow{3}{*}{85} & \multirow{3}{*}{36} & \multirow{3}{*}{36} & \multirow{3}{*}{/} & 1.1 & 22.0 & 19.5 & 23.8 & 22.6 & \\  & & & & & & & 3.5 & 18.8 & 32.2 & 44.0 & 38.2 & \\  & & & & & & & 7.0 & 17.4 & 46.7 & 54.0 & 51.8 & / \\  & & & & & & & 10.5 & 16.9 & 61.1 & 61.3 & 62.8 & \\  & & & & & & & & 14.0 & 19.0 & 52.2 & 62.5 & 80.4 & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Prototype observations on the Nuozhadu Dam and the Jinping-I Dam (Fr \(<\) 8.0). Figure 15: Comparison of air demand: experimental measurements data (\(Q_{a,\text{obs}}\)) versus predicted data (\(Q_{a,\text{pred}}\)); dashed lines = \(\pm\) 15%The solution for the air demand of spillway tunnels in free-surface flow presented in this paper comprehensively includes the main hydraulic factors and structural factors on the basis of the interfacial shear stress formulation and can be considered a preliminary design method.