# CFD-DEM Simulation of the Transport of Manganese Nodules in a Vertical Pipe
## 3 Mathematical Models  
### Computational Model  
A two-way coupling method for the solid-liquid two-phase flow was established based on the Euler-Lagrange model. The commercial software ANSYS Fluent 18.0 was used to solve the flow field based on the Euler coordinate system. Meanwhile, the commercial software EDEM 2018 was used to describe the motion of solid particles in the Lagrange coordinate system. ANSYS Fluent and EDEM are connected using compile UDF (User Defined Function, which can be loaded into ANSYS Fluent to control the coupling between ANSYS Fluent and EDEM). Figure 2 shows the flow chart of the coupling between EDEM and Fluent to describe the simulation process. In ANSYS Fluent, for each time step, after convergence was accomplished, the flow data was conveyed to EDEM. Then iterations were performed in EDEM at the same time step to calculate the interaction forces based on the parameters of fluid and solid particles in the same computational element. Subsequently, the information of solid particles was updated and transmitted to ANSYS Fluent. The above steps were repeated until the total simulation time \(t_{total}\) was accomplished [19].  
For the flow simulation in the coupling process, the Euler model is used to calculate flow parameters. The effect of the solid phase is considered through introducing the volume fraction of the liquid phase, \(\epsilon_{l}\).  
The continuity equation of the liquid phase is expressed as:  
\[\frac{\partial(\epsilon_{l}\rho_{l})}{\partial t}+\
abla\cdot(\epsilon_{l} \rho_{l}\mathbf{u}_{l})=0 \tag{10}\]  
The momentum equation of the liquid phase is given by [20]:  
\[\frac{\partial(\epsilon_{l}\rho_{l}\mathbf{u}_{l})}{\partial t}+\
abla\cdot( \epsilon_{l}\rho_{l}\mathbf{u}_{l}\mathbf{u}_{l})=-\epsilon_{l}\
abla\mathbf{P}+\
abla\cdot (\epsilon_{l}\mu_{l}\
abla\mathbf{u}_{l})+\epsilon_{l}\rho_{l}\mathbf{g}+\mathbf{F}_{sl} \tag{11}\]  
\[\mathbf{F}_{sl}=-\frac{\sum\mathbf{F}_{t_{sl}}}{\Delta V_{cell}} \tag{12}\]  
where \(t\) denotes time (s), \(\mathbf{u}_{l}\) denotes the velocity of the liquid phase (m/s), \(\mathbf{P}\) denotes the static pressure of the liquid phase (Pa), \(\mu_{l}\) denotes the dynamic viscosity of the liquid phase (Pa-s), \(\mathbf{g}\) denotes the acceleration of gravity (m/s\({}^{2}\)), \(\mathbf{F}_{sl}\) denotes the sum of the forces acting on the liquid phase by solid particles in the computational cell (N), \(\mathbf{F}_{ls,i}\) denotes the force acting on particle \(i\) by the liquid phase (N), and \(\Delta V_{cell}\) denotes the volume of the cell (m\({}^{3}\)).  
For the EDEM part, the Lagrange model is used to track the trajectories of solid particles. The translation and rotation of solid particles are governed by [PERSON]'s equation, which is expressed as [21]:  
\[m_{i}\frac{d\mathbf{v}_{i}}{dt}=\sum_{j}\mathbf{F}_{c,ij}+\sum_{k}\mathbf{F}_{nc,jk}+\mathbf{F }_{ls,i}+\mathbf{F}_{g,i} \tag{13}\]  
\[\mathbf{I}_{i}\frac{d\mathbf{\omega}_{i}}{dt}=\sum_{j}\bigl{(}\mathbf{M}_{t,ij}+\mathbf{M}_{r, ij}\bigr{)} \tag{14}\]  
where \(m_{i}\) denotes the mass of solid particle \(i\) (kg), \(\mathbf{v}_{i}\) denotes the translational velocity of solid particle \(i\) (m/s), \(\mathbf{F}_{c,ij}\) denotes the contact forces between particles \(i\) and \(j\) (N), \(\mathbf{F}_{nc,ik}\) denotes the non-contact forces between particles \(i\) and \(k\) (N, e.g., electromagnetic force), \(\mathbf{F}_{ls,i}\) denotes the force acting on particle \(i\) by the liquid phase (N, e.g., pressure gradient force, drag force, etc.), \(\mathbf{F}_{g,i}\) denotes the volumetric force acting on particle \(i\) (N, e.g., gravity),  
Figure 2: Flow chart for EDEM-Fluent coupling.  
denotes the inertia moment of particle \(i\) (N\(\cdot\)m), \(\mathbf{\omega}_{i}\) denotes the angular velocity of particle \(i\) (rad/s), \(M_{i,ij}\) denotes the tangential friction moment between particles \(i\) and \(j\) (N\(\cdot\)m), and \(M_{r,ij}\) denotes the normal friction moment between particles \(i\) and \(j\) (N\(\cdot\)m).