# Improved Calculation Method for Siphon Drainage with Extended Horizontal Sections
## 3 Analysis of the Formation Mechanism of Wet and Dry Slug Flow
### Improved Calculation Method for Siphon Flow Rate with Long Horizontal Pipe Sections  
The standard siphon velocity calculation is based on [PERSON]'s equation,  
\[v=\sqrt{\frac{2g\Delta H}{1+\lambda\frac{I}{D}+\Sigma\zeta}} \tag{50}\]  
which is based on the assumption of continuous single-phase flow and shows that the siphon velocity was influenced by frictional resistance, including along-stream and local lift losses. The effect of lift and height difference on the selection of siphon velocity parameters was further demonstrated by [PERSON]'s study [36], but only a semi-empirical solution was given without a more detailed explanation of the mechanism. The different force transfer characteristics of the fluid in the siphon were analyzed separately by setting the apex of the siphon as the critical analysis plane by [PERSON] [37]. Different calculation lengths were selected according to the acceleration direction of fluid motion around the critical analysis surface, and smaller values were taken in the results to control the calculation error between 3% and 19%. Although the calculation errors were significantly reduced, the research focused on the flow velocity calculation of \"straight-up- straight-down\" siphon drainage systems. Although in long or short siphon systems with a high lift, the fluid will precipitate much air due to the reduction of air pressure. However, in a siphon with a long horizontal section, the effect of frictional resistance from air precipitation increases with the length of the top horizontal section. Therefore, it is urgent and necessary to study the mechanism of siphon flow velocity in siphons with extended horizontal sections and to calculate it accurately in engineering practice. In his study of short pipe siphoning, [PERSON] found that the magnitude of the fluid thrust at the top of the siphon plays a crucial role in how the flow velocity is calculated. That is, the ability to maintain continuous flow when the fluid moves to the highest position determines [PERSON]'s equation's scope of application. However, long horizontal siphons have more complex and stochastic fluid motion than short siphons and have different velocity loss characteristics. For short siphons, the critical analysis surface can be intuitively chosen from the highest point of the siphon based on the energy loss characteristics, but for long horizontal siphons, the fluid 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. Therefore, to extend the formula of siphon flow velocity to the operating mode of long siphons, the analytical calculation of the siphon drainage velocity for the horizontal section should be added to the original formula of segmental integration. Therefore, the siphon velocity loss in the \"horizontal section\" should be added to the siphon velocity equation considering the effect of the long horizontal section. Based on the above reasoning, and assuming that the siphoned fluid in the horizontal section is a segmental slug flow, i.e., a cross-section at any location with a single-phase composition, this simplifies the computational model to simulate the movement of a homogeneous fluid in the pipe. Similarly, to simulate the most unfavorable state, the calculated length of the horizontal section of the siphon is taken to be the entire length of the horizontal section, as follows. \[\left\{