## 3 Numerical Model  
### Basic Governing Equations  
The basic governing equations are closed by introducing reasonable constitutive equations based on the two-fluid model (see Table 2).  
Where \(\sigma_{g}\) and \(\sigma_{s}\) being the stress tensors of the gas phase and solids phase, respectively.  
\[\sigma_{g}=\alpha_{g}u_{g,eff}[\
abla\mathbf{v}_{g}+(\
abla\mathbf{v}_{g})^{\rm T}]+ \alpha_{g}(\lambda_{g}-\frac{2}{3}\mu_{g,eff})(\
abla\cdot\mathbf{v}_{g})I \tag{1}\]  
\[\mu_{g,eff}=\mu_{g}+\mu_{g,t} \tag{2}\]  
\[\begin{split}&\mu_{g,t}=\rho_{g}\zeta_{\mu}\frac{\lambda_{g}^{2}} {\varepsilon_{g}}\end{split} \tag{3}\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Number** & \(\mathbf{Q_{s}}\) **(m\({}^{3}\)/h)** & \(\mathbf{U_{g}}\) **(m/s)** & \(\mathbf{M_{s}}\) **(kg/s)** & \(\mathbf{\alpha_{s,inlet}}\) & \(\mathbf{u_{s,inlet}}\) **(m/s)** & \(\mathbf{P_{out}}\) **(MPa)** \\ \hline
1 & 0.40 & 4.71 & 0.213 & 0.318 & 4.43 & 2.91 \\
2 & 0.60 & 5.62 & 0.206 & 0.286 & 5.30 & 2.91 \\
3 & 0.80 & 6.43 & 0.194 & 0.245 & 6.09 & 2.92 \\
4 & 1.00 & 7.24 & 0.181 & 0.199 & 6.79 & 2.93 \\
5 & 1.20 & 8.10 & 0.168 & 0.184 & 7.72 & 2.93 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Conveying experiment parameters.  
Figure 1: Experimental system schematic diagram of dense-phase pneumatic conveying under high pressure. 1. Gas cylinders; 2. Gas buffer tank; 3. Fluidizing gas flowmeter; 4. Pressurizing gas flowmeter; 5. Supplementary gas flow meter; 6. Material storage vessels; 7. Conveying pipeline loop; 8. Differential pressure transmitter; 9. Pressure transducer; 10. Sight-glass section; 11. Temperature sensor; 12. On-line sampler; 13. Load cell; 14. Electric control valve; 15. Control cabinet.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Items** & **Phase** & **Formula** \\ \hline Continuity equations & Gas and Solids & \(\frac{2}{\mu\sigma}(\alpha_{g}\rho_{g})+\
abla\cdot(\alpha_{g}\rho_{g}\rho_{g})= 0,\sum\alpha_{i}=1\) \\ \multirow{3}{*}{Momentum equations} & Gas & where \(i=s\) for the solids phase and \(i=g\) for the gas phase. \\  & Gas & \(\frac{2}{\alpha}(\alpha_{g}\rho_{g}\sigma_{g})+\
abla\cdot(\alpha_{g}\rho_{g} \sigma_{g})=-\alpha_{g}\
abla\mathbf{v}_{g}+\
abla\cdot\mathbf{v}_{g}-\mathbf{F}_{sg}+ \alpha_{g}\rho_{g}\mathbf{g}\) \\ \cline{1-1}  & Solids & \(\frac{2}{\alpha}(\alpha_{g}\rho_{g}\sigma_{g})+\
abla\cdot(\alpha_{g}\rho_{g} \sigma_{g})=-\alpha_{s}\
abla\mathbf{v}_{g}+\
abla\cdot\mathbf{\sigma}_{s}+\mathbf{F}_{sg} +\alpha_{g}\rho_{g}\mathbf{g}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Basic governing equations of the present two-fluid model.  
where \(\mu_{g}\) is the gas viscosity, Pa-s; \(\mu_{g,t}\) is the gas turbulent viscosity, Pa-s; \(\mu_{g,t}\) is the effective gas viscosity, Pa-s; \(k_{g}\) is the gas turbulent kinetic energy, J; \(\varepsilon_{g}\) is the gas turbulent dissipation rate; and C\({}_{\mu}\) is the turbulence model parameter.  
\[\mathbf{\sigma}_{s}=(-p_{s}+\alpha_{s}\lambda_{s}(\
abla\cdot\mathbf{v}_{s}))\mathbf{I}+2 \alpha_{s}\mu_{s,eff}\mathbf{S}_{s} \tag{3}\]  
\[\begin{array}{l}\mu_{s,eff}=\mu_{s}+\mu_{s,t}\\ \mathbf{S}_{s}=\frac{1}{2}(\
abla\mathbf{v}_{s}+(\
abla\mathbf{v}_{s})^{T})-\frac{1}{2}( \
abla\cdot\mathbf{v}_{s})\mathbf{I}\end{array} \tag{4}\]  
where \(p_{s}\) is the solids pressure, Pa; \(\mu_{s}\) is the solids viscosity Pa-s; \(\mu_{s,t}\) is the solids turbulent viscosity, Pa-s; \(\mu_{s,eff}\) is the effective solids viscosity Pa-s; \(\lambda_{s}\) is the solids bulk viscosity, Pa-s; \(\mathbf{S}_{s}\) is the deviatoric part of strain tensor rate.