# Improved Calculation Method for Siphon Drainage with Extended Horizontal Sections
## 3 Analysis of the Formation Mechanism of Wet and Dry Slug Flow
### Mechanism and Calculation for Pressure Drop of Dry Slug Flow  
Unlike single-phase flow motion, the mechanical properties of slug flow depend heavily on the bubble-liquid interaction. The mechanical transfer properties of wet slug flow have been discussed when a liquid film is present, while the calculation of dry slug flow is more intuitive. When the bubble is composed of the gas-liquid interface at both ends and the tube wall, instead of only the gas-liquid interface, this flow type can be considered a dry slug flow, while the gas-liquid interface and the tube wall form a gas-liquid-solid three-phase contact line. The most widely used theoretical models regarding the relationship between contact angle and contact line velocity are fluid dynamics [30] and molecular dynamics theory [31]. Among them, the molecular dynamics theory model suggests that the motion of the contact line is determined by the statistical dynamics of collisions between gas, solid, and liquid molecules in the three-phase region [31], as defined by  
\[U=2\kappa^{0}\lambda\text{sinh}\left[\frac{\sigma(\cos\theta_{S}-\cos\theta_{ D})\vartheta^{2}}{2k_{B}T}\right] \tag{34}\]  
where \(\vartheta\) is the average displacement length, \(k_{B}\) is [PERSON]'s constant, \(h\) is [PERSON]'s constant, \(T\) is temperature; \(\kappa^{0}\) is the equilibrium frequency of random displacement of molecules, which can be expressed in terms of the activation energy of the wetting process  
\[\kappa^{0}=\left(\frac{k_{B}}{h}\right)\exp\left(-\frac{\Delta E}{Nk_{B}T}\right) \tag{35}\]  
where \(N\) is [PERSON]'s constant. During the siphon drainage process, the fluid in the rising section pushes the fluid in the horizontal section toward the drain. In a millimeter-diameter siphon, the horizontal segment fluid undergoes a discontinuous motion under the action of thrust. Assuming that the fluid does not compress, the forces during the entire motion include fluid thrust, slug bubble interface pressure, resistance along the wall, and negative pressure on the drain side of the slug bubble. Further, the primary energy conversion process of its motion process is gravitational potential energy \(\rightarrow\) kinetic energy \(\rightarrow\) pressure potential energy, \(\rightarrow\) kinetic energy \(\ldots\) finally transformed into pressure potential energy, pushing the horizontal segment terminal fluid to vertical segment drop, forming a continuous siphon flow. For siphons in this size range, in the process of resuming siphoning through water level lift, the thrust provided by the lift change is consumed step-by-step by the slug bubble and liquid slug in the horizontal section, plus the resistance effect of the interface between the two ends of the slug bubble and the pipe wall, resulting in a certain degree of reduction in the critical lift that causes siphon restart. A simplified force model of the dry slug flow unit is shown in Figure 6. In this form of flow, the bubbles have the same velocity as the liquid and flow alternately through the pipe without forming a liquid film on the pipe wall. According to the simplified model, it is known that the pressure drop of the slug flow can be divided into three parts, namely, the frictional pressure drop of the bubble and fluid movement and the frictional pressure drop generated by the movement of the gas-liquid-solid three-phase contact line, as shown in Figure 7 for the simplified pressure drop model. In the dry slug flow mode, the gas, liquid, and solid phases are entirely separated by the gas-liquid interface and moving contact line, and the bubbles have the same velocity as the liquid and flow alternately through the pipe and do not form a liquid film on the pipe wall. Therefore, to satisfy the continuity equation, the velocities of bubbles and slugs should be equal, as shown in the following equation. \[U_{L}=U_{G}=U \tag{36}\]  
The subscripts \(L\) and \(G\) denote the liquid and gas phases, respectively.