Theory & Method
Decay Heat Simulator · AntraSIM-0090 · version 1.0.2 · build 2026-09-20. This document describes the model as implemented in this application. Section 2 documents the assessment of the two source models (ANS 5.1 table and ANS-79 power law) and why they were replaced.
How to read this document. Each topic follows the progression Intuition → Physical mechanism → Mathematical formulation → Engineering interpretation → Assumptions and limitations. A newcomer can stop after the first two layers of each section; an experienced engineer will find equations, units, numerical implementation, data provenance and limits in the rest.
Model status. The application implements the summation form of ANSI/ANS-5.1-1979 (23 exponential groups per fissioning nuclide) with operating history, neutron-capture correction and U-239/Np-239 decay. The U-235 data are verified against published standard values (§7). The Pu-239 and U-238 data and the interpretation of one formula are not independently verified (§10). The application is not validated against calorimetric or plant data, is not a licensing-grade implementation of any edition of the standard, and shall not be used as a safety-grade tool.
1. Physical Background
Intuition When a reactor is shut down the chain reaction stops within seconds, but the fuel keeps producing heat. The fission fragments are radioactive, and each decay releases beta and gamma energy that ends up as heat in the fuel. This decay heat starts at about 6.9 % of the pre-shutdown thermal power (default case) and has fallen to 0.61 % after one day, but it never stops abruptly: after one year it is still 0.08 %.
Physical mechanism Fission produces hundreds of different radioactive nuclides. Each has its own half-life, from fractions of a second to years. The decay heat at time t after shutdown is the sum over all nuclides of (number present) × (decay rate) × (energy per decay). Nuclides with short half-lives dominate early and disappear; long-lived ones dominate late. Two additional sources exist: neutron capture in fission products (which creates further radioactive nuclides) and the decay of U-239 and Np-239, formed when U-238 captures a neutron.
Power history→fission-product inventory→radioactive decay→deposited energy→decay heat Pd(t)
Engineering interpretation Decay heat is the thermal load for post-shutdown heat removal, pool cooling and core-uncovery analyses. It depends on how long and at what power the reactor had been running: a reactor that has operated for years carries a full inventory of long-lived fission products, whereas one started a day ago does not. This is why the standard is formulated as a response to an operating history.
Assumptions and limitations Decay heat is the heat from radioactive decay only. The residual fission power that follows a scram (delayed-neutron tail, tens of seconds) is a separate contribution and is not included; it is calculated by the AntraSIM Point Kinetics Simulator. Total thermal power after a scram is the sum of the two.
2. Assessment of the Source Models
Question The source notebook and version 0.0.0 of this application compared (a) an eight-point table labelled “ANSI/ANS 5.1-2005, U-235”, interpolated log–log, and (b) a three-segment power law labelled “ANS-79”. Are these adequate, or is there a better way to model them?
Finding Yes. The ANS-5.1 standard is not a table of eight values or a set of power laws; it is a summation of 23 exponentials per fissioning nuclide, evaluated for a given operating history [3][4][6]. That form is finite at t = 0, covers 0.1 s to 10⁹ s, represents operating history, isotopic mix, neutron capture and actinides, and has a documented uncertainty. Both legacy curves deviate substantially from it (table below), and they cannot represent any of those effects. They have therefore been replaced by the summation model. The legacy curves remain available as hidden, clearly labelled comparison series because they show the size of the deviation.
Table 2.1 — Legacy curves compared with the ANS-5.1-1979 summation model (generated from this code). “FP only” is the same basis as the legacy curves: U-235, infinite operation, 200 MeV/fission, no capture correction, no actinides. “Reference” is the default case of this application (with capture φ = 1 and actinides R = 0.6).
| t [s] | FP only [%] | Reference total [%] | Legacy table [%] | Legacy ANS-79 [%] | Table vs FP only | ANS-79 vs FP only | Table vs Reference | ANS-79 vs Reference |
|---|
| 1 | 6.159 | 6.461 | 6.6 | 7.66 | +7.2 % | +24.4 % | +2.1 % | +18.5 % |
| 10 | 4.75 | 5.044 | 4.4 | 5.049 | -7.4 % | +6.3 % | -12.8 % | +0.1 % |
| 100 | 3.102 | 3.381 | 2.9 | 3.328 | -6.5 % | +7.3 % | -14.2 % | -1.5 % |
| 1e3 | 1.901 | 2.126 | 1.76 | 1.915 | -7.4 % | +0.7 % | -17.2 % | -9.9 % |
| 1e4 | 0.9569 | 1.094 | 0.85 | 1.077 | -11.2 % | +12.5 % | -22.3 % | -1.6 % |
| 1e5 | 0.4874 | 0.5846 | 0.41 | 0.4855 | -15.9 % | -0.4 % | -29.9 % | -17.0 % |
| 1e6 | 0.2783 | 0.2866 | 0.2 | 0.1645 | -28.1 % | -40.9 % | -30.2 % | -42.6 % |
| 1e7 | 0.1257 | 0.1275 | 0.06 | 0.05574 | -52.3 % | -55.6 % | -52.9 % | -56.3 % |
Audit of the source material Findings from reading the notebook and version 0.0.0 completely, and their resolution:
| ID | Finding | Resolution |
|---|
| A-1 | The eight-point “ANS 5.1-2005” table does not reproduce the ANS-5.1 data for any constant recoverable energy or irradiation time: it is 7 % high at 1 s, 7 % low from 10 s to 1000 s and up to 52 % low at 10⁷ s relative to U-235 infinite-operation fission-product decay heat (Table 2.1). Its origin is not stated in the supplied material and could not be established. | Replaced by the summation model. Retained only as a labelled legacy curve. |
| A-2 | The “ANS-79” power law is +24 % at 1 s, +6 % to +7 % at 10–100 s, +0.7 % at 10³ s, +12.5 % at 10⁴ s and −41 % at 10⁶ s relative to the same basis. The first segment t−0.181 grows without bound as t → 0 (11.6 % at 0.1 s); the day-scale segment decays too fast for t > 10⁵ s. The origin of its coefficients is not established in the supplied material. | Replaced. Retained as a labelled legacy curve, reproduced exactly as executable in the notebook (V17). |
| A-3 | The notebook text (markdown cell) gives 1.39·t(h)−0.27 up to 10 h and 0.74·t(d)−0.47 beyond, whereas the executable function uses 1.39·(t/3600)−0.25 and 0.52·(t/86400)−0.47. The text version is discontinuous at 10 h (0.746 % → 1.12 %); the executable version is continuous (0.782 % → 0.785 %). | Executable version kept for the legacy curve (as in version 0.0.0); text version discarded. |
| A-4 | “ANS79” is not defined in the notebook. It presumably refers to ANSI/ANS-5.1-1979, which is the exponential-summation standard itself, not a power law. This is an inference, not confirmed. | Model now based on the 1979 data (§10). |
| A-5 | Both legacy models ignore operating history, although the notebook itself remarks that the last days of operation matter on the hour scale. They also ignore fission-fraction mix, neutron capture and actinide decay. | All four are inputs now (§3, §4). |
| A-6 | The table interpolator extrapolates silently log–log below 1 s and above 10⁷ s (SciPy “extrapolate”). | The summation model needs no extrapolation inside its range; the legacy curve is flagged as comparison only; the tabulated range of the standard (0.1 s – 10⁹ s) is shaded and warned about. |
| A-7 | The notebook clamps the minimum time at 10⁻³ s and plots a linear view of log-spaced samples; the linear view therefore has very few samples at late times. | Linear plot uses its own linear grid starting at t = 0, where the model is finite. |
| A-8 | Version 0.0.0 stored input configuration schema 1 (power, time window) for a different model. | Schema incremented; unsupported older files are rejected with an explicit message (§4). |
3. Governing Equations
3.1 Response of the fission products to one fission
Intuition Imagine a single fission event. Its fragments produce a burst of decay heat that starts high and dies away.
Mathematical formulation The standard represents this pulse response per fissioning nuclide i as a sum of 23 decaying exponentials [3][4][6]:
with fi in MeV/(fission·s) at time t [s] after the fission, αij in MeV/(fission·s) and λij in s⁻¹. Integrating over all time gives the total decay energy per fission Qd = Σj αij/λij (13.18, 10.93 and 16.23 MeV for U-235, Pu-239 and U-238 with the data of §8).
3.2 Operating history
Intuition Decay heat at a given time is the sum of the responses of all fissions that ever took place, each weighted by how long ago it occurred. Long-lived groups accumulate over long operation; short-lived groups only reflect the last minutes or hours.
Mathematical formulation The power history is a sequence of intervals k, each with start ak and duration dk measured backwards from shutdown (t = 0) and a relative power level (fraction of P₀) that is constant or varies linearly between a near-shutdown value Lk and a far value L′k (a ramp). The fission rate is proportional to the level (level × P₀/Q). Superposition of the pulse response gives, for one nuclide,
Fi is in MeV per fission-rate-normalised unit; for one constant interval of level 1 starting at the shutdown, W = 1 − e−λT and the expression reduces to the standard’s F(t,T) = Σ (α/λ) e−λt(1 − e−λT) [3][8]. For infinite operation (T → ∞) W = 1. The weights Wij do not depend on t, so a history only changes the 23 coefficients of each nuclide. The ramp term is the exact integral of a linearly varying level (verified in V20 against an independent quadrature); xE1(x) is evaluated by a series for x < 0.5 to avoid cancellation.
3.3 Isotope mix and normalisation
The fission fractions xi (Σxi = 1; U-235 is the remainder after Pu-239 and U-238, as directed by the standard [3]) and the recoverable energy per fission Q [MeV] give the fission-product decay heat as a fraction of P₀:
P₀ is the thermal power that corresponds to 100 %; the fission rate is P₀L/Q. The standard leaves Q to the user [4]; 200 MeV is the value recommended in [4] (“may be conservative by at least 2 %”).
3.4 Neutron capture in fission products
Intuition Some fission products absorb neutrons during operation and turn into other radioactive nuclides, adding a little decay heat. The effect grows with the burnup of the fuel and is small (about 1 %) at 10³ s.
Mathematical formulation The standard provides a multiplicative correction [3][4]:
with cooling time t ≤ 10⁴ s, irradiation time T ≤ 1.2614×10⁸ s and φ ≤ 3 fissions per initial fissile atom. In this application T is the irradiation span (from the start of the oldest interval with power to shutdown), limited to 1.2614×10⁸ s, and G is held at G(10⁴ s) for t > 10⁴ s. The printed source [3] shows the second term without its variable; it is interpreted here as 5.23×10⁻¹⁰·t (§10).
3.5 Actinides: U-239 and Np-239
Mechanism U-238 captures neutrons to form U-239, which decays (half-life 23.5 min) to Np-239, which decays (half-life 2.36 d) to Pu-239. Both decays release heat. In a thermal reactor about R U-239 atoms are formed per fission.
Mathematical formulation With λU = 4.91×10⁻⁴ s⁻¹, λNp = 3.41×10⁻⁶ s⁻¹ and energies per decay qU = 0.474 MeV, qNp = 0.419 MeV [3], the heat released by one U-239 atom created at time zero, including the Np-239 ingrowth, is
This is again a sum of exponentials and is handled by the same history superposition as §3.2, multiplied by R/Q per unit P₀. The total is
where (ag, λg) are the three pseudo-groups of pact (one has a negative amplitude) and Wg is the history weight of §3.2. The constants are those of the standard as quoted in [3]; the closed-form ingrowth is verified against direct integration of the two-nuclide chain (V10).
3.6 Uncertainty band
Reference [3] tabulates one-sigma uncertainty factors σij (%) per group and nuclide, fitted to the uncertainty data of the 1979 standard and described there as a conservative representation. The application forms upper and lower bounds by scaling the fission-product coefficients, αij → αij(1 ± 2σij/100), which treats all groups as fully correlated (a conservative simplification). Capture and actinide terms are not scaled. This is not the uncertainty procedure of the standard itself.
3.7 Integrated decay energy
The energy released between shutdown and time t, i.e. the energy that the heat-removal systems must absorb, is
with P₀ in MW this gives MJ (reported in GJ and MWh).
4. Inputs and Operating Conditions
Inputs Reference thermal power P₀ [MWth]; recoverable energy per fission Q [MeV]; operating history (mode; constant-power time; time-point table; history time unit d/h); fission fractions of Pu-239 and U-238 [%]; capture correction (on/off, φ); actinide decay (on/off, R); time unit and plot windows of the decay-heat curves; inspection time.
History semantics There are three ways to define the history, and all are normalised to the same canonical list of intervals (start, duration, near and far level) before evaluation. Infinite (default): one interval at 100 % of P₀ with T → ∞. Constant: one interval at 100 % of P₀ for time T. Table (pop-out window): a power-versus-time table of 2–200 time points (time since the start of the history, power in % of P₀, 0–200 %). The first point is the start of the history (time 0); the last point is the shutdown (SCRAM), so the history plot runs from 0 on the left to the shutdown time on the right. Power varies linearly between consecutive points, which gives ramps; two points with the same time form a step. Times must not decrease, and all intermediate times can be edited freely. While “Table” is selected it takes over the history definition; the Infinite and Constant settings and the table are all preserved and stored in the configuration. A zero-duration spike (equal times) changes nothing and a 0 % period before the first fission carries no fissions (V6); equivalent definitions give identical results (V5, V21). The equivalent full-power time is the integral of the power level over the history.
Time unit and CSV exchange The time unit of the history (days or hours) is selectable in the main window and in the table window; it applies to the table, to the constant-power time and to the history plot, and is stored in the configuration (times are stored in days). In the table window the history is exported as time_d,power_pct (or time_h,power_pct when hours are selected) and imported from either form, comma or semicolon delimited (decimal comma with semicolon), with # comment lines; a first time other than 0 is shifted to 0 and reported. The hold/ramp CSV format of the previous release (power_pct,hold_d,ramp_to_next_d) is still accepted and converted to time points. An invalid file is rejected as a whole and leaves the table unchanged (V22). Up to 200 points are supported.
Units and conversions Times are stored in seconds (1 min = 60 s, 1 h = 3600 s, 1 d = 86 400 s); histories are entered in days. Percent in the interface, fractions internally (1 % = 0.01). Decay heat is shown in % of P₀ on the left axis and in MWth (= % × P₀/100) on the right axis, correct for linear and logarithmic scales.
Configuration files The Input Configuration (IC) is a JSON file with format “AntraSIM-input-configuration”, schemaVersion 4 (time-point table and history time unit). Schema-3 files (hold/ramp table, release 1.1.0) are migrated explicitly to time points and reproduce the same history (V23); schema-1 (version 0.0.0, different model) and schema-2 files (six-interval table, release 1.0.0) are rejected, not migrated. Loading is all-or-nothing (V18).
5. Calculated Quantities
Components Total decay heat, fission-product part (including the capture factor), the U-239 + Np-239 part, and upper and lower bounds. Milestones The time at which the total falls below 5 %, 2 %, 1 %, 0.5 % and 0.2 % of P₀ is found by bisection in ln t (V15). For the default case: 5 % of P₀ after 10.6 s; 2 % of P₀ after 21.2 min; 1 % of P₀ after 3.84 h; 0.5 % of P₀ after 2.02 d; 0.2 % of P₀ after 33.5 d. Energy The integrated decay energy of §3.7 up to the inspection time. Legacy The two legacy curves can be shown in the decay-heat plots (hidden by default); their value and relative deviation from the total at the inspection time are listed in the results panel.
Effect of the history Table 5.1 shows how strongly finite histories reduce the decay heat relative to infinite operation (fission-product part, capture and actinides on, same code).
Table 5.1 — Total decay heat [% of P₀] after different operating histories (default corrections; generated from this code).
| History | 10 s | 60 min | 24 h | 30 d | 365 d |
|---|
| Infinite | 5.044 | 1.473 | 0.6086 | 0.208 | 0.08233 |
| Constant 365 d | 4.951 | 1.387 | 0.5237 | 0.1269 | 0.01381 |
| Constant 30 d | 4.817 | 1.259 | 0.3994 | 0.04503 | 0.001927 |
| Constant 1 d | 4.417 | 0.8675 | 0.1062 | 0.002352 | 6.875e-05 |
| Power-history table (§8) | 4.886 | 1.323 | 0.4763 | 0.1191 | 0.01315 |
6. Numerical Method
Approach There is no time-stepping. Every quantity is a closed-form sum of exponentials in t, evaluated directly at the requested time; a change of input triggers an immediate recalculation (about 2.5 million evaluations per second in Node, 2 ms for both plot grids). The default plot grids hold about 60 points per decade (log) and 401 points (linear).
Details The history weight uses expm1, so that 1 − e−λd is accurate for the slowest groups (λ ≈ 10⁻¹⁴ s⁻¹, where a naive difference of exponentials would lose all digits) and for infinite operation (e−∞ = 0). The energy integral uses 8-point Gauss–Legendre panels, four panels per decade from 10⁻³ s, with a panel edge at 10⁴ s where G(t) has a kink; the analytic integral of the exponential sum is reproduced to about 1e-14 relative (V11). Milestone times use 100 bisection steps on ln t.
Stiffness and non-smooth terms Not applicable to the exponential sum (no stability limit). The only non-smooth term is the hold of G(t) at t = 10⁴ s; it is inactive for t ≤ 10⁴ s and is marked in the plots. Non-finite values stop the calculation with a message; they are never plotted.
7. Verification and Numerical Accuracy
Scope Verification shows that the equations are solved correctly and that the published reference values are reproduced. It is not validation against measurements. The suite below runs in the application (footer → Verification) and in Node; the table is generated from an actual run of the same code.
Table 7.1 — Release verification suite (23 of 23 passed; runtime 81 ms in Node).
| ID | Test | Criterion | Result | Measured |
|---|
| V1 | U-235 infinite operation vs. ANS-5.1-1979 standard values (9 cooling times, WCAP-16009 Table 8-2) | max |ΔF/F| ≤ 0.03 % | PASS | 0.0074 % |
| V2 | U-235, 10⁶ s irradiation from zero vs. standard values (9 cooling times) | max |ΔF/F| ≤ 0.03 % | PASS | 0.0105 % |
| V3 | Closed-form F(t,T) vs. independent adaptive quadrature of the pulse response (3 nuclides, 5 cases) | max relative difference ≤ 1e-7 | PASS | 6.54e-12 |
| V4 | Infinite-operation mode equals Constant with T = 10¹⁷ s (saturated limit); the 10¹³ s convention differs by the reported amount | max relative difference ≤ 1e-12 (10¹⁷ s); 10¹³ s convention reported | PASS | 0; 10¹³ s: 4.63 % |
| V5 | Equivalent history modes: Table (0 d and 365 d at 100 %) = Constant (365 d); intermediate time points at 100 %, 265 d and 365 d = Constant 365 d | max relative difference ≤ 1e-12 | PASS | 0 |
| V6 | Zero-magnitude inputs change nothing: a zero-duration power spike (equal times) and a leading 0 % period before the first fission | max relative difference ≤ 1e-12 | PASS | 1.11e-16 |
| V7 | Fission products scale linearly with power level (365 d at 50 % = 0.5 × 365 d at 100 %) | max relative difference ≤ 1e-12 | PASS | 0 |
| V8 | Isotope mixture is the fission-fraction-weighted sum of the single-nuclide results (70/25/5 %) | max relative difference ≤ 1e-12 | PASS | 2.22e-16 |
| V9 | Independent NRC calculation (ANS-94 data, 90 % U-235 / 10 % U-238, 200 MeV, capture, R = 0.7), 25 cooling times 1 s – 10⁴ s | max |Δ| ≤ 1.5 % (data-edition difference 1979 vs 1994 expected) | PASS | 0.94 % |
| V10 | U-239 / Np-239 decay power vs. RK4 integration of the two-nuclide chain (5 cases, finite irradiation) | max relative difference ≤ 1e-6 | PASS | 2.35e-14 |
| V11 | Decay-energy integral (Gauss–Legendre panels) vs. analytic integral of the exponential sum | max relative difference ≤ 1e-9 | PASS | 1.23e-14 |
| V12 | Uncertainty band brackets the nominal curve (lower ≤ total ≤ upper) at six cooling times | always; relative half-width reported (max) | PASS | 6.6 % |
| V13 | Finite, non-negative and monotonically decreasing total decay heat, 10⁻³ s – 10¹⁰ s (default and demonstration table history) | all 401 points × 2 cases | PASS | pass |
| V14 | Capture correction G(t) = 1 + (3.24e-6 + 5.23e-10 t) T^0.4 φ: value at 1000 s, φ = 0 gives 1, held beyond 10⁴ s | G(1000 s) = 1.006 5 ± 1e-4; G(φ=0) = 1; G(5e4 s) = G(1e4 s) | PASS | G(1000 s) = 1.00654 |
| V15 | Time-to-level root finder: total(t*) equals the requested level | |total(t*)/level − 1| ≤ 1e-9 | PASS | 0 |
| V16 | Default-case regression values (total decay heat at 1 s … 10⁷ s) | max relative difference ≤ 1e-9 to stored values | PASS | 0 |
| V17 | Legacy notebook curves reproduced (comparison only): ANS-79 power law at 150 s / 150.5 s; table node at 100 s | 7.66·150^-0.181 = 3.0928 %; 1.39·(150.5/3600)^-0.25 = 3.0740 %; table node 0.029 | PASS | 3.0928 % / 3.0740 % |
| V18 | IC round trip restores every input exactly (incl. time points and history unit); invalid JSON, wrong format, schema 2, single-point table, non-numeric, inconsistent fractions and decreasing times are rejected | round trip exact; 7 of 7 malformed files rejected | PASS | exact; rejected 7/7 |
| V19 | Finite results at t = 0 and over 10⁻³ s – 10¹⁰ s (fit is finite at t = 0, unlike a t^−0.181 power law) | all finite | PASS | total(0) = 6.896 % |
| V20 | Linear ramps between time points (0→100 % from 10 d to 30 d; 100→0 % over the last 5 d to shutdown) vs. independent adaptive quadrature of the pulse response | max relative difference ≤ 1e-6 | PASS | 2.20e-12 |
| V21 | A ramp between equal power levels equals a hold: points at 0, 100, 150 and 200 d, all 100 % = Constant 200 d | max relative difference ≤ 1e-12 | PASS | 2.22e-16 |
| V22 | Power-history CSV: time–power round trip exact in d and in h; v1.1.0 hold/ramp format converts to identical results; ';' delimiter with decimal comma; first time ≠ 0 is shifted to 0; 8 malformed files rejected | exact; max relative difference ≤ 1e-12; 8 of 8 rejected | PASS | ok; 0 |
| V23 | Schema-3 configuration (v1.1.0 hold/ramp table) is migrated explicitly to time points and reproduces the default table history | migratedFrom = 3; vertices equal the default table; max relative difference ≤ 1e-12 | PASS | 0 |
7.1 Published standard values (V1, V2)
Reference [3], Table 8-2, quotes the ANSI/ANS-5.1-1979 U-235 values for infinite irradiation and for 10⁶ s irradiation from zero concentration. The implemented data reproduce all 18 values to within 0.011 %:
Table 7.2 — U-235, ANSI/ANS-5.1-1979: standard values [3] versus this implementation, MeV/fission (fission products only, no correction).
| t [s] | F(t,∞) standard | F(t,∞) here | Δ [%] | F(t,10⁶ s) standard | F(t,10⁶ s) here | Δ [%] |
|---|
| 0 | 13.183 | 13.1826 | -0.0032 | 12.626 | 12.6260 | +0.0001 |
| 1 | 12.318 | 12.3178 | -0.0019 | 11.761 | 11.7612 | +0.0017 |
| 10 | 9.500 | 9.4993 | -0.0074 | 8.943 | 8.9427 | -0.0030 |
| 20 | 8.461 | 8.4612 | +0.0018 | 7.905 | 7.9046 | -0.0052 |
| 40 | 7.465 | 7.4645 | -0.0063 | 6.908 | 6.9080 | -0.0004 |
| 100 | 6.204 | 6.2038 | -0.0033 | 5.647 | 5.6472 | +0.0043 |
| 200 | 5.374 | 5.3744 | +0.0067 | 4.818 | 4.8178 | -0.0037 |
| 400 | 4.673 | 4.6731 | +0.0016 | 4.117 | 4.1166 | -0.0105 |
| 1000 | 3.801 | 3.8013 | +0.0068 | 3.245 | 3.2449 | -0.0046 |
The values agree with the mathematical limit T → ∞ rather than with F(t, 10¹³ s), which is 0.04 %, 0.15 % and 0.30 % lower (0 s, 10³ s, 10⁴ s) and 4.6 % lower at 10⁸ s (V4). A later edition is reported [8] (weak evidence) to define F(t,∞) as F(t,10¹³ s); this application follows the published 1979 values.
7.2 Independent calculation (V9)
Reference [4], Table 2, column 4, is an independent calculation with the 1994 standard data (mean values, 90 % U-235 / 10 % U-238, 200 MeV, capture correction with T = 2×10⁸ s and φ = 1, actinides with R = 0.7). Using the same options with the 1979 data, this application agrees to within 0.94 % at all 25 cooling times between 1 s and 10⁴ s. The deviation is largest at 1 s (early times, where the two data editions differ most); it is not attributed to a specific cause beyond the data revision, which is expected but not confirmed.
Table 7.3 — Independent NRC calculation [4] versus this implementation (decay heat as fraction of P₀).
| t [s] | NRC [4] | This model | Δ [%] | t [s] | NRC [4] | This model | Δ [%] |
|---|
| 1 | 0.06670 | 0.06607 | -0.94 | 150 | 0.03180 | 0.03167 | -0.39 |
| 1.5 | 0.06470 | 0.06428 | -0.65 | 200 | 0.03009 | 0.02996 | -0.42 |
| 2 | 0.06306 | 0.06275 | -0.50 | 400 | 0.02638 | 0.02626 | -0.46 |
| 4 | 0.05845 | 0.05826 | -0.33 | 600 | 0.02431 | 0.02420 | -0.46 |
| 6 | 0.05541 | 0.05527 | -0.25 | 800 | 0.02281 | 0.02271 | -0.44 |
| 8 | 0.05317 | 0.05307 | -0.19 | 1000 | 0.02162 | 0.02154 | -0.39 |
| 10 | 0.05141 | 0.05134 | -0.14 | 1500 | 0.01944 | 0.01937 | -0.35 |
| 15 | 0.04824 | 0.04819 | -0.11 | 2000 | 0.01790 | 0.01784 | -0.32 |
| 20 | 0.04603 | 0.04598 | -0.11 | 4000 | 0.01449 | 0.01444 | -0.32 |
| 40 | 0.04089 | 0.04082 | -0.18 | 6000 | 0.01285 | 0.01280 | -0.37 |
| 60 | 0.03796 | 0.03786 | -0.26 | 8000 | 0.01185 | 0.01180 | -0.38 |
| 80 | 0.03593 | 0.03582 | -0.32 | 10000 | 0.01115 | 0.01111 | -0.36 |
| 100 | 0.03440 | 0.03429 | -0.33 | | | | |
7.3 Regression values
Default-case regression values (total decay heat, % of P₀), stored in the test suite and printed from the same run:
| t [s] | 1 | 10 | 100 | 1e3 | 1e4 | 1e5 | 1e6 | 1e7 |
|---|
| Total [% of P₀] | 6.4614251 | 5.04365817 | 3.38074476 | 2.12570722 | 1.0944152 | 0.584569639 | 0.286566423 | 0.127507479 |
Default case: total decay heat at shutdown (t = 0) 6.90 % of P₀; falls below 5 % after 10.6 s; 2 % after 21.2 min; 1 % after 3.84 h. The reference transient is physically sensible: monotonically decreasing, finite at t = 0, no sign changes (V13, V19).
8. Model Parameters and Constants
Evidence classes: externally sourced derived assumption not verified.
Table 8.1 — Defaults (reference case: light-water reactor, 3000 MWth, long-term operation approximated as infinite, U-235 fissions only, capture and actinide corrections on).
| Parameter | Default | Unit | Class | Basis | Editable |
|---|
| P₀ | 3000 | MWth | assumption | Set by the owner (the source notebook and release 1.0.0 used 3900 MWth). Within the range of large LWRs. Percent results do not depend on P₀. | yes |
| Q | 200 | MeV/fission | externally sourced | Recommended in [4] (bounding, possibly conservative by ≥ 2 %). | yes |
| Fission mix | 100 % U-235 | % | externally sourced | Simplified basis of the standard; recommended in [4] as bounding. | Pu-239, U-238 % |
| History | infinite | – | assumption | Highest decay heat for given power; matches the legacy curves and the standard’s tabulated F(t,∞). | yes |
| φ (capture) | 1.0 | – | externally sourced | Value adopted in [4] (“high plutonium production”); φ ≤ 3 validity limit [3]. | yes |
| R (U-239 per fission) | 0.6 | – | assumption | [4] reports 0.514 (ORIGEN, 17×17 PWR), 0.6 (Regulatory Guide 1.157, Example 1), 0.7 (bounding), 1.0 (RELAP5 default); 0.6 is mid-range. | yes |
| T for G(t) | ≤ 1.2614×10⁸ | s | externally sourced | Validity limit of the capture formula [3]; [4] used 2×10⁸ s. | no |
| Power-history table (inactive by default) | time points (d, %): (0, 100), (200, 100), (200, 0), (230, 0), (232, 100), (382, 100), (385, 70), (388, 70), (389, 100), (390, 100) | – | assumption | Illustrative only: 200 d at full power, a 30 d outage with a 2 d ramped restart, 150 d at full power and a short load-following episode; shutdown at 390 d. | yes |
| History time unit | d | – | assumption | Days or hours (presentation only; times are stored in days). | yes |
| Time window | 1 s – 10⁸ s; linear 0 – 3600 s | s | assumption | Presentation choice within the tabulated range of the standard (0.1 s – 10⁹ s). | yes |
Table 8.2 — Reasonableness check (previous version 0.0.0 → this version).
| Constant | Previous | New default | Reasoning / literature | Assessment |
|---|
| P₀ | 3900 MWth (v1.0.0 and notebook) | 3000 MWth | Owner-defined reference value; no plant is identified in the supplied material. Only MWth values and the energy scale with P₀. | Changed by request; plausible; not verified |
| Decay-heat curve | 8-point table / power law | 23-group summation | Table 2.1: legacy curves deviate by up to +24 % / −52 % from the standard basis. | Replaced |
| Q | implicit (fraction × P₀) | 200 MeV | [4]. Only relevant for finite histories and for actinides. | Explicit now |
| Capture, actinides | not modelled | φ = 1, R = 0.6 | Required by the standard and by RG 1.157 as reported in [4]; adds about 5 % (1 s) to 20 % (10⁵ s) (Table 2.1, “Reference” vs “FP only”). | Added; R is an assumption |
8.1 Group data (ANSI/ANS-5.1-1979)
externally sourced α [MeV/(fission·s)], λ [s⁻¹], σ [%, one sigma] as printed in [3] (Tables 8-1 and 8-14), with text-extraction artefacts corrected as listed in §10. The U-235 set is verified (V1, V2); the Pu-239 and U-238 sets are not independently verified.
Table 8.2 (data) — Group data: U-235 thermal | Pu-239 thermal | U-238 fast. Generated from the numbers in the application code.
| Group | U-235 α | U-235 λ | σ % | Pu-239 α | Pu-239 λ | σ % | U-238 α | U-238 λ | σ % |
|---|
| 1 | 6.5057e-1 | 2.2138e1 | 20.00 | 2.0830e-1 | 1.0020e1 | 30.00 | 1.2311e0 | 3.2881e0 | 12.00 |
| 2 | 5.1264e-1 | 5.1587e-1 | 18.00 | 3.8530e-1 | 6.4330e-1 | 25.00 | 1.1486e0 | 9.3805e-1 | 14.00 |
| 3 | 2.4384e-1 | 1.9594e-1 | 3.90 | 2.2130e-1 | 2.1860e-1 | 7.00 | 7.0701e-1 | 3.7073e-1 | 19.50 |
| 4 | 1.3850e-1 | 1.0314e-1 | 3.10 | 9.4600e-2 | 1.0040e-1 | 4.60 | 2.5209e-1 | 1.1118e-1 | 19.80 |
| 5 | 5.5440e-2 | 3.3656e-2 | 2.60 | 3.5310e-2 | 3.7280e-2 | 4.20 | 7.1870e-2 | 3.6143e-2 | 20.20 |
| 6 | 2.2225e-2 | 1.1681e-2 | 2.25 | 2.2920e-2 | 1.4350e-2 | 3.90 | 2.8291e-2 | 1.3272e-2 | 11.20 |
| 7 | 3.3088e-3 | 3.5870e-3 | 1.95 | 3.9460e-3 | 4.5490e-3 | 3.80 | 6.8382e-3 | 5.0133e-3 | 6.80 |
| 8 | 9.3015e-4 | 1.3930e-3 | 1.85 | 1.3170e-3 | 1.3280e-3 | 4.00 | 1.2322e-3 | 1.3655e-3 | 5.70 |
| 9 | 8.0943e-4 | 6.2630e-4 | 1.75 | 7.0520e-4 | 5.3560e-4 | 4.00 | 6.8409e-4 | 5.5158e-4 | 5.50 |
| 10 | 1.9567e-4 | 1.8906e-4 | 1.70 | 1.4320e-4 | 1.7300e-4 | 4.20 | 1.6975e-4 | 1.7873e-4 | 5.30 |
| 11 | 3.2535e-5 | 5.4988e-5 | 1.65 | 1.7650e-5 | 4.8810e-5 | 4.50 | 2.4182e-5 | 4.9032e-5 | 5.10 |
| 12 | 7.5595e-6 | 2.0958e-5 | 1.65 | 7.3470e-6 | 2.0060e-5 | 4.50 | 6.6356e-6 | 1.7058e-5 | 5.00 |
| 13 | 2.5232e-6 | 1.0010e-5 | 1.80 | 1.7470e-6 | 8.3190e-6 | 4.90 | 1.0075e-6 | 7.0465e-6 | 4.70 |
| 14 | 4.9948e-7 | 2.5438e-6 | 2.00 | 5.4810e-7 | 2.3580e-6 | 5.00 | 4.9894e-7 | 2.3190e-6 | 3.80 |
| 15 | 1.8531e-7 | 6.6361e-7 | 2.00 | 1.6710e-7 | 6.4500e-7 | 5.00 | 1.6352e-7 | 6.4480e-7 | 3.40 |
| 16 | 2.6608e-8 | 1.2290e-7 | 2.00 | 2.1120e-8 | 1.2780e-7 | 5.00 | 2.3355e-8 | 1.2649e-7 | 3.60 |
| 17 | 2.2398e-9 | 2.7213e-8 | 2.00 | 2.9960e-9 | 2.4660e-8 | 5.00 | 2.8094e-9 | 2.5548e-8 | 3.90 |
| 18 | 8.1641e-12 | 4.3714e-9 | 2.00 | 5.7030e-11 | 9.3780e-9 | 5.00 | 3.6236e-11 | 8.4782e-9 | 4.70 |
| 19 | 8.7797e-11 | 7.5780e-10 | 2.00 | 5.7030e-11 | 7.4500e-10 | 5.00 | 5.7030e-11 | 7.5130e-10 | 5.00 |
| 20 | 2.5131e-14 | 2.4786e-10 | 2.00 | 4.1380e-14 | 2.4260e-10 | 5.00 | 4.4963e-14 | 2.4188e-10 | 5.00 |
| 21 | 3.2176e-16 | 2.2384e-13 | 2.00 | 1.0880e-15 | 2.2100e-13 | 5.00 | 3.6654e-16 | 2.2739e-13 | 5.00 |
| 22 | 4.5038e-17 | 2.4600e-14 | 2.00 | 2.4540e-17 | 2.6400e-14 | 5.00 | 5.6293e-17 | 9.0536e-14 | 5.00 |
| 23 | 7.4791e-17 | 1.5699e-14 | 2.00 | 7.5570e-17 | 1.3800e-14 | 5.00 | 7.1602e-17 | 5.6098e-15 | 5.00 |
Table 8.3 — Actinide and capture constants [3]: qU-239 = 0.474 MeV, λU-239 = 4.91×10⁻⁴ s⁻¹; qNp-239 = 0.419 MeV, λNp-239 = 3.41×10⁻⁶ s⁻¹; G(t) constants 3.24×10⁻⁶ and 5.23×10⁻¹⁰ s⁻¹, exponent 0.4.
9. Model Scope and Limitations
Represented Fission-product decay heat of thermal U-235 and Pu-239 fissions and fast U-238 fissions for a piecewise-constant power history; neutron capture in fission products (t ≤ 10⁴ s); U-239/Np-239 decay; group-wise uncertainty band; integrated energy; times to reach stated levels.
Not represented Residual fission power after shutdown; time-varying fission fractions and burnup dependence (one fraction set is applied to the whole history); spatial and axial distribution, partially uncovered cores; energy deposition location (beta vs gamma); actinides other than U-239 and Np-239 (which reach about 3–4 % at 10³–10⁴ s in [4]); capture effects beyond 10⁴ s (G is held); dependence of Q on nuclide and burnup; fission-product decay heat outside 0.1 s – 10⁹ s (extrapolation, flagged); non-LWR spectra; the uncertainty procedure of the standard; ANS-5.1-1994, -2005 or -2014 data. The standard assumes a fixed ratio of total to fission power; at start-up of fresh fuel this is inexact [6].
Sensitivity to user choices The results of the standard depend strongly on the choices of fission mix, operating history, capture, actinides and energy per fission [4][5]; two analyses using “ANS 5.1” can therefore differ significantly. The application exposes all these choices.
10. Data Provenance and Open Points
What was found The 23-group data of the 1979 standard for U-235, Pu-239 and U-238 are printed in the NRC-approved topical report [3], Table 8-1, together with the standard’s own U-235 values that were used for verification (Table 8-2), the actinide constants (Table 8-6), the capture formula (Eq. 8-45) and group uncertainty factors (Table 8-14). An independent NRC calculation [4] provides 25 further comparison values.
What was not found The standard itself (any edition) was not available; the ANS-5.1-1994, -2005 and -2014 coefficient tables were not found in an open, quotable source (an open-source code that supports the 2014 tables does not distribute them). The origin of the notebook’s eight-point table and of the “ANS-79” coefficients could not be established.
Unverified items
- Text-extracted data. [3] was consulted as extracted text. Obvious extraction artefacts were corrected (e.g. “5.1587E-0O” → 5.1587E−01). The U-235 table is confirmed by 18 published values (V1, V2). Pu-239 and U-238 have no such check: only positivity, monotone λ ordering, monotone decay curves and plausible ratios to U-235 (Pu-239/U-235 = 0.83 at 1 s, 0.93 at 10³ s; U-238/U-235 = 1.15 at 1 s) were checked, and a 90/10 U-235/U-238 mix agrees with [4] (V9). Pu-239 groups 18 and 19 both print α = 5.703×10⁻¹¹ and the printed text contains an unexplained “(7)”; they affect only t ≳ 10⁷ s.
- U-235 group 22. λ = 2.4600×10⁻¹⁴ s⁻¹ is used as printed; an alternative reading 2.2460×10⁻¹⁴ s⁻¹ would change results by at most 0.3 % near 10⁹ s.
- Capture formula. The printed formula lacks the variable after 5.23×10⁻¹⁰; it is read as t (cooling time). Agreement with [4] (V9) supports, but does not prove, this reading: without the term the deviation would grow to -1.3 % at 10⁴ s.
- Edition. The data are from 1979, whereas the source notebook names the 2005 edition. The NRC comparison shows ≤ 1 % difference to 1994 data for the U-235/U-238 case up to 10⁴ s; later times and Pu-239 have not been compared.
- R and P₀ are assumptions (Table 8.1).
11. References
External standards and reports (evidence grade stated for each)
- [1] American Nuclear Society, ANSI/ANS-5.1-1979, Decay Heat Power in Light Water Reactors, 1979. Further reading — not consulted; its data were taken from [3].
- [2] American Nuclear Society, ANSI/ANS-5.1-2005 (also editions 1994 and 2014), Decay Heat Power in Light Water Reactors. Further reading — not consulted. The source notebook names the 2005 edition.
- [3] Westinghouse, WCAP-16009-NP-A, Revision 0, January 2005, Section 8 “WCOBRA/TRAC Reactor Kinetics and Decay Heat Models”, Tables 8-1, 8-2, 8-6, 8-14, Equations 8-3 and 8-45 (NRC ADAMS ML050910161). Consulted in full for Section 8, as extracted text.
- [4] U.S. NRC, “Attachment 1, Appendix K Decay Heat Standards” (decay-heat changes to 10 CFR 50.46 and Appendix K), including Table 2 (ADAMS ML021720702). Consulted in full.
- [5] U.S. NRC information notice “Estimates of Decay Heat Using ANS 5.1 Decay Heat Standard May Vary Significantly”, 5 July 1996 (ADAMS ML031060021). Consulted at excerpt level.
- [6] Argonne National Laboratory, SAS4A/SASSYS-1 documentation, ANL/NSE-SAS/5.7.1, §4.4 “Decay Heat” (sas-doc.nse.anl.gov). Consulted in full.
- [7] OECD/NEA, JEF Report 17, Chapter 9 “Decay heat validation studies” (oecd-nea.org). Consulted at excerpt level: describes the 23-exponential representation of the ANS standard and the ±4 % difference between the ANS-5.1 1993 and Tobias standards.
- [8] US Patent 8,238,509, “Neutron monitoring systems including gamma thermometers and methods of calibrating nuclear instruments using gamma thermometers”. Weak evidence (patent text), used only for the statements that ANSI/ANS-5.1-2005 Tables 9–12 give αi, λi for four nuclides in 23 groups and define F(t,∞) = F(t,10¹³ s). Consulted in full.
Antranova model basis
- [A1] Antranova Engineering AB, Antranova_DecayHeat_ANS79_ANS5.1-2005_v1.0.0, Jupyter notebook, version 1.0.0.
- [A2] Antranova Engineering AB, AntraSIM-0090 Decay Heat Simulator, previous release (file version 0.0.0).