# Improved Calculation Method for Siphon Drainage with Extended Horizontal Sections
## 2 Preliminary Analysis of the Flow Pattern in the Siphon  
According to [PERSON]'s law, the solubility of air is positively related to the ambient air pressure. Under high-lift siphon conditions, the flow velocity in the descending section was greater than the flow velocity in the back of the ascending section, resulting in a significant drop in air pressure in the tube, which gradually exceeded the critical range for air precipitation. The bubbles' height in the rising section was influenced by several factors, including the total length of the siphon, the diameter of the pipe, and the siphon lift. All other things being equal, the height of the bubbles in the rise section decreased as the siphon lift increased. Similarly, the gravity flow length at the top of the descending section of the siphon during siphon flow was not constant but was influenced by the pipe diameter, pipe length, lift, and descending section length, among other factors. All other things being equal, the gravity flow length of the descending section increased with increasing pipe diameter and increased with increasing lift. Furthermore, due to the difference in flow velocity between the ascending and descending sections, very obvious bubbles were observed in the horizontal section of the high-lift long-pipe siphon, where the 3-5 mm pipe diameter was characterized mainly by segmental slug flow, while the 6.5-8 mm pipe diameter was mainly characterized by elastic or annular flow with occasional bubble flow. [PERSON] showed that the maximum diameter of the bubble that can maintain a stable structure is about 4.6 mm under stagnant water without boundary conditions. In contrast, in flowing water, the size of the bubble is determined by its force; that is, when its surface tension and the shear stress and mass forces acting on its surface are balanced, the bubble will not break, and its size is unstable. Therefore, the equation proposed by [PERSON] [26] can be used to estimate the critical size of bubbles in water flow. \[d_{95}=0.725\bigg{(}\frac{\sigma}{\rho_{l}}\bigg{)}^{\frac{3}{5}}\bigg{(}\frac {1}{\varepsilon}\bigg{)}^{\frac{3}{5}} \tag{1}\]  
where \(d_{95}\) is the sum of the volume of bubbles more minor than the critical diameter occupies 95% of the total volume of bubbles, \(\rho_{l}\) is the density of the liquid, \(\sigma\) is the surface tension of the bubble interface, and \(\varepsilon\) is the energy consumption per unit mass. In this study, a circular pipe was used for siphon drainage, so the energy consumption in a pipe of length L can be considered as  
\[P_{u}=Q\gamma h_{l} \tag{2}\]  
where \(Q\) is the flow rate, \(h_{l}\) is the head loss of the section, \(\gamma\) is the heaviness of the fluid. Then the mass of water in the pipe section is  
\[M=\rho_{l}AL=\frac{\rho_{l}QL}{v_{l}} \tag{3}\]where \(A\) is the cross-sectional area of the pipe and \(v_{l}\) is the average flow velocity of the test section. Dividing Equation (2) by Equation (3), the energy consumption per unit mass for pipe flow can be expressed as  
\[\epsilon=\frac{gh_{l}v_{l}}{L} \tag{4}\]  
According to the Darcy formula,  
\[h_{l}=\lambda\frac{L}{d}\frac{v_{l}^{2}}{2g} \tag{5}\]  
where \(D\) is the siphon diameter, \(v_{l}\) is the average flow velocity of the test section, and \(\lambda\) is the along-travel resistance coefficient (determined by the Reynolds number Re). Substituting Equation (5) into Equation (4) yields  
\[\epsilon=\frac{\lambda v_{l}^{3}}{2D} \tag{6}\]  
Therefore,  
\[d_{95}=0.1497449\times\left(\frac{D}{v_{l}}\right)^{\frac{4}{5}} \tag{7}\]  
[PERSON]'s theory states that the presence of bubbles in a siphon pipe is directly related to the pipe diameter and the average fluid flow velocity in the pipe, which is further influenced by the lift, pipe length, and other factors.