## 2 Mathematical Modeling  
### Slagging of the Boiler Furnace at the Macro-Level  
The object of the study was the steam boiler furnace BKZ-210-140F, made in Russia, with following main parameters of superheated steam [30; 31; 32; 33]: \(D_{\mathrm{ss}}=210\) t/h, \(P_{\mathrm{ss}}=14\) MPa, \(T_{\mathrm{ss}}=560\)\({}^{\circ}\)C. The inner surfaces of the furnace, with sizes of 7424 mm \(\times\) 7808 mm \(\times\) 24,000 mm (Figure 1a), are completely shielded by tubes with a diameter of 60 mm (wall thickness of 5.5 mm) with step of 64 mm. The front and rear walls form a slope of a cold funnel in the lower part of the furnace. At the top of the furnace, the rear wall forms aerodynamic protrusion (Figure 1a). The platen superheater is located above the aerodynamic protrusion inside the furnace space. The furnace is equipped with four direct-flow burners with five secondary air channels and four primary air channels alternating in height (Figure 1b). The burners are located in one tier at the corners of the furnace and are directed tangentially to a conditional circle in the center of the furnace with a diameter of 900 mm (Figure 1c). The fuel and air supply devices are installed in an area with a size of 670 mm \(\times\) 2270 mm (Figure 1b). Numerical modeling of the combustion process in the furnace was carried out for the following three fuels:  
* lignite of the Bolshesyrskoye deposit, Kansk-Achinsk coal mine, Krasnoyarsk region, Russian Federation (hereinafter--coal);
* coal water slurry (hereinafter--CWS), a fuel blend consisting of 50 wt% of coal and 50 wt% of water;
* coal water slurry with petrochemicals (hereinafter--CWSP), a fuel blend consisting of 45 wt% of coal, 45 wt% of water and 10 wt% of used engine oil. The thermal characteristics of these fuels are presented in Table 1[34; 35; 36]. Within the framework of the developed mathematical model, the water added during the preparation of CWS and CWSP, when fuels are sprayed by nozzles, is modeled as a liquid that is concentrated on the surface of coal particles. The evaporation of this moisture from the surface of coal particles is described by the droplet evaporation model [12]. An oil in the composition of CWSP is described in the mathematical model as a component of volatile substances in the composition of solid fossil fuel. \begin{table}
\begin{tabular}{l c c c} \hline \hline  & **Coal** & **CWS** & **CWSP** \\ \hline \multicolumn{4}{c}{Thermal characteristics} \\ \hline Operating humidity, \% & 20.8 & 60.4 & 54.36 \\ Operating ash content, \% & 6.2 & 3.1 & 2.8 \\ Volatile yield, \% & 32.56 & 16.28 & 24.65 \\ High calorific value, MJ/kg & 21.41 & 10.7 & 14.03 \\ \hline \multicolumn{4}{c}{Ultimate analysis (for dry ash-free state)} \\ \hline \(\mathcal{C}^{\text{daf}}\) \% & 74.1 & 74.1 & 77.4 \\ \(H^{\text{daf}}\) \% & 5 & 5 & 5.5 \\ \(N^{\text{daf}}\) \% & 0.9 & 0.9 & 0.74 \\ \(S^{\text{daf}}\) \% & 0.3 & 0.3 & 0.24 \\ \(O^{\text{daf}}\) \% & 19.7 & 19.7 & 16.12 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main fuel characteristics. Figure 1: Scheme of the steam boiler furnace BKZ-210-140F (**a**), burner configuration (**b**), furnace cross-section (**c**). The fusibility characteristics of the coal ash (Table 2) are taken in accordance with the reference data [35]. The initial deformation temperature, hemispherical temperature and flow temperature were determined in accordance with the procedures described in [37; 38; 39]. In the boiler BKZ-210-140F, to reduce emissions of nitrogen oxides and reduce furnace slagging, \"burner not in operation\" technologies were applied. Primary air with fuel is supplied through burners B, C and D (Figure 1c), while secondary air is supplied to all four burners. This creates a lack of oxygen for the burners \"in operation\". The flow rate and temperature of primary and secondary air in all calculation options are taken constant. The consumption of each type of fuel was set based on the required steam output of the boiler at an identical load.