# Uncertainty Propagation through a Point Model for Steady-State Two-Phase Pipe Flow
## 2 Materials and Methods
### Flow Model
#### 2.1.2 Bubble Zone  
Let \(H_{\mathrm{g}}=1-H_{\mathrm{l}}\) be the gas fraction in the bubble zone. The liquid holdup in the bubble zone is the solution to the momentum balance for both phases, that is \(H_{\mathrm{l}}\) such that  
\[\rho_{\mathrm{g}}f_{\mathrm{g}}S_{\mathrm{g}}u_{\mathrm{g}}^{2}H_{\mathrm{g}}^ {3}-\rho_{\mathrm{l}}f_{\mathrm{l}}S_{\mathrm{l}}u_{\mathrm{l}}^{2}H_{\mathrm{ l}}^{3}+\rho_{\mathrm{g}}f_{\mathrm{i}}S_{\mathrm{i}}(u_{\mathrm{l}}H_{ \mathrm{l}}-u_{\mathrm{g}}H_{\mathrm{g}})|u_{\mathrm{l}}H_{\mathrm{l}}-u_{ \mathrm{g}}H_{\mathrm{g}}|+(\rho_{\mathrm{l}}-\rho_{\mathrm{g}})\tfrac{\pi}{4} D^{2}H_{\mathrm{g}}^{3}H_{\mathrm{l}}^{3}g\sin\theta=0, \tag{7}\]  
where the perimeters \(S_{p}\) and \(S_{\mathrm{i}}\) are defined in Figure 1, and the friction factors \(f_{p}\) and \(f_{\mathrm{i}}\) are defined below. The superficial velocities \(u_{p}\) must be chosen according to the flow regime. The superficial velocities equal the average superficial velocities in Equation (2) for stratified flow, while they are functions of the bubble zone holdup for slug flow, namely  
\[\text{Stratified flow:}\quad\quad u_{p} =U_{p} \tag{8a}\] \[\text{Sug flow:}\quad\begin{cases}u_{\mathrm{g}}=U_{\mathrm{g}}+ \Delta\\ u_{\mathrm{l}}=U_{\mathrm{l}}-\Delta.\end{cases} \tag{8b}\]where \(\Delta=(H_{\rm s}-H_{\rm I})u_{\rm b}+(1-H_{\rm s})U_{\rm I}-H_{\rm s}U_{\rm g}\). Furthermore, the interface friction factor is a modified version of the expression proposed in [PERSON] and [PERSON] (2016) and is given by  
\[f_{\rm i}=f_{\rm i0}\left(1+10h\left(\frac{2\mu_{\rm g}}{H_{\rm g}D}\sqrt{\frac{ S_{\rm i}\rho_{\rm g}}{\pi H_{\rm g}g(\rho_{\rm I}-\rho_{\rm g})\cos\theta}}-0.36 \right)^{0.67}\right)\,, \tag{9}\]  
where \(h\) is the line fraction approximated by  
\[h=1-\cos\left(\frac{1}{3}+\left(\frac{3\pi}{2}\right)^{\frac{1}{3}}+\left(\pi -\frac{2}{3}\right)H_{\rm I}+H_{\rm I}^{\frac{1}{3}}+H_{\rm g}^{\frac{1}{3}}\right) \tag{10}\]  
and we impose a minimum of \(f_{\rm i0}=f_{\rm g}(\varepsilon=0)\). The friction factors \(f_{\rm i0}\), \(f_{\rm g}\), \(f_{\rm I}\) and \(f_{\rm s}\) are interpolated from laminar and turbulent values, as described in Section 2.1.5. We use the Hagen-Poiseuille and Haaland formulas found in [PERSON] (2016) given as  
\[f_{p}^{\rm I} =16\text{Re}_{p}^{-1} \tag{11a}\] \[f_{p}^{\rm I} =0.07716\cdot\log_{10}^{-2}\left(6.9\text{Re}_{p}^{-1}+0.234 \left(\frac{\varepsilon}{D}\right)^{1.11}\right)\,, \tag{11b}\]  
where  
\[\text{Re}_{p}^{-1}=\begin{cases}\frac{\mu_{\rm g}(S_{\rm g}+S_{ \rm i})}{\pi D^{2}\rho_{\rm g}|u_{\rm g}|}\,,&p=\rm g\\ \frac{\mu_{\rm I}S_{\rm I}}{\pi D^{2}\rho_{\rm I}|u_{\rm I}|}\,,&p=\rm I\\ \frac{\mu_{\rm I}}{D\rho_{\rm m}^{\rm s}U_{\rm m}}\,,&p=\rm s\end{cases} \tag{12}\]  
and \(\rho_{\rm m}^{p}=(1-H_{p})\rho_{\rm g}+H_{p}\rho_{\rm I}\) is the mix density where \(p\) is \(\rm I\) or \(\rm s\) for bubble zone or slug zone, respectively. The comparative study in [PERSON] and [PERSON] (2016) suggests a more accurate and computationally efficient approximation than Equation (11b) for the Colebrook turbulent friction factor. However, model tuning is not the main objective for this work. Instead, we use the Haaland approximation to allow for comparison to [PERSON] et al. (2018). The friction factor is a small contributor to the computational cost of the point model. Thus, it is not essential to find the most efficient approximation.