Theory & Method

Fuel Burnup Simulator · AntraSIM-0150 · version 1.0.0 · build 2026-09-24. This document describes the model as implemented in the application; the numbers and tables that depend on the model (verification results, regression values, parameter defaults, expected exercise results) were generated at build time from the same core code that the application runs.

1. Physical Background

Intuition A fuel assembly is fresh, then loses fissile atoms and gains fission products and plutonium as it burns. The multiplication factor of the infinite lattice, k_∞, therefore changes with burnup (energy released per unit mass of initial heavy metal, MWd/kgU) — the “core age” of the K/A catalog. A reactor is loaded with more fissile material than it needs for criticality at the start; the excess reactivity is held down by control rods, soluble boron in a PWR, and in a BWR mainly by burnable poison in the fuel — gadolinia (Gd₂O₃) mixed into some of the pellets.

Physical mechanism

Gadolinium has two isotopes with enormous absorption cross-sections. A pellet containing it is so absorbing that the neutrons cannot get into its interior: the pin acts as a black absorber, and only the outer skin burns away, layer by layer (self-shielding). Therefore the absorption per pin is roughly constant during most of the life of the poison and then collapses when the last layer is gone. The poison holds reactivity down in the beginning and gives it back gradually while the fuel is being depleted, so that k_∞ first stays low or even falls, then rises to a maximum at the burnout, and only then follows the depletion of the fuel down.

Engineering interpretation The burnup at which k_∞P_NL falls to one (batch-average, without rod or boron worth) is the reactivity-limited burnup of the fuel; with several fuel batches in the core the discharge burnup is larger than the single-batch value, and each cycle uses only part of it. Assumptions and limitations A zero-dimensional model with fitted constants stands for the lattice (Section 9).

2. Assessment of the Source Material and Learning Objectives

The objectives are taken from NUREG-1021, BWR catalog, Reactor Theory, section 292007 (fuel burnup, core age, burnable poisons) [1], consulted at excerpt level. No earlier simulator was the source.

Table 2.1 — Coverage. Finding A-1: all one-group constants are fitted and illustrative (Section 8); results are qualitatively right and quantitatively of the correct order only.

3. Governing Equations

Intuition Every nuclide is created by capture in its predecessor and removed by capture and fission; the reaction rates are proportional to the neutron flux, which can be replaced by the fluence.

Mathematical formulation Atom numbers per initial uranium atom, N_i [–]; fluence τ = ∫φ dt [b⁻¹]; one-group cross-sections σ_f,i, σ_c,i, ν_i; i = 5 (U-235), 8 (U-238), 9 (Pu-239), 0 (Pu-240), 1 (Pu-241):

Burnup follows the fission energy: with the fissions per initial atom F = ∫ΣσfN dτ and 200 MeV per fission, B = (B/F)·F with B/F = E_fN_A/M = 938.21 MWd/kgU per fission per atom, so dτ/dB = 1/[(B/F)Σ_iσ_f,iN_i]. The lattice reactivity, with fission-product absorption σ_FP per fission, parasitic absorption X per U atom and gadolinium absorption A_Gd:

The burnable poison: n₁₅₅, n₁₅₇ [–] are the remaining fractions of the two isotopes, c_j their natural abundances, σ_th,j [b] the 2200 m/s cross-sections, s [–] the ratio of the one-group to the 2200 m/s value, g [–] the Gd atoms per U atom in the cell and ζ [–] the optical thickness of a pin:

Limits: for ζ ≪ 1 the poison burns like an ordinary absorber, n_j = exp(−sσ_th,jτ); for ζ ≫ 1, f_s ≈ 1/ζ, so A_Gd ≤ g s Σ_jc_jσ_th,j/ζ₀ — a ceiling independent of the amount of gadolinium. The number of Gd atoms per U atom in a pellet with w weight-percent Gd₂O₃ (in UO₂) is 2(w/362.5)(270.03)/(1 − w/100)·(1/100) ×100 = 1.4897 w/(100 − w), and g is this value times the fraction of pins that carry gadolinia. The cycle length is the root of k_eff = k_∞(B)P_NL = 1 where P_NL [–] is the non-leakage probability. With n batches the linear reactivity model gives the discharge burnup B_d = 2n/(n+1)·B₁, B₁ the single-batch root, and B_d/n per cycle.

Axially: 24 nodes with the relative power q(z) = cos[π(z − ½)/1.1] normalised to mean 1; a node with local burnup B_i = B q_i takes its k_∞ from the depletion curve of its zone, and the core average is the importance-weighted mean k = Σ w_ik_i with w_i ∝ q_i² (first-order perturbation weight).

Engineering interpretation The Gd absorption stays at its ceiling until the surface has burned through, then falls within a few MWd/kgU; this produces the knee and the peak of k_∞. Assumptions and limitations Constant spectrum; constant power shape; U-236, Np and Am are not tracked; no burnup dependence of the fuel-temperature or void feedback.

4. Inputs and Operating Conditions

Fuel (enrichment, specific power, parasitic and fission-product absorption), core (non-leakage, batches, inspection burnup), burnable poison (pin fraction, Gd₂O₃ weight-percent in the bottom, middle and top zone, zone heights, ζ₀, s). Specific power converts burnup to days only. A zero pin fraction or zero weight-percent switches the poison off in the zone. Rods, boron, xenon transients and temperature are not inputs.

Input Configuration (IC) and CSV The Input Configuration (IC) is a JSON file (format “AntraSIM-input-configuration”, schemaVersion 1) that stores the complete canonical configuration; every key carries its unit. Loading is all-or-nothing: the file is parsed, identified (format, schema, application id), compared structurally with the reference configuration (same keys, same types, admissible array lengths), range-checked, and only then committed. Older schemas would be migrated by an explicit registered migration or rejected with a message; a migration hook is exercised by the verification suite. Restore defaults returns every input, table and exercise-dependent setting to the reference configuration. CSV exports carry metadata lines (#), the complete IC as one # line and ASCII-only column names.

5. Calculated Quantities

Depletion curves of three lattice types (bottom, middle, top zone) and of a Gd-free lattice; the core-average k_∞(B) with axial zoning; the slope; fresh k_∞; Gd worth; burnup of the peak; Gd burnout; single-batch and discharge burnup; cycle length; axial profiles at the inspection burnup; Pu fission share; the absorption budget. Default case: k_∞ without gadolinia 1.2859, with gadolinia 1.0712, peak 1.1156 at 13 MWd/kgU, Gd worth -15590 pcm, Gd-157 burnout at 11.5 MWd/kgU in the middle zone, single-batch burnup 21.89 MWd/kgU (k_∞ = 1.0526).

Table 5.1 — Gd-free lattice, 4 % enrichment (generated from the code).

6. Numerical Method

The equations are integrated with the classical fourth-order Runge–Kutta method with burnup as the independent variable (step 0.02 MWd/kgU, output every 0.5 MWd/kgU up to 100 MWd/kgU), which removes the need to state a flux level. Convergence is fourth-order (T4). Zone curves are interpolated linearly. The root of k_eff = 1 is found by bisection on the interpolated core curve. The default case (three lattices plus a Gd-free lattice, 24 axial nodes) takes 21 ms.

7. Verification and Numerical Accuracy

Verification shows that the equations are solved correctly; it is not validation. Findings during development: (1) the first conservation test failed by 7·10⁻³ because U-235 capture produces U-236, which was not tracked as an “other heavy” atom; the term was added to the equation and to the independent matrix of test T1; (2) an interpolation routine assumed a burnup grid of 0.5 MWd/kgU and gave wrong core-average values on the 1 MWd/kgU grid until the spacing was taken from the grid; (3) the first poison calibration gave a gadolinia worth of only about −5 % and no peak — the optical thickness and the pin fraction were re-chosen (ζ₀ = 40, 25 % of the pins) after it was seen that the total number of Gd atoms sets the amount of reactivity that can be held down while ζ₀ sets the time of burnout; (4) a first step-convergence test compared a nearly extinguished Gd fraction relatively and appeared to fail; the criterion was changed to an absolute error for these fractions with a check of fourth-order behaviour (error ratio 16.6 for step doubling), and the production step was reduced from 0.05 to 0.02 MWd/kgU; (5) a first version of the burnup constant used 937 MWd/kgU per fission per atom from a rough hand calculation; the code value 938.2 is derived and the test now checks the derivation. The suite runs in the application and in Node.

Table 7.1 — Release verification suite (13 of 13 passed; generated from an actual run).

8. Model Parameters and Constants

Evidence classes: externally sourced derived assumption not verified. There is no earlier release. All one-group cross-sections were obtained by a random search that made the Gd-free k_∞ follow a target curve (1.30, 1.19, 1.10, 1.03, 0.97, 0.92 at 0, 10, …, 50 MWd/kgU) that was itself taken from general knowledge, not from a data set. They are not evaluated cross-sections; in particular the Pu-239 fission share (about 63 % at 40 MWd/kgU in the default case) is larger than typical lattice-code values.

9. Model Scope and Limitations

Represented Depletion of five heavy nuclides with a common one-group spectrum; lumped fission products, xenon, samarium and structure; a self-shielded two-isotope gadolinium poison with burnout; three axial zones burned in a fixed cosine power shape; single-batch and linear-reactivity-model cycle length. Not represented Spectrum changes with burnup and with void, resonance shielding changes of U-238 and Pu, spatial self-shielding inside the fuel pin apart from the lumped factor for gadolinia, the true radial and axial power redistribution as the poison burns, the reduction of the power shape by the Gd in the same node, U-236, Np and Am, decay of Pu-241, xenon and samarium transients, temperature and void feedback, control-rod and boron worth, and the loading pattern of a real multi-batch core. The linear reactivity model assumes a linear k_∞(B) between the batches. The constants are fitted; the model must not be used to predict a real core.

10. Data Provenance and Open Points

What was found The energy per fission (200 MeV), Avogadro’s number, the molar masses and the abundances of natural gadolinium are standard values. What was not found Evaluated one-group cross-sections, a lattice-code depletion curve for comparison and a re-check of the Gd 2200 m/s cross-sections (60 900 b and 254 000 b, recalled). Unverified items All fitted constants, the Gd cross-sections and their scale, ζ₀, X, σ_FP, the axial power shape and the target curve.

11. References

External standards and reports

1. [1] U.S. Nuclear Regulatory Commission, Operator Licensing Examination Standards for Power Reactors, NUREG-1021 (BWR K/A catalog, Reactor Theory, section 292007). Consulted at excerpt level (project copy).
2. [2] J. R. Lamarsh and A. J. Baratta, Introduction to Nuclear Engineering, 3rd ed., Prentice Hall, 2001. Further reading — not consulted (fission-product poisoning, burnup).
3. [3] J. J. Duderstadt and L. J. Hamilton, Nuclear Reactor Analysis, Wiley, 1976. Further reading — not consulted (depletion equations, burnable poisons, linear reactivity model).
4. [4] U.S. DOE, DOE Fundamentals Handbook — Nuclear Physics and Reactor Theory, DOE-HDBK-1019/2-93. Further reading — not consulted.
5. [5] M. Driscoll, T. Downar and E. Pilat, The Linear Reactivity Model for Nuclear Fuel Management, American Nuclear Society, 1990. Further reading — not consulted.

Antranova model basis

1. [A1] Antranova Engineering AB, AntraSIM Production System Prompt v1.0.3 and AntraSIM Template v1.0.0.

12. Exercises and Laboratory Work

Six exercises of 15–20 minutes. Each has a Load set-up button; the numbers under Expected results were generated from the same code. Read the middle-zone lattice curve where a single lattice is meant; the core-average curve includes the axial zoning.