## 3 Numerical Model
### Solids Stress Model
#### 3.4.2 Frictional Stress  
Frictional stress is one of the most significant interparticle forces in dense gas-solid flow, revealing the fundamental properties of such flows from a mechanical perspective. A comprehensive frictional stress model comprises both frictional pressure and frictional viscosity. In general, different yield criterions can be employed to develop the corresponding frictional stress model. For instance, [PERSON] introduced the Pitman-Schaeffer-Gray-Stiles yield criterion to develop the Dartevelle frictional stress model [25]. Compared to conventional frictional stress models, this model simultaneously accounts for both the dilatancy and consolidation of the solids phase in dense-phase gas-solid flow [31]. Dartevelle frictional stress model [25]:  
\[p_{f}=\left\{ \begin{array}{cc}1000\frac{(\alpha_{s}-\alpha_{s,\min})^{3}}{( \alpha_{s,\max}-\alpha_{s})^{3}}&\alpha_{s,\min}<\alpha_{s}<\alpha_{s,\max}\\ 0&\alpha_{s}\leq\alpha_{s,\min}\end{array} \right. \tag{13}\]  
\[\mu_{f}=\frac{p_{f}\sin^{2}\phi_{i}}{\sqrt{2\sin^{2}\phi_{i}(\boldsymbol{S}_{ s}:\boldsymbol{S}_{s}+\Theta_{s}/d_{s}^{2})+\left(\
abla\cdot\boldsymbol{v}_{s} \right)^{2}}} \tag{14}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Setting** \\ \hline Granular viscosity & Gidaspow [43] \\ Granular bulk viscosity & [PERSON] et al. [44] \\ Solids pressure & [PERSON] et al. [44] \\ Granular conductivity & Gidaspow [43] \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameter settings of KTGF. [PERSON] and [PERSON] [33] introduced the extended von Mises/Coulomb yield criterion, incorporating the fluctuation strain rate term \(\Theta_{s}/{d_{s}}^{2}\) proposed by [PERSON] to modify and improve the [PERSON] and Jackson frictional pressure model and Schaeffer frictional viscosity. As a result, the Srivastava and Sundaresan frictional stress model was obtained. [PERSON] and [PERSON] frictional stress model [33]:  
\[p_{c}=\left\{ \begin{matrix}F\frac{(\alpha_{s}-\alpha_{s,\min})^{\prime}}{( \alpha_{s,\max}-\alpha_{s})^{\prime}}&\alpha_{s,\min}<\alpha_{s}<\alpha_{s, \max}\\ 0&\alpha_{s}\leq\alpha_{s,\min}\end{matrix} \right. \tag{15}\]  
\[p_{f}=\left[1-\frac{\
abla\cdot\mathbf{v}_{s}}{\sqrt{2}n\sin\phi_{i}\sqrt{S_{s}: \mathcal{S}_{s}+\Theta_{s}/d_{s}^{2}}}\right]^{n-1}p_{c} \tag{16}\]  
\[\mu_{f}=\frac{p_{f}\sin\phi_{\mathrm{I}}}{\sqrt{2}\sqrt{S_{s}:\mathcal{S}_{s} +\Theta_{s}/d_{s}^{2}}}\left\{n-(n-1)(\frac{p_{f}}{p_{c}})^{\frac{1}{n-1}}\right\} \tag{17}\]  
The coefficient '\(n\)' in the above equation mainly depends on the state of the solids phase. When the solids phase expands, i.e., \(\
abla\cdot\mathbf{v}>0\), \(n=\sqrt{3/(2\mathrm{sin}\phi_{\mathrm{I}})}\); when the solids phase compresses, i.e., \(\
abla\cdot\mathbf{v}<0\), \(n=1.03\). Therefore, the [PERSON] and Sundaresan frictional stress model considers both the expansion and compression characteristics of the solids phase. A frictional pressure model is usually obtained by fitting material properties to experimental data. For example, [PERSON] and [PERSON], [PERSON] et al., and [PERSON] et al. [45] have developed frictional pressure models based on the physical properties of glass beads with different diameters. The Dartevelle frictional pressure model suggests that frictional pressure is only dependent on the solids volume fraction, which is obviously not comprehensive. Although the [PERSON] and [PERSON] frictional pressure model introduced three empirical constants (Fr, r, s) to characterize material frictional properties, it did not propose a method for determining these constants. Therefore, in general, these empirical constants (Fr, r, s) need to be adjusted based on simulation results. This not only increases the computational cost but also affects the accuracy, reliability, and applicability of the simulations. To address these issues, [PERSON] et al. [46] incorporated two material property parameters, particle stiffness \(k_{n}\) and particle diameter \(d_{s}\), into the frictional pressure model.