## 2 Materials and Methods
### Research Methods
#### 2.2.1 Gas and Particle Motion Models  
As noted in the introduction, in the CFD-DEM model, gas is modeled using the Euler approach, and particles are modeled using the Lagrange approach. The gas flow is described by two basic equations: the law of conservation of mass (or the continuity Equation (2)) and the law of conservation of momentum (or the Navier-Stokes Equation (3)) [63; 64; 65]:  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot\left(\rho\overset{\rightarrow}{u} \right)=S_{m}, \tag{2}\]  
where \(\rho\) is the gas density; \(\overset{\rightarrow}{u}\) is the gas velocity; \(S_{m}\) is the source term that determines the mass added from the dispersed second phase or another source (for instance, evaporation of liquid droplets); and \(t\) is the time;  
\[\frac{\partial}{\partial t}\left(\rho\overset{\rightarrow}{u}\right)+\
abla \cdot\left(\rho\overset{\rightarrow}{u}\overset{\rightarrow}{u}\right)=- \
abla p+\
abla\cdot\left(\overline{\overline{\overline{\epsilon}}}\right)+ \rho\overset{\rightarrow}{g}+\overset{\rightarrow}{F}, \tag{3}\]  
where \(p\) is the static pressure; \(\overline{\overline{\pi}}\) is the stress tensor; \(\overset{\rightarrow}{g}\) is acceleration of gravity; and \(\rho\overset{\rightarrow}{g}\) and \(\overset{\rightarrow}{F}\) are the gravitational and external volumetric forces. The equation of the stress tensor has the following form:  
\[\overline{\overline{\tau}}=\mu\left[\left(\
abla\overset{\rightarrow}{u}+ \
abla\overset{\rightarrow}{u}^{T}\right)-\frac{2}{3}\
abla\cdot\overset{ \rightarrow}{u}I\right], \tag{4}\]  
where \(\mu\) is the molecular viscosity; \(T\) is the transposition of the vector; \(I\) is the unit tensor; and the second term of the right side of the equation is responsible for the effect of volume dilation. However, DEM can be divided into resolved (RDPM) and unresolved (UDPM). For RDPM, the gas flow is calculated on a computational mesh around particle boundaries which become solid boundaries. This approach is very detailed, which requires significant computing power for simulating systems of practical scale [66]. For the possibility of simulating larger systems, UDPM is used [67]. This model calculates the average volume  
Figure 5: The dependence of the area-weighted average pressure at the inlet of the distribution device cell on time. values of the velocity and density of the gas, while the volume of the mesh element should exceed the particle size by several times. Equations (2) and (3) averaged by volume have the following form [68]:  
\[\frac{\partial(\rho\varepsilon_{p})}{\partial t}+\
abla\cdot\left(\rho \varepsilon_{p}\overset{\rightarrow}{u}\right)=S_{m}, \tag{5}\] \[\frac{\partial}{\partial t}\left(\rho\varepsilon_{p}\overset{ \rightarrow}{u}\right)+\
abla\cdot\left(\rho\varepsilon_{p}\overset{ \rightarrow}{u}\overset{\rightarrow}{u}\right)=-\varepsilon_{p}\
abla p+ \
abla\cdot\left(\overline{\overline{\epsilon}}\right)\cdot\varepsilon+\rho \varepsilon_{p}\overset{\rightarrow}{g}+\overset{\rightarrow}{F}+\overset{ \rightarrow}{F}_{drag}, \tag{6}\]  
where \(\varepsilon_{p}\) is the porosity of the particle bed and \(\overset{\rightarrow}{F}_{drag}\) are the drag forces. It should be noted that to calculate the drag forces, closing expressions are required; the type depends on the chosen DEM approach: soft or hard spheres [68]. The porosity of the particle bed is determined by the Particle Centroid Method (PCM). The particle affects the porosity value of the particle bed only of that cell of the computational mesh where its center is located:  
\[\varepsilon_{p}=1-\frac{\sum n_{p,i}\cdot V_{p,i}}{V_{\Omega}}, \tag{7}\]  
where \(n_{p,i}\) is the volume fraction of the \(i\)-th particle of the cell; \(V_{p,i}\) is the volume of the \(i\)-th particle; and \(V_{\Omega}\) is the volume of the cell for which the porosity value of the particle bed \(\varepsilon_{p}\) is calculated.