# Embedded One-Dimensional Orifice Elements for Slosh Load Calculations in Volume-Of-Fluid CFD
## 2 Materials and Methods
### Orifice Flow Losses
#### 2.1.3 Two-Phase Flow Losses  
For two-phase flow, additional losses occur through the orifice. We use the separated flow model [35; 42], which takes the total pressure drop over a pipe section or orifice for a single-phase flow and adds a two-phase multiplier \(\phi^{2}\) (determined experimentally). This multiplier accounts for mixing effects and is calculated at every time step according to the flow conditions. A multiplier coefficient is calculated for viscous losses as well as for contracting and expanding losses. The Baroczy-Chisolm method [35] is used to determine the two-phase multiplier for viscous losses, and the method of [PERSON] [34] (based on the Baroczy-Chisolm method) is used to determine the two-phase losses due to contracting and expanding flow. For the Baroczy-Chisolm method [35], the dryness fraction is defined as  
\[\mathrm{x}=\frac{\rho_{g}\bar{u}_{g}A_{g}}{\rho_{l}\bar{u}_{l}A_{l}+\rho_{g} \bar{u}_{g}A_{g}}\,, \tag{9}\]  
indicating the ratio of the mass flow rate of the gas to the liquid-gas mixture, where the subscripts \(l\) and \(g\) are used to distinguish between the liquid and gas phase, respectively. Further, \(A_{l}\) and \(A_{g}\) are the cross-sectional areas of the orifice occupied by the liquid and gas, respectively. The void fraction \(\xi\) is the ratio between the volume of the orifice occupied by the gas \(V_{g}\) to the total orifice volume \(V_{lg}\):  
\[\xi=\frac{V_{g}}{V_{lg}}\,. \tag{10}\]  
In general, we employ the subscript \(lg\) to indicate the two-phase mixture. The mass velocity \(G\) is used extensively in one-dimensional two-phase flow literature and is given as  
\[G=\rho_{l}\tilde{u}_{l}\frac{1-\xi}{1-\mathrm{x}}=\rho_{g}\tilde{u}_{g}\frac{ \xi}{\mathrm{x}}\,. \tag{11}\]  
Finally, the physical property coefficient is given as  
\[\Gamma^{2}=\frac{\Delta p_{g0}}{\Delta p_{l0}}\,, \tag{12}\]  
where subscripts \(g0\) and \(l0\) refer to the gas or liquid phase, respectively, if it were to flow through the orifice at the mixture's mass flow rate. The total pressure differential due to viscous effects of the two-phase mixture \(\Delta p_{lg}\) is given as  
\[\Delta p_{lg}=\phi_{l0}^{2}\Delta p_{l0}\,. \tag{13}\]  
The two-phase multiplier, using the Baroczy-Chisolm method, is defined as  
\[\phi_{l0}^{2}=1+(\Gamma^{2}-1)(B\mathrm{x}(1-\mathrm{x})+\mathrm{x}^{2})\,, \tag{14}\]  
where the coefficient \(B\) can be found in Table 1[35] and the physical property coefficient \(\Gamma\) can be determined using Equation (12). \(\Delta p_{l0}\) and \(\Delta p_{g0}\) are found using Equation (3) for the liquid  
\[\Delta p_{l0}=\frac{K_{f0}L}{D_{H}}\frac{G^{2}}{2\rho_{l}} \tag{15}\]  
and gas phase  
\[\Delta p_{g0}=\frac{K_{fg0}L}{D_{H}}\frac{G^{2}}{2\rho_{g}}\,, \tag{16}\]  
with the mass velocity \(G\) calculated using (11). \(K_{fg0}\) and \(K_{f0}\) are found using Equation (4). Note that the Reynolds number in Equation (4) is, for two-phase flow problems, determined as  
\[Re=\frac{GD_{H}}{\tilde{\mu}_{u}}\,, \tag{17}\]  
where \(\tilde{\mu}_{u}\) is the viscosity of the respective phase being dealt with. Since the two-phase effects are accounted for separately, the hydraulic diameter \(D_{H}\) is calculated using the full perimeter of the orifice. \begin{table}
\begin{tabular}{c c c} \hline \hline \(\mathbf{\Gamma}\) & \(\mathbf{G}\left(\frac{\mathbf{kg}}{\mathrm{m}^{3}}\right)\) & \(\mathbf{B}\) \\ \hline \multirow{4}{*}{\(\Gamma\leq 9.5\)} & \(G\leq 500\) & 4.8 \\ \cline{2-3}  & \(500<G<1900\) & \(\frac{2400}{G}\) \\ \cline{2-3}  & \(G\geq 1900\) & \(\frac{85}{G^{3}}\) \\ \hline \multirow{2}{*}{\(9.5<\Gamma<28\)} & \(G\leq 600\) & \(\frac{520}{G^{3}}\) \\ \cline{2-3}  & \(G>600\) & \(\frac{21}{G}\) \\ \hline \(\Gamma\geq 28\) & & \(\frac{15000}{G^{3}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Values of \(B\) in the Baroczy–Chisolm method.