## 3 Dynamic Heterogeneity Investigation into Debris Bed Safety Analysis
### Results and Discussion
#### 3.3.2 Criticality Analysis  
This section examines how changes in the bed's heterogeneous configurations impact the assessment of its criticality, including analyzing the \(k_{eff}\) alteration during particle redistribution and the safety margin for criticality safety after particles have ceased moving, using re-criticality accidents as an indicator. The effects of various transformations in heterogeneous configurations can be categorized into (1) the Particle-Centralized fissile material effect (PC effect), which compares the Mixed model to the Homogeneous model, and (2) the effect of Bed-Centralized fissile material at the bottom of the debris bed (BC effect), which compares the Stratified model to the Mixed model. Regarding the PC effect, the Mixed and Homogeneous models display a decrease in \(k_{eff}\) eigenvalues during particle redistribution, as demonstrated in Figure 9. This decline continues until the redistribution of particles stops, indicating that the decreased height of the debris bed during particle redistribution effectively suppresses the non-leakage probability (\(P_{NL}\) [-]). To delve further into the PC effect on the \(k_{eff}\) change, Figure 10 illustrates the relationship between the change in \(k_{eff}\) (\(\Delta k_{eff}(t)\) [-]) and the change in debris bed height (\(\Delta H_{bd}(t)\) [m]) for the Mixed and Homogeneous models. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**Homogeneous Model**} & **Mixed Model** & **Stratified Model** \\ \hline \(H_{bd}\) [m] & 0.589 & 0.536 & 0.640 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Height of the debris bed after particle redistribution stopped among three models. Figure 8: Historical data of the debris bed’s height during particle redistribution among three models. The definitions of \(\Delta k_{eff}(t)\) and \(\Delta H_{bd}(t)\) are as below:  
\[\Delta k_{eff}(t)=k_{eff}(t)-k_{eff}(t=0), \tag{29}\]  
\[\Delta H_{bd}(t)=H_{bd}(t)-H_{bd}(t=0), \tag{30}\]  
where \(t\) [s] stands for the time after particle redistribution starting. Figure 10a shows that the sensitivity of \(\Delta k_{eff}(t)\) to \(\Delta H_{bd}(t)\) in the Mixed model is higher than in the Homogeneous model, showing that the accumulated fissile materials' movements would strengthen the effect of the declined \(P_{NL}\) when the debris bed height is suppressed during particle redistribution. In addition, the Mixed model also exhibited a greater decreased range in the debris bed height than the Homogeneous model during the particle redistribution (Figure 10b); therefore, the PC effect in the Mixed model caused a higher decreased range of \(k_{eff}\) than the Homogeneous model after particle redistribution, as in Figure 10c. Nevertheless, since the PC effect in the Mixed model also contributes to the higher thermal utilization factor [-] (due to the higher concentrated fissile material in Fuel particles) and the greater resonance escape probability \(P_{p}\) [-] (due to the self-shielding effect from the Structure-Fuel-Structure spatial arrangement [49]), the value of \(k_{eff}\) in the Mixed model was still higher than the Homogeneous model after particle redistribution, in which the results are listed in Table 7.  
As for the BC effect, the Stratified model also presented a declining trend during particle redistribution until particles had ceased, as shown in Figure 9. Following this, in the initial stage of the particle redistribution, the Stratified model displayed a higher sensitivity of \(\Delta k_{eff}(t)\) to \(\Delta H_{bd}(t)\) than the Mixed model (Figure 10a), showing that the downward movement of the accumulated fissile material layer in the Stratified model  
Figure 10: Neutronic analysis during particle redistribution among three models: (**a**) ratio of the ’s change to the bed height’s change, (**b**) the bed height’s change, and (**c**) \(k_{eff}\)’s change. Figure 9: History data of the \(K_{eff}\) eigenvalues during particle redistribution among three debris bed models.