## 2 Mathematical Model
### Coarse Grain DEM Model for Solid Phase
#### 2.1.1 Drag Force  
For calculating \(\mathbf{F}_{drag,i}\), a proper drag force model should be selected. In the literature, the Gidaspow type drag models [40; 41; 42] and the EMMS drag models [43; 44; 45; 46; 47; 48] are most used. Generally, the Gidaspow type drag models neglect any heterogeneities induced by the complex gas-solid flow. They lie on the assumption that in each computational cell homogeneous conditions exist. Research work by [PERSON] and [PERSON] [44] has demonstrated that the momentum transfer between solid phase and gas phase is over estimated by the Gidaspow type drag models. They are incapable of describing the drag force drop caused by mesoscale effect. By contrast, the EMMS drag models take the flow heterogeneity induced by the particle cluster formation into account though dividing the gas-solid flow into a dilute phase (gas dominated zone) and a dense phase (particle dominated zone). Because the heterogeneity is considered, the EMMS drag force models can predict the drag force drop in case of particle clustering which has been observed in experiments [39]. For the above reason, the following EMMS drag force model [44] is adopted in this study:  
\[\mathbf{F}_{drag,i}=V_{i}\frac{\beta_{i}}{1-\varepsilon_{i}}(\mathbf{u}_{i}- \mathbf{v}_{i}) \tag{2}\]where \(\varepsilon_{i}\), \(\beta_{i}\) and \(\mathbf{u}_{i}\) are the void fraction (volume fraction of gas), interphase momentum exchange coefficient and gas velocity associated with particle \(i\), respectively. The \(\beta_{i}\) is given by:  
\[\beta_{i}=\frac{3}{4}\frac{\varepsilon_{i}(1-\varepsilon_{i})\rho_{g}|\mathbf{ u}-\mathbf{v}|}{d_{p}}\varepsilon_{i}^{-2.65}C_{d}H_{d} \tag{3}\]  
where \(d_{p}\), \(\rho_{g}\), \(C_{d}\) and \(H_{d}\) are the particle diameter, gas density, drag coefficient and heterogeneity index, respectively. The drag coefficient \(C_{d}\) for an isolated particle depends on the particle Reynolds number \(Re_{p}\) and is given by:  
\[C_{d}= \begin{cases}\frac{24(1+0.15R_{p}^{0.687})}{Re_{p}}&Re_{p}\leq 1000\\ 0.44&Re_{p}>1000\end{cases}  \tag{4}\]  
where \(Re_{p}\) is expressed as:  
\[Re_{p}=\frac{\varepsilon_{i}\rho_{g}|\mathbf{u}_{i}-\mathbf{v}_{i}|d_{p}}{ \mu_{g}} \tag{5}\]  
Here, \(\mu_{g}\) is the viscosity of the fluidizing gas phase. The heterogeneity index \(H_{d}\) depends on both locally transient and globally averaged information of two-phase velocities and void fraction. It may be expressed as:  
\[H_{d}=a(Re_{p}+b)^{c} \tag{6}\]  
where \(a\), \(b\) and \(c\) are parameters associated with the void fraction \(\varepsilon\) as listed in Table 1[49; 50]. These parameters are obtained by researchers through comprehensive optimization. They have been shown to have the ability to produce good results over a wide range of applications, although they may not be optimal for many special cases. Because the parameter optimization is not an easy task, the widely applicable parameters given in Table 1 are preferred in this work. Generally, the value of \(H_{d}\) is between 0 and 1. When \(H_{d}\) trends to 1, it represents that the particles are locally homogenously distributed and the corresponding fluidizing state is stable. Whereas if \(H_{d}\) trends to 0, severe particle clustering exists and the system shows strong heterogeneity. \begin{table}
\begin{tabular}{l l} \hline \(a=0.8526-\frac{0.5846}{1+(\varepsilon/0.4325)^{22.6279}}\) & \(0.4\leq\varepsilon<0.46\) \\ \(c=0\) & \\ \(a=0.0320-\frac{0.7399}{1+(\varepsilon/0.4912)^{54.265}}\) & \(\frac{1}{4265}\) \\ \(b=0.00225+\frac{772.0074}{1+10^{66.3224(\varepsilon-0.3987)}}+\frac{0.02404}{1+ 10^{53.8948(0.5257-\varepsilon)}}\) & \(0.46\leq\varepsilon<0.545\) \\ \(c=0.1705-\frac{0.1731}{1+(\varepsilon/0.5020)^{37.7091}}\) & \(\frac{3}{37.7091}\) \\ \(b=(0.8223-0.1293\varepsilon)^{13.0310}\) & \(0.545\leq\varepsilon<0.99\) \\ \(c=\frac{0.0800}{1-\exp{(-\varepsilon-0.9942)}/0.00218}\) & \(0.99\leq\varepsilon<0.9997\) \\ \(a=1,c=0\) & \(0.9997\leq\varepsilon\leq 1\) \\ \hline \end{tabular}
\end{table}