## 2 Mathematical Modeling
### Slagging of the Boiler Furnace at the Macro-Level
Accounting for the radiation component of the heat flux is performed under the following conditions: the particle emissivity factor \(\varepsilon=0.85\); the radiation scattering coefficient by particles is assumed to be 0.6; the absorption coefficient of the gaseous medium corresponded to a mixture of \"gray\" gases. The kinetic parameters of chemical reactions are presented in the Table 6. The ratio of carbon, hydrogen and oxygen in the volatile matter (first reaction, Table 6) was determined in accordance with the chemical composition of fuel. \begin{table}
\begin{tabular}{c c c c} \hline
**Reaction** & \(A_{r}\) & \(E_{r}\) **(kJ/(mol-K))** & **Ref.** \\ \hline \multicolumn{4}{c}{Homogeneous combustion} \\ \hline Volatile + 1.706O\({}_{2}\) = CO\({}_{2}\) + 1.543H\({}_{2}\)O & 2.1 \(\times\) 10\({}^{11}\) & 202.7 & [43; 44] \\ H\({}_{2}\) + \(\frac{1}{2}\)O\({}_{2}\) = H\({}_{2}\)O & 2.1 \(\times\) 10\({}^{14}\) & 129.8 & [44; 45; 46] \\
2 CO + O\({}_{2}\) = 2 CO\({}_{2}\) & 1.4 \(\times\) 10\({}^{13}\) & 96.8 & [44; 45; 46] \\ \hline \multicolumn{4}{c}{Heterogeneous combustion} \\ \hline C + O\({}_{2}\) = CO\({}_{2}\) & 2 \(\times\) 10\({}^{12}\) & 60.6 & [45; 47; 48] \\ C + \(\frac{1}{2}\)O\({}_{2}\) = CO & 2 \(\times\) 10\({}^{12}\) & 60.6 & [44; 45; 48; 49] \\ C + CO\({}_{2}\) = 2 CO & 1.3 \(\times\) 10\({}^{7}\) & 259 & [44; 45; 48; 49; 50; 51] \\ \hline \end{tabular}
\end{table} 
Table 6: Kinetic parameters of chemical reactions. \begin{table}
\begin{tabular}{l l} \hline
**Process** & **Model** \\ \hline Turbulence & \(k\)-\(\varepsilon\) model \\ \hline Radiative heat transfer & Model of discrete ordinates and interaction of particles with radiation \\ \hline Volatile yield & One-stage model according to the Arrhenius equation \\ \hline Homogeneous combustion & Combination of the kinetic model of combustion of gaseous components with the model of \\ of turbulence & “vortex shedding” \\ \hline Heterogeneous combustion & Diffusion-kinetic model \\ \hline \end{tabular}
\end{table} 
Table 5: Numerical models. Slagging of furnace walls is described by the original mathematical model using the user-defined function in Ansys Fluent. In the function, the probability of an ash particle build-up to the furnace wall depends on its temperature and burnout degree, as well as the ash fusibility characteristics [27]. The incompleteness of fuel combustion (0.5%) was taken as a criterion for particle burnout; i.e., when the degree of burnout is more than 99.5%, the adhesion of a particle to the wall becomes possible, otherwise it is reflected from the wall according to the elastic collision law. For the burnt fuel particles, the probability of sticking to the furnace walls depends on temperature and ash fusibility characteristics, which are presented in Table 2. When developing the mathematical model, a linear increase in the probability of particle build-up was assumed depending on temperature (Table 2): at particle temperature \(T=T_{A}\), the probability of build-up is equal to 0; at particle temperature \(T=T_{C}\) the probability is equal to 1 [52]. Building a finite element mesh is an important step for obtaining reliable simulation results. 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. By refining the meshes near the burners, the calculations were carried out for grids consisting of \(0.25\times 10^{6}\), 0.5 \(\times\) 10\({}^{6}\) and 1 \(\times\) 10\({}^{6}\) elements (Table 7). The grid convergence index (GCI) is calculated according to recommended procedure for estimation of discretization error [53; 54]. The number of elements and the characteristic 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).