# Air Entrainment and Air Demand in the Spillway Tunnel at the Jinping-I Dam
## 4 Discussion
### Estimation Method of Air Demand
#### 4.2.3 Relationship between Dimensionless Air Entrainment Rate and Unit Width Flow Rate  
According to the above-mentioned analysis on air demand formulas, it is still difficult to directly estimate air demand only based on some parameters known in advance, such as aerator parameters and flow rate. Typical prototype observation data on air demand of twelve projects are collected and the profile of dimensionless air entrainment rate \(\beta\) with unit width flow rate \(q_{\text{w}}\) is shown in Figure 15 in log-log coordinates.  
For the spillway tunnels of the Jinping-I Dam and the Xiaowan Dam, their discharging capabilities are huge and flow velocities are extremely high; in order to meet the requirements of air entrainment under high velocity conditions, relatively large sizes of aeration devices were employed. As a result, their air entrainment ratios are relatively greater than those of other projects at the same flow rates.  
From a group-as-a-whole perspective, the air entrainment rate is inversely proportional to the unit width flow rate, and the changing trend is consistent for different aeration devices. All the curvesroughly follow a power-law relationship. For different aerators, the decreasing rate with flow rate is distinguished, indicating the decreasing rate of dimensionless air entrainment rate is related to aeration devices layout and size. For a particular aerator, the decreasing rate is also not always a constant and it tends to slightly increase with increasing flow rate. In the range of the unit width flow rate \(<\)-50 m\({}^{3}\)/(s\(\cdot\)m), the decreasing rate nearly does not change; while in the range of unit width flow rate \(>\)-100 m\({}^{3}\)/(s\(\cdot\)m), the decreasing rate has a clear growth trend as the unitwidth flow rate increases. This indicates that the decreasing rate of dimensionless air entrainment rate is also related to the unit width flow rate. Therefore, the following formula can be established to relate the dimensionless air entrainment rate with a unit width flow rate:  
\[\beta=\text{a}\cdot q_{\text{w}}\text{b}^{+\text{c}(q_{\text{w}}-\text{m})} \tag{9}\]  
where a, b, c and m are coefficients, depending on aerator device parameters.  
By nonlinear least square approximation, values of coefficients a, b, c, m in the Jinping-I spillway tunnel are obtained. They are 1.16, \(-\)0.32, \(-\)0.0005, and 67.5, respectively. The fitting curve is shown in Figure 15 (the red solid line). As the prototype data are limited, it is unrealistic to definitely establish the relationships of these coefficients with aerator device parameters. By fitting previous prototype data with Equation (9), distribution ranges of coefficient a, b, c, and m can be obtained. For a, it varies from 0.5 to 3.0. For b, it varies from \(-\)0.85 to \(-\)0.25. For c, it varies from \(-\)0.0006 to 0. For m, it varies from 30 to 80. Before drawing a confirm conclusion on how these coefficients vary with aerator parameters, more systemic prototype observations are needed. Apart from being used to establish air demand formula, the collected prototype observation data shown in Figure 15 could also be used as a significant reference for aerator devices design in projects with similar operating conditions.  
Figure 15: Relationship between dimensionless air entrainment rate and the unit-width flow rate in some projects. Here, the aerator’s location is described in the parentheses of the legend, the abbreviations before and after the space, respectively, represent the discharge structure and the aerator. The specific meaning of these abbreviations are illustrated in Table A2 in Appendix A.