# Improved Calculation Method for Siphon Drainage with Extended Horizontal Sections
## 2 Preliminary Analysis of the Flow Pattern in the Siphon
It can be seen that in a 200 m-long distance siphon drain with a pipe diameter of 5 mm when the lift is less than 3.3 m, more than 95% of the bubbles in the flow state can reach a stable diameter of 5 mm or more, as shown in Figure 2. In other words, the air accumulation problem caused by \"non-segmental slug flow\" can be effectively avoided in a 5 mm-long siphon, and this is confirmed by the flow pattern observed in the test. For liquid-transport processes, single-phase flow of a whole tube of liquid is the most desirable and most efficient but impossible form of flow in a siphon drainage system. This is determined by the unique physical characteristics of the siphon drainage system: the presence of reverse gravitational motion of the liquid. As liquid gradually rises in the vertical section, the air pressure inside decreases, which, according to [PERSON]'s law, leads to a decrease in the solubility of the gas solute in the liquid, which in turn leads to air precipitation and the formation of bubbles in the tube. Therefore, gas-liquid two-phase flow  
Figure 2: Bubble \(d_{95}\) curve in 200 m 5 mm siphon pipe. is inevitable in the siphon drainage process, which also greatly enhances the complexity of siphon drainage theory calculations. In addition, the most direct effect caused by bubbles in the tube is the formation of a non-homogeneous fluid, which leads to momentum transfer with jump characteristics. However, the segmental slug flow formed under different conditions will have different pressure transfer paths and pressure drop magnitudes. Different types of flow patterns in a circular tube will exhibit different characteristics, and the different forms of gas present in the fluid will directly affect the energy and momentum exchange rates. Therefore, it is essential to distinguish between different flow types and choose an appropriate gas-liquid two-phase motion model to describe and predict the siphoning process accurately. This study focused on improving the computational accuracy of long-pipe siphons, which have more complex and stochastic fluid motion than short-pipe siphons and different flow-rate losses. While the critical analysis surface of the short siphon can be intuitively chosen as the highest point of the siphon based on the energy loss characteristics, the long horizontal siphon will be in a state of continuous energy loss at its highest horizontal position. Therefore, in the process of calculating the siphon flow velocity in the long horizontal section, although the process of air pressure change in the rising section is similar to that in the short tube, the interaction between the fluid and the tube wall in the horizontal section has a considerable effect on the change of flow velocity. [PERSON] [27] developed a slug bubble theory model based on lubrication theory and capillary effect theory that can be used to predict the shape and pressure drop of long bubbles in a circular tube in pressure-driven flow. The pressure drop before and after the bubble is listed as follows. \[\Delta P_{H}=\frac{2\sigma}{R_{H}}=\frac{2\sigma}{R}\Big{(}1+2.675 Ca^{\frac{ 2}{3}}\Big{)} \tag{8}\]  
\[P_{u}=Q\gamma h_{l} \tag{2}\]0  
The pressure drop generated through the entire bubble is  
\[P_{u}=Q\gamma h_{l} \tag{2}\]1  
The pressure jumps at the front and rear ends of the bubble are shown in Figure 3, and the theoretical results agree with the numerical simulations obtained through other research theories. Moreover, such a theoretical formulation for the low capillary number case can predict the pressure jump at higher capillary numbers. As can be seen from the figure, a pressure change curvature inversion occurred for \(Ca\approx 1\) due to the balance between the capillary and viscous forces.