## 3 Results
### The Effect of Cross-Section Shapes  
For the estimation of the frictional pressure drop for horizontal or vertical flow in conventional large circular tubes, the Lockhart-Martinelli correlation (1949) [2] is often used with the approximation of the two-phase frictional multiplier \({\Phi_{\mathrm{L}}}^{2}\) (-) suggested by [PERSON] (1969) [5]. This correlation especially agrees well in the annular flow. The Lockhart-Martinelli correlation is expressed as  
\[{\Phi_{\mathrm{L}}^{2}}=\frac{{\Delta P_{\mathrm{f}}}/{\Delta L}}{{\Delta P_{ \mathrm{L}}}/{\Delta L}} \tag{4}\]  
\[\frac{{\Delta P_{\mathrm{L}}}}{{\Delta L}}=\lambda\frac{1}{D}\frac{{G^{2}}{(1 -x)}^{2}}{{2\rho_{\mathrm{L}}}} \tag{5}\]  
where \({\Delta P_{L}}/{\Delta L}\) is the liquid single-phase flow frictional pressure drop gradient estimated by Equation (5), and \(\lambda\) (-) is the coefficient of pipe friction. For the laminar flow (Hagen-Poiseuille correlation):  
\[\lambda=\frac{64}{Re_{\mathrm{L}}} \tag{6}\]  
For the turbulent flow (Blasius correlation):  
\[\lambda=\frac{0.316}{{R_{\mathrm{L}}^{0.25}}} \tag{7}\]  
The approximation of the two-phase frictional multiplier \(\Phi\) (-) by [PERSON] is expressed as  
\[{\Phi_{\mathrm{LChisholm}}^{2}}=1+\frac{C}{\chi}+\frac{1}{\chi^{2}} \tag{8}\]  
where \(\chi\) (-) denotes the Lockhart-Martinelli parameter, and \(\chi\) (-) needs to be used properly by the flow regime situations. \(C\) (-) indicates [PERSON]'s parameter. For the laminar (liquid)-laminar (vapor or gas) regime (vv):  
\[\chi_{\mathrm{vv}}=\left(\frac{1-x}{x}\right)^{0.5}\left(\frac{{\rho_{\mathrm{ G}}}}{{\rho_{\mathrm{L}}}}\right)^{0.5}\left(\frac{{\mu_{\mathrm{L}}}}{{\mu_{ \mathrm{G}}}}\right)^{0.5} \tag{9}\]  
For the laminar (liquid)-turbulent (vapor or gas) regime (vt):  
\[\chi_{\mathrm{vt}}=\left(\frac{16 Re_{\mathrm{L}}^{-1}}{0.046 Re_{\mathrm{G}}^{ -0.2}}\right)^{0.5}\left(\frac{1-x}{x}\right)\left(\frac{{\rho_{\mathrm{G}}}}{ {\rho_{\mathrm{L}}}}\right)^{0.5} \tag{10}\]  
For the turbulent (liquid)-laminar (vapor or gas) regime (tv):  
\[\chi_{\mathrm{tv}}=\left(\frac{0.046}{16}\right)^{0.5}Re_{\mathrm{L}}^{0.4} \left(\frac{1-x}{x}\right)^{0.5}\left(\frac{{\rho_{\mathrm{G}}}}{{\rho_{ \mathrm{L}}}}\right)^{0.5}\left(\frac{{\mu_{\mathrm{L}}}}{{\mu_{\mathrm{G}}}} \right)^{0.5} \tag{11}\]For the turbulent (liquid)-turbulent (vapor or gas) regime (tt):  
\[\chi_{\rm tt}=\left(\frac{1-x}{x}\right)^{0.9}\left(\frac{\rho_{\rm G}}{\rho_{ \rm L}}\right)^{0.5}\left(\frac{\mu_{\rm L}}{\mu_{\rm G}}\right)^{0.1} \tag{12}\]  
Figures 13 and 14 show the relation between \(\Phi_{\rm L}\)2 and the inverse of \(\chi\) flowing vertically upward and downward with the calculation from Equation (8) using \(C=5\) denoted by the dash line and \(C=12\) denoted by the solid line. The error bars indicate the measurement errors. For \(C\) in Equation (8), the values of 5 and 12 are originally given by [PERSON] for the laminar (liquid)-laminar (vapor or gas) and laminar (liquid)-turbulent (vapor or gas) regimes. Figure 13: Two-phase frictional multiplier vs. inverse of Lockhart–Martinelli parameter flowing upward in circular, rectangular, and triangular tubes. In the high \(\chi^{-1}\) region, the characteristics were well correlated by the relation between \(\rm\Phi_{L}\)\({}^{2}\) and \(\chi^{-1}\) independently of the mass flux. Especially at 400 kg\(\cdot\)m\({}^{-2}\cdot\)s\({}^{-1}\), Equation (8) with the parameter C = 12 well reproduced the measurements, and no difference among the tubes was observed, which indicates that the two-phase frictional pressure drop at high mass flux can correlate considering the effect of different hydraulic diameters only flowing in the liquid phase in the tube. Therefore, the cross-section shape is a minor effect at the high mass flux. Likewise, there was no difference in pressure drop at low mass fluxes of 30 and 50 kg\(\cdot\)m\({}^{-2}\cdot\)s\({}^{-1}\). On the other hand, the pressure drop in the non-circular tubes became smaller at 100 and 200 kg\(\cdot\)m\({}^{-2}\cdot\)s\({}^{-1}\), especially in downward flow.