# A Mathematical Model of Industrial Waste-Derived Fuel Droplet Combustion in High-Temperature Air
## 2 Materials and Methods
### Experimental Data  
The experimental research [23] into the ignition and combustion of single CLF droplets was conducted on two different test installations. The first condition [23] (Figure 1a) created the radiation-supported heating conditions of a suspended droplet in a tubular muffle furnace (\(V_{g}\approx 0\)) with a temperature varying in the range of 723-1273 K. Using the second condition (Figure 1b), the same fuel droplet was heated convectively with hot air at a velocity of \(V_{g}\approx 3\) m/s, the air temperature varied within a range of 723-973 K [23]. The second experimental condition (Figure 1b), in laboratory conditions, simulated the interaction processes of fuel droplets with a stream of hot air, which are the basis of industrial technologies for the flaring of liquid and solid fuels [39].  
Carrying out experiments on the demonstrated conditions [23] made it possible to comprehensively analyze the effect of various heating conditions on the ignition characteristics of CLF droplets (Figure 2). The characteristics of ignition under near-critical conditions substantially depend on the patterns of phase transitions and the mechanism of heat and mass transfer for various methods of heat supply to the CLF droplet [40]. Fuel droplets exposed to radiant (Figure 2a) and convective (Figure 2b) heating feature identical patterns of physical and chemical processes. Conditionally the following sequence of basic reaction phases of a CLF droplet and warmed-over air (moving and motionless) can be distinguished: 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, heating of the solid remainder and finally its heterogeneous ignition and combustion. The main difference between the processes presented in the video frames (in Figure 2a,b) is in the significantly longer passive warming of the fuel droplet and formation of a combustible gas mixture mainly due to the diffusion of volatiles in the oxidizer medium in the vicinity under radiant heating conditions (Figure 2a).  
Figure 1: Schematic of experimental conditions: (**a**) first condition; (**b**) second condition.  
In the experimental research (Figure 2) [23], the ignition-delay times (\(t_{d}\)) were determined using the commercial Tema Automotive software (Image Systems AB, Sweden) and the built-in Threshold algorithm by the variation of the luminance around the droplet over time [11]. The systematic error of the measuring system in determining the ignition-delay time is less than 3%. Random errors of a series of five repeated measurements under the same experimental conditions are less than 10%. The main experimental values [23] of the average \(t_{d}\) of droplets (with the initial diameter \(d_{p}=1\) mm) of CLF with the addition of various MSW at different air temperatures under conditions of dominant radiant and convective heating are presented in Tables 3 and 4. The calculation results for the developed model will be validated by comparison with these experimental data.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**No. of Composition** & \multicolumn{5}{c}{**Air-Flow Temperature (\(V_{g}\approx 3\) m/s)**} \\ \hline  & **723 K** & **773 K** & **823 K** & **873 K** & **923 K** & **973 K** \\ \hline No. 1 & 7.48 & 6.00 & 4.86 & 3.93 & 3.14 & 2.56 \\ No. 2 & 7.03 & 5.62 & 4.53 & 3.65 & 2.89 & 2.36 \\ No. 3 & 6.65 & 5.38 & 4.38 & 3.53 & 2.81 & 2.31 \\ No. 4 & 6.54 & 5.28 & 4.26 & 3.43 & 2.71 & 2.19 \\ No. 5 & 6.37 & 5.15 & 4.14 & 3.34 & 2.65 & 2.14 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Average ignition-delay times of CLF exposed to convective heating (\(V_{g}\approx 3\) m/s) at different air temperatures [23].  
Figure 2: Video frames 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).