## 3 Results
### Mathematical Model
#### 3.2.1 Determination of Thermal Conductivity of the Vapor Phase  
Diesel fuel is made up of a mixture of several different hydrocarbons, such as paraffins, aromatics, and naphthene, and biodiesel is composed of mono-alkyl esters of long-chain fatty acids. It is possible to determine the thermal conductivity of a gas from the process in which the excited state of a molecule transfers energy to another molecule, or alternatively, from the internal friction values or from the heat diffusivity. At low pressures and constant volume, it depends on the distance molecules travel between collisions, the average molecular speed, the gas density, and the specific heat capacity of a substance:  
\[\lambda^{gas}=\frac{1}{3}\cdot c_{V}\cdot\rho\cdot\bar{w}\cdot\bar{I} \tag{6}\]  
The dynamic viscosity of the gas was determined with the following formula:  
\[\mu=\frac{1}{3}\cdot\rho\cdot\bar{w}\cdot\bar{I} \tag{7}\]  
Dividing Equation (6) by Equation (7), the following expression was obtained:  
\[\lambda^{gas}=\mu\cdot c_{V}\rightarrow\ \lambda^{gas}=K\cdot\mu\cdot c_{V} \tag{8}\]  
The formula derived by [PERSON] and [PERSON] [44], which was used to determine the thermal conductivity of methane, petroleum and aromatic hydrocarbons at low pressures (\(0.2<p<5\)_at_) and temperatures below its critical values, can be written as  
\[\lambda^{gas}=0.445\cdot 10^{-5}\cdot\frac{C_{p}^{gas}}{\Lambda}\cdot\frac{T}{T _{cr}} \tag{9}\]  
For all other hydrocarbons with temperatures exceeding \(0.6\cdot T_{cr}\), thermal conductivity was calculated using the formula  
\[\lambda^{gas}=\left(14.52\cdot\frac{T}{T_{cr}}-5.14\right)^{\frac{2}{3}}\cdot \frac{c_{p}^{gas}}{\Lambda}\cdot 10^{-6} \tag{10}\]  
According to the authors, the average error in calculating the thermal conductivity using Formula (10) varies from 1.8% to 2.6%. Furthermore, the method proposed by [PERSON] and [PERSON] [44] was also used to establish the thermal conductivity value of diesel fuel gases at \(T=293\) K, whose value was equal to \(\lambda^{gas}=95.72\cdot 10^{-4}\) W/(m\(\cdot\)K). In practice, the thermal conductivity of gases is often determined from the expression of the Prandtlcriterion \(Pr=c_{p}^{gas}\cdot\mu/\lambda^{gas}\). Alternatively, this criterion can be determined using the formula given in [15]:  
\[Pr=\frac{c_{p}^{gas}}{c_{p}^{gas}+2.46} \tag{11}\]  
When the molecular weight of the gas is relatively small (M \(<\) 15), the values calculated using this formula agree quite accurately with the results of the experimental studies. [PERSON] [45] derived an analogous relationship that allows one to calculate the thermal conductivity of materials of high molecular weight:  
\[Pr=\frac{c_{p}^{gas}}{1.204\cdot c_{p}^{gas}+1.47} \tag{12}\]  
The average error in calculating the molecular weight using Formulas (11) and (12) does not exceed 15% [15]. The result of the calculation using Equation (12) obtained for RME was \(Pr=0.819\). The mathematical expression \(\lambda^{gas}=\frac{c_{p}^{gas}\cdot\mu}{Pr}\) used for the assessment of the thermal conductivity of RME at \(T=293\) K shows the calculated result of \(\lambda^{gas}=64.35\cdot 10^{-4}\) W/(m\(\cdot\)K), whose numerical value is very similar to that obtained using the equation proposed in [15]. Refs.