## 3 Dynamic Heterogeneity Investigation into Debris Bed Safety Analysis
### Results and Discussion
#### 3.3.2 Criticality Analysis
could further strengthen the phenomenon of the \(P_{NL}\)'s declination when debris bed height was suppressed. However, in the latter stage of the particle redistribution, the BC effect in the Stratified model for \(P_{NL}\)'s declination was weakened when the bottom fissile material layer became more flattened, and the Stratified model demonstrated a lower sensitivity of \(\Delta k_{eff}(t)\) to \(\Delta H_{bd}(t)\) than the Mixed model, in the end. Meanwhile, in Figure 10b, the Stratified model also showed a minor decreased range of the debris bed height compared to the Mixed model; hence, the BC effect in the Stratified model brought a lower decreased range of \(k_{eff}\) than the Mixed model, as shown in Figure 10c. Nonetheless, since the BC effect in the Stratified model also provided a negative effect on \(P_{p}\), due to the loss of the spatial heterogeneity on the bottom fissile material layer, the Stratified model was grounded on the lower \(k_{eff}\) value compared to the Mixed model after particle redistribution in the end, and the results are also listed in Table 7.  
Based on the results above, the PC and BC effects on the \(k_{eff}\)'s change (\(\Delta k_{eff}\)) and safety margin of criticality safety (\(1-k_{eff}\)) after particle redistribution can be evaluated in the form used below:  
\[\text{PC effect}=\frac{X_{\text{Mixed model}}-X_{\text{Homogeneous model}}}{X_{\text{Homogeneous model}}}, \tag{31}\]  
Figure 8:0  
where \(X\) means the target result (such as \(\Delta k_{eff}\) or \(1-k_{eff}\)), and the subscripts of \(X\) stand for the corresponding debris bed models. The results of PC and BC effects in the criticality assessment, applying Equations (31) and (32), are listed in Figure 8:1  
In summary, PC and BC effects can contribute to the positive and negative effects on \(\Delta k_{eff}\) (in Table 8) due to the strengthening (in PC effect) and weakening (in BC effect) of \(P_{NL}\)'s declination during particle redistribution, respectively. For the criticality safety margin of \(1-k_{eff}\), however, PC and BC effects were the negative and positive dominant effects, respectively (in Table 8), since the \(P_{p}\) was strengthened by the heterogeneous spatial arrangement of the Fuel-Structure particles in the PC effect, and vice versa in BC effect. Meanwhile, all the models were kept at sub-critical states during the particle redistribution process (as shown in Figure 9). Thus far, it can be proven that the CFD-DEM-MCN coupled solver is capable of assessing the impact of various transformations of the heterogeneous configuration of the bed on the criticality safety assessment of hypothetical SAs in SFRs. \begin{table}
\begin{tabular}{c c c c} \hline  & **Homogeneous Model** & **Mixed Model** & **Stratified Model** \\ \hline \(\Delta k_{eff}/\Delta H_{bd}\) [1/m] & 0.100 & 0.130 & 0.119 \\ \(\Delta H_{bd}\) [m] & \(-\)0.845 & \(-\)0.898 & \(-\)0.794 \\ \(\Delta k_{eff}\) [-] & \(-\)0.084484 & \(-\)0.116748 & \(-\)0.094490 \\ \(k_{eff}\) [-] & 0.865216 & 0.879252 & 0.841981 \\ \hline \end{tabular}
\end{table} 
Table 7: Neutronic analysis results after particle redistribution had been stopped (at 10 s) among three models. \begin{table}
\begin{tabular}{c c c} \hline  & **PC Effect [\%]** & **BC Effect [\%]** \\ \hline \(\Delta k_{eff}\) & 38.19 & \(-\)19.06 \\ \(1-k_{eff}\) & \(-\)10.41 & 30.87 \\ \hline \end{tabular}
\end{table} 
Table 8: PC and BC effects on the criticality assessment results after particle redistribution had been stopped (at 10 s).