# 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
\begin{array}{c}P_{veri}=\rho g\bigg{(}h_{max}-\frac{v_{opt}^{2}}{2 \delta}-\sum_{i=1}^{i}\bigg{(}\frac{v_{opt}^{2}-1\lambda_{i}-1l_{i}}{d}+v_{ veri-1}^{2}\zeta_{veri}+l_{i}\bigg{)}\bigg{)}\\ v_{veri}=v_{ver0}\frac{h_{e}(P_{int}-P_{max})RT+P_{min}}{P_{min}},for0<\sum_{ i=1}^{i}l_{i}<l_{ver};\\ P_{horii}=\rho g\bigg{(}h_{atm}-\frac{v_{opt}^{2}}{2\delta}-\sum_{i=1}^{i} \bigg{(}\frac{v_{uni-1}^{2}\lambda_{i}-1l_{i}}{d}+v_{horii-1}^{2}\zeta_{upi}+h _{i}\bigg{)}\bigg{)}\\ v_{horii}=v_{ver0}\frac{h_{e}(P_{up}-P_{i})RT+P_{i}}{P_{i}},forl_{ver}<\sum_{ i=1}^{i}l_{i}\leq L_{ver+hori}\end{array} \right. \tag{51}\]  
The calculation of siphon drainage with a long horizontal section can be divided into two parts: the vertical upward section (whose main feature is the decrease in air pressure with increasing height) and the horizontal section (whose main feature is the accumulation of losses along the range). In the vertical upward section of the siphon, the air pressure increases with the height of the position and a \"parabolic\" type decline occurs, the fluid kinetic energy is converted to gravitational potential energy, and the fluid moves from acceleration to deceleration, but still maintains a significant flow rate. When the fluid moves to the horizontal section, the air pressure value inside shows a state of height, while the change in air pressure is significantly reduced. Based on this, in the fluid motion formula, if we continue to use the calculation of the vertical section of the air pressure, then the value of \(P_{hori}\) will continue to decrease, which is not in line with the actual situation. Therefore, to simplify the calculation, the air pressure at the initial position of the horizontal section was approximated by taking the air pressure at the top of the vertical section. Take \(P_{hori}=P_{top}\), where \(P_{top}\) is a constant value approximately equal to the air pressure at the top of the vertical section, as shown in Equation (51). In this study, since there was no applied pressure throughout, the air pressure fluctuation inside the tube did not exceed two times the atmospheric pressure, so the air could be precipitated by default with the liquid out of the siphon and no longer dissolved in water. Therefore, when calculating the void ratio and the total length of the air slug, the descending section length of the tube should be included in the calculation of the equation. Therefore, \(P_{i}\) in Equation (51) can be expressed as  
\[\Delta P_{hori}=P_{horii}-P_{horii-1}=\Delta P_{L}+\Delta P_{G}+\Delta P_{MCL} \tag{52}\]  
Before starting the calculation, some known data need to be given. The input data include the diameter of the circular tube D; the physical properties of each phase \(\rho_{1}\), \(\rho_{2}\), \(\mu_{1}\), \(\mu_{2}\), \(\sigma\); and the projection of gravitational acceleration in the flow direction \(g_{x}\). The calculation algorithm is described as follows, and the calculation flow chart is given in Figure 8.  
First, we calculated the value of the siphon flow rate in the ideal state without considering air release by using the whole tube as the calculation segment, assigned it to the initial value of the flow rate, and calculated the instantaneous flow rate at the corresponding position according to Equation (51) combined with the change of air pressure occurring in the tube with height up to the horizontal segment. When the iteration proceeds to the horizontal section, a determination of the type of dry or wet slug flow is required. It can be seen that the pressure drop at both ends of the bubble decreased sharply when the capillary number was less than \(10^{-3}\) from Figure 3 as well as Equation (44). Therefore, to simplify the calculation, the wet slug flow or dry slug flow can be judged according to whether the _Ca_ is greater than \(10^{-3}\).