## 3 Modeling  
### Slug Unit Pressure Model in Inclined Pipe  
Slug flow is extremely common in gas-liquid mixed transportation pipelines, and it is mainly dominated by liquid slug filling the pipeline section or alternating flow of large bubbles. The experimental pipeline in this study is along a hilly terrain pipe, where the liquid tends to accumulate in the lowest places to block the gas, resulting in slug flow in the up-dip pipeline. The physical model of the inclined pipe slug unit is shown in Figure 3. A liquid slug unit (\(L_{\mathrm{u}}\)) is divided into two areas: the liquid film area (\(L_{\mathrm{f}}\)) and the liquid slug area (\(L_{\mathrm{s}}\)) are displayed in Figure 3. The motion of the slug flow is related to these two areas. If the flow velocity in the liquid slug area is greater than that in the liquid film area, the liquid slug overtakes the liquid film in front, so that part of the liquid slug occupies the liquid film area. Due to gravity and the shear force between the pipe wall and the gas-liquid interphase, the liquid in the tail part of the liquid slug falls off and merges into the liquid film as it cannot catch up with the accelerated part. Under this movement, the liquid slug area is pushed forward, forming a slug flow. In a slug unit, the continuity equations of liquid and gas phases are [45]:  
\[\frac{V_{\mathrm{TS}}\mathrm{A}\rho_{\mathrm{L}}}{f}=\mathrm{A}\rho_{\mathrm{ L}}\alpha_{\mathrm{s}}t_{\mathrm{s}}V_{\mathrm{II}}+\mathrm{A}\rho_{\mathrm{L}} \alpha_{\mathrm{f}}t_{\mathrm{f}}V_{\mathrm{f}} \tag{1}\]  
\[\frac{V_{\mathrm{GS}}\mathrm{A}\rho_{\mathrm{G}}}{f}=\mathrm{A}\rho_{\mathrm{ G}}(1-\alpha_{\mathrm{s}})t_{\mathrm{s}}V_{\mathrm{lg}}+\mathrm{A}\rho_{\mathrm{G}}(1- \alpha_{\mathrm{f}})t_{\mathrm{f}}V_{\mathrm{f}} \tag{2}\]  
where \(V_{\mathrm{LS}}\) and \(V_{\mathrm{GS}}\) are the superficial velocities of liquid and gas phases in m/s. \(V_{\mathrm{II}}\) and \(V_{\mathrm{lg}}\) are the true average velocity of liquid and gas in the liquid slug area in m/s. \(\alpha_{\mathrm{s}}\) and \(\alpha_{\mathrm{f}}\) are the liquid holdup of the liquid slug area and liquid film area. \(V_{\mathrm{m}}\) is the mixing velocity of liquid and gas in the liquid slug area in m/s. \(V_{\mathrm{f}}\) is the average velocity in the liquid film area in m/s. \(V_{\mathrm{t}}\) is the average bubble velocity in the liquid film area in m/s. \(\rho_{\mathrm{L}}\) and \(\rho_{\mathrm{G}}\) are the density of the liquid and the gas in kg/m\({}^{3}\), respectively. \(f\) is the slug frequency in Hz. \begin{table}
\begin{tabular}{c c c} \hline  & **Water** & **Air** \\ \hline Density & 997.05 kg.m\({}^{-3}\) & 1.184 kg.m\({}^{-3}\) \\ \hline Kinetic Viscosity & 8.9008 \(\times\) 10\({}^{-4}\) Pa\(\cdot\)s & 1.849 \(\times\) 10\({}^{-5}\) Pa\(\cdot\)s \\ \hline Surface Tension & \(\sigma_{\mathrm{A/W}}\) = 0.07197 N\(\cdot\)m\({}^{-1}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Physical parameters of fluids. \(t_{\rm s}\) and \(t_{\rm f}\) are the time taken for the liquid slug and bubble, respectively, to pass through a certain point in s.  
Time for the liquid slug to pass through the liquid slug area and liquid film area:  
\[t_{\rm s}=\frac{L_{\rm s}}{V_{\rm m}},\ t_{\rm f}=\frac{L_{\rm f}}{V_{\rm f}}, \tag{3}\]  
Time for bubbles to pass through the liquid slug area and liquid film area:  
\[t_{\rm s}=\frac{L_{\rm s}}{V_{\rm f}},\ t_{\rm f}=\frac{L_{\rm f}}{V_{\rm f}} \tag{4}\]  
Substituting Equation (3) into Equation (1), and Equation (4) into Equation (2), we obtained:  
\[\frac{V_{\rm LS}}{f}=\frac{\alpha_{\rm s}V_{\rm II}L_{\rm s}}{V_{\rm m}}+\alpha _{\rm f}L_{\rm f} \tag{5}\]  
\[\frac{V_{\rm GS}}{f}=\frac{(1-\alpha_{\rm s})V_{\rm Ig}L_{\rm s}}{V_{\rm f}}+( 1-\alpha_{\rm f})L_{\rm f} \tag{6}\]  
where \(L_{\rm s}\) is the liquid slug area length in m and \(L_{\rm f}\) is the liquid film area length in m.  
It is assumed that the slug liquid level in the liquid film area is fully developed and uniform.