## 3 Results
### Mathematical Model  
A droplet of liquid fuel with radius \(r_{0}\) was analyzed in a stationary heated stationary oxidizing environment with its gas temperature \(T\infty\) (Figure 4). The ignition process of a spherically symmetric droplet begins at the point \(r=r_{1}\) (see Figure 4b) where the speed of a chemical reaction exceeds its maximum \(W(T)\). A second-order chemical reaction of fuel oxidation can be written as follows [6; 15]:  
\[W=k_{0}\cdot c_{1}\cdot c_{2}\exp\left(-\frac{E}{R\cdot T}\right) \tag{1}\]  
Figure 3: Duration of the main phases of the chemical reaction kinetics for fossil fuel and biofuel droplets (\(d_{0}=2.2\) mm), including their blend over time: ignition delay (**a**), combustion (**b**), and ignition delay + combustion (extinction) (**c**). It was assumed that until the ignition vapor of the droplet and the oxidizer mix in the correct stoichiometric ratio. The flame separates the fuel and the oxidizer (which is usually air) and occurs at the \(\phi=1\) (stoichiometric condition). The reaction rate is controlled by the molecular diffusion of the reactants toward the reaction zone. For the case of hydrocarbon fuel droplet combustion in air, the reaction is respectively described by  
\[C_{x}H_{y}+\Big{(}x+\frac{y}{4}\Big{)}(O_{2}+3.76N_{2})\to xCO_{2}+ \Big{(}\frac{y}{2}\Big{)}H_{2}O+3.76\Big{(}x+\frac{y}{4}\Big{)}N_{2} \tag{2}\]  
In this situation, the air properties are obtained from the databases listed in Ref. [44]. The mole fractions can be estimated from the reactants. The coordinates of the ignition point of the evaporating droplet could be established from the common solution of a two-equation system [6; 15]:  
\[\frac{\partial W}{\partial r}=0;\ p_{1}=\phi\cdot p_{2} \tag{3}\]  
Differentiating Equation (1) and taking an expression of the stoichiometric ratio (Equation (3)), we derive a sum of two components. The first could be established from the partial pressure around the droplet and the second from the distribution of the vapor temperature [6; 15]. \[2\cdot p_{1}\cdot\frac{k_{0}}{\phi}\cdot\exp\biggl{(}-\frac{E}{R\cdot T} \biggr{)}\frac{\partial p_{1}}{\partial r}+p_{1}^{2}\cdot\frac{E}{R\cdot T^{ 2}}\cdot\frac{k_{0}}{\phi}\cdot\exp\biggl{(}-\frac{E}{R\cdot T}\biggr{)}\frac {\partial T}{\partial r}=0\ \text{or}\ \frac{2\cdot\partial p_{1}}{p_{1}\cdot \partial r}+\frac{E\cdot\partial T}{R\cdot T^{2}\cdot\partial r}=0 \tag{4}\]  
In this model, because of the infinite rate assumption, the flame zone is of infinitesimal thickness and would thus be represented by a surface rather than an extended reaction zone. Generally, it is necessary to ensure that the fuel and oxidant diffuse to the reaction zone in stoichiometric proportions, and again, because of the infinite reaction rate assumption, their concentrations become zero at the reaction interface. When these approximations are  
Figure 4: Hydrocarbon fuel droplet. (**a**) Diagram of all phenomena involved in the evaporation and combustion processes: 1—surface of a droplet; 2—vapor; 3—heat; 4—oxygen; 5—chemical reaction zone; 6—products formed from chemical reaction. (**b**) Diagram of relationships between process parameters: 1—temperature distribution in space \(T(r)\); 2—distribution of vapor concentration around droplet \(c_{1}(r)\). The dotted line indicates the zone where the chemical reaction starts (separating fuel vapor from the oxidizer). made, an analytic solution may be obtained.