## 3 Results
### Mathematical Model
#### 3.2.2 Determination of the Critical Temperature and Pressure of Hydrocarbon Fuels  
Methods for calculating the thermophysical properties of liquids and gases are often described in the works of scientists [45; 48]. For these calculations, it is necessary to know their critical parameters. The critical temperatures for hydrocarbon fuels are usually found using the Thomas and Smith method [48]. The formula derived by the authors is often used for viscosity assessment when the liquid temperature \(T_{liq}<0.7\cdot T_{cr}\). Knowing the viscosity of the liquid when the temperature values satisfy the conditions mentioned above, we calculate the critical temperature by solving the inverse problem (Table 2):  
\[T_{cr}=T_{liq}\cdot\left[\frac{1}{\theta}\cdot lg\bigg{(}\frac{8.569\cdot\mu} {\sqrt{\rho}}\bigg{)}+1\right] \tag{17}\]  
Using this method, [PERSON] and [PERSON] [49] have found that there was a \(\pm 15\%\) probability of calculation error for large molecular compounds such as polycyclic aromatic hydrocarbon molecules and ethers. The critical temperature \(T_{cr}=716.5\) K for RME was calculated according to the method proposed by [PERSON] and [PERSON] [48]. Other input parameters were \(\mu=5\) cP, \(\rho=0.9\) g/cm\({}^{3}\), \(T=313.2\) K and \(\theta=1.285\). The boiling point of a substance can also be calculated using the Law of Additivity (\(\theta=\frac{T_{b}}{T_{cr}}\)), so it is easy to find the value of the critical temperature of a liquid, knowing its boiling point and structural formula. When applying this method, the coefficient \(\theta\) was calculated using empirical formulas. In the present work, the Lydersen method [50] was used to determine the boiling point of the RME, as this method is highly accurate (Table 3):  
\[\theta=0.567+\sum\Delta\theta+\left(\sum\Delta\theta\right)^{2} \tag{18}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Atoms, Groups, Bonds** & **Parts\(\Delta\theta\)** & **Number of Parts** & \(\sum\Delta\theta\) \\ \hline C & \(-0.462\) & 18 & \(-8.316\) \\ \hline H & +0.249 & 36 & +8.964 \\ \hline O & +0.054 & 1 & +0.054 \\ \hline Double bonds & +0.478 & 1 & +0.478 \\ CO (ketones, ethers) & +0.105 & 1 & +0.105 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Data to calculate the coefficient \(\theta\) according to Formula (17). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Atoms, Groups, Bonds** & **Parts\(\Delta\theta\)** & **Number of Parts** & \(\sum\Delta\theta\) \\ \hline \(-\)CH\({}_{3}\) & 0.020 & 2 & 0.040 \\ \hline \(-\)CH\({}_{2}\)– & 0.020 & 14 & 0.280 \\ \hline \(-\)O– & 0.021 & 1 & 0.021 \\ \hline \(=\)CO & 0.040 & 1 & 0.040 \\ \hline \(-\)CH & 0.012 & 2 & 0.024 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Data to calculate the coefficient \(\theta\) according to Formula (18). Considering that the boiling point of RME is \(T_{b}=610\) K and \(\theta=0.8080\), the critical temperature \(T_{cr}=755\) K was determined from the relationship \(\theta=\frac{T_{b}}{T_{cr}}\). The value \(T_{cr}=785.7\) K was determined according to the recommendations provided in [48; 50]. It has close analogies to the results obtained by other researchers who also investigated different types of fuel, which consisted of molecules of the same chemical composition, but different molecular structure [51]:  
\[T_{cr}=355+0.97\cdot a-0.00049\cdot a^{2} \tag{19}\]  
where \(a=(1.8\cdot T_{50}-359)\cdot\rho_{15}^{15}\). Equation (19) was used to determine the critical temperature of a diesel fuel fraction that led to the result of \(T_{cr}=739.4\) K. In the calculation, the evaporation temperature of the 50% mineral diesel fraction was chosen. The evaporation temperature of the 50% diesel fuel fraction was assumed to be equal to \(T_{50}=553\) K and \(\rho_{15}^{15}=0.8607\).