## 4 Research Methodology  
The total aerodynamic resistance \(\Delta P\) is defined as a set of resistances of the degrees in the shelf device (pressure loss on the shelf) \(\Delta p_{i}\)[53]:  
\[\Delta P=\sum\limits_{i=1}^{n}\Delta p_{i} \tag{2}\]  
The aerodynamic resistance of each shelf contact depends on the nature of the gas flow interaction with the shelves, installed at a certain angle, the shelf perforation degree \(\psi\), the dispersed material amount in the inter-shelf space, characterized by the layer porosity \(\varepsilon\) or the concentration of the solid particles in the workspace of the device \(\beta\). One should note that the specified pressure drop in the holes will decrease along the shelf. Therefore, the efficiency of the perforation holes with the same diameter will also decrease approaching the outloading space. The pressure loss for elementary jet moving along a perforated shelf can be considered as the sum of the losses along the shelf length with the current coordinate \(X\) and local loss through the perforation holes:  
\[\Delta p_{i}=const=\lambda\cdot\frac{X}{d_{e}}\cdot\frac{\rho_{gas}\cdot W_{ gas}^{2}}{2}+\xi_{hol}\cdot\frac{\rho_{gas}\cdot W_{hol}^{2}}{2}, \tag{3}\]  
where \(\lambda\cdot\frac{X}{d_{e}}\cdot\frac{\rho_{gas}\cdot W_{gas}^{2}}{2}=\Delta p _{l}\)--pressure loss along the shelf length with current coordinate, Pa (where \(\lambda\) -- friction coefficient); \(d_{e}\) is the equivalent diameter of the shelf contact, m; \(d_{e}=\frac{4L_{in}B_{gas}}{2(L_{sh}+B_{gas})}\)[54]; \(L_{sh}\) is the shelf length, m; \(B_{dev}\) is the width of the device, m; \(W_{gas}\) is the cross section velocity of the gas flow, m/s; \(\rho_{gas}\)--gas flow density, kg/m\({}^{3}\); \(\xi_{hol}\cdot\frac{\rho_{gas}\cdot W_{hol}^{2}}{2}=\Delta p_{hol}\)--local loss in the perforation hole, Pa (where \(\xi_{hol}\) is the coefficient of the hole resistance); \(W_{hol}\)--flow velocity in the perforation holes, m/s. Thus, the total pressure loss between the control cross-sections consists of two components:  
\[\Delta p_{i}=\Delta p_{hol}+\Delta p_{l}, \tag{4}\]  
where  
\[\Delta p_{hol}=\Delta p_{i}-\Delta p_{l}. \tag{5}\]  
The nature of the gas flow jet lines along the shelf was experimentally confirmed [55]. It shows the parallel flow of two phenomena--the shelf bending and passing through its holes. Air velocity in the perforation holes:  
\[W_{hol}=\varphi\cdot\sqrt{\frac{\Delta p_{hol}}{\rho_{gas}}}=\varphi\cdot \sqrt{\frac{\Delta p_{i}-\Delta p_{l}}{\rho_{gas}}}=\varphi\cdot\sqrt{\frac{ \Delta p_{i}-zX}{\rho_{gas}}}, \tag{6}\]  
where \(\varphi\) is the velocity coefficient, which is 0.97-0.98 according to the experimental data [56]; \(z\)--specific pressure loss along the length, Pa/m. The air flow loss through the holes of the shelf perforation is the following:  
\[V_{sh}=\int\limits_{0}^{L_{sh}}W_{hol}df_{sh}=\varphi\cdot B_{dev}\int \limits_{0}^{L_{sh}}\psi\cdot\sqrt{\frac{\Delta p_{i}-zX}{\rho_{gas}}}dX, \tag{7}\]  
where \(df_{sh}=dX\cdot B_{sh}\); \(\psi\)--degree of shelf perforation (free shelf cross-section); \(f_{sh}\)--shelf area, m\({}^{2}\). In the outloading space \(L_{cl}\) (Figure 2) the pressure drop \(\Delta p_{cl}\) will be minimal:  
\[\Delta p_{cl}=\Delta p_{i}-zL_{sh}, \tag{8}\]and the air flow loss is defined as  
\[V_{cl}=\varphi\cdot L_{cl}\cdot B_{dev}\cdot\sqrt{\frac{\Delta p_{i}-zL_{sh}}{ \rho_{gas}}}. \tag{9}\]  
Total air flow rate is \(V_{sh}+V_{cl}\).