## 2 Mathematical Modeling
### Slagging of the Boiler Furnace at the Micro-Level  
Figure 2 shows the area for solving the task of the heating surface slagging (on the example of a tube) in the boiler furnace when burning three different fuels (coal, CWS, and CWSP). It was assumed (Figure 2a) that a tube with a diameter of \(d=60\) mm at the initial time \(t=0\) is in a heated gaseous medium \(T=T_{g}\). During fuel combustion (\(t>0\)), a slag layer with a thickness \(\delta_{s}\) gradually forms on the tube surface (Figure 2b). The limiting value of the slag layer thickness was set to 7 mm, because further increase in the layer is limited by the inter-tube distance [55]. The rates of a slag layer formation when burning different fuels were taken from the results of numerical studies via the model described in Section 2.1. Temperature (\(T_{s}=560\)\({}^{\circ}\)C) and pressure (\(P_{s}=14\) MPa) of the steam-water mixture inside the shielding tubes were set according to [35]. The heat transfer coefficient \(\alpha_{1}\) from the steam-water medium to the inner wall of the tube was taken according to [56]. The thermophysical properties of the tube material (steel) corresponded to the data of [35]. The wall thickness \(\delta_{s}\) of the steel tubes was 5.5 mm. The ash thermophysical properties were taken according to the data of [57]; the thermophysical properties of the air corresponded to the data of [58]. The heat transfer coefficient \(\alpha_{2}\) from the flue gases to the outer wall of the shielding tubes and the flue gases' emissivity factor were taken according to the recommendations [35]. Heat transfer through the heat exchange surface was considered in a planar formulation (for an infinite plate), since the ratio of the outer and inner diameters of the tube is less than 1.8 [56]. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Parameters** & **Grid 1** & **Grid 2 (Examined)** & **Grid 3** \\ \hline Number of cells & 0.25 \(\times\) 10\({}^{6}\) & 0.5 \(\times\) 10\({}^{6}\) & 1 \(\times\) 10\({}^{6}\) \\ \hline Minimum element size, mm & 70 & 50 & 10 \\ \hline Average element size, mm & 500 & 260 & 130 \\ \hline Slagging rate of coal, g/(m\({}^{2}\cdot\)s) & 35.618 & 38.061 & 41.308 \\ \hline GCI (\%) & 0.082 & 0.046 & 0.001 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Grid convergence analysis. The solution area of the formulated task included the following three zones (Figure 2b): steel, slag and flue gases. The total size of the entire system \(L\) was 10 mm. The complexity of solving the formulated task consisted in the dynamic change in the slag layer thickness over time (the position of a border between ash layer and heated flue gases shifted). The processes of heat and mass transfer in the region of solving the task of slagging tubes in the boiler furnace (Figure 2) are described by a system of partial differential equations written in the Cartesian coordinate system. The mathematical model is formulated in a one-dimensional formulation due to the uniformity of heat supply to the tube surface. The algorithm for solving the formulated problem assumed the implementation of several computational schemes. Each of the schemes was determined by the geometry of the solution area corresponding to the different stages of the slagging of the pipe surface over time (Figure 2). Within each step mentioned below, the coordinates of the different regions' boundaries of the steel tube wall/ash layer/heated flue gases system were numbered from the origin of coordinates in the direction of the solution area outer boundary (\(x=L\)). First stage at 0 \(<t\leq t_{1}\) (Figure 2a).