Multiphysics Modelling of the Molten Salt Fast Reactor using NekRS and the Fission Matrix Method

This paper proposes a multiphysics computational model for the Molten Salt Fast Reactor using the Cardinal framework to integrate NekRS for thermal hydraulics with the Fission Matrix method for efficient neutronics, replacing the standard OpenMC solver to enable fast and accurate simulations of the tightly coupled fuel-coolant system.

The Gist

Imagine you are trying to design a super-efficient, self-cleaning car engine that never stops running. But instead of oil and metal parts, this engine uses liquid metal salt that is so hot it glows, and the fuel itself is mixed right into the liquid. This is the concept of a Molten Salt Fast Reactor (MSFR).

The problem is that this engine is incredibly complex. The heat changes how the liquid flows, and how the liquid flows changes where the heat is generated. It's a giant, swirling dance of physics where everything depends on everything else. If you try to simulate this on a computer, it's like trying to predict the weather while simultaneously calculating the trajectory of a billion tiny billiard balls. It's too much for a standard computer to handle quickly.

This paper is about a team of researchers at Penn State who built a new, smarter way to simulate this "liquid engine" using a clever shortcut.

The Cast of Characters

To understand their solution, let's meet the tools they used, imagined as characters in a movie:

1. NekRS (The Flow Expert): Think of this as a high-speed camera that can see exactly how the hot liquid salt swirls, spins, and carries heat. It's incredibly detailed and fast because it runs on powerful graphics cards (GPUs), just like the ones in video game computers.
2. Cardinal (The Director): This is the stage manager. It doesn't do the heavy lifting itself; instead, it tells the Flow Expert and the Physics Expert when to talk to each other and makes sure they are on the same page.
3. OpenMC (The Old Way): Usually, to figure out where the nuclear reactions happen, scientists use a method called "Monte Carlo." Imagine this as throwing a million darts at a board to guess where the bullseye is. It's very accurate, but it takes a long time to throw all those darts.
4. The Fission Matrix Method (The New Shortcut): This is the star of the show. Instead of throwing a million darts every time, the researchers created a giant cheat sheet (a database). They pre-calculated the results for different temperatures and stored them in a grid. When the simulation needs an answer, it just looks up the closest values in the cheat sheet and draws a line between them. It's like using a GPS map instead of driving around the city to find the route.

How They Put It Together

The researchers built a digital twin of the reactor. Here is how their simulation works, step-by-step:

1. The Setup: They created a 3D model of the reactor loop. It's a simplified version, ignoring the pumps and heat exchangers to focus on the main flow, kind of like looking at a river without worrying about the bridges crossing it.
2. The Cheat Sheet (FMDBs): Before the main simulation started, they used a supercomputer to calculate the nuclear reactions for five different "temperatures" (800K, 900K, etc.). These became their Fission Matrix Databases.
3. The Dance (The Loop):
   • The Director (Cardinal) tells the Flow Expert (NekRS): "Here is the heat source. Go calculate how the liquid moves and where it gets hot."
   • NekRS runs the simulation, calculating the speed and temperature of the liquid. It finds "stagnation zones"—pockets of liquid that get stuck and overheat, like a car engine idling in traffic.
   • NekRS sends the new temperature data back to the Director.
   • The Director asks the Fission Matrix (The Cheat Sheet): "Based on these new temperatures, what is the nuclear reaction doing now?"
   • The Cheat Sheet quickly interpolates (guesses) the answer based on the pre-calculated data and sends a new heat map back to the Flow Expert.
   • They repeat this dance thousands of times until the numbers stop changing.

What They Found

The results were promising!

• The Shortcut Worked: The "Cheat Sheet" method (Fission Matrix) gave results that were almost identical to the slow, dart-throwing method (OpenMC), but much faster.
• Hot Spots: They found that in the corners of the reactor, the liquid slows down and gets trapped. Because it's not moving, it keeps getting hotter, creating "stagnation zones." This is a crucial safety insight.
• Temperature Match: The final temperature of the liquid coming out of the reactor was 997.4 K. When they compared this to their old, slower method, the difference was only 0.1 degrees. That's like measuring the temperature of a cup of coffee and being off by less than a single grain of sand.

Why This Matters

This research is a big step forward because it proves we can simulate these complex, future nuclear reactors without needing to wait weeks for a computer to finish the math. By using a "reduced-order model" (the cheat sheet), they can run simulations faster, allowing engineers to test more designs and ensure safety more efficiently.

In short, they took a puzzle that usually takes a month to solve and found a way to solve it in a day, without losing any of the important details. This brings us one step closer to building the clean, safe, and efficient nuclear power plants of the future.

Technical Summary

1. Problem Statement

The Molten Salt Fast Reactor (MSFR) is a Generation-IV concept where the coolant and fuel are the same fluid. This unique characteristic creates a tight coupling between neutronics (nuclear reactions) and thermal-hydraulics (fluid flow and heat transfer), as the fuel circulates through the primary system.

• Challenge: Developing accurate computational models for the MSFR requires a multiphysics approach. Traditional high-fidelity coupling often relies on Monte Carlo particle transport codes (like OpenMC), which can be computationally expensive for transient or iterative multiphysics simulations.
• Goal: The authors aim to develop a reduced-order neutronic model coupled with high-fidelity thermal-hydraulics to efficiently simulate the MSFR while maintaining accuracy.

2. Methodology

The study utilizes the Cardinal framework (a wrapper within the MOOSE environment) to couple two distinct solvers:

A. Thermal-Hydraulics Solver: NekRS

• Code: NekRS, a GPU-accelerated Spectral Element Method (SEM) CFD code.
• Physics: Solves unsteady incompressible Navier-Stokes and energy equations using a RANS $k-\tau$ turbulence model.
• Geometry: A 3D model of the MSFR primary loop (simplified inlet/outlet, no pumps/heat exchangers) generated with Gmsh.
• Mesh: $1.35 \times 10^6$ hexahedral elements with polynomial order 3, resulting in $\approx 8.6 \times 10^7$ quadrature points.
• Boundary Conditions:
  • Inlet: Uniform temperature (898 K), parabolic velocity profile.
  • Walls: Adiabatic, non-slip.
  • Turbulence: Parabolic profiles for $k$ and $\tau$.

B. Neutronics Solver: Fission Matrix (FM) Method

• Concept: Instead of using the standard Monte Carlo OpenMC code, the authors employ the Fission Matrix Method. This treats the reactor as an $n \times n$ matrix where elements represent neutron production between spatial cells.
• Database Generation (FMDBs):
  • Generated using the Monte Carlo code Serpent.
  • Created for five uniform temperature states: 800 K, 900 K, 1000 K, 1100 K, and 1200 K.
  • Geometry: 3D cylinder with a fertile blanket and boron-carbide layer, discretized into a $13 \times 13 \times 13$ Cartesian grid.
• Interpolation: To handle non-uniform temperature distributions from the CFD solver, the FM elements ($a_{ij}$) are linearly interpolated between the nearest FMDBs based on the local fuel temperature.
• Implementation: Custom C++ functions were integrated into MOOSE to load databases, perform interpolation, solve the eigenvalue problem ($F = \frac{1}{k_{eff}} A F$), and return the fission source and $k_{eff}$.

C. Multiphysics Coupling (Cardinal)

• Scheme: A Picard iteration "in time."
• Workflow:
  1. FM model calculates fission source distribution based on initial temperature/density.
  2. Source is scaled to total power ($3 \times 10^8$ W) to generate volumetric heat source ($q'''_{fis}$).
  3. Heat source is passed to NekRS.
  4. NekRS performs time-step calculations ($N=10,000$ steps, $\Delta t = 2.0 \times 10^{-4}$) to update fluid temperature and velocity.
  5. Updated temperatures are fed back to the FM model for the next iteration.
• Convergence: Achieved when maximum, average, and outlet temperatures reach steady-state values.

3. Key Contributions

1. Reduced-Order Neutronics in Cardinal: Successfully integrated the Fission Matrix method into the Cardinal/MOOSE framework as a replacement for OpenMC, demonstrating a viable path for faster multiphysics simulations.
2. Dynamic FMDB Interpolation: Implemented a linear interpolation scheme to generate FM matrices for arbitrary, non-uniform temperature fields derived from CFD, bridging the gap between reduced-order models and high-fidelity thermal-hydraulics.
3. GPU-Accelerated Workflow: Leveraged the Frontier supercomputer to run NekRS on GPUs while the FM model ran on CPUs, optimizing the computational cost of the coupled system.
4. Validation: Provided a direct comparison between the new NekRS-FM model and a previously established NekRS-OpenMC model.

4. Results

• Multiplication Factor ($k_{eff}$): The FM model yielded a $k_{eff}$ of 1.04632.
• Thermal-Hydraulic Performance:
  • Outlet Temperature: 997.4 K (NekRS-FM) vs. 997.3 K (NekRS-OpenMC). Difference: 0.1 K.
  • Average Temperature: 973.0 K (NekRS-FM) vs. 983.1 K (NekRS-OpenMC). Difference: 10.1 K.
  • Note: The authors attribute the 10.1 K difference in average temperature to the fact that total energy is conserved in both models, but the spatial distribution of heat generation varies slightly between the FM and OpenMC methods.
• Flow Dynamics:
  • Velocity distributions showed stagnation regions in the periphery of the central cylinder.
  • Temperature maps confirmed that these low-velocity stagnation zones act as heat traps, reaching maximum temperatures.
• Heat Source Distribution: The instantaneous heat source exhibited a sinusoidal shape, consistent with previous OpenMC-based models.

5. Significance and Future Work

• Significance: This work proves that the Fission Matrix Method can serve as an effective, reduced-order surrogate for Monte Carlo transport in multiphysics MSFR simulations. It offers a potential pathway to reduce computational time while maintaining sufficient accuracy for thermal-hydraulic feedback analysis.
• Future Directions:
  • Mesh Sensitivity: Conduct sensitivity analyses on the FM mesh resolution (coarser vs. finer) to find the optimal discretization.
  • Geometry Optimization: Investigate the influence of mesh shape (e.g., switching from Cartesian to cylindrical meshes) on the accuracy of the results.

In conclusion, the paper presents a successful proof-of-concept for a high-fidelity, GPU-accelerated multiphysics simulation of an MSFR using a reduced-order neutronic solver, validating its performance against established high-fidelity benchmarks.