Small Modular Reactors (SMRs) push fission physics to the edge—compact cores where neutron leakage, heat buildup, and stability swings amplify in tight quarters. The standard approach—solving neutron transport, Navier-Stokes, and kinetics separately—is a computational slog, validated but unwieldy: Monte Carlo codes like MCNP chew hours to model a 1-meter core, and CFD for coolant flow chokes on turbulence. These tools work, but they’re siloed, missing the forest for the trees. What if we model the reactor as a coupled system with discrete states, blending physics rigor with dynamic simplicity? Not a replacement, but a complement—streamlining design and control without losing the plot.

The Physics Case

In a 1 m³ SMR core (e.g., 10 MW, 5% U-235), smallness rewrites the rules:

• Neutrons: Leakage spikes—mean free path (~10 cm) is a fat chunk of the radius. \(k_{eff}\) teeters near 1, hypersensitive to geometry.
• Heat: Surface area drops (scales as \(r^2\)), but power’s still volumetric (\(r^3\)). Coolant channels (say, 1 cm wide) turn turbulent, gradients soar.
• Stability: Neutron lifetime shrinks (~10⁻⁴ s vs. 10⁻³ s in big cores), feedback (Doppler, coolant density) kicks faster.

The old math—e.g., \(\frac{1}{v} \frac{\partial \psi}{\partial t} + \vec{\Omega} \cdot \nabla \psi + \Sigma_t \psi = \int \Sigma_s \psi' d\Omega' + S\)—nails this, but it’s a beast, solved in pieces then stitched together. In reality, it’s one system: a 50°C fuel temp jump cuts reactivity (\(\rho \approx -10^{-5} \Delta T\)), slows neutrons, shifts flow. We’re proposing a layer above: map the system states, not every particle.

The Framework

Define states from measurable physics:

• Stable: \(k_{eff} = 1 \pm 0.01\), \(\nabla T < 10°C/cm\), power steady (\(P = 10 MW\)).
• Overheated: \(T_{\text{fuel}} > 600°C\), \(\rho < -0.001\), flow rate drops 10%.
• Neutron-Starved: \(k_{eff} < 0.98\), boundary flux (\(\phi_b\)) doubles core avg.
• Oscillating: \(P(t)\) varies ±5% over 1 s, driven by delayed neutrons (\(\beta = 0.0065\)).

Model transitions with a hybrid system:

• \(A\): Transition matrix, e.g., \(A_{12} = \frac{\partial T}{\partial t} / T_{\text{crit}}\) from heat-rate equations, tied to \(\alpha_T = -2 \times 10^{-5}/°C\) (Doppler coeff).
• \(B u\): Control inputs—rod insertion (\(\Delta \rho = -0.01\)), flow boost (\(\Delta Q = 0.1 m^3/s\)).
• \(N\): Noise term, stochastic neutron scatter (~1% flux variance).

Data comes cheap: flux (\(\phi\)), temp (\(T\)), pressure (\(p\)) from sensors, not full-field solves.

Countering the Whiz

• “Hand-Waving”: Not a replacement—think of this as a reduced-order model atop detailed sims. \(A\)’s entries derive from transport/kinetics (e.g., \(\frac{\partial k}{\partial T}\)), validated against benchmarks like TRIGA pulses.
• “We’ve Got This”: True, MCNP works—but a 1-hour run per config won’t scale for rapid prototyping. This cuts to minutes, guiding where to aim the big guns.
• “Oversimplifies Coupling”: States aren’t bins; they’re coarse-grained snapshots of continuous fields. Spatial dependence? Bake it into \(A\) via zonal averages (core vs. edge), not 3D grids.
• “No Validation”: Test it: perturb a 10 MW core (drop flow 10%, \(T_{\text{fuel}} + 50°C\)). Match criticality (\(k_{eff}\)) and power to ORNL’s HFIR data within 5%. Numbers, not sci-fi.
• “Not the Bottleneck”: Fair—materials and costs rule deployment. But better models shrink design cycles, slashing R&D bleed.
• “Elegance Doesn’t Matter”: Accuracy first, yes—but if this predicts a meltdown as well as a 10⁶-cell mesh in 1% of the time, it’s a win.

The Pitch

Simulate it: a cylindrical core, 1 m³, water-cooled, 5% enrichment. Hit it with a rod pull (\(\rho + 0.005\))—does \(P_s\) track power spikes like a TRIGA log? If it holds within 5% of MCNP, you’ve got a tool: not the whole answer, but a damn good compass. Scale to Kilopower’s 1 kW rig—same states, tighter margins. Physicists, tear it apart: where’s the crack?

Concept via a collaboration between Grok, built by xAI and Silvia Hartmann

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