## 2 Mathematical Model
### CFD Model for Gas Phase  
In a gas-solid fluidized beds, it is difficult to directly calculate the mutual interaction between the continuum gas and discrete particles. An alternative way to solve this problem is the local volume average technique which has been widely used in the continuum approach of modelling fluidization process [42]. According to this volume average technique, the governing equations of the gas phase are the fluid continuity and Navier-Stokes equations expressed in terms of the local mean variables over the computational cell:  
\[\left\{\begin{array}{l}\frac{\partial}{\partial t}\Big{(}\varepsilon\rho_{g }\Big{)}+\
abla\cdot\Big{(}\varepsilon\rho_{g}\mathbf{u}\Big{)}=0\\ \frac{\partial}{\partial t}\Big{(}\varepsilon\rho_{g}\mathbf{u}\Big{)}+\
abla \cdot\Big{(}\varepsilon\rho_{g}\mathbf{u}\Big{)}=-\varepsilon\
abla p-\mathbf{ S}_{p}+\
abla\cdot\Big{(}\varepsilon\boldsymbol{\tau}_{g}\Big{)}+\varepsilon \rho_{g}\mathbf{g}\end{array}\right. \tag{26}\]  
where \(\varepsilon\), \(\rho_{g}\), \(\mathbf{u}\), \(p\) and \(\mathbf{g}\) have the same meaning as mentioned above. \(\mathbf{S}_{p}\) and \(\boldsymbol{\tau}_{g}\) are the volumetric gas-particle interaction force and viscous stress tensor, respectively. According to [PERSON]'s third law of motion, the value of \(\mathbf{S}_{p}\) at a given CFD computational cell is defined as the volume average of the drag forces acting on all the particles located in this cell:  
\[\mathbf{S}_{p}=\frac{1}{V_{cell}}\sum_{i=1}^{N_{CGM}^{c}}\mathbf{F}_{drug,\,CGM _{i}}=\frac{1}{V_{cell}}\sum_{i=1}^{N_{O}^{c}}\mathbf{F}_{drug,\,i} \tag{27}\]  
where \(V_{cell}\) and \(N^{c}\) are the volume of the computational cell and the number of particles located in this cell, respectively. The viscous stress tensor \(\boldsymbol{\tau}_{g}\) is given by:  
\[\boldsymbol{\tau}_{g}=\Big{(}-\frac{2}{3}\mu_{g}\Big{)}\big{(}\
abla\cdot \mathbf{u}\big{)}\mathbf{I}+\mu_{g}\Big{(}\
abla\mathbf{u}+(\
abla\mathbf{u} )^{T}\Big{)} \tag{28}\]  
By definition, the void fraction \(\varepsilon\) in a computational cell is the ratio of the void volume (gas volume) to the volume of the cell. If \(V_{i}\) is the volume of particle \(i\) inside a computational cell, then the void fraction in this cell is:  
\[\varepsilon=1-\frac{1}{V_{cell}}\sum_{i=1}^{N_{CGM}^{c}}V_{CGM_{i}}=1-\frac{1}{ V_{cell}}\sum_{i=1}^{N_{O}^{c}}V_{i} \tag{29}\]  
The void fraction \(\varepsilon_{i}\) and gas velocity \(\mathbf{u}_{i}\) at the center of mass of particle \(i\) can be calculated by interpolation from the flow variables stored at the grid nodes. As illustrated in Figure 2, particle \(i\) divides the cell, which it belongs to, into four parts. Each part has the area \(S_{ij}\) (\(j=1,\,2,\,3,\,4\)). If the void fraction and gas velocity valued at the four vertexes of this cell are \(\varepsilon_{ij}\) and \(\mathbf{u}_{ij}\left(j=1,\,2,\,3,\,4\right)\), then \(\varepsilon_{i}\) and \(\mathbf{u}_{i}\) can be given by:  
\[\varepsilon_{i} =\frac{S_{4}\varepsilon_{i1}+S_{3}\varepsilon_{i2}+S_{2} \varepsilon_{i3}+S_{i1}\varepsilon_{i4}}{S_{i1}+S_{i2}+S_{i3}+S_{i4}} \tag{30}\] \[\mathbf{u}_{i} =\frac{S_{4}\mathbf{u}_{i1}+S_{3}\mathbf{u}_{i2}+S_{2}\mathbf{u}_ {i3}+S_{1}\mathbf{u}_{i4}}{S_{i1}+S_{i2}+S_{i3}+S_{i4}} \tag{31}\]  
in the two-dimensional cases.