## 2 Analytical Methodology
### Coupling Methodology and Process of the CFD-DEM-MCN Solver  
The CFD-DEM-MCN coupled solver built in this research was executed automatically in a cyclical manner, composed of two separately executing codes (STAR-CCM+ code for CFD-DEM solver, MVP3 code for MCN solver) and coupled externally using the originally created interface (MVP3 and STAR-CCM+ interface). The MVP3 and STAR-CCM+ interface, based on the Python language with the assistance of Java script, is in charge of data exchange and workflow control between two codes. The flowchart of the CFD-DEM MCN coupled solver is also presented in Figure 3.  
After initializing the MVP3 and STAR-CCM+ models, the MVP3 input file is called by the interface and transferred to the MVP3 code to solve the \(k_{eff}\) eigenvalue problem. The solved \(k_{eff}\) value saved in the MVP3 output file will be extracted and transformed to the CSV file type to be collected by the interface. Subsequently, if the debris bed is under a sub-critical state (\(k_{eff}<1\)), the calculation will keep processing forwards; otherwise, the power excursion due to re-criticality can be treated as an occurrence, and the calculation of the CFD-DEM-MCN coupled solver will be terminated here. For the case of the sub-critical state, the Java macro file, which is the Javascript-based file executable for the STAR-CCM+ workflow being controlled externally, will be automatically created and called in the MVP3 and STAR-CCM+ interface to activate the STAR-CCM+ calculation of the current CFD timestep (\(t_{CFD}^{n}\)) with the parameters updating, such as particle heat source. In this research, based on the conservative concerns in the cooling characteristic analysis, the particle heat source transferred from the interface to STAR-CCM+ is assigned as the constant value without utilizing the decreasing decay heat with time passing. After the STAR-CCM+ calculation has been activated, the DEM solver in the STAR-CCM+ will start to calculate the force and the heat transfer from the fluid phase to particles by referring to the fluid phase information in the CFD solver (referring to the initial condition if at the first timestep), subsequently estimating the force and heat transfer between particles. This process in one elapse of the DEM timestep (\(\Delta t_{DEM}\)) will be repeated continuously until the accumulated \(\Delta t_{DEM}\) is equal to one elapse of the CFD timestep (\(\Delta t_{CFD}\)), eventually transferring the solved results of particle information and the momentum/energy feedback to the CFD solver. After updating the fluid volume fraction and momentum/energy source terms in the CFD cells based on the transferred data from the DEM solver, the fluid information will be evaluated until the result has converged in the CFD solver, transferring the converged fluid phase information to the DEM solver as the reference for the next CFD timestep (\(t_{CFD}^{n+1}\)). In addition, the solved particle information (e.g., particle index, particle centroid position, velocity, temperature) and the fluid phase information will also be extracted in the CSV file type and transferred to the MVP3 and STAR-CCM+ interface, temporarily stopping the STAR-CCM+ calculation here. Before stepping into the next timestep, if the end time (\(t_{end}\)) has been reached, the CFD-DEM-MCN coupled solver will be stopped here; otherwise, the calculation will proceed to the next timestep with the updating of the heterogeneous bed configuration. Based  
Figure 3: The flowchart of the CFD–DEM–MCN coupled solver. on each particle's index and the corresponding particle centroid position obtained from the STAR-CCM+, the MVP3 and STAR-CCM+ interface will automatically generate the new MVP3 input file to reflect the updated heterogenous bed configuration in the MCN solver.