## 3 Modeling
### Slug Unit Pressure Model in Inclined Pipe
Under this assumption, considering the incompressibility of gas and liquid, the overall liquid phase mass balance on the slug unit is obtained:  
\[V_{\rm LS}L_{\rm u}=\alpha_{\rm s}V_{\rm II}L_{\rm s}+\alpha_{\rm f}V_{\rm L }L_{\rm f} \tag{7}\]  
For a stable slug unit, the length can be known \(L_{\rm u}=L_{\rm s}+L_{\rm f}\). Combined with Equation (7), the slug flow unit length and frequency are obtained:  
\[L_{\rm u}V_{\rm LS}-L_{\rm u}V_{\rm t}\alpha_{\rm f}=\alpha_{\rm s}V_{\rm II} L_{\rm s}+\alpha_{\rm f}V_{\rm L}L_{\rm f}-(L_{\rm s}+L_{\rm f})V_{\rm t}\alpha_{ \rm f} \tag{8}\]  
\[L_{\rm u}=L_{\rm s}\frac{V_{\rm II}\alpha_{\rm s}-V_{\rm t}\alpha_{\rm f}}{V_{ \rm LS}-V_{\rm t}\alpha_{\rm f}} \tag{9}\]  
\(L_{\rm s}\) can be obtained according to [PERSON] and [PERSON] [46] as:  
\[ \begin{cases}L_{\rm s}=30{\rm D}&{\rm D}<0.038{\rm m}\\ \ln L_{\rm s}=-26.6+28.5(\ln{\rm D}+3.67)^{0.1}&{\rm D}\geq 0.038{\rm m} \end{cases}  \tag{10}\]  
\[f=0.048h_{\rm s}V_{\rm II}\frac{2.02+Fr}{{\rm D}} \tag{11}\]  
\[Fr=\frac{{V_{\rm s}}^{2}}{g{\rm D}} \tag{12}\]  
where \(F_{\rm r}\) is the Froude number. It is expressed as a dimensionless parameter of the relative magnitude of fluid inertia force and gravity. Figure 3: Physical model of the slug unit in inclined pipe. For the liquid phase  
\[\text{A}\rho_{\text{L}}\alpha_{\text{s}}(V_{\text{t}}-V_{\text{l1}})=\text{A} \rho_{\text{L}}\alpha_{\text{f}}(V_{\text{t}}-V_{\text{f}}) \tag{13}\]  
After simplification, it can be arranged as:  
\[\alpha_{\text{s}}(V_{\text{t}}-V_{\text{l1}})=\alpha_{\text{f}}(V_{\text{t}}-V_ {\text{f}}) \tag{14}\]  
By analyzing Equation (14), it can be shown that on any section of a slug unit, the total volume flow is constant. For the cross-sections of the liquid slug area and the liquid film area:  
\[V_{\text{m}}=V_{\text{GS}}+V_{\text{LS}}=V_{\text{lg}}(1-\alpha_{\text{s}})+V_ {\text{l1}}\alpha_{\text{s}} \tag{15}\]  
\[V_{\text{m}}=V_{\text{t}}(1-\alpha_{\text{f}})+V_{\text{f}}\alpha_{\text{f}} \tag{16}\]  
\[V_{\text{t}}=CV_{\text{ll}}+V_{\text{d}} \tag{17}\]  
In the above equation, \(C\) and \(V_{\text{d}}\)[47] can be calculated:  
\[C= \begin{cases}1.05+0.15\sin^{2}\theta&Fr^{\prime}<3.5\\ 1.20&Fr^{\prime}\geq 3.5\end{cases}  \tag{18}\]  
\[V_{\text{d}}= \begin{cases}V_{\text{dH}}\cos\theta+V_{\text{dV}}\sin\theta&Fr^ {\prime}<3.5\\ V_{\text{dV}}\sin\theta&Fr^{\prime}\geq 3.5\end{cases}  \tag{19}\]  
\[V_{\text{dH}}=0.54\sqrt{g\text{D}} \tag{20}\]  
\[V_{\text{dV}}=0.35\sqrt{g\text{D}} \tag{21}\]  
\[Fr^{\prime}=\frac{V_{\text{s}}}{\sqrt{g\text{D}}} \tag{22}\]  
where \(C\) is the proportional coefficient, \(V_{\text{d}}\) is the bubble drift velocity in m/s; \(V_{\text{dH}}\) and \(V_{\text{dV}}\) are the floating velocities of bubbles in static liquid in vertical and horizontal pipelines in m/s, respectively; \(\theta\) is the pipeline inclination angle in radians; D is the inner diameter of the pipe in m; and \(V_{\text{m}}\) is the mixing speed of liquid phase and gas phase in liquid slug area in m/s. In the above equations, the expression of the average velocity \(V_{\text{f}}\) in the liquid film area is provided in Equation (14), the liquid true average velocity \(V_{\text{l1}}\) in the liquid slug area can be calculated from Equation (15), and the bubble's average velocity \(V_{\text{t}}\) in the liquid film area can be obtained from Equation (16). Furthermore, the slug average liquid holdup \(\alpha\)L can be defined as:  
\[\alpha_{\text{L}}L_{\text{u}}=\alpha_{\text{s}}L_{\text{s}}+\alpha_{\text{f} }L_{\text{f}} \tag{23}\]  
Combining Equation (7), Equation (14), and Equation (15), it can be shown that,  
\[\alpha_{\text{L}}=\frac{V_{\text{t}}\alpha_{\text{s}}+V_{\text{lg}}(1-\alpha _{\text{s}})-V_{\text{GS}}}{V_{\text{t}}} \tag{24}\]  
\(\alpha_{\text{s}}\) can be obtained as suggested by [PERSON] et al.