## 3 Results
### Mesured Frictional Pressure Drop
#### 3.1.1 Circular Tubes  
This section examines the frictional pressure drop in circular tubes, whose data were obtained at wide ranges of the inside diameter. Figures 5 and 6 show the relation between the measured frictional pressure drop gradients \(\Delta P_{\rm f}/\Delta L\) (Pa\(\cdot\)m\({}^{-1}\)) and quality x (-) flowing vertically upward and downward, respectively. The parameter of these figures is the mass flux. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & \(N\) & **vv** & **vt** & **tv** & **tt** \\ \hline \multirow{2}{*}{Circular} & Up & 66 & 14 & 48 & 2 & 2 \\  & Down & 131 & 41 & 90 & 0 & 0 \\ \hline \multirow{2}{*}{Rectangular} & Up & 40 & 8 & 28 & 1 & 3 \\  & Down & 36 & 15 & 21 & 0 & 0 \\ \hline \multirow{2}{*}{Triangular} & Up & 39 & 10 & 27 & 1 & 1 \\  & Down & 39 & 20 & 19 & 0 & 0 \\ \hline \multirow{2}{*}{Total} & Number & 351 & 108 & 233 & 4 & 6 \\  & \% & 100.0 & 30.8 & 66.4 & 1.1 & 1.7 \\ \hline \multicolumn{7}{l}{laminar: \(Re_{\rm L}\), \(Re_{\rm G}<2000\), turbulent: \(Re_{\rm L}\), \(Re_{\rm G}>2000\).} \\ \end{tabular}
\end{table} 
Table 2: All measured frictional pressure drop data. Figure 4: Distribution of all data in plot of \(Re_{\rm L}\) vs. \(Re_{\rm G}\). Figure 5: Frictional pressure drop vs. quality in upward flow. (Mass fluxes are parameters). Figure 6: _Cont._  
As in the case of the conventional large circular tube, the measured frictional pressure drop in small circular tubes increased with increasing quality in the wide range with a maximum at around 0.8; after that, it mildly decreased with increasing quality to around quality 1.0. Moreover, increasing mass flux increased the frictional pressure drop. To examine the effect of the different inner diameters, Figure 7 shows the relation between the measured frictional pressure drop gradients \(\Delta P_{\rm f}/\Delta L\) and quality x flowing vertically downward; the inner diameter is the parameter. As in the case of the conventional large tubes, decreasing the inner diameter of tube increased the frictional pressure drop. However, the frictional pressure drop showed a different tendency with mass flux and the inner diameter. For the high mass flux condition (200 kg\(\cdot\)m\({}^{-2}\cdot\)s\({}^{-1}\) and 400 kg\(\cdot\)m\({}^{-2}\cdot\)s\({}^{-1}\)), the frictional pressure drop monotonically increased with increasing quality in the range of from 0.05 to 0.8. On the other hand, for the low mass flux (50 kg\(\cdot\)m\({}^{-2}\cdot\)s\({}^{-1}\) and 100 kg\(\cdot\)m\({}^{-2}\cdot\)s\({}^{-1}\)), the frictional pressure drop decreased with increasing quality in the low-quality region; its values were not affected by the different mass flux or inner diameter, and these are given in Figures 6 and 7. After that, the smaller the inner diameter, the higher the increase in frictional pressure drop at low quality. This physical phenomenon could also be confirmable in the upward flow, as shown in Figure 5. The phenomenon was confirmed by [PERSON] and [PERSON] (1965) [13], [PERSON] and [PERSON] (1990) [1] and [PERSON] et al. (1991, 1992) [4; 5]. [PERSON] and [PERSON] performed an experiment on air-water mixtures flowing vertically upward in a circular tube with diameters ranging from 5 to 28.8 mm. They reported that the frictional pressure drop decreased with increasing quality in the low-quality region, and it became evident at a lower mass flux. [PERSON] and [PERSON] ran experiments using 10 rectangular tubes with different aspect ratios (1-40) and arranging them in an inclination angle (from horizontally to vertically). For the case of small channels with a large aspect ratio and hydraulic diameters smaller than about 10 mm, they noted that the frictional pressure drop made a bump in the vicinity of the transition from bubble to bubbly slug flow. [PERSON] et al. measured the frictional pressure drop of air-water mixtures flowing horizontally in a rectangular channel (19.05, 3.18 mm).