## 2 Basic Analysis of Prediction Models and Model Verification
### Prediction Models of Saturated Vapor Pressure and Saturated Density
#### 3.2.1 Prediction Models of Saturated Vapor Pressure and Saturated Density  
Peng-Robinson Equation of State and Its Modified Models  
The Peng-Robinson equation of state [86] (hereinafter referred to as the PR equation) has been published for more than 40 years and has become one of the best two-parameter cubic equations of state for the calculation of the vapor-liquid equilibrium, which refers to the volumetric properties and thermodynamic properties of pure substances and mixtures. Nowadays, the PR equation has a very wide range of applications in the chemical industry [87; 88; 89; 90; 91; 92]. The PR equation is shown in Equations (5)-(10):  
\[p=\frac{RT}{V\ -\ b}-\frac{a}{V(V+b)+b(V\ -\ b)} \tag{5}\]  
\[a=a_{c}\alpha \tag{6}\]  
\[a_{c}\ \approx\ 0.475235\frac{(RT_{c})^{2}}{p_{c}} \tag{7}\]  
\[b\ \approx\ 0.07796\frac{RT_{c}}{p_{c}} \tag{8}\]  
\[\alpha=[1+m(1\ -\ T_{r}^{0.5})]^{2} \tag{9}\]  
\[\left\{ \begin{array}{ll}m=0.37646+1.54226\omega\ -\ 0.26992\omega^{2}&,\omega\leq 0. 491\\ m=0.379642+1.48503\omega\ -\ 0.164423\omega^{2}+0.016666\omega^{3}&,\omega>0.491 \end{array} \right. \tag{10}\]  
where \(p\) is the pressure, Pa; \(R\) is the gas constant, which equals to \(8.3145\) J/(mol\(\cdot\)K); \(T\) is the temperature, K; \(V\) is the specific volume, m\({}^{3}\)/kg; \(a\) and \(b\) are the parameters of the PR equation; \(\alpha\) is the \(\alpha\) function of the PR equation; \(T_{c}\) is the critical temperature, K; \(p_{c}\) is the critical pressure, Pa; \(m\) is the coefficient related to the eccentricity factor \(\omega\); and \(T_{r}\) is the contrast temperature, \(T_{r}=T/T_{c}\), K.  
However, according to the constants of the two-parameter cubic equation of state determined by the critical condition, only a fixed critical compressibility factor can be obtained. For the PR equation, the critical compressibility factor is \(0.3074\). Meanwhile, \begin{table}
\begin{tabular}{c c c c} \hline
**Working Fluid** & **FC-72** & **Novec 649** & **HFE-7100** \\ \hline Molecular structure & & & \\ CAS \# & 355-42-0 & 756-13-8 & 163702-08-7 \\ Molar mass (g/mol) & 338.04 [83] & 316.04 [69] & 250 [71] \\ Critical temperature (K) & 451.4 [84] & 441.81 [69] & 468.45 [71] \\ Critical pressure (kPa) & 1859 [84] & 1869 [69] & 2230 [71] \\ Critical density (kg/m\({}^{3}\)) & 621 [84] & 606.81 [69] & 555 [71] \\ Eccentricity factor & 0.486 [83] & 0.471 [69] & 0.436 [77] \\ Dipole moment & 0 [83] & 0.43 [69] & 2.4 [71] \\ Boiling point (K) & 330.27 [83] & 322.15 [85] & 333.15 [7] \\ \hline \end{tabular}
\end{table} 
Table 2: Basic thermophysical properties of FC-72, Novec 649 and HFE-7100. the actual critical compressibility factor of fluids mostly varies between 0.21 and 0.31 [88], which may lead to a large deviation in the calculated results of the saturated liquid density. This is a great inevitable limitation of the PR equation. Therefore, the PR equation has been modified by many researchers [87; 88; 89; 90; 91; 92; 93; 94; 95]. For pure substances, the modification of the PR equation can be divided into three categories: the modification of the \(\alpha\) function, the specific volume translation modification, and the modification of the \(a\) and \(b\) parameters. 1 **Modification of \(\alpha\) function**  
In most cases, the modification of the \(\alpha\) function is more beneficial for improving the predictive accuracy of the saturated vapor pressure, but for the liquid density, this modification does not seem to work [89; 90; 91; 92]. [PERSON] et al. [89] optimized the coefficient \(m\) in the \(\alpha\) function based on the physical property data of 1721 pure substances so as to enhance the general applicability of the PR equation and improve the predictive ability of the thermophysical properties such as the saturated vapor pressure, latent heat of vaporization and specific heat. The optimized form of \(m\) is shown in Equation (11) (hereinafter referred to as MAPR1):  
\[m=0.3919+1.4996\omega\ -\ 0.2721\omega^{2}\ +\ 0.1063\omega^{3} \tag{11}\]  
[PERSON] et al.