## 2 Mathematical Model
### Particle-Group Crushing and Polymerization Model
Note that Equation (32) only takes the influence of the external gas on the equivalent diameter of a particle-group into consideration. In fact, the mutual interactions between particle-groups also have considerable influence on the equivalent diameter of a particle-group. For this reason, the energy input from other particle-groups \(E_{cc}\) is considered in our coarse grain DEM-CFD model. In this situation, we have:  
\[E_{cc}=\frac{\sum\limits_{j=1}^{N_{p}}\left(\mathbf{F}_{cont,\,CGM_{ij}}\cdot \mathbf{v}_{CGM_{ij}}\right)}{m_{CGM_{i}}} \tag{38}\]  
where \(\mathbf{F}_{cont,\,CGM_{ij}},\mathbf{v}_{CGM_{ij}}\,m_{CGM_{i}}\) and \(N_{p}\) are the contact force exerted by particle-group \(j\) to particle-group \(i\), relative velocity of particle-groups \(i\) and \(j\), mass of particle-group \(i\) and number of particle-groups colliding with particle-group \(i\). Finally, the equivalent particle-group diameter \(d_{cl}\) for the coarse grain DEM-CFD model can be given by:  
\[d_{cl}=\frac{K}{E_{input}}=\frac{K}{E_{st}+E_{cc}-\left(E_{st}\right)_{mf}} \tag{39}\]  
From Equations (35)-(39), if \(\varepsilon_{\max}\) is given, then \(d_{cl}\) can be calculated. The analysis of Matsen [51] indicated that all the particles would be completely dispersed in gas-solid fluidized beds at a critical void fraction 0.9997. So it is reasonable to set \(\varepsilon_{\max}\) as this critical value 0.9997. Now, a particle-group crushing criterion can be presented based on the mismatch between the equivalent diameter \(\left(d_{cl}\right)_{i}\) and actual diameter \(\left(d_{CGM}\right)_{i}\) of a particle-group \(i\). Specifically, if:  
\[\left(d_{CGM}\right)_{i}>\left(d_{cl}\right)_{i} \tag{40}\]holds, then particle-group \(i\) will break into several child particle-groups. The number of the child particle-groups \(n_{new}\) is determined by:  
\[n_{new}=\left\lfloor\frac{(d_{CGM})_{i}^{2}}{(d_{cl})_{i}^{2}}\right\rfloor+1 \tag{41}\]  
where \(\lfloor\cdot\rfloor\) refers to the operation of rounded down. For simplicity, it assumes that there are \(n_{new}-1\) child particle-groups having the same diameter \((d_{cl})_{i}\) and mass \(m_{k}\), and the \(n_{new}\)th child particle-group consisted by the remainder mass has the following mass and diameter:  
\[m_{i,new}=m_{CGM_{i}}-(n_{new}-1)m_{k} \tag{42}\]  
\[d_{i,new}=2\bigg{(}\frac{3m_{new}}{4\pi\rho_{p}}\bigg{)}^{\frac{1}{3}} \tag{43}\]  
The velocity of the child particle-groups is inherited from their parent particle-group \(i\). Then, as to the polymerization of two colliding particle-groups, their velocity difference after collision is chosen as a criterion. Considering the collision between particle-groups \(i\) and \(j\), if their velocities after collision have the following relationship:  
\[\left(E_{st}\right)_{mf}=g\frac{\rho_{p}-\rho_{g}}{\rho_{p}}\left(U_{mf}+\frac {\varepsilon_{mf}U_{p}}{1-\varepsilon_{mf}}\right) \tag{35}\]0  
then the two particle-groups are regarded as aggregated. After polymerization, the mass and diameter of the new particle-group are given by:  
\[\left(E_{st}\right)_{mf}=g\frac{\rho_{p}-\rho_{g}}{\rho_{p}}\left(U_{mf}+\frac {\varepsilon_{mf}U_{p}}{1-\varepsilon_{mf}}\right) \tag{35}\]1  
\[\left(E_{st}\right)_{mf}=g\frac{\rho_{p}-\rho_{g}}{\rho_{p}}\left(U_{mf}+\frac {\varepsilon_{mf}U_{p}}{1-\varepsilon_{mf}}\right) \tag{35}\]2  
The velocity of the new particle-group is set as either \(\mathbf{v}_{CGM_{i}}\) or \(\mathbf{v}_{CGM_{j}}\).