# CFD-DEM Simulation of the Transport of Manganese Nodules in a Vertical Pipe
## 5 Results and Discussion
### Effect of Initial Solid Volume Fraction
#### 5.1.2 Average Parameters  
At \(V_{mix,0}\) = 4.0 m/s and \(d_{s}\) =15.0 mm, the average rising velocity of seawater is plotted as a function of non-dimensional radial position in Figure 6. With increasing initial SVF, the average seawater velocity decreases from about 4.7 m/s to 4.5 m/s in the central zone, but the variation of the average seawater velocity with initial SVF is negligible near the wall. This trend agrees well with the experimental observation [29]. At a given particle size, the increase in initial SVF leads to the participation of more particles. Consequently, more energy will be consumed and both seawater and particle velocities decrease as initial SVF increases, as indicated in Figure 6. In the central zone and as initial SVF increases from 5% to 10%, the seawater velocity decreases by about 0.7 m/s while the velocity of manganese nodules decreases by about 0.5 m/s. Due to the effect of wall shear, the seawater velocity decreases remarkably near the wall, while manganese nodules suffer less from this effect. In addition, seawater in the central zone accelerates much easier than manganese nodules due to considerably lower density. There is a jump in the value of average solid velocity for \(C_{v,0}\) = 1% between \(r\)/\(R\) = 0.05 and \(r\)/\(R\) = 0.15. Compared with \(C_{v,0}\) = 5% and 10%, the number of particles at the center of the pipe is not dominant for \(C_{v,0}\) = 1%. The smaller the particle number, the higher the slip velocity needed to create the drag force to balance the gravity and the buoyancy force in a vertical direction. Therefore, average solid velocity is low at the pipe center. The average slip velocities for different initial SVFs are plotted as a function of \(r\)/\(R\) in Figure 7. It can be seen that the average slip velocity decreases generally with increasing radial distance from the pipe axis. It is also evidenced that the slip velocities for \(C_{v,0}\) = 5% and 10% decrease remarkably compared to that corresponding to \(C_{v,0}\) = 1%. For particles with diameters of 5.0 mm, 15.0 mm, and 30.0 mm, the settling velocities are about 0.353 m/s, 0.612 m/s, and 0.865 m/s, respectively, which were calculated through \(u_{t}=\sqrt{\frac{8}{3}d_{s}g}\frac{(\rho_{s}-\rho_{l})}{\rho_{l}}\). For a particle falling or rising in both static and dynamic fluid, it mainly suffers from the drag force, the buoyancy force, and the gravity force in a vertical direction. As the liquid velocity increases, the drag coefficient of particle is reduced. The drag force is related to the drag coefficient and the slip velocity. After the balance of forces is attained, the slip velocities of the three cases are all greater than their settling velocities. Figure 6: Effects of initial SVF on average phase velocities (\(V_{mix,0}\) = 4.0 m/s, \(d_{s}\)= 15.0 mm). The distribution of average solid volume fraction \(C_{\text{up,ave}}\) in a radial direction is shown in Figure 8, where \(r\) is the distance between the center of the particle and the cross-sectional center of the pipe and \(R\) denotes the radius of the pipe. It is evidenced that the average SVF decreases with radial position \(r\) generally. Furthermore, the reduction of initial SVF promotes the uniformity of the distribution of manganese nodules. The lift force and the pressure gradient force have insignificant influence on the lateral movement of particles, which is ascribed to the low rotation intensity of particles and uniform pressure gradients over the cross section. Therefore, the contact force plays an important role in determining the particle distribution along the radial direction. At high initial SVFs, some particles at the cross-sectional center are exposed to contact forces from all directions because they are surrounded by other particles. However, the contact forces counteract each other; consequently, central particles can move stably with large average SVF, and lateral movement is not influential. Surrounding particles will be bounced back towards the wall after they interact with the central particles.