## 2 Basic Analysis of Prediction Models and Model Verification
### Prediction Models of Surface Tension  
#### 3.3.1 Prediction Models of Surface Tension  
A correlation between the dimensionless surface tension \(\sigma_{r}=\sigma{p_{c}}^{-2/3}{T_{c}}^{-1/3}\) (or \(\sigma_{r}=\sigma{V_{c}}^{2/3}{T_{c}}^{-1}\)) and the contrast temperature \(T_{r}\) based on the principle of corresponding states (PCS) was proposed by [PERSON] et al. [99], as shown in Equation (54):  
\[\sigma=p_{c}^{2/3}T_{c}^{1/3}\text{ST}(1-T_{r})^{11/9} \tag{54}\]  
where \(ST\) is the function obtained by fitting the physical property data of many substances. Because the correlation proposed by [PERSON] et al. [99] was only for non-polar fluids, some researchers subsequently modified the parameter \(ST\) in Equation (54) to make it adapt to polar fluids. [PERSON] et al. [100] fitted \(ST\) to a function of the eccentricity factor \(\omega\), shown in Equation (55) (hereinafter referred to as the PST model). In addition, the surface tensions of more than 1700 kinds of fluids by genetic algorithms were calculated by [PERSON] et al. [101], and the corresponding Equation (56) (hereinafter referred to as the GST model) was obtained.  
\[\Sigma=p_{c}^{2/3}T_{c}^{1/3}\frac{1.86+1.18\omega}{19.05}\left[\frac{3.75+0.9 1\omega}{0.291-0.08\omega}\right]^{2/3}(1-T_{r})^{11/9} \tag{55}\]  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Electronic Fluoride Liquid** & **PR** & **MAPK1** & **MAPK2** & **VTPR1** & **VTPR2** & **VTPR3** & **MPPR** & **PT** & **NEOS** \\ \hline \multirow{2}{*}{FC-72} & ARD (\%) & 3.74 & 3.81 & 3.77 & 1.00 & 0.41 & 3.06 & 8.64 & 9.60 & 2.95 \\  & MRD (\%) & 4.19 & 4.25 & 4.23 & 1.34 & 0.65 & 3.44 & 9.69 & 10.06 & 3.93 \\  & ARD (\%) & 2.99 & 3.05 & 3.00 & 2.67 & 1.67 & 4.32 & 7.25 & 8.35 & 2.82 \\  & MRD (\%) & 3.28 & 3.34 & 3.31 & 2.85 & 2.33 & 4.55 & 8.36 & 8.64 & 4.09 \\  & ARD (\%) & 3.86 & 3.91 & 3.87 & 1.35 & 0.67 & 3.47 & 7.72 & 8.07 & 5.68 \\  & MRD (\%) & 4.17 & 4.23 & 4.17 & 2.00 & 1.52 & 3.86 & 7.98 & 8.47 & 6.85 \\ \hline \hline \end{tabular}
\end{table}
Table 6: ARDs and MRDs between the predicted and reference values of different prediction models of the saturated liquid densities of electronic fluoride liquids.  
Figure 6: Relative deviations between the predicted and reference values [66] of the saturated liquid densities of Novec 649.  
\[\sigma=p_{c}^{2/3}T_{c}^{1/3}[7.728729\,*10^{-4}T_{br}+2.476318\,*10^{-4}(T_{br}^{3 }+V_{c})](1-T_{r})^{11/9} \tag{56}\]  
where \(T_{br}\) is the contrast temperature under the boiling point, K.  
[PERSON] et al. [102] proposed a prediction model of the surface tension suitable for N\({}_{2}\), O\({}_{2}\), rare gases, alkanes and refrigerants based on the previous studies [103; 101], as shown in Equation (57) (hereinafter referred to as the MST model). The model proposed by [PERSON] et al. [104] was obtained by fitting the surface tension data of methane, ethane and propane halogenated hydrocarbons, as shown in Equation (58) (hereinafter referred to as the DST model).  
\[\Sigma=kT_{c}\bigg{(}\frac{N_{A}}{V_{c}}\bigg{)}^{2/3}(4.35+4.14\,\omega)(1-T_{ r})^{1.26}(1+0.19\,(1-T_{r})^{0.5}-0.25(1-T_{r})) \tag{57}\]  
\[\sigma=0.57088p_{c}^{2/3}T_{c}^{1/3}(1+0.7467\,\omega)(1-T_{r})^{1.26}\,\,(1+0.0973\,(1-T_{r})^{0.5}-0.1885(1-T_{r})) \tag{58}\]  
In addition, [PERSON] et al. [105; 106] optimized all exponents based on the surface tension data of some refrigerants and two prediction models were successively proposed, as shown in Equations (59) and (60) (hereinafter referred to as the NST1 model and the NST2 model, respectively). In the NST1 model, the molecular dipole moment \(\mu\) is used to reflect the influence of molecular polarity on the surface tension of the fluids.  
\[\Sigma=0.813p_{c}^{0.565}T_{c}^{0.364}(4.259\,\omega-0.645\frac{4300\mu^{2}}{T _{c}V_{c}})^{0.143}(1-T_{r})^{1.266} \tag{59}\]  
\[\sigma=0.658p_{c}^{0.618}T_{c}^{0.34}(1+\omega)^{0.77}\,(1-T_{r})^{1.262} \tag{60}\]  
The prediction models of the surface tensions of electronic fluoride liquids are summarized in Table 7.