# 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
In this section, we study the velocity of bubbles and liquid slugs based on the average interfacial velocity of liquid slugs based on the visualized data. Under experimental conditions, the two-phase flow is dry slug flow when the liquid slug passes through a small diameter channel with hydrophobic properties. According to previous studies, the pressure drop of the two-phase flow consists of the pressure drop of the liquid slug, the bubble, and the moving contact line [32] as follows  
\[\Delta P_{TP}=\Delta P_{G}+\Delta P_{L}+\Delta P_{MCL} \tag{37}\]  
where \(\Delta P\) is the pressure drop, the subscripts \(TP\), \(G\), \(L\), and \(MCL\) are the two-phase integrated gas phase, liquid phase, and contact line, respectively. As the measured flow velocities at each lift of the 200 m length siphon involved in the test and at the medium and high lift of the 100 m length siphon proved that the siphon flow in the 3-8 mm diameter tube is laminar, the frictional pressure drop of each single phase bubble or liquid slug can be calculated by the following equation  
\[\Delta P_{G}=f_{G}\frac{L_{G}}{D}\frac{1}{2}\rho_{G}u^{2} \tag{38}\]  
\[\Delta P_{L}=f_{L}\frac{L_{L}}{D}\frac{1}{2}\rho_{L}u^{2} \tag{39}\]  
The following equation can express the pressure drop of the slug flow  
\[\Delta P_{TP}=f_{L}\frac{L_{L}}{D}\frac{1}{2}\rho_{L}u^{2}+f_{G}\frac{L_{G}}{ D}\frac{1}{2}\rho_{G}u^{2}+\Delta P_{MCL} \tag{40}\]  
In addition, pressure loss is also induced when the gas-liquid-solid three-phase contact line moves, i.e., the third term on the right-hand side of the Equation (38). The difference in density and surface tension between different fluids will create a curved  
Figure 6: Deformation of the static liquid section slug in a circular tube by force. Figure 7: Frictional pressure drop of moving contact line. interface, and any curved interface between the two phases of the fluid will create a pressure difference that depends on the exact shape of the interface. However, in many cases, the gas-liquid interface can be assumed to be a curved lunar surface described using only the contact angle and the hydraulic diameter. The contact angle depends on the hydrodynamics at the contact line; thus, that part of the pressure drop can be referred to as the contact angle or the pressure drop due to the contact line. The pressure drop at the contact line of the \"bubble-liquid slug\" cell can be expressed by applying [PERSON]'s equation to the contact line  
\[\kappa^{0}=\left(\frac{k_{B}}{h}\right)\exp\left(-\frac{\Delta E}{Nk_{B}T}\right) \tag{35}\]0  
where \(\theta_{A}\) and \(\theta_{R}\) are the forward and backward contact angles, respectively. Therefore, as long as the forward and backward contact angles differ, a net pressure drop is generated in the contact line. Moreover, the difference between the forward and backward contact angles is caused by two phenomena, i.e., contact angle hysteresis and the effect of contact line velocity on contact angle. Although previous models considered the effect of contact line velocity, the pressure drop caused by contact angle hysteresis was not considered. [PERSON] et al. [33] proposed that the magnitude of the dynamic contact angle is related to the capillary number Ca, and these correlations have the following form. \[\kappa^{0}=\left(\frac{k_{B}}{h}\right)\exp\left(-\frac{\Delta E}{Nk_{B}T}\right) \tag{35}\]1  
where \(\theta_{S-A}\) and \(\theta_{D-A}\) are the static advancing contact angle, and dynamic advancing contact angle, respectively, and both \(c\) and \(d\) are constants. Therefore, the dynamic advancing contact angle can be calculated as  
\[\kappa^{0}=\left(\frac{k_{B}}{h}\right)\exp\left(-\frac{\Delta E}{Nk_{B}T}\right) \tag{35}\]2  
Regarding the values of \(c\) and \(d\), according to [PERSON] [34], the way the dynamic contact angle varies and the values of the relevant constants are obtained, as follows Equation  
\[H= \begin{cases}4.47 Ca^{0.42}&(Ca<1\times 10^{-3})\\ 2.24 Ca^{0.54}&(Ca>1\times 10^{-3})\end{cases}