## 2 Mathematical Model
### Particle-Group Crushing and Polymerization Model  
Because of the complex particle-particle and gas-particle interactions inside a gas-solid fluidized beds, the particle-groups of the coarse grain DEM-CFD model can hardly to maintain their initial configurations during the whole simulation process. Frequent crushing and polymerization of the particle-groups should be taken into consideration. However, the current coarse grain DEM-CFD model ignores this important physical process. Consequently, its ability for describing the heterogeneity caused by the complex mesoscale structures of gas-solid systems may be not enough. The system is likely to be overly averaged due to the unchangeable statistical averaging done at the beginning of the simulation. In order to rationalize the coarse grain DEM-CFD model by overcoming the unrealistic coarse grain treatment of particles, a dynamic particle-group crushing and polymerization model is present in this section. For capturing the crushing of a particle-group, the effective size of each particle-group should be measured. Following the definition of the equivalent cluster diameter used by the EMMS model [37; 38], we use \(d_{cl}\) to represent the equivalent diameter of a particle-group. In a gas-solid fluidized bed, \(d_{cl}\) is generally inversely proportional to the energy input from the gas \(E_{input}\):  
\[d_{cl}=\frac{K}{E_{input}} \tag{32}\]  
where \(K\) is a coefficient to be determined and \(E_{input}\) equals to the difference between the total energy input \(E_{st}\) and the energy input for the state of generalized minimum fluidization \(\left(E_{st}\right)_{mf}\):  
\[E_{input}=E_{st}-\left(E_{st}\right)_{mf} \tag{33}\]  
For the EMMS model, the energy consumption is divided into three parts [38]. That is:  
\[ \begin{array}{ll}E_{st}&=\left(E_{st}\right)_{c}+\left(E_{st}\right)_{f}+ \left(E_{st}\right)_{b}\\ &=\frac{1}{\left(1-\varepsilon\right)\rho_{p}}\left[n_{c}F_{c}U_{c}f+n_{f}F_{ f}U_{f}(1-f)+n_{b}F_{b}U_{f}(1-f)\right]\end{array}  \tag{34}\]  
Figure 2: Illustration of the interpolation from gas to particles (two dimensional cases). where the subscripts \(c\), \(f\) and \(b\) refer to the dense phase, dilute phase and inter phase, respectively. \(\rho_{p}\), \(n\), \(F\), \(f\) and \(U\) are the particle density, estimated particle number of each phase, drag force acting on single particle or cluster in suspension, volume fraction of dense phase and apparent gas velocity, respectively. Since the drag force and apparent gas velocity have the same direction, only their norms are considered in Equation (34). At the state of generalized minimum fluidization, the energy input \(\left(E_{st}\right)_{mf}\) can be calculated by:  
\[\left(E_{st}\right)_{mf}=g\frac{\rho_{p}-\rho_{g}}{\rho_{p}}\left(U_{mf}+\frac {\varepsilon_{mf}U_{p}}{1-\varepsilon_{mf}}\right) \tag{35}\]  
where \(U_{mf}\), \(U_{p}\), \(\varepsilon_{mf}\) and \(g\) are the minimum fluidization velocity, apparent particle velocity, void fraction at the minimum fluidization and gravitational constant, respectively. According to the relationship \(d_{cl}=d_{p}\) at the limiting condition \(\varepsilon=\varepsilon_{\max}\), the coefficient \(K\) is determined by:  
\[K=gd_{p}\frac{\rho_{p}-\rho_{g}}{\rho_{p}}\left(\frac{U_{p}}{1-\varepsilon_{ \max}}-(U_{mf}+\frac{\varepsilon_{mf}U_{p}}{1-\varepsilon_{mf}})\right) \tag{36}\]  
For the DEM model, because the motion of each particle can be directly tracked, there is no need to divide the system into dense, dilute and inter phases. The number of particles in any local area can be exactly calculated. Therefore, the expression of \(E_{st}\) for a single particle-group \(i\) may be simplified as:  
\[E_{st}=\frac{1}{(1-\varepsilon)\rho_{p}}(\mathbf{F}_{dmg,\,CGM}\cdot\mathbf{u} )=\frac{1}{(1-\varepsilon)\rho_{p}}(n_{i}\mathbf{F}_{dmg,\,O}\cdot\mathbf{u}) \tag{37}\]  
where \(n_{i}\) is the number of original particles inside the particle-group \(i\).