Theory & Method
Two-Group Diffusion Core Simulator · AntraSIM-0120 · 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.
How to read this document. Each topic follows the progression Intuition → Physical mechanism → Mathematical formulation → Engineering interpretation → Assumptions and limitations . Section 12 contains laboratory exercises with expected results; each exercise has a Load set-up button that configures the simulator exactly as described.
Model status. Software-verified (Section 7) against closed-form rod-worth integrals, an independent full-core enumeration of the rod map and an independent Bessel-function representation. Not validated against plant or core-simulator data; the rod worths, the shadowing coefficient and the reactivity excess are illustrative inputs (assumption / not verified, Sections 8 and 10). Operator-level principle simulator for training; it is deliberately not a two-group diffusion solver. Not safety-grade.
1. Physical Background
Intuition The neutron flux in a reactor core is not uniform. Neutrons leak out through the surface, so the flux is highest in the middle and falls towards the edges; in a bare cylinder the axial shape is a cosine and the radial shape is a Bessel function. A control rod removes neutrons where it sits, and a neutron that is absorbed where the flux is high is one that would otherwise have been very likely to cause a fission. That is why the same rod is worth much more in the middle of the core than at the edge, and why a rod that is inserted from the bottom changes reactivity slowly at first, fastest when its tip passes the mid-plane, and slowly again at the end — an S-shaped curve.
Physical mechanism
neutron leakage → flux shape (cosine × Bessel) → importance of a position ∝ flux² → rod worth vs. insertion and vs. position → rod patterns and sequences
When many rods are inserted next to each other they compete for the same neutrons — the flux around them is depressed — so each of them is worth less than it would be alone (shadowing ). An isolated rod surrounded by withdrawn rods has the highest worth. This is the reason for rod-pattern and sequence rules: they prevent a situation in which a single rod that is dropped or withdrawn carries too much reactivity.
Engineering interpretation Operators use these ideas to understand differential rod worth during rod movements, why a shallow rod near the bottom of a bottom-entry BWR moves reactivity little, why a top-peaked or bottom-peaked flux (from void or xenon) changes the rod worth, and why rods are withdrawn in predetermined groups. Assumptions and limitations This is a deliberately operator-level model: fixed one-group flux shapes and first-order perturbation theory; it is not a two-group diffusion solver (Section 9).
2. Assessment of the Source Material and Learning Objectives
The objectives are taken from NUREG-1021, BWR catalog, Reactor Theory, sections 292001 (flux distribution and leakage), 292005 (control-rod worth) and 292008 (rod patterns and sequences during operation) [1], consulted at excerpt level. The original plan for this simulator was a two-group finite-difference diffusion core; on reflection it was replaced by the operator-level model of this release because the diffusion solver adds numerical detail (group constants, boundary conditions, iteration) that a course for reactor operators does not need, while the physical points — flux shape, S-curve, shadowing, sequence — are all obtained from closed-form shapes. That is a deliberate deviation from the original plan and is stated here so that it can be reversed if a full solver is wanted.
Objective (paraphrased) Support in AntraSIM-0120
Describe axial and radial flux distributions and the effect of leakage and reflectors Axial flux plot, radial factor J0 , reflector saving δ, buckling and peaking factors. Explain integral and differential rod worth and the S-curve Worth plot with both curves; closed-form fraction x − sin(2πx)/(2π). Effect of flux tilt (void, xenon) on rod worth Axial skew input. Rod patterns, shadowing and insertion sequences; highest single-rod worth Clickable radial rod map, coupling κ, sequence plot. Two-group energy dependence, moderator and temperature effects on rod worth Not modelled ; see AntraSIM-0140 for reactivity coefficients.
Table 2.1 — Coverage. Finding A-1: two-group diffusion was replaced by an operator-level model (see text above).
3. Governing Equations
Intuition For a bare reactor the flux is the solution of the wave equation ∇²φ + B²φ = 0 with the flux extrapolated to zero just outside the core; a small absorber changes the reactivity in proportion to the flux squared at its position (first-order perturbation theory).
Mathematical formulation With extrapolated height He = H + 2δ and extrapolated radius Re = R + δ [cm]:
ϕ ( z ) = cos π ( z − H / 2 ) H e [ 1 + s ( 1 − 2 z H ) ] , ϕ ( r ) = J 0 ( 2.405 r R e ) , B 2 = ( 2.405 R e ) 2 + ( π H e ) 2
with s [–] an empirical axial skew (s = 0 is the pure cosine) and B² [cm⁻²] the geometric buckling. A rod inserted from the bottom to height zr has the integral worth
W ( z r ) = W t o t ∫ 0 z r ϕ 2 ( z ) d z ∫ 0 H ϕ 2 ( z ) d z , W W t o t = x − sin 2 π x 2 π ( s = 0 , δ = 0 , x = z r / H )
and the differential worth dW/dz ∝ φ²(z) has its maximum at the flux peak. For the radial rod map each control cell j has a worth density proportional to the local flux squared and saturating with the absorption aj offered by the rods in and around the cell:
W p a t t e r n = C ∑ j ϕ j 2 ( 1 − e − a j ) , a j = s j + κ ∑ i ∈ N ( j ) s i , ρ = ρ A R O − W p a t t e r n
where sj ∈ {0, 1} is the state of the rod in cell j (0 = withdrawn), N(j) the eight neighbouring cells, κ [–] the neighbour coupling and C [pcm] a constant fixed by requiring that the average worth of an isolated rod equals w0 . The saturating factor 1 − e−a produces the shadowing: the more absorption a cell already receives, the less an additional rod adds. The total worth is a state function that never decreases when a rod is added and is smaller than the sum of the isolated rod worths. Engineering interpretation An isolated rod in the centre has by far the highest worth; the same rod in an inserted group is worth a fraction of that. Assumptions and limitations First-order perturbation theory (small rods); one energy group; fixed flux shapes that do not change when rods move (the flux redistribution around inserted rods is represented only by the empirical saturation).
4. Inputs and Operating Conditions
Core height, equivalent radius and reflector saving; the axial skew, the rod insertion (per cent of the height from the bottom) and the total worth of the rod group; the reactivity with all rods out, the average isolated rod worth and the coupling κ; and the state (in/out) of every rod of the quadrant, set by clicking the map or by the pattern buttons. Rods are counted with their symmetry multiplicity (1 for the central rod, 2 on the symmetry lines, 4 elsewhere). Everything is a state, there is no time-dependent operating input, and a zero worth or an empty pattern changes nothing (ρ = ρARO ).
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
Flux normalised to the core average and its peak factor; integral and differential worth versus insertion; worth at the chosen insertion; buckling and the radial peak factor; the worth of every rod (isolated, and in the current pattern); the total worth and reactivity of the pattern; cumulative reactivity for a centre-out and a periphery-in insertion sequence with the largest single-rod increment. Default case: the flux peak-to-average ratio is 1.516 (π/2 = 1.571 for a bare core), the axial worth at 50 % insertion is 8500 pcm, the core contains 137 rods, the checkerboard pattern with 69 rods inserted is worth 15540 pcm (ρ = -1539.7 pcm), and all rods in are worth 17026 pcm (ρ = -3026.1 pcm).
Table 5.1 — Axial integral worth versus insertion for a bare core (δ = 0; generated from the code).
Insertion fraction of total worth worth [pcm] ρ [pcm] 10 % 0.0064511 109.67 13890 25 % 0.090845 1544.4 12456 50 % 0.5 8500 5500 75 % 0.90915 15456 -1455.6 90 % 0.99355 16890 -2890.3
6. Numerical Method
Axial integrals are evaluated in closed form when s = 0 (using ∫cos² = u/2 + sin(2au)/4a) and by Simpson quadrature otherwise; the curves use 100 intervals of 1 % of the stroke. J0 is summed from its power series (40 terms). The rod map is evaluated on the seven-by-seven quadrant using mirror indices for the neighbours across the symmetry lines; the individual rod worths are obtained as differences of the total worth with and without the rod. There is no iteration, tolerance or time stepping. The default case takes 1 ms, so results update on every valid input change.
7. Verification and Numerical Accuracy
Verification shows that the equations are solved correctly; it is not validation. The suite runs in the application and in Node. Two failures occurred during development and were traced to the model, not to the tests: a first shadowing rule (each rod’s worth multiplied by e−sn for n inserted neighbours) allowed the total worth to decrease when a rod was added, because the neighbours lost more than the new rod gained; it was replaced by the saturating cell-absorption rule above, which is monotone by construction. The default reactivity budget was also recalibrated so that the all-rods-in pattern is subcritical (all-rods-in worth 17026 pcm).
Table 7.1 — Release verification suite (12 of 12 passed; generated from an actual run).
ID Test Criterion Result Measured T1 Axial rod-worth fraction: closed form x − sin(2πx)/(2π) (δ = 0), closed form with reflector saving vs Simpson quadrature, and end points 0 and 1 max difference ≤ 1e-9 PASS 1.13e-14 T2 Symmetric cosine: flux symmetric, differential worth peaks at mid-height, worth monotonic; positive skew moves the worth peak downward symmetry ≤ 1e-12; peak index 50; skewed peak < 50 PASS peak 50 → 44 with skew 0.3; W(100 %) = 17000.00 T3 Integral of the differential worth over the full stroke equals the total worth relative difference ≤ 1e-4 (trapezoid, 100 intervals) PASS 1.11e-5 T4 J₀ power series vs. the integral representation (1/π)∫cos(x sin θ)dθ, and J₀(2.4048) = 0 max difference ≤ 1e-10 PASS 1.61e-15 T5 Rod-pattern worth on the symmetric quadrant (mirror indices, multiplicities 1/2/4) and its normalisation vs. independent enumeration of every rod of the full core relative difference ≤ 1e-12; rod counts equal PASS 2.22e-16; 137 rods T6 Shadowing: with κ = 0 worths add exactly; with κ > 0 inserting a rod never lowers the total worth, the all-rods-in worth is below the sum of isolated worths, and the isolated worth of the central rod equals its pattern worth additivity ≤ 1e-9; monotonic; shadowed < isolated PASS all rods in: 17026 pcm vs isolated sum 64390 pcm T7 Insertion sequences: cumulative reactivity decreases monotonically and both sequences end at the all-rods-in pattern; the centre-out sequence has the larger maximum incremental rod worth end difference ≤ 1e-9; maxInc(centre) > maxInc(periphery) PASS max increment 1537 vs 298 pcm/rod T8 Zero-magnitude semantics: no rods inserted, zero rod worth or zero insertion change nothing (ρ = ρ_ARO exactly); axial worth is linear in the bank worth exact; linear to 1e-12 PASS ρ(all out) = 14000 T9 IC round trip restores every input (including all 49 rod states); 7 malformed files are rejected; an explicit migration is applied and reported exact; 7 of 7 rejected; migratedFrom = 0 PASS round trip true; rejected 7/7; migrated from 0 T10 All laboratory exercise set-ups are valid and produce finite results 6 of 6 PASS 6 set-ups T11 Default-case regression values (pattern ρ, axial worth at 50 %, axial peaking, maximum incremental rod worth of the centre-out sequence) max relative difference ≤ 1e-9 to the stored values PASS 0 (ρ = -1539.671) T12 Invalid inputs are refused: radius below range, fractional rod state, skew out of range, wrong number of states 4 of 4 rejected PASS 4 of 4
8. Model Parameters and Constants
Evidence classes: externally sourced derived assumption not verified . Reference case: a large BWR-type core with about 137 control rods, cold, xenon-free (as in AntraSIM-0110). There is no earlier release.
Parameter Default Unit Class Basis and assessment Editable
H, R 370, 200 cm assumption Active height and equivalent radius of a large BWR-type core (about 130 control cells of pitch 30.5 cm); order of magnitude of a 3000 MWth core, not verified against a specific plant. yes δ (reflector saving) 7 cm assumption Extrapolation of the flux zero beyond the core boundary; typical value of a few cm to 10 cm for water-reflected LWR cores; not verified. yes Axial skew 0 – assumption Empirical tilt (0 = symmetric cosine). yes Wbank 17000 pcm assumption Total worth of the rod group, equal to the 17 %Δk/k total rod worth of AntraSIM-0110; assumption. yes w0 470 pcm assumption Average worth of one isolated rod; chosen so that the shadowed all-rods-in worth is about 17 %Δk/k (Table in exercise 4). The central rod then has an isolated worth of about 1.5 %Δk/k; both are of the order of design values, not verified. yes κ (coupling) 0.5 – assumption Weight with which a rod acts on each of its 8 neighbouring cells relative to its own cell; the saturation 1 − exp(−a) of the cell absorption a produces the shadowing. Empirical, no source. yes ρARO 14000 pcm assumption Same k-excess as AntraSIM-0110 (cold, xenon-free, all rods out). yes Cell pitch 30.5 cm externally sourced Twelve inches (two fuel assemblies) — the usual BWR control-cell pitch; rounded, not re-checked. no
Consistency check: the sum of the isolated worths of all rods (64390 pcm) is about four times the shadowed all-rods-in worth (17026 pcm); the highest isolated rod worth is 1536.7 pcm.
9. Model Scope and Limitations
Represented Fixed one-group flux shapes; first-order rod worth; shadowing between neighbours by a saturation rule; symmetric quarter-core patterns. Not represented Two-group spectrum effects, moderator density and temperature effects, xenon, burnup, flux redistribution by the rods, partially inserted rods in the radial map (rods are in or out), rod interaction beyond the nearest neighbours, and rod-drop dynamics. The rod worths are illustrative and not plant data.
10. Data Provenance and Open Points
What was found Learning objectives from NUREG-1021 [1] (excerpt level); the flux shapes and perturbation formula follow textbook derivations that were re-derived and verified by quadrature. What was not found No core-simulator or plant rod-worth data were used. Unverified items Core dimensions, w0 , ρARO , κ and the skew are assumptions of the right order of magnitude; the control-cell pitch of 30.5 cm (twelve inches) is recalled.
11. References
External standards and reports
[1] U.S. Nuclear Regulatory Commission, Operator Licensing Examination Standards for Power Reactors , NUREG-1021 (BWR K/A catalog, Reactor Theory, sections 292001, 292005, 292008). Consulted at excerpt level (project copy).[2] J. R. Lamarsh and A. J. Baratta, Introduction to Nuclear Engineering , 3rd ed., Prentice Hall, 2001. Further reading — not consulted (bare-reactor flux shapes, control-rod worth).[3] J. J. Duderstadt and L. J. Hamilton, Nuclear Reactor Analysis , Wiley, 1976. Further reading — not consulted (perturbation theory).[4] U.S. DOE, DOE Fundamentals Handbook — Nuclear Physics and Reactor Theory , DOE-HDBK-1019/2-93. Further reading — not consulted (rod worth and flux shapes).
Antranova model basis
[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.
Exercise 1 — Flux shape, peaking and reflector saving
Objective: connect leakage to the flux shape.
Learning outcomes. Explain why the flux is peaked; state the peak-to-average ratio of a cosine; explain what a reflector does.
Load set-up no reflector saving, symmetric axial shape.
Read the flux peak/average in the Inspection panel and the buckling. Set the reflector saving δ to 15 cm and read both again. Expected results. Peak/average = 1.571 = π/2 for the pure cosine; radial peak/average of the rod cells 2.345 (the value 2.32 is the bare-cylinder continuum limit; the discrete cells give a slightly different number). Buckling B² = 0.0002167 cm⁻². With δ = 15 cm the peak/average falls to 1.463 and B² to 0.0001868 cm⁻²: the flux is flatter and leakage smaller.
Discussion. What is the effect on power peaking limits?
Exercise 2 — The S-curve of rod worth
Objective: read integral and differential rod worth.
Learning outcomes. Explain the shape of the curves; predict the worth of a given notch range.
Load set-up bottom-entry rods at 25 % insertion, δ = 0.
Read the integral worth at 25 % and at 50 %; click the worth plot at 75 %. Find the position of the maximum differential worth. Compare the worth of the first 10 % of the stroke with that of the 10 % around the mid-plane. Expected results. At 25 % insertion the worth is 1544.4 pcm, 0.09085 of the total (closed form 0.25 − sin(π/2)/2π = 0.0908); at 50 % it is 8500 pcm, half of the total. The differential worth is practically zero at the very bottom and has its maximum 340 pcm per % of stroke at the mid-plane. Table 5.1 lists the fractions at other insertions.
Discussion. Why do the last notches of a full withdrawal change reactivity so little?
Exercise 3 — Bottom-peaked flux
Objective: see how a tilted flux changes the worth of shallow rods.
Learning outcomes. Relate a void-induced flux tilt to rod worth in a bottom-entry BWR.
Load set-up skew = +0.3, rods at 25 %.
Read the worth at 25 % insertion and the position of the flux peak. Set the skew to 0 and compare. Expected results. With skew 0.3 the flux peak moves to 162.8 cm (mid-plane 185 cm), the peak-to-average ratio is 1.599 and the worth at 25 % is 2159.6 pcm (0.127 of the total) instead of 1544.4 pcm (0.09085) without skew.
Discussion. Which operating change (recirculation flow, power) would tilt the flux the other way?
Exercise 4 — Rod patterns and shadowing
Objective: compare patterns; separate the isolated worth from the pattern worth.
Learning outcomes. Explain shadowing; compare a distributed and a clustered pattern.
Load set-up central group of 21 rods inserted.
Read the total worth and ρ; then press Checkerboard and read again. Hover a rod in the map and compare its isolated worth with its worth in the pattern. Press All in . Expected results. Central group: 21 rods, worth 10546 pcm, ρ = 3454.1 pcm — not enough to shut the core down although these are the most valuable rods. Checkerboard: 69 rods, worth 15540 pcm, ρ = -1539.7 pcm (subcritical). All rods in: 17026 pcm, ρ = -3026.1 pcm, whereas the sum of the isolated worths would be 64390 pcm.
Discussion. Why is a distributed pattern more effective per rod than a clustered one?
Exercise 5 — Insertion sequence and the highest single-rod worth
Objective: understand why sequences are prescribed.
Learning outcomes. Explain the sequence limits on rod worth; identify the worst step of a sequence.
Load set-up all rods out.
Study the sequence plot: the slopes of the two curves are the worths of the rods being inserted. Read the largest single-rod increments. Expected results. The centre-out sequence (41 steps) starts with a rod worth 1537 pcm (the isolated central rod) and its largest increment is 1537 pcm; the periphery-in sequence starts with 38.8 pcm and its largest increment is 298.4 pcm. Both end at -3026.1 pcm.
Discussion. What would happen to the reactivity change if the highest-worth rod dropped out during startup?
Exercise 6 — The isolated central rod
Objective: quantify the effect of neighbours on one rod.
Learning outcomes. Explain why the highest possible single-rod worth occurs for an isolated central rod.
Load set-up only the central rod inserted.
Read its worth; then insert its three nearest neighbour cells and read the worth of the central rod again (hover it). Expected results. Isolated central rod: 1536.7 pcm. With the three nearest neighbour cells also inserted the central rod contributes 119.87 pcm, i.e. 0.07801 of its isolated worth.
Discussion. How does this relate to the withdrawal sequence rules in the plant procedures?