Theory & Method
Reactor Startup Simulator · AntraSIM-0130 · 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 an independent matrix-exponential solution of the full kinetics equations, the inhour equation, the 1/M law and step-size convergence. Not validated against plant start-up data; reactor, detector and rod parameters are illustrative inputs classified as assumptions; the six-group delayed-neutron data are externally sourced but were confirmed only through secondary tables (Sections 8 and 10). Principle simulator for training; not a start-up procedure and not safety-grade.
1. Physical Background
Intuition Before start-up the reactor is subcritical, but it is not silent: a neutron source (spontaneous fission, photoneutrons or an installed start-up source) keeps a small neutron population alive, and every source neutron is amplified by the chain reaction that the fuel can still sustain. The closer the core is to criticality, the larger the amplification, so the count rate in the source-range detector rises as control rods are withdrawn. The operator therefore does not step blindly into criticality: after each withdrawal he waits until the count rate has settled, plots the inverse multiplication 1/M (the initial count rate divided by the present one), and extrapolates the trend to 1/M = 0, which is the estimated critical rod position.
Physical mechanism
rod withdrawal → reactivity change → subcritical multiplication M = 1/(1−keff ) → count rate (source-range detector) → 1/M extrapolation → critical position → supercritical: period and start-up rate → point of adding heat
Once the core is supercritical the neutron level grows exponentially; the growth is slowed by delayed neutrons, which are emitted by fission products seconds to minutes after fission. The rate of growth is expressed as the reactor period (time for the level to increase by a factor e) or the start-up rate in decades per minute. When the power has risen enough to heat the coolant measurably, the operator has reached the point of adding heat (POAH); beyond that point the temperature feedback from the heated moderator and fuel starts to matter.
Engineering interpretation The method is a safety practice: approach criticality in decreasing steps so that criticality is never reached unexpectedly, keep the period long enough for the operator and the protection system to react, and recognise the POAH. Assumptions and limitations The model is point kinetics; it has no spatial effects, xenon, boiling or protection system (Section 9).
2. Assessment of the Source Material and Learning Objectives
The objectives are taken from NUREG-1021, BWR catalog, Reactor Theory, sections 292003 (approach to criticality and the effect of neutron sources) and 292008 (startup, rod manipulation and heat-up) [1], consulted at excerpt level. The existing AntraSIM point-kinetics simulator of the family was not re-used; this model is an independent implementation with an added source term, rod worth, detector and heat balance.
Objective (paraphrased) Support in AntraSIM-0130
Subcritical multiplication, the role of the source, 1/M Count-rate plot, 1/M versus rod position with extrapolation. Count-rate statistics and counting time Poisson noise, adjustable counting time. Reactor period, start-up rate, delayed neutrons, reactivity in dollars Period, start-up rate and ρ in pcm and $ at every time. Rod manipulation: step size, waiting time Editable rod schedule, exercises 1, 3, 5. Point of adding heat, heat-up rate and temperature feedback Lumped heat balance, POAH marker, heat-up rate plot, exercise 4. Xenon, spatial effects, protection-system actions (SCRAM, period trips) Not modelled.
Table 2.1 — Coverage. Finding A-1: the assumed period limit of 30 s and the 55.6 K/h heat-up limit are illustrative and not taken from any plant procedure.
3. Governing Equations
Intuition The neutron level changes because of net production by prompt neutrons, a slow additional supply from the decay of delayed-neutron precursors and the external source. The precursors integrate the recent fission history, which is why the level after a reactivity change relaxes over tens of seconds to minutes.
Mathematical formulation Point kinetics with six delayed-neutron groups and a source Q:
d n d t = ρ − β Λ n + ∑ i = 1 6 λ i C i + S , d C i d t = β i Λ n − λ i C i
where n is the neutron level (proportional to the count rate) [cps], Ci the precursor concentration of group i in the same units, ρ the reactivity [–], β = Σβi the delayed-neutron fraction, λi [s⁻¹] the precursor decay constants, Λ [s] the prompt generation time and S the source [cps/s]. The prompt neutron lifetime is many orders of magnitude shorter than the other time scales, so the prompt-jump approximation is used: n follows the precursors instantaneously,
n = ∑ i λ i y i + Q β − ρ , d y i d t = β i n − λ i y i , y i = Λ C i , Q = Λ S = − ρ 0 n 0
with n0 the initial count rate and ρ0 the initial (negative) reactivity. For a constant negative reactivity the level settles at n = Q/(−ρ), so the multiplication is M = n/n0 = ρ0 /ρ and the inverse multiplication is linear in the reactivity:
1 M = n 0 n = ρ ρ 0
Supercritical, the asymptotic exponent ω = 1/τ is the positive root of the inhour equation (here without the Λω term of the prompt-jump model), and the start-up rate follows from the period:
ρ = ∑ i = 1 6 ω β i ω + λ i , S U R = 60 τ ln 10 [ decades/min ]
The reactivity consists of the rod contribution with an S-shaped integral worth (Section 3 of AntraSIM-0120 gives the physics) and an isothermal temperature term, and the temperature follows a lumped heat balance:
ρ ( t ) = ρ i n + W [ u − sin 2 π u 2 π ] + α T ( T − T 0 ) , u = x 48 , C t h d T d t = P r a t e d n κ − U A ( T − T 0 )
with x the rod position [notch], W the bank worth, αT [pcm/K] the temperature coefficient, Cth [MJ/K] the heat capacity, κ the count rate corresponding to rated power Prated and UA [MW/K] the heat-loss coefficient. Rod movements are increments added to the preceding position and each takes (notches × time per notch); coincident events add. Engineering interpretation The 1/M law shows why the trend must be extrapolated only from settled count rates and why the last points are the most reliable. Assumptions and limitations Prompt-jump kinetics is valid for ρ well below β; the run is stopped when ρ exceeds β/2 or the power exceeds 10 % of rated.
4. Inputs and Operating Conditions
Source-range settings (initial count rate, Poisson noise and counting time, seed, equivalent count rate at rated power); rod bank (initial reactivity with all rods in, bank worth, initial position, time per notch, βeff , Λ); the rod-withdrawal schedule as a table of (time, movement) rows; heat-up parameters (rated power, heat capacity, loss coefficient, initial temperature, temperature coefficient, detectable temperature rise); and the simulated time. The schedule is the time-dependent operating input: every row is an increment added to the preceding rod position (positive = withdraw, negative = insert), a movement of zero changes nothing, coincident rows add, and the position must stay within 0…48 notches. The table is edited in the page (an add/delete row control with per-row validation) rather than in a pop-out window.
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. The event table is stored as an array of (time_s, movement) pairs and may vary in length.
5. Calculated Quantities
Count rate and power; reactivity components in pcm and dollars; reactor period and start-up rate (central difference of ln n over 2 s); temperature and heat-up rate; the count rate and 1/M before every rod movement (with Poisson noise when enabled); the extrapolated critical notch from the last two subcritical points; the critical notch of the rod worth alone; the first time ρ ≥ 0; the time the level reaches 10⁵ cps; the POAH; and the shortest stable period (evaluated at least 20 s after the last rod movement has ended). For the default schedule: critical notch by the rod worth 32.958, extrapolated critical notch 33.03, ρ ≥ 0 first at 1142 s, final position 34 notch with ρ = 86.2 pcm (βeff = 550 pcm).
Table 5.1 — Default start-up: state before each rod movement (noise off; generated from the code).
Before pull t [s] rod position [notch] ρ [pcm] count rate [cps] 1/M start 0 0 -2500 5 1 pull 1 60 0 -2500 5 1 pull 2 240 16 -1913 6.5279 0.7658 pull 3 420 24 -1000 12.469 0.4008 pull 4 600 28 -511.3 24.305 0.2055 pull 5 780 30 -287.4 42.955 0.1161 pull 6 960 31 -183.7 66.597 0.0748 pull 7 1140 32 -86.5 131.15 0.03767 pull 8 1320 33 3.62 511.93 0.009212 end 1800 34 86.2 5513600 6.945e-7
6. Numerical Method
Approach The six precursor equations and the temperature are integrated with the classical fourth-order Runge–Kutta method at Δt = 0.25 s; the level follows algebraically from the prompt-jump relation. Rod positions are evaluated exactly at every stage time, so the kinks at the start and end of a movement are resolved only to the step size. Output is stored every second. Stability The fastest precursor constant is λ6 = 3.01 s⁻¹ and the fastest mode of the reduced system stays of that order below ρ = β/2, so Δt·|λ| ≈ 0.75, well inside the RK4 stability limit of about 2.8; the run is stopped if ρ exceeds β/2 or the power exceeds 10 % of rated, and the step-size test (T4) shows agreement to 1e-3 or better between Δt = 0.25 s and 0.03125 s. Poisson counts use Knuth’s method below 30 counts and a rounded normal approximation above. The default case takes 9 ms.
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 with the same code. Findings during development: (1) the sign of the extrapolation formula was wrong (xpred = xb + …, not xb − …); it was found because the predicted critical position came out below the last measured point, and it is now locked by regression value T12; (2) the period was NaN because the helper read the array length from the wrong object; (3) a comparison of the prompt-jump kinetics with the full kinetics at +300 pcm and 20 s failed at 5 % — the cause is the prompt-jump approximation itself (its exponent error of about 0.3 % is amplified by ωt) at a reactivity beyond its validity limit of β/2; the test was moved to +200 pcm and the run stops at β/2; (4) the shortest-period statistic was dominated by the momentary transient right after a rod step and is now evaluated only 20 s after the movement has ended; (5) coincident events initially did not equal a single event because each had its own ramp; they now merge.
Table 7.1 — Release verification suite (13 of 13 passed; generated from an actual run).
ID Test Criterion Result Measured T1 Subcritical equilibrium N = −Λ S/ρ is preserved (rods held, no heating) max relative drift ≤ 1e-9 over 600 s PASS 1.79e-12 T2 Prompt-jump kinetics (RK4, Δt = 0.05 s) vs. full kinetics by matrix exponential (steps to −500 pcm at 1 s, −100 pcm at 30 s, +100 pcm at 60 s and +200 pcm at 10 s from a −2500 pcm equilibrium) max relative difference ≤ 1e-2 (prompt-jump approximation error) PASS 7.47e-3 T3 Asymptotic reactor period (ρ = +50, +100, +200 pcm): (a) simulation vs. the inhour equation of the prompt-jump model (Λ = 0); (b) simulation vs. the full inhour equation with Λ (a) ≤ 1e-6 (numerical accuracy); (b) ≤ 1e-2 (approximation error) PASS 2.27e-12 ; 3.49e-3 T4 Step-size convergence of the default start-up (Δt = 0.25 s and 0.125 s vs. 0.03125 s) at 300, 600 and 780 s Δt = 0.25 s: ≤ 1e-3; error decreases with Δt PASS 2.60e-12 ; 1.55e-13 T5 1/M law: after a long hold N∞ = CR₀ ρ₀/ρ, i.e. 1/M = ρ/ρ₀ is linear in the reactivity relative difference ≤ 1e-6 PASS 2.50e-11 (N∞ = 8.396 cps) T6 Event semantics: two coincident +2 notch events equal one +4 event; a zero-movement event changes nothing relative difference ≤ 1e-12 PASS 0 T7 Critical-notch solver: ρ_rods(x_crit) = 0 |ρ| ≤ 1e-9 pcm; monotone S-curve PASS 0 ; x_crit = 32.9582 T8 Poisson count noise: mean of 1 s counts at the initial level within 4 σ of the true level; identical seed gives identical counts, a different seed different counts ≤ 4 σ; deterministic PASS mean 4.320 vs 5 (2.15 σ) T9 Energy balance of the lumped heat model: ∫P dt − ∫UA ΔT dt = C ΔT (trapezoid on the 1 s output) relative difference ≤ 5e-3 PASS 1.19e-6 (ΔT = 3.924 K) T10 IC round trip restores every input including the event table; 7 malformed files (missing field, wrong type, decreasing times, rod out of range, …) 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 T11 All laboratory exercise set-ups are valid and compute 5 of 5 PASS slowest 89 ms T12 Default-case regression values (noise off): neutron level at 600 s and 1200 s, temperature at 1499 s, extrapolated critical notch max relative difference ≤ 1e-9 to the stored values PASS 0 (N(1200 s) = 299.874 cps) T13 Invalid inputs are refused: movement out of range, empty schedule, decreasing times, positive initial reactivity 4 of 4 rejected PASS 4 of 4
8. Model Parameters and Constants
Evidence classes: externally sourced derived assumption not verified . Reference case: a cold, xenon-free BWR-type core of 3000 MWth with all rods in and a 2.5 %Δk/k shutdown margin. There is no earlier release.
Parameter Default Unit Class Basis and assessment Editable
βi , λi (six groups) 0.000215, 0.001424, 0.001274, 0.002568, 0.000748, 0.000273 / 0.0124, 0.0305, 0.111, 0.301, 1.14, 3.01 – / s⁻¹ externally sourced Thermal fission of U-235 (Keepin-type data). The values were transcribed from published secondary tables and compared between two independent web tables; the primary attribution was not confirmed. The relative fractions are scaled to βeff . no βeff 0.0055 – assumption Illustrative BWR-type value (0.0055 = 550 pcm) below the total 0.0065; real values depend on fuel, burnup and importance weighting; not verified. yes Λ 4.000e−5 s assumption Prompt neutron generation time of the order of 10⁻⁵ s for LWR cores; enters only through the prompt-jump approximation validity, not the results below prompt critical. yes ρinitial -2500 pcm assumption Cold, xenon-free shutdown with the bank fully in (SDM 2.5 %Δk/k); assumption. yes Wbank 3000 pcm assumption Worth of the rod bank pulled during the start-up; +500 pcm at full withdrawal stays below βeff . Critical notch derived: see Table 5.1. yes Notches, time per notch 48, 2 –, s assumption Full stroke divided into 48 notches; 2 s per notch is of the order of BWR control-rod drive speed; assumption. time per notch: yes αT -1 pcm/K assumption Isothermal temperature coefficient during heat-up; assumption; not verified. yes Prated 3000 MWth assumption Reference BWR-type core. yes Cth 2000 MJ/K assumption About 400 t water plus 150 t metal: estimated from masses and specific heats; not verified for a specific plant. Gives 55 K/h at about 1.2 % power. yes UA 0.01 MW/K assumption Heat loss to the surroundings; small; assumption. yes κ 1.000e12 cps assumption Conventional equivalent count rate at rated power; the source-range detector itself saturates near 105 cps (assumption). yes CR0 5 cps assumption Initial source-range count rate (of the order of 3–10 cps for a start-up source). yes Period limit 30 s not verified Assumed limit used for advisories; not taken from any plant procedure. no
Consistency check: 1 % of rated power heats the assumed 2000 MJ/K at 54 K/h, of the order of the usual 55.6 K/h heat-up limit; the rod bank reaches +500 pcm at full withdrawal, below βeff ; the critical notch is 32.958 of 48.
9. Model Scope and Limitations
Represented Point kinetics with a neutron source and six delayed-neutron groups; S-shaped rod-bank worth; a single isothermal temperature coefficient; a lumped heat balance; source-range counting statistics. Not represented Spatial effects and flux tilts after rod movements (which make real count rates depend on detector position), xenon, boiling and void feedback, several rod groups with different worths, the decay of photoneutron sources and of decay heat, the intermediate-range monitor scales, the SRM saturation, and the reactor protection system. The prompt-jump approximation is invalid close to prompt critical.
10. Data Provenance and Open Points
What was found The six-group delayed-neutron data for thermal fission of U-235 were transcribed from two independent secondary web tables that agree with each other (a course-note table citing Hetrick and a general encyclopaedia table); Keepin’s original report [2] was not consulted. What was not found Plant start-up data; a primary source for the group constants. Unverified items βeff , Λ, all rod, heat and detector parameters, the period limit of 30 s and the heat-up limit of 55.6 K/h are assumptions.
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 292003 and 292008). Consulted at excerpt level (project copy).[2] G. R. Keepin, Physics of Nuclear Kinetics , Addison-Wesley, 1965. Further reading — not consulted ; the group data were taken from secondary tables that cite it.[3] D. L. Hetrick, Dynamics of Nuclear Reactors , American Nuclear Society, 1993. Consulted at excerpt level only through a course-note table that reproduces its delayed-neutron data (weak secondary evidence).[4] J. R. Lamarsh and A. J. Baratta, Introduction to Nuclear Engineering , 3rd ed., Prentice Hall, 2001. Further reading — not consulted (subcritical multiplication, inhour equation).[5] U.S. DOE, DOE Fundamentals Handbook — Nuclear Physics and Reactor Theory , DOE-HDBK-1019/2-93. Further reading — not consulted (reactor operations, period).
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
Five exercises of 15–25 minutes. Each has a Load set-up button; the numbers under Expected results were generated from the same code (noise-free unless stated).
Exercise 1 — Subcritical multiplication and 1/M
Objective: predict the critical rod position from count-rate data.
Learning outcomes. Explain subcritical multiplication; construct a 1/M plot; explain why the extrapolation improves as the core approaches criticality.
Load set-up default schedule, noise off.
Read the count rate before each pull (Table 5.1 or the count-rate plot). Watch the 1/M plot: which points does the extrapolation use and where does it cross zero? Compare the extrapolated critical position with the milestone “Critical notch (rod worth alone)”. Expected results. Starting at 5 cps with ρ = -2500 pcm the count rate is 24.3 cps at 600 s and 300 cps at 1200 s. 1/M falls monotonically towards zero; the critical notch of the rod worth is 32.958. Extrapolating successive pairs of points gives the predicted critical notches 68.32, 32.78, 32.21, 32.6, 32.81, 33.01: the first prediction is far off (the S-curve is strongly non-linear at low insertion), later ones converge on 32.958, the final one (33.03 notch in the default schedule) is only a few tenths of a notch beyond it. The core becomes critical at 1142 s (during the pull to notch 33).
Before pull t [s] rod position [notch] ρ [pcm] count rate [cps] 1/M start 0 0 -2500 5 1 pull 1 60 0 -2500 5 1 pull 2 240 16 -1913 6.5279 0.7658 pull 3 420 24 -1000 12.469 0.4008 pull 4 600 28 -511.3 24.305 0.2055 pull 5 780 30 -287.4 42.955 0.1161 pull 6 960 31 -183.7 66.597 0.0748 pull 7 1140 32 -86.5 131.15 0.03767 pull 8 1320 33 3.62 511.93 0.009212 end 1800 34 86.2 5513600 6.945e-7
Discussion. Why does the prediction lie beyond the true critical position?
Exercise 2 — Count-rate noise and counting time
Objective: relate statistical scatter to the counting time.
Learning outcomes. Estimate the relative scatter 1/√counts; choose a counting time.
Load set-up Poisson noise on, counting time 5 s, seed 3.
Observe the scatter of the grey 1 s counts and of the 1/M points; note the extrapolated critical notch. Increase the counting time to 60 s (keep the seed). Expected results. With 5 s counting the extrapolated notch is 33.09; with 60 s it is 33.06 (noise-free 33.03). At the initial 5 cps the relative scatter of a 5 s count is 1/√25 = 20 %, of a 60 s count 1/√300 = 5.8 %, and it falls as the count rate rises.
Discussion. Why is the low count rate at the start of the start-up the limiting statistical case?
Exercise 3 — Too large a step: period and start-up rate
Objective: connect a rod step to the reactor period.
Learning outcomes. Read period and start-up rate; judge a rod step against a period limit; recognise the transient period after a step.
Load set-up pull to 30 notches, then +5 notches at 400 s.
Note the reactivity before the last pull and after it. Read the period and start-up rate in the Inspection panel after the pull (click the plot at about 500 s). How does the power evolve, and what does the calculation do at the end? Expected results. Before the last pull ρ = -287.4 pcm at notch 30; after it the rods are at notch 35 and ρ = 160.8 pcm (0.29 $). The shortest stable period is 16.66 s (at 430 s), the highest start-up rate 4.98 dpm, well beyond the assumed 30 s limit; the level rises to 97550000000 cps and the calculation stops when the power has reached 10 % of rated.
Discussion. Which step size would have kept the period above 30 s?
Exercise 4 — Point of adding heat
Objective: find the POAH and see temperature feedback.
Learning outcomes. Define the POAH; relate heat-up rate to power; explain the first effect of temperature feedback.
Load set-up default schedule, one notch inserted at 2100 s, 3600 s simulated.
Find the POAH marker in the temperature plot and the corresponding power. Read the heat-up rate after the rod insertion and the temperature feedback at the end. Expected results. A temperature rise of 0.1 K (the assumed detectable value) is first reached at 2076 s. After the insertion the power holds near 0.15–0.2 % of rated (final 0.1841 %, 1841000000 cps); the peak heat-up rate is 10.55 K/h, well below 55.6 K/h; the temperature at the end is 43.924 °C and the temperature feedback has grown to -3.924 pcm, which is what brings the reactivity back from +3 pcm to about zero.
Discussion. Why is the feedback negligible before the POAH?
Exercise 5 — Waiting long enough
Objective: show the cost of reading the count rate too early.
Learning outcomes. Explain why the count rate takes minutes to settle; explain why the resulting error is non-conservative.
Load set-up 40 s between pulls, 20 s counting time.
Note the 1/M points and the extrapolated critical position; compare with the critical notch. Lengthen the interval between pulls in the schedule and repeat. Expected results. The true (noise-free) 1/M values before each pull are 1, 1, 0.777, 0.415, 0.221, 0.0947, 0.0113 (the last is a supercritical-side point, excluded from the extrapolation). The extrapolated critical notch is 33.39 against a true 32.958, i.e. beyond the true position: the operator would believe the core to be farther from criticality than it is. After a single 28-notch pull (ρ = -511.3 pcm, settled level 24.45 cps) the level has reached only 0.8087 of its final value after 60 s and 0.9199 after 100 s.
Discussion. Which delayed-neutron groups determine the settling time?