# A Mathematical Model of Industrial Waste-Derived Fuel Droplet Combustion in High-Temperature Air
## 2 Materials and Methods
### Mathematical Model
The scheme for a reacting particle in the multiple surface reactions model is shown in \[\
abla\cdot\left[\overset{\rightarrow}{u}(\rho H+p)\right]=-\
abla\cdot \left(\sum_{i}h_{i}l_{i}\right) \tag{3}\]4  
The rate of chemical reactions:  
\[\
abla\cdot\left[\overset{\rightarrow}{u}(\rho H+p)\right]=-\
abla\cdot \left(\sum_{i}h_{i}l_{i}\right) \tag{3}\]5  
\[\
abla\cdot\left[\overset{\rightarrow}{u}(\rho H+p)\right]=-\
abla\cdot \left(\sum_{i}h_{i}l_{i}\right) \tag{3}\]6  
where \(\overline{R}_{j,r}\)--rate of particle surface species depletion, kg/s; \(A_{p}\)--particle surface area, m\({}^{2}\); \(\eta_{r}\)--effectiveness factor; \(R_{j,r}\)--rate of particle surface species reaction per unit area, kg/(m\({}^{2}\)-s); \(p_{n}\)--bulk partial pressure of the gas phase species, Pa; \(R_{\text{kin},r}\)--kinetic rate of reaction, units vary; \(D_{0,r}\)--diffusion rate coefficient for reaction; and \(N_{r}\)--apparent order of reaction. The diffusion rate coefficient \(D_{0,r}\) is defined as:  
\[\
abla\cdot\left[\overset{\rightarrow}{u}(\rho H+p)\right]=-\
abla\cdot \left(\sum_{i}h_{i}l_{i}\right) \tag{3}\]7  
The kinetic rate of multiple reactions is defined as:  
\[\
abla\cdot\left[\overset{\rightarrow}{u}(\rho H+p)\right]=-\
abla\cdot \left(\sum_{i}h_{i}l_{i}\right) \tag{3}\]8  
where \(A_{r}\)--pre-exponential factor; \(\beta_{r}\)--temperature exponent; \(E_{r}\)--activation energy for reaction, J/(mol\(\cdot\)K); \(p_{r,d}\)--bulk partial pressure of the diffusion-limited species, Pa; and subscripts: \(n\)--number of gas specie. The main kinetic parameters of chemical reactions used in the modeling are presented in \[\
abla\cdot\left[\overset{\rightarrow}{u}(\rho H+p)\right]=-\
abla\cdot \left(\sum_{i}h_{i}l_{i}\right) \tag{3}\]9  
\[\text{div}\left(\rho k\overset{\rightarrow}{u}\right)=\text{div}\left[\left( \mu+\frac{\mu_{t}}{\sigma_{k}}\right)\text{grad}\,k\right]+G_{k}-\rho\varepsilon \tag{4}\]0 A reacting particle in the multiple surface reactions model [42]: \(T_{p}\)—droplet temperature; \(T_{\infty}\)—ambient gas temperature; \(C_{d,b}\) and \(C_{d,s}\)—concentration of the diffusion-limited species; \(C_{k}\)—concentrations of all other species. The thermophysical and transport properties of multi-component CLF droplets, which are a heterogeneous mixture of coal, water and MSW, were set as effective values, which were determined analytically using the method [70] based on the data presented in Tables 3 and 4. The obtained values are presented in \[\text{div}\left(\rho k\overset{\rightarrow}{u}\right)=\text{div}\left[\left( \mu+\frac{\mu_{t}}{\sigma_{k}}\right)\text{grad}\,k\right]+G_{k}-\rho\varepsilon \tag{4}\]1  
For the developed model of CLT droplet ignition under different heating conditions, a structured rectangular grid 30 \(\times\) 30 mm in size was generated using ANSYS Fluent [42]. To assess the influence of the grid element sizes on the simulation results, a series of calculations was carried out with a gradual increase in the number of elements. The calculations were carried out for grids consisting of 5625, 14,400 and 360,000 elements (Table 7). The grid convergence index (GCI) was calculated according to the recommended procedure for the estimation of discretization errors [71; 72]. The number of elements and the charac  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Reaction** & \(A_{r}\) & \(E_{r}\) **(kJ/(mol\(\cdot\)K))** & **Ref.