# A Mathematical Model of Industrial Waste-Derived Fuel Droplet Combustion in High-Temperature Air
## 4 Effect of Temperature Gradient Inside the Droplet and in Its Vicinity  
This section presents the first results on the modernization of the developed mathematical model for a more reliable description of the experimental data. Compared to the base model (1)-(28), the advanced version of the model additionally takes into account the non-uniform temperature distribution inside the CLF droplet according to the heat transfer equation:  
\[\frac{\partial T_{p}}{\partial t}=a_{p}\left(\frac{\partial^{2}T_{p}}{ \partial\tau^{2}}+\frac{2}{r}\frac{\partial T_{p}}{\partial r}\right) \tag{30}\]  
where \(r\)--droplet radius in spherical coordinates, m; \(T_{p}\)--droplet temperature, K; and \(a_{p}\)--thermal diffusivity of CLF, m\({}^{2}\)/s.  
The complexity of the mathematical model made it possible to construct typical temperature fields for solving the problem of ignition of a CLF droplet for qualitative comparison with videograms of experimental studies [23] (Figure 11). The first case (Figure 11a) was radiant heating (\(V_{g}\approx 0\)). The second case (Figure 11b) was convective heating (\(V_{g}\approx 3\) m/s). As can be seen, there is a qualitative agreement between experimental data and simulation results in the stages of industrial waste-derived fuel droplet combustion: passive warming, near-surface water vaporization, thermolysis of coal and municipal waste (solid combustible components), blending of ignitable gases with the oxidizing medium, gas blend firing and burnout, and heating of the solid remainder followed by its heterogeneous ignition and combustion.  
Figure 11: Temperature fields and videoframes obtained in experiments of ignition and combustion of CLF droplets No. 5 (FC 90% + cardboard 10%) at \(T_{g}=873\) K [23] under the following conditions: (**a**) radiant heating (\(V_{g}\approx 0\)); (**b**) convective heating (\(V_{g}\approx 3\) m/s).  
Using the advanced mathematical model, ignition-delay times \(t_{d}\) are determined depending on the temperature of the heating source \(T_{g}\). Satisfactory agreement between experiment and theory was obtained (Figure 12). In this case, the difference in the results for \(t_{d}\) over the entire range of temperatures under study does not exceed 19% both under conditions of radiant heating (Figure 12a) and convective heating (Figure 12b), which does not exceed the confidence limit for the obtained experimental data [23].  
It should be noted that when the non-uniform temperature distribution inside a single CLF droplet and in its vicinity is taken into account, the PC calculation times increase by a factor of 5-10. On the scale of the steam boiler, there will be an exponential increase in the PC calculation times. On the scale of the steam boiler, there will be an exponential increase in the PC calculation times. The relatively simple modernization performed on the mathematical model, on the one hand, leads to a satisfactory agreement between the results of numerical simulation and experimental data, but, on the other hand, requires large computing power. The most rational approach in practical applications is the development of a mathematical model with maximum simplifications but with acceptable reproducibility of experimental data.