## 4 Results and Discussion
### Derivation of the General Correlation for Condensation Heat Transfer in Small-Diameter Smooth Tube  
From Figure 6, it is found that the experimental values for the small-diameter smooth tube could not be correlated with sufficiently high accuracy, and could not be easily predicted by using the existing correlation. Therefore, we tried to derive an empirical correlation to predict the heat transfer coefficients, for many types of refrigerants, based on the [PERSON] et al. [2] correlation. In their correlation, [PERSON] et al. defined the \(Nu\) number as  
\[Nu=(Nu_{\text{F}}{}^{\text{a}}+Nu_{\text{B}}{}^{\text{a}})^{\frac{1}{2}}, \tag{15}\]  
where \(Nu_{\text{F}}\) is the forced convection condensation Nusselt number, and \(Nu_{\text{B}}\) is the free convection condensation Nusselt number. [PERSON]'s modified Froude number \(Fr_{\text{so}}\) was used to distinguish the wavy flow and annular flow pattern in the small-diameter smooth tube. [PERSON] and [PERSON] [3] have reported that the flow is annular when \(Fr_{\text{so}}>20\), and wavy when \(Fr_{\text{so}}<20\), based on their experimental results for a small-diameter smooth tube. The [PERSON]'s modified Froude number \(Fr_{\text{so}}\) is found by  
\[Fr_{\text{so}}=0.025 Re_{\text{L}}^{1.59}\Big{\{}\Big{(}1+1.09\chi_{\text{tt}}^ {0.039}\Big{)}/\chi_{\text{tt}}\Big{\}}^{1.5}/Ga^{0.5}(Re_{\text{L}}\leq 1250), \tag{16}\]  
Figure 6: Comparison of measured and predicted heat transfer coefficient values. \[Fr_{\text{so}}=1.26 Re_{\text{L}}^{1.04}\Big{\{}\Big{(}1+1.09\lambda_{\text{tt}}^{0.039}\Big{)}/\lambda_{\text{tt}}\Big{\}}^{1.5}/Ga^{0.5}(Re_{\text{L}}>1250), \tag{17}\]  
where \(Re_{\text{L}}\) is the liquid Reynolds number of the refrigerant, and is defined as follows:  
\[Re_{\text{L}}=\left\{G(1-x)\ d_{\text{i}}\right\}/\mu_{\text{L}}, \tag{18}\]  
where it conducts the optimization of the free convection condensation Nusselt number \(Nu_{\text{B}}\), which is wavy when \(Fr_{\text{so}}<20\), as follows:  
\[Nu_{\text{B}}=0.297\ Bo^{0.11}H(\xi)(GaPr_{\text{L}}/Ph_{\text{L}})^{0.25}, \tag{19}\]  
where \(H(\xi)\) and a void fraction \(\xi\) are found by Equations (20) and (8). \[H(\xi)=\xi+\Big{\{}10(1-\xi)^{0.1}-10+1.7\times 10^{-4}Re_{\text{L}}\Big{\}} \sqrt{\xi}\Big{(}1-\sqrt{\xi}\Big{)} \tag{20}\]  
\(Nu_{\text{F}}\) is defined as  
\[Nu_{\text{F}}=\sqrt{0.5f_{\text{V}}}Re_{\text{L}}\Phi_{\text{V}}(\rho_{\text{ L}}/\rho_{\text{V}})^{0.5}\{x/(1-x)\}(Pr_{\text{L}}/T_{\text{i}}^{*}), \tag{21}\]  
where \(\Phi_{\text{V}}\) and \(f_{\text{V}}\) are the friction factor and the two-phase pressure drop multiplier from Equations (11) and (13), respectively. (\(Pr/T_{\text{i}}^{*}\)) is obtained from the following equation, by using the Nusselt number \(Nu\) from Equation (23). \[(Pr_{\text{L}}/T_{\text{i}}^{*})=\frac{(Nu^{\text{a}}-Nu_{\text{B}}{}^{\text {a}})^{\frac{1}{\text{a}}}}{\sqrt{0.5f_{\text{V}}}Re_{\text{L}}\Phi_{\text{V} }(\rho_{\text{L}}/\rho_{\text{V}})^{0.5}\{x/(1-x)\}}, \tag{22}\]  
\[Nu=\alpha d_{\text{i}}/\lambda_{\text{L}}, \tag{23}\]  
where \(\lambda_{\text{L}}\) is the thermal conductivity of the refrigerant. Figure 7a shows the relationship between (\(Pr/T_{\text{i}}\)*) and \(Re_{\text{L}}\) at the annular flow area (\(Fr_{\text{so}}>20\)). The following correlations were finally obtained within \(\pm 30\%\), then, the minimal error between the experimental and predicted values was lowest by \(a=5\).