## 3 Results
### Comparison of Various Modified Chisholm Parameters  
In this section, the various modified [PERSON]'s parameter in Lockhart-Martinelli correlation for small tubes are examined, and it is determined whether or not they reproduce the measurement data. As stated previously, [PERSON]'s parameter in Lockhart-Martinelli correlation is often used for the estimation of frictional pressure drop in conventional large circular tubes. However, several researchers reported that the original [PERSON]'s parameter showed a disagreement with the measurements in small channels. I also confirmed this fact in Figure 15. Therefore, the various modified [PERSON]'s parameter were suggested. Thus, in this section, the calculated values using five modified Chisholm's parameters are compared with the measurements data, except for the data of the decreasing frictional pressure drop with increasing quality, because the correlation including this tendency has not yet been proposed. [PERSON] et al. (1991, 1992) [4; 5] measured the frictional pressure drop of air-water mixtures flowing horizontally in a rectangular channel (19.05, 3.18 mm). Their experimental results were compared with [PERSON]'s correlation. However, at the lower mass flux condition, the measurement data generally followed the form of [PERSON]'s correlation, and the optimum [PERSON]'s parameter increased with increasing mass flux. Figure 16: Comparison of the measured frictional pressure drop between upward and downward flow in circular, rectangular, and triangular tubes. As a consequence, they deemed [PERSON]'s parameter to be not a constant but a function of the Lockhart-Martinelli parameter and mass flux. The modified parameter is as follows:  
\[C=f(\chi,Re_{\rm LO})=a\chi^{b} \tag{13}\]  
\[Re_{\rm LO}=\frac{GD}{\mu_{\rm L}} \tag{14}\]  
where \(Re_{\rm LO}\) is Reynolds number for mixture flowing as liquid, a = \(-2.44+0.00939\)\(Re_{\rm LO}\) and b = \(-0.938+0.000432\)\(Re_{\rm LO}\). This correlation is applicable for \(Re_{\rm LO}\) < 2200 and <1.0. The reported average error of using this parameter was \(\pm 19\%\) for \(G\leqq 400\) kg\(\cdot\)m\({}^{-2}\cdot\)s\({}^{-1}\). [PERSON] and co-workers (1993, 1996) [7; 8] reported on air-water mixtures flowing vertically in circular and rectangular tubes with a diameter range of from 1 to 5 mm. They reported that if the inner diameter is less than 5 mm, [PERSON]'s parameter should consider the effect of the inner diameter, because they found [PERSON]'s parameter to change from 21 to 0 as the hydraulic diameter decreased. Therefore, they suggested that the modified parameter is given by  
\[C=f(d)=21\left(1-e^{-0.333d}\right) \tag{15}\]  
where \(d\) in mm is the hydraulic diameter of the channel, and this parameter is the dimension. The comparison of this parameter with the database demonstrated a good agreement, except for ammonia vapor flow within an error of \(\pm 12\%\). [PERSON] and [PERSON] (2001) [10] proposed the modified [PERSON]'s parameter based on 305 data points of horizontal rectangular channels with an air-water mixture flowing through them ((0.4, 1.0, 2.0, 4.0) 20 mm). They stated that [PERSON]'s parameter depends on the flow regimes, and that the flow regimes depend on the mass flux and hydraulic diameter. Hence, the parameter \(C\) should consider the effects of the mass flux and hydraulic diameter. On the other hand, the surface tension is an important factor in the laminar (liquid)-laminar (gas or vapor) regime (vv); therefore, they proposed the non-dimensional parameter considering the effect of surface tension.