## 3 PIV Experimental Apparatus
### Validation against PIV Experiment  
The reliability of the simulation was verified by the self-conducted PIV experiment. The system becomes opaque at higher volume fractions and impeller speeds, which affects the application of optical measurement techniques. Thus, the impeller speed was set at 240 rpm and the solid volume fraction was 2%. The simulation and experiment were carried out under the same conditions. Figure 5 compares the liquid phase flow field of the PIV experiment and CFD simulation. A large single circulation loop existed in the stirred tank. The liquid was ejected obliquely downward from the impeller to the bottom of the tank with a maximum velocity magnitude and then divided into two streams. Most of the liquid was fed into the main circulation of the whole tank, which forms the downward main loop; and a small part went down, forming the inverted conical secondary induced circulation region in the center of the bottom of the tank. A high-speed flow was generated near the tip of the blade due to the rotation of the impeller, and the downward flow formed a high-speed jet. The liquid velocity was relatively high in the impeller region and along the jet, whereas it was lower at the upper part of the stirred tank and at the center of the  
\begin{table}
\begin{tabular}{c c} \hline
**Description** & **Equation** \\ \hline Stress tensor of liquid phase & \(\overline{\tau}_{l}=\alpha_{l}\mu_{l}(\
abla\overset{\rightarrow}{u}_{l}+ \
abla\overset{\rightarrow}{u}_{l}^{T})+\alpha_{l}\Big{(}\lambda_{l}-\frac{2}{ 3}\mu_{l}\Big{)}\
abla\overset{\rightarrow}{u}_{l}^{\overset{\leftarrow}{l}}\) \\ Stress tensor of solid phase & \(\overline{\tau}_{s}=\alpha_{s}\mu_{s}(\
abla\overset{\rightarrow}{u}_{s}+ \
abla\overset{\rightarrow}{u}_{s}^{T})+\alpha_{s}\Big{(}\lambda_{s}-\frac{2}{ 3}\mu_{s}\Big{)}\
abla\overset{\rightarrow}{u}_{s}^{\overset{\leftarrow}{l}}\) \\ Solids bulk viscosity & \(\lambda_{s}=\frac{4}{3}\alpha_{s}^{2}p_{s}d_{s}g_{0s,8s}(1+e_{ss})\Big{(}\frac{ \theta_{s}}{\pi}\Big{)}^{1/2}\) \\ Solids pressure & \(p_{s}=\alpha_{s}\rho_{s}d_{s}+2p_{s}(1+e_{ss})\alpha_{s}^{2}\frac{2}{3}\rho_{0s }\theta_{s}\) \\ Solids shear viscosity & \(\mu_{s}=\frac{4}{3}\alpha_{s}^{2}p_{s}d_{s}g_{0s,8s}(1+e_{ss})\sqrt{\frac{ \theta_{s}}{\pi}}\) \\  & \(+\frac{100\mu_{s}\sqrt{\pi\rho_{s}}}{64\pi\rho_{s}\rho_{s}\rho_{s}}\Big{[}1+ \frac{4}{3}g_{0s,8s}\alpha_{s}(1+e_{ss})\Big{]}^{2}\) \\ Conductivity of granular energy & \(k_{\mathrm{gs}}=\frac{25\rho_{s}d_{s}\sqrt{\pi\rho_{s}}}{64(1+e)\beta_{0s}}\Big{[} 1+\frac{6}{3}g_{0s,8s}\alpha_{s}\Big{]}^{2}\) \\ Rate of kinetic energy dissipation & \(\gamma_{\mathrm{gs}}=\frac{12\left(1-\alpha_{s}\right)g_{0s}}{d_{s}\sqrt{\pi}} \rho_{s}\alpha_{s}^{2}\theta_{s}\frac{3}{2}\) \\ Rate of energy exchange & \(\varphi_{\mathrm{ls}}=\frac{4\rho_{s}}{4\sqrt{\pi\rho_{s}}\rho_{s}}\Big{[}\frac{ 18\mu_{s}}{d_{s}^{2}P_{s}}\Big{]}\overset{\rightarrow}{u}_{l}-\overset{ \rightarrow}{u}_{s}^{\overset{\leftarrow}{l}}\) \\ Radial distribution function & \(g_{0,ss}=\left[1-\left(\frac{\alpha_{s}}{\alpha_{s,max}}\right)^{1/3}\right]^{-1}\) \\ \hline \end{tabular}
\end{table}
Table 3: Equations for the KTGF model.  
bottom of the tank. The simulation and experimental results are consistent, which verifies the reliability of the EE-KTGF model.