# Improved Calculation Method for Siphon Drainage with Extended Horizontal Sections
## 3 Analysis of the Formation Mechanism of Wet and Dry Slug Flow
### Basis Modeling of Slug Flow
\[\tau_{2}=\mu_{2}\bigg{(}\frac{\partial u_{x}}{\partial r}+\frac{\partial u_{r}}{ \partial x}\bigg{)}=0 \tag{14}\]  
This equation can be combined with Equation (11) to obtain  
\[\left.\frac{\partial u_{2}}{\partial r}\right|_{r=0}=0 \tag{15}\]  
Meanwhile, the tube wall obeys the no-slip boundary condition at \(r=R\)  
\[\left.u_{1}\right|_{r=R}=0 \tag{16}\]  
At the gas-liquid interface (\(r=R_{B}=R-\delta\)), the mixed boundary condition is used to characterize the continuity of the velocity and shear stress distribution in the system,  
Figure 4:0  
Figure 4:1  
The above equation can be further written as  
Figure 4:2  
Double integration of the gas-liquid phase in Equation (13) yields  
Figure 4:3  
Figure 4:4  
where the integration constants \(C_{1i}\) and \(C_{2i}\) can be calculated by the boundary conditions (Equations (16)-(19)). The velocity and stress distribution of the continuous phase (liquid film) in the velocity distribution of each phase of the mixture is shown below  
Figure 4:5  
Figure 4:6  
The velocity and stress distribution in the dispersed phase (bubbles or droplets) are shown below  
Figure 4:7  
Figure 4:8  
Each parameter in the formula is calculated as follows  
\[\left\{ \begin{array}{c}C_{1}=(\rho_{1}-\rho_{2})g_{x}\frac{R_{B}^{2}}{2\
u_ {1}}\\ C_{1}^{\prime}=\frac{\xi_{1}^{\prime}}{4\
u_{1}}=\frac{1}{4\
u_{1}}\Big{(}g_{x }-\frac{1}{\rho_{1}}\frac{\partial p}{\partial x}\Big{)}\\ C_{2}^{\prime}=\frac{g_{1}^{\prime}}{4\
u_{2}}=\frac{1}{4\
u_{2}}\Big{(}g_{x }-\frac{1}{\rho_{2}}\frac{\partial p}{\partial x}\Big{)}\end{array} \right. \tag{26}\]The instantaneous flow rate of the liquid in the liquid film can be obtained by integrating Equation (22)  
\[q_{lf}=2\pi\int_{R_{B}}^{R}u_{1}(r)dr \tag{27}\] \[=\pi R^{2}\Big{(}C_{1}^{\prime}\frac{R^{2}}{2}-\frac{C_{1}}{2} \Big{)}-\pi R_{B}^{2}\Big{[}C_{1}^{\prime}\Big{(}R^{2}-\frac{R_{B}^{2}}{2} \Big{)}-C_{1}\Big{(}\ln\Big{(}\frac{R}{R_{B}}\Big{)}+\frac{1}{2}\Big{)}\Big{]}\] \[=\frac{\pi}{8r_{1}}\Big{(}g_{x}-\frac{1}{P_{1}}\frac{2p}{\alpha x }\Big{)}\big{(}R^{2}-R_{B}^{2}\big{)}^{2}-\pi C_{1}\bigg{[}\frac{R^{2}-R_{B}^{ 2}}{2}-R_{B}^{2}\ln\Big{(}\frac{R}{R_{B}}\Big{)}\bigg{]}\]  
The instantaneous flow rate of the gas in the bubble can be found by integrating Equation (24)  
\[q_{G}=2\pi\int_{0}^{R_{B}}u_{2}(r)dr \tag{28}\] \[=\pi R_{B}^{2}\bigg{[}C_{1}^{\prime}\big{(}R^{2}-R_{B}^{2}\big{)} -C_{1}\ln\Big{(}\frac{R}{R_{B}}\Big{)}+C_{2}^{\prime}\frac{R_{B}^{2}}{2}\bigg{]}\] \[=\frac{\pi}{4}g_{x}\bigg{[}\frac{R_{B}^{2}\big{(}R^{2}-R_{B}^{2} \big{)}}{v_{1}}+\frac{R_{B}^{4}}{2v_{2}}\bigg{]}-\frac{\pi}{4}\frac{\partial p }{\partial x}\bigg{[}\frac{R_{B}^{2}\big{(}R^{2}-R_{B}^{2}\big{)}}{\mu_{1}}+ \frac{R_{B}^{4}}{2\mu_{2}}\bigg{]}-\pi C_{1}R_{B}^{2}\ln\Big{(}\frac{R}{R_{B}} \Big{)}\]  
[PERSON] [29] provided the instantaneous flow rate relationship for a gas-liquid two-phase mixture in a tube during flow based on the assumption that the direction of flow through the liquid film is always opposite to the direction of the bubble velocity  
\[q_{G}=q_{LS}+q_{lf} \tag{29}\]  
The difference is that [PERSON] extends the assumptions to  
\[U_{B}A_{B}+U_{lf}A_{lf}=U_{LS}A \tag{30}\]  
Or  
\[q_{G}+q_{lf}=q_{LS} \tag{31}\]  
The expressions (17)-(19) have the meaning of balanced equations: the left side of its equal sign is the sum of the volume fluxes of each phase entering or leaving the control volume V through the cross-section \(x=0\), while the right side is the sum of the volume fluxes of each phase entering or leaving the control volume V through the cross-section \(x=\mathrm{L}\). Equation (31) assumes that the flow direction of the film should always be opposite to the flow direction of the bubbles. It is known that the velocity of the bubble is higher than the velocity of the  
However, when the fluid velocity or siphon diameter decreases, the film thickness decreases until it disappears, and the pressure drop of the fluid inside the tube can no longer be calculated using such equations based on the film thickness.