# CFD-DEM Simulation of the Transport of Manganese Nodules in a Vertical Pipe
## 2 Geometry Model and Operating Parameters
### Flow Parameters  
The initial mixture velocity, \(V_{mix,0}\), represents the velocity of seawater and manganese nodules released at the inlet of the pipe and is defined as:  
\[V_{mix,0}=\frac{Q_{l}}{A} \tag{1}\]  
where \(Q_{l}\) represents the volume flow rate of seawater at the inlet (m\({}^{3}\)/s) and \(A\) denotes the cross-sectional area of the passage in the pipe (m\({}^{2}\)).  
The initial solid volume fraction, \(C_{v,0}\), is defined as the ratio of the volume flow rate of solid particles to the volume flow rate of the liquid released at the inlet of the pipe:  
\[C_{v,0}=\frac{Q_{s}}{Q_{l}}\times 100\% \tag{2}\]  
Due to the velocity gap between solid particles and seawater, local solid volume fraction might be higher than the initial solid volume fraction \(C_{v,0}\). The definition of local solid volume fraction, \(C_{v,t}\), is given by:  
\[C_{v,t}=\frac{B_{s,t}}{B_{pipe}}\times 100\% \tag{3}\]  
where \(B_{s,t}\) denotes the total volume of manganese nodules in the pipe at time \(t\) (m\({}^{3}\)) and \(B_{pipe}\) is the volume of the pipe (m\({}^{3}\)).  
For a deep-sea mining system, the mechanisms of the total pressure drop as manganese nodules are transported have not been elucidated. Here, the formulae proposed in [10] are  
Figure 1: Schematic of the hydraulic mineral lifting system.  
used to provide a reference for the numerically obtained results. A non-dimensional total pressure drop in a vertical pipe, \(I_{v}\), is defined as:  
\[I_{v}=\frac{\Delta P}{\rho_{l}SL} \tag{4}\]  
where \(\Delta P\) denotes the pressure difference between the inlet and outlet of the vertical pipe (Pa), \(\rho_{l}\) denotes the density of liquid (kg/m\({}^{3}\)), \(g\) denotes the acceleration of gravity (m/s\({}^{2}\)), and \(L\) denotes the pipe length (m).  
The total pressure drop in the pipe is composed of three parts:  
\[I_{v}=I_{l}+I_{s}+I_{c} \tag{5}\]  
where \(I_{l}\) represents the pressure drop due to the friction between liquid and wall, \(I_{s}\) denotes the pressure drop due to the presence of solid particles, and \(I_{c}\) is related to particle-particle and particle-wall collisions.  
\(I_{l}\) is given by:  
\[I_{l}=\lambda_{l}\frac{V_{mix,0}^{2}}{2 gD} \tag{6}\]  
where \(V_{mix,0}\) denotes the initial mixture velocity (m/s), \(\lambda_{l}=0.11(\Delta/D+68/Re)^{0.25}\) is the friction coefficient, and \(\Delta/D\) is the pipe-wall relative roughness.  
\(I_{s}\) is given by:  
\[I_{s}=C_{v,l}\left(\frac{\rho_{s}}{\rho_{l}}-1\right) \tag{7}\]  
where \(\rho_{s}\) is the density of solid particles (kg/m\({}^{3}\)).  
The average liquid velocity \(V_{l}\) (m/s) at the radial position \(r\) (m) is obtained by averaging the axial velocity magnitudes of the liquid \(V_{l,i}\) (m/s, in vertical direction) in the region [\(r-0.05r\), \(r+0.05r\)] of the pipe.  
\[V_{l}(r)=\frac{1}{n}\sum_{l=1}^{n}V_{l,i}\in[r-0.05r,\;r+0.05r] \tag{8}\]  
Based on Equation (8), the average solid velocity vs. (m/s) and average slip velocity \(V_{slip}\) (m/s) can be attained. The radial position \(r\) denotes the distance between the axis of the pipe and the center of the particle, so there is an excluded region for large \(r\). It should be noted that the slip velocity is the difference between the velocities of liquid and particles at the position where particles exist. The average slip velocity is not equal to \(V_{l}-V_{s}\).  
The average solid volume fraction (SVF) at the radial position \(r\) refers to the ratio of the volume of the solid particles to that of the pipe in the region [\(r-0.05r\), \(r+0.05r\)].  
\[C_{v,ave}(r)=\frac{\sum B_{s,i}\in[r-0.05r,r+0.05r]}{B_{pipe}\in[r-0.05r,r+0. 05r]}\times 100\%=\frac{\sum B_{s,i}\in[r-0.05r,r+0.05r]}{\pi\{\left(r+0.05r \right)^{2}-\left(r-0.05r\right)^{2}\}L}\times 100\% \tag{9}\]