## 3 Results
### Mathematical Model
#### 3.2.4 Determination of the Specific Heat Capacities of Hydrocarbon Fuel Vapors  
Determining the specific heat capacities of the hydrocarbon fuel vapors at constant pressure does not cause any major difficulties if the dependence of enthalpy on temperature \(H(T)\) is known. Then  
\[c_{p}^{gas}=\left(\frac{\partial H}{\partial T}\right)_{p} \tag{27}\]  
The variables required to build the dependence of \(H(T)\) in the temperature range of 323 K to 823 K and having different values of relative density \(\rho_{15}^{15}\) can be found in [PERSON] (1963); [PERSON] (1963). After performing \(H(T)\) calculations for RME vapors having a relative density value (\(\rho_{15}^{15}=0.91\)), an outcome of \(c_{p}^{gas}=1.630\) J/(g\(\cdot\)K) was obtained for a temperature of 323 K. For the assessment of the specific heat capacities of paraffinic combustible hydrocarbon liquids that are derived from petroleum and have relative densities \(\rho_{15}^{15}=0.68\div 0.90\), the following equation can be used [PERSON] (1963); [PERSON] (1963):  
\[c_{p}^{gas}=\frac{4-\rho_{15}^{15}}{1541}\cdot\left(1.8\cdot T+211\right) \tag{28}\]  
According to the proposed expression, the specific heat capacity values of \(c_{p}^{gas}=2.12\) J/(g\(\cdot\)K) were obtained for diesel fuel at \(T=323\) K. Very good correspondence between the obtained value and the calculation output presented in ref. [PERSON] (1963) is more than evident. 3.2.5. Derivation of the Simplified Equation for the Assessment of the Ignition Delay  
Assuming that \(t\)\(\rightarrow\)0, the distance from the ignition point \((r_{1}/r_{0})\) could be written as follows [28]:  
\[\frac{r_{1}}{r_{0}}=\frac{\left(1-\frac{T_{0}}{T_{\infty}}\right)}{1-\frac{E}{4 \cdot R\cdot T_{\infty}}\cdot\left(\sqrt{1+\frac{8\cdot R\cdot T_{\infty}}{E }}-1\right)} \tag{29}\]  
Based on Equations (5) and (29):  
\[\frac{r_{1}}{r_{0}}\cdot exp\bigg{(}\frac{L}{R\cdot T_{1}}-\frac{L}{R\cdot T_ {b}}\bigg{)}=\frac{1}{\phi}+1 \tag{30}\]  
It is evident that the ignition delay depends on both the distance from the ignition point \((r_{1}/r_{0})\) and the temperature of the droplet surface at the moment of ignition \((T_{1})\):  
\[T_{1}=T_{\infty}-(T_{\infty}-T_{0})\cdot exp\left(-\frac{3\cdot\lambda^{gas} \cdot\tau_{ign}}{\rho\cdot c_{p}^{liq}\cdot r_{0}^{2}}\right) \tag{31}\]  
Appreciating this, the fact that \(\tau_{ign}\) serves as the basis of the Equation (31) could be maintained [28]:  
\[\frac{L}{R\cdot T_{\infty}\cdot(1-Y)}+ln\left(\frac{Y}{1-\frac{T(r_{1})}{T_{ \infty}}}\right)=ln\left(\frac{1}{\phi}+1\right)+\frac{L}{R\cdot T_{b}} \tag{32}\]  
where \(Y=\left(1-\frac{T_{0}}{T_{\infty}}\right)\cdot exp\left(-\frac{3\cdot\lambda^ {gas}\cdot\tau_{ign}}{\rho\cdot c_{p}^{liq}\cdot r_{0}^{2}}\right)\). Taking advantage of comparable occasions of the classical theory of combustion-based Q-S droplet ignition models [6; 15], we have derived a basic expression for calculations of ignition delay of proposed types of fuel and their blends:  
\[\tau_{ign}=\frac{\rho\cdot c_{p}^{liq}\cdot r_{0}^{2}}{3\cdot\lambda^{gas}} \left(1-\frac{1}{\theta_{0}}\cdot\left(\frac{1-\frac{T_{0}}{T_{\infty}}-\frac {R\cdot T_{\infty}}{L}ln\frac{\phi\cdot E\cdot\theta_{0}}{R\cdot T_{\infty}}} \right)\right) \tag{33}\]  
Assuming that \(\frac{R\cdot T_{b}}{L}ln\frac{\phi\cdot E\cdot\theta_{0}}{R\cdot T_{\infty}}<<1\), Equation (33) can be rewritten as  
\[\tau_{ign}=\frac{\rho\cdot c_{p}^{liq}\cdot r_{0}^{2}}{3\cdot\lambda^{gas}} \bigg{(}\frac{T_{b}-T_{0}}{T_{\infty}-T_{0}}\bigg{)} \tag{34}\]  
Calculation can be simplified by leaving out the exponential dependence Y(\(\tau_{ign}\)) out:  
\[\tau_{ign}=\frac{\rho\cdot c_{p}^{liq}\cdot r_{0}^{2}}{3\cdot\lambda^{gas}} ln\left(\frac{1-\frac{T_{0}}{T_{\infty}}}{1-\frac{T_{b}}{A^{*}\cdot T_{\infty}}}\right) \tag{35}\]  
where \(A^{*}=1-\frac{R\cdot T_{b}}{L}ln\left(\frac{1-\frac{T_{0}}{T_{\infty}}}{\left( 1-\frac{T(r_{1})}{T_{\infty}}\right)}\cdot\frac{\phi}{\left(1+\phi\right)}\right)\).