# 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
 \tag{44}\]  
Although the model does not contain any empirical parameters, it only applies to the forward contact line velocity solution, i.e., the dynamic forward contact angle. In order to be able to apply the above equation to the dynamic backward contact angle, in previous work on the model calculation, it was assumed that the difference between the static and dynamic backward contact angle is the same as the difference between the static and dynamic forward contact angle. Thus, the dynamic backward contact angle can be calculated according to the following equation  
\[\theta_{S-R}-\theta_{D-R}=\theta_{D-A}-\theta_{S-A} \tag{45}\]  
where \(\theta_{S-R}\) and \(\theta_{D-R}\) are the static and dynamic backward contact angles, and \(\theta_{S-A}\) and \(\theta_{D-A}\) are the static and dynamic forward contact angles, respectively. According to the above formula, it is known that the larger the flow velocity is, the more significant difference between the static contact angle and dynamic contact angle is; conversely, the smaller it is. Combined with the formula of the critical radius, the calculation results within the application range of this formula show a monotonic decreasing trend with the increase of the contact angle. Therefore, during siphon drainage with a low lift and high flow rate, the end of the liquid slug in the horizontal section of the siphon (corresponding to the head of the slug bubble) is prone to rupture, while the head of the liquid slug is relatively intact, and this phenomenon is also observed in the experiment. The relationships between the four dynamic static contact angles were obtained from Equation (42) to Equation (45) can be further combined with experimental correlations. First,to eliminate the empirical parameters in the molecular dynamics equations, the independent variables of hyperbolic sine are assumed to be minor, and then the hyperbolic sine function is substituted into its Taylor expansion to reduce Equation (35) to a linear equation  
\[U=\frac{\kappa^{0}\theta^{3}\sigma(\cos\theta_{S}-\cos\theta_{D})}{k_{B}T} \tag{46}\]  
Using Equations (35) and (45) to find U, respectively, and joint, the same equation can be written for the backward contact angle (i.e., \(-\)U) to obtain the relationship equation for the contact angle, respectively. \[\cos\theta_{S-R}-\cos\theta_{D-R}=\cos\theta_{D-A}-\cos\theta_{S-A} \tag{47}\]  
The single contact line pressure drop is solved by substituting the dynamic contact angle correlation Equation (44) into Equation (47) and then substituting the resulting dynamic forward and backward contact angles into Equation (41), which obtains the contact line pressure drop associated with the capillary number. Thus, with the pipe material and fluid type determined, the pressure drop of the moving contact line is then determined by the speed and number of its movements. The slug frequency is the number of fluid slugs that flow through a specific pipe location during a specific time interval. The primary defining equation for slug frequency is  
\[f=\frac{U_{B}}{L_{B}+L_{LS}}=\frac{U_{B}}{L_{unit}} \tag{48}\]  
[PERSON] [35] obtained an improved formula for calculating the slug frequency  
\[f=0.0434\bigg{[}\frac{U_{spL}}{U_{LS}}\bigg{(}\frac{2.02}{D}+Fr_{S}\bigg{)} \bigg{]}^{1.02} \tag{49}\]  
where \(Fr_{S}=\frac{U_{B}^{2}}{gD}\).