## 3 Dynamic Heterogeneity Investigation into Debris Bed Safety Analysis
### Simulation Settings
\begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Debris bed shape & Conical shape \\ Debris bed height [m] & 1.434 \\ Debris bed bottom radius [m] & 2.485 \\ Slope angle [\({}^{\circ}\)] & 30 \\ Total debris bed mass (MOX fuel + stainless steel) [kg] & 37,902.05 \\ MOX fuel mass [kg] & 27,045.45 \\ Stainless steel mass [kg] & 10,856.60 \\ Average porosity [-] & 0.59 \\ Volume ratio of MOX fuel to stainless steel & 0.64: 0.36 \\ Debris shape & Spherical shape \\ Debris particle radius [m] & 0.065 \\ \hline \end{tabular}
\end{table} 
Table 2: Geometry parameters of the three debris bed models’ initial configurations. generation rate for each Fuel particle is derived by dividing the total volume of the MOX fuel in the debris bed (without setting the heat source in Structure particles). Finally, the Homogeneous particle physical properties are based on the conception that the MOX fuel and stainless steel in the debris bed are evenly distributed to each particle. Hence, the input parameters of each Homogeneous particle can be derived using the volume-weighted average value of the total Fuel particles and Structures from either the Mixed model or the Stratified model (via the same approach as Equation (28), applying the volume ratio of MOX fuel to stainless steel in Table 2). The results of the derived Homogeneous particle's input parameters are also listed in Table 3. Here, based on the information in Tables 2 and 3, the initial configuration of the Homogeneous model can be created by randomly filling Homogeneous particles into the assigned conical space in the calculating domain, and the Mixed model's initial configuration can be created by randomly filling all the Fuel and Structure particles into the exact geometry of the assigned conical space. For preparing the initial configuration of the Stratified model, the Fuel particles in the Mixed model are filled into the bottom layer of the same assigned conical space, and the Structure particles left in the Mixed model are also filled into the upper layer of the conical space. A mesh-independent test for the CFD-DEM solver was also conducted via different structures of the computational meshes in the fluid region based on the Homogeneous model, as shown in Figure 6.  
The polyhedral meshes were utilized to discretize the fluid region. Based on the time discretization scheme of the implicit unsteadiness (timestep of 0.01 s in the CFD solver, five sub-steps in the DEM solver), at the time after 100 s of initializing the CFD-DEM solver in the Homogenous model, the temperature distribution of the liquid sodium due to the decay heat generated from the Homogeneous particles became stable. The different average temperatures of the liquid sodium in the calculating domain (\(T_{f,Ag. }\) [K]) corresponding to different CFD mesh structures are also listed in Table 4. In this study, it was noticed that refining the CFD meshes excessively to a certain number (53,149) did not have a significant impact on the deviation of \(T_{f,Ag. }\) from the coarser mesh structure. As a result, the Mixed and Stratified models will be further investigated, employing the same mesh structure of the 53,149 mesh number. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Physical Properties** & **Fuel Particle** & **Structure Particle** & **Homogeneous Particle** \\ \hline Density, [kg/m\({}^{3}\)] & \(1.10\times 10^{4}\) & \(7.85\times 10^{3}\) & \(9.84\times 10^{3}\) \\ Young’s modulus [GPa] & 220 & 200 & 212.8 \\ Poisson’s ratio [-] & 0.33 & 0.29 & 0.32 \\ Static friction coef. [-] & 0.61 & 0.61 & 0.61 \\ Rolling resistance coef. [-] & 0.001 & 0.001 & 0.001 \\ Heat capacity [J/kg\(\cdot\)K] & 330 & 500 & 319.2 \\ Thermal conductivity [W/m\(\cdot\)K] & 3 & 16.2 & 7.75 \\ Decay heat generation rate [W/m\({}^{3}\)] & \(3.388\times 10^{7}\) & 0 & \(2.168\times 10^{7}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Input parameters of the physical properties among different particles in the DEM solver.