# A Mathematical Model of Industrial Waste-Derived Fuel Droplet Combustion in High-Temperature Air
## 2 Materials and Methods
### Mathematical Model
** \\ \hline \multicolumn{4}{c}{**Water Evaporation**} \\ \hline H\({}_{2}\)O() = H\({}_{2}\)O(g) & 5.13\(\cdot\)10\({}^{6}\) & 87.9 & [49; 50] \\ \hline \multicolumn{4}{c}{**Pyrolysis**} \\ \multicolumn{4}{c}{Biomass pyrolysis} \\ \hline lignin = 0.209 CO\({}_{2}\) + 0.396 CO + 0.109H\({}_{2}\) + 0.249H\({}_{2}\)O + 0.037 vol & 2.202\(\cdot\)10\({}^{12}\) & 181 & [50] \\ hemicellulose = 0.209 CO\({}_{2}\) + 0.396 CO+0.109H\({}_{2}\) + 0.249H\({}_{2}\)O + 0.037 vol & 2.527\(\cdot\)10\({}^{11}\) & 147 & [50] \\ cellulose = 0.209 CO\({}_{2}\) + 0.396 CO + 0.109H\({}_{2}\) + 0.249H\({}_{2}\)O + 0.037 vol & 1.379\(\cdot\)10\({}^{14}\) & 193 & [50] \\ \hline \multicolumn{4}{c}{Rubber pyrolysis} \\ \hline rubber = 0.8H\({}_{2}\) + 0.0009C\({}_{2}\)H\({}_{4}\) + 0.194 CH\({}_{4}\) + 0.0025C\({}_{3}\)H\({}_{6}\) + 0.0018C\({}_{4}\)H\({}_{6}\) + 0.0008C\({}_{2}\)H\({}_{2}\) & 5.5\(\cdot\)10\({}^{18}\) & 181 & [51; 52] \\ \hline \multicolumn{4}{c}{Plastic pyrolysis} \\ \hline polyethylene = 0.825H\({}_{2}\) + 0.07C\({}_{2}\)H\({}_{4}\) + 0.05 CH\({}_{4}\) + 0.03C\({}_{3}\)H\({}_{6}\) + 0.02C\({}_{2}\)H\({}_{6}\) + 0.005C\({}_{3}\)H\({}_{8}\) & 15\(\cdot\)10\({}^{3}\) & 40 & [53] \\ \hline \multicolumn{4}{c}{**Gasification**} \\ \hline C + H\({}_{2}\)O = CO + H\({}_{2}\) & 2.07\(\cdot\)10\({}^{7}\) & 220 & [50; 54; 55; 56] \\ C + CO\({}_{2}\) = 2 CO & 1.32\(\cdot\)10\({}^{7}\) & 259 & [50; 55; 56; 57; 58; 59] \\ C + 2H\({}_{2}\) = CH\({}_{4}\) & 5\(\cdot\)10\({}^{6}\) & 30 & [50; 55; 56; 58] \\ C + 2H\({}_teristic size of the computational grid were considered optimal when the corresponding convergence criteria were met (GCI \(\leq\) 0.05%). Grid 2 satisfies this condition (Table 7). For the computational model, the grid was built in such a way that the value of the maximum size of the grid element was 0.25 mm (14,400 elements in total, 14,641 nodes). In this area of the problem's solution (Figure 4), the ongoing physical and chemical processes were controlled. A single droplet injection system was used to generate a CLF droplet with the initial temperature \(T_{p,0}\) = 300 K and size \(d_{p}\) = 1 mm. It was placed in the geometric median of the solution domain in Figure 4 (X-position = 0.015 m; Y-position = 0.015 m). At the initial point of time, a uniform distribution of the oxidizer temperature \(T_{g}\) was set. The boundaries of \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Characteristics** & \multicolumn{5}{c}{**No. of Composition**} \\ \hline  & **No. 1** & **No. 2** & **No. 3** & **No. 4** & **No. 5** \\ \hline \(Q\) (MJ/kg) & 10.78 & 11.29 & 13.05 & 11.88 & 11.43 \\ Specific heat capacity (kJ/(kg\(\cdot\)K)) & 2.4600 & 2.4175 & 2.3549 & 2.4136 & 2.4136 \\ Thermal conductivity (W/(m\(\cdot\)K)) & 0.4796 & 0.4582 & 0.4550 & 0.4620 & 0.4547 \\ Viscosity (Pa-s) & 0.557 & 0.557 & 0.557 & 0.557 & 0.557 \\ \(W^{\text{a}}\) (\%) & 40.00 & 38.00 & 36.20 & 36.20 & 36.50 \\ \(A^{\text{d}}\) (\%) & 26.50 & 24.05 & 24.03 & 23.87 & 24.15 \\ \(V^{\text{daf}}\) (\%) & 23.10 & 29.10 & 27.53 & 30.74 & 30.45 \\ \(C^{\text{daf}}\) (\%) & 87.20 & 83.51 & 88.27 & 85.15 & 83.11 \\ \(H^{\text{daf}}\) (\%) & 5.10 & 5.19 & 4.71 & 5.38 & 5.22 \\ \(N^{\text{daf}}\) (\%) & 2.10 & 1.91 & 1.92 & 1.89 & 1.92 \\ \(S^{\text{daf}}\) (\%) & 1.10 & 1.00 & 1.05 & 0.99 & 1.01 \\ \(O^{\text{daf}}\) (\%) & 4.50 & 8.39 & 4.05 & 6.59 & 8.74 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The main characteristics of different CLF compositions. Figure 4: Scheme of the solution domain for ignition in the CLF droplet/air system. the solution domain (furnace chamber walls) were described by the first type (Dirichlet) of boundary conditions. The temperatures at the solution domain boundaries were set as constant and equal to \(T_{g}\) according to the experimental data. In the modeling of the composite fuel droplet ignition under the convective heating conditions, the velocity of heated air \(V_{g}\) was 3 m/s. The oxidizer (O\({}_{2}\)) concentration was assumed to be equal to the mass content of oxidizer in the air \(\varphi_{\text{O}_{2}}=0.21\).