## 3 Verification of the CFD-DEM Solver
### Numerical Model on Particle Sedimentation  
As illustrated in Figure 2, a three-dimensional numerical model incorporating the multiple spherical particles and the viscous fluid was established in the CFD-DEM solver _SediFoam_. The fluid and the particle were simulated by the CFD and the DEM, respectively. For the DEM discretization domain, the wall boundary conditions were adopted. While for the CFD discretization domain, the fluid velocity and pressure were set as zero and zero-gradient on the top surface of the model, respectively, i.e., the patch boundary condition. Moreover, on the remaining boundary surfaces, the slip-wall and zero-gradient conditions were assigned to the fluid velocity and the pressure, respectively, i.e., the wall boundary condition. The length, width, and height of the numerical model were 0.01, 0.01, and 0.2 m, respectively. The even-sized particles were of diameter 0.001 m. Based on the initial concentration \(c\) of particles, four cases were set up, which were Case 1 (\(c=0\)), Case 2 (\(c=0.05\)), Case 3 (\(c=0.1\)), and Case 4 (\(c=0.25\)). Parameters of the numerical model are summarized in Table 1.  
Figure 2: Numerical model on settlement of multiple particles.  
By substituting the settlement rate of a single suspended particle \(V_{(s,0)}\) = 0.141 m/s and the diameter of the model vessel \(D\) = 0.02 m into Equation (1), the predictions of the Richardson empirical formula on various particle concentrations are presented in Table 2.  
Figure 3 illustrates that the velocity of uniformly sized spherical particles varies the particle concentration during the settlement process. The simulated values agree well with the predictions of the empirical formula, which verifies the correctness of the CFD-DEM solver _Sedifoam_.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Parameter** & **Symbol** & \multicolumn{4}{c}{**Value**} \\ \hline  & & **Case1** & **Case2** & **Case3** & **Case4** \\ \hline \multicolumn{8}{c}{**Particle-DEM**} \\ \hline Number & \(N\) & \(1\) & \(955\) & \(1910\) & \(4775\) \\ Concentration & \(c\) & \(0\) & \(0.05\) & \(0.1\) & \(0.25\) \\ Diameter/(m) & \(d_{s}\) & \(0.001\) & \(0.001\) & \(0.001\) & \(0.001\) \\ Mass & \(\rho_{s}\) & \(2650\) & \(2650\) & \(2650\) & \(2650\) \\ \(\mathrm{density/(N/m)}\) & \(k_{m}/k_{t}\) & \(500/143\) & \(500/143\) & \(500/143\) & \(500/143\) \\ Dumping/(N/ms) & \(\tau_{m}/\tau_{I}\) & \(20,000/10,000\) & \(20,000/10,000\) & \(20,000/10,000\) & \(20,000/10,000\) \\ Friction coefficient & \(\mu\) & \(0.4\) & \(0.4\) & \(0.4\) & \(0.4\) \\ \hline \multicolumn{8}{c}{**Fluid-CFD**} \\ \hline Mass & \(\rho_{f}\) & & & \(1000\) & & \\ \(\mathrm{density/(kg/m^{2})}\) & \(\rho_{f}\) & & & \(1000\) & & \\ Viscosity/(m\({}^{2}\)/s) & \(v\) & & & \(10^{-6}\) & & \\ Gravity (m/s\({}^{2}\)) & \(g\) & & & \(9.8\) & & \\ \hline \multicolumn{8}{c}{**Domain dimensions**} \\ \hline Length/(m) & \(L\) & & \(0.01\) & & \\ \(\mathrm{Wide/(m)}\) & \(W\) & & \(0.01\) & & \\ Height/(m) & \(H\) & & \(0.2\) & & \\ Mesh size/(m) & & & \(0.002\times 0.002\times 0.002\) & \\ \hline \end{tabular}
\end{table}
Table 1: Parameters of sedimentation model of multiple particles.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Parameter** & **Symbol** & \multicolumn{4}{c}{**Value**} \\ \hline Diameter/(m) & \(D\) & & & & & 0.02 & & & \\ Particle size/(m) & \(d\) & & & & & 0.001 & & & \\ Reynolds number & \(Re\) & & & & 141 & & & \\ Empirical coefficient & \(n\) & & & & 3.8103 & & & \\ Particle concentration & \(c\) & \(0\) & \(0.05\) & \(0.10\) & \(0.25\) & \(0.50\) & \(0.75\) & \(0.90\) & \(0.95\) & \(1.00\) \\ Velocity/(m/s) & \(V_{(s,c)}\) & \(0.141\) & \(0.116\) & \(0.0944\) & \(0.0471\) & \(0.0101\) & \(0.000717\) & \(0.0000218\) & \(0.00000156\) & \(0\) \\ \hline \end{tabular}
\end{table}
Table 2: Predictions of the Richardson empirical formula.