## 2 Analytical Methodology
### Cfd-Dem
#### 2.1.3 Momentum and Energy Exchange between Phases  
This section aims to explain the momentum/energy exchange source terms in the continuous fluid phase (\(\overset{\rightarrow}{S}_{m}\) and \(S_{E}\)) and discrete particle phase (\(\overset{\rightarrow}{F}_{f,i}\) and \(Q_{f,i}\)). It should be noted in advance that the momentum/energy exchange between fluid and particle phases in this research is based on the conception of the two-way coupling, which means that the momentum/energy transferred to a specific particle from the fluid phase of the CFD mesh where the particle centroid is located will provide the equal and opposite feedback to the fluid phase of this CFD mesh [32]. In addition, the CFD mesh coupling with DEM simulation is always unresolved in STAR-CCM+, which means that the momentum/energy exchanges of a CFD mesh are calculated by collecting the feedback of all the particles whose centroids are located at this CFD mesh, without solving the detail flow redirection around the solid particles. In other words, the shape of the solid particle is unresolved in the CFD mesh [32; 36]. Firstly, the forces exerted from the fluid phase to the particle \(i\) considered can be expressed as below:  
\[\overset{\rightarrow}{F}_{f,i}=\overset{\rightarrow}{F}_{p,i}+\overset{ \rightarrow}{F}_{d,i}, \tag{9}\]  
where \(\overset{\rightarrow}{F}_{p,i}\) [kg:m/s\({}^{2}\)] and \(\overset{\rightarrow}{F}_{d,i}\) [kg:m/s\({}^{2}\)] are pressure gradient and viscous drag forces, respectively. \(\overset{\rightarrow}{F}_{p,i}\) can be counted as the buoyancy, and formulated as below [32]:  
\[\overset{\rightarrow}{F}_{p,i}=-V_{i}\
abla p, \tag{10}\]  
where \(V_{i}\) [m\({}^{3}\)] is the volume of the particle \(i\). \(\overset{\rightarrow}{F}_{d,i}\) is due to the relative velocity between particle and viscous fluid, as follows [32]:  
\[\overset{\rightarrow}{F}_{d,i}=\frac{1}{2}C_{d,i}\rho_{f}A_{i}\left|\overset{ \rightarrow}{V}_{f}-\overset{\rightarrow}{V}_{i}\right|(\overset{ \rightarrow}{V}_{f}-\overset{\rightarrow}{V}_{i}), \tag{11}\]  
where \(A_{i}\) [m\({}^{2}\)] is the projected area of the particle \(i\). \(C_{d,i}\) [-] is the drag coefficient of particle \(i\). In our previous study, the different drag coefficient models provided by the STAR-CCM+ were verified and validated with experiments which aimed to simulate the binary-density particle redistribution under the consideration of the postulated SA scenario of the prototype SFR [35]. In the experiment, a binary-density particle bed, composed of lighter-density Fluorinated Ethylene Propylene (FEP, density = 2000 kg/m\({}^{3}\)) and a heavier-density stainless steel sphere (density = 7870 kg/m\({}^{3}\)), was settled in a transparent container, as in Figure 2a. The initial position of the stainless steel particle was located on the center top of the particle bed. Then, the water was injected from the bottom to observe the movement of the binary-density particle bed. Since the refractive index of the FEP particle was close to the water at visible wavelengths, it appears almost transparent in the water, and the historical data of the stainless steel particle's movement can be tracked even in the middle of the FEP particles. Subsequently, the CFD-DEM model was created. Two types of simulating particles were placed into the flow region, as shown in Figure 2b. In both simulations and experiments, the historical data of the heavier steel particle's movement were transformed via Fast Fourier Transform (FFT) to observe the particle movement pattern on the frequency domain, as shown in Figure 2c. By comparing the data between the STAR-CCM+s simulation and the experiments, the validated results reveal that the Gidaspow drag coefficient model yields a modest error deviation, which is applied in this research. Further details of the model validation and the Gidaspow drag coefficient model were explained in our previous study [35].