Theory & Method
BWR Power Flow Simulator · AntraSIM-0160 · 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 a closed-form void integral, an independent algebraic equilibrium solver, a matrix-exponential solution of the full six-group kinetics, mass-balance closure and step-size convergence. Not validated against plant data; the feedback coefficients, the slip ratio, the fuel and void time constants and the pressure-regulator constants are illustrative inputs classified as assumptions (Sections 8 and 10). Operator-level principle simulator for training; not safety-grade.
1. Physical Background
Intuition In a boiling water reactor (BWR) the coolant boils in the core, and steam bubbles (voids) in the water reduce the moderation. The reactor power can therefore be changed without moving a control rod: more recirculation flow sweeps the steam out of the core, the void fraction falls, reactivity rises and the power follows until the Doppler and void feedback stop it at a new level. Less flow does the opposite. Control rods are used to select the power-flow line and to shape the power distribution; flow is used for the daily power changes over a range of about 60–100 % of rated power.
Physical mechanism
core flow ↑ → boiling boundary and quality fall → void ↓ → reactivity ↑ → power ↑ → void ↑ and fuel hotter (Doppler) → new equilibrium
steam-dome pressure ↑ → steam denser, more subcooling → void ↓ → power ↑ (a pressure regulator failure or a bypass-valve closure gives a power increase)
The heat from the fuel makes steam whose flow, times the enthalpy that must be added to the feedwater to reach saturated steam, equals the thermal power. The steam leaves through the turbine or, if the turbine takes less, through the bypass valves; the pressure regulator keeps the dome pressure constant by adjusting the turbine flow. When the steam demand changes faster than the regulator can follow, the pressure moves, and the pressure changes the void. The response to a change is not instantaneous: the void follows the heat flux with a delay of the order of a second, the heat flux follows the fission power with the fuel time constant of several seconds, and the delayed neutrons hold back the power for tens of seconds.
Engineering interpretation The power-flow map shows the equilibrium power for each flow along a rod line; the operator moves along it with flow and between lines with rods. A rapid flow increase can overshoot the final power because the void collapses faster than the Doppler feedback builds up (Exercise 4). Assumptions and limitations A lumped core with uniform heating and a single steam volume; no stability (density-wave) oscillations, xenon, recirculation loop hydraulics or scram (Section 9).
2. Assessment of the Source Material and Learning Objectives
The objectives are taken from NUREG-1021, BWR catalog, section 292008 (power and flow effects, reactor operations reactivity manoeuvres) [1], consulted at excerpt level. No earlier simulator was the source.
Objective (paraphrased) Support in AntraSIM-0160
Effect of recirculation flow on void, reactivity and power Flow manoeuvres, power-flow map, reactivity components, Exercises 1, 2, 4. Rod movement and power at constant flow; power and void coefficients Rod manoeuvre, feedback plots, Exercise 3. Steam-pressure changes and their effect on void and power; bypass valves Pressure model, bypass and set-point manoeuvres, Exercises 5 and 6. Heat balance: power, steam flow, subcooling and quality Steam balance, void-quality plot, Section 3. Speed of the power response after a manoeuvre Ramp times, delayed neutrons, Exercise 4. Xenon effects on manoeuvring, stability, thermal limits, recirculation pump trips, scram Not modelled.
Table 2.1 — Coverage. Finding A-1: the feedback coefficients, the slip ratio and the regulator constants are illustrative.
3. Governing Equations
Intuition Neutron power responds to reactivity through the delayed neutrons; reactivity is the sum of the rods, the void and the fuel temperature; the void follows from the heat balance of the coolant.
Mathematical formulation Kinetics with the prompt-jump approximation (six delayed groups i with decay constants λi [s⁻¹] and fractions βi , β = Σβi = βeff ; n [–] is the fission power relative to rated, ci [–] the scaled precursors, ρ [–] the reactivity):
n = ∑ i β i c i β − ρ , d c i d t = λ i ( n − c i ) , ρ = ρ r o d ( t ) + α v ( α ¯ − α ¯ 0 ) + α D Δ T f ( q − q 0 )
with ρrod [–] the reactivity from rod movement, αv [pcm per % void] the void coefficient, ᾱ [%] the core-average void, αD [pcm/K] the Doppler coefficient, ΔTf [K] the fuel-temperature rise at rated power and q [–] the heat flux to the coolant relative to rated. The fuel heat flux and the void follow first-order lags:
τ f d q d t = n − q , τ v d α ¯ d t = α ¯ e q ( q , W , P ) − α ¯
with τf and τv [s] the fuel and void time constants, W [–] the core flow and P [kPa] the steam-dome pressure. Heat and steam balance (Q = q·Qrated , Wc the core flow in kg/s, hf , hfg , hg [kJ/kg] the saturation enthalpies, hfw the feedwater enthalpy):
m ˙ s = Q h g − h f w , Δ h = Q W c , h s u b = m ˙ s W c ( h f − h f w ) , x e = Δ h − h s u b h f g , z b L = h s u b Δ h
where ṁs is the steam flow, Δh the enthalpy added in the core, hsub the inlet subcooling (feedwater mixing in the downcomer), xe the exit quality and zb /L the height of the boiling boundary with uniform heating. The void fraction with slip S [–] and ρg /ρf the steam-to-water density ratio, a = S·ρg /ρf , and its average over the boiling length:
α ( x ) = x a + ( 1 − a ) x , α ¯ e q = ( 1 − z b L ) 1 x e ∫ 0 x e α d x = ( 1 − z b L ) 1 x e [ x e 1 − a − a ( 1 − a ) 2 ln a + ( 1 − a ) x e a ]
The pressure obeys a steam mass balance with a proportional-integral regulator (Cs [kg/kPa] steam-volume capacity, G and Ki the gains, Pset the set-point, ṁbp the bypass flow):
C s d P d t = m ˙ s − m ˙ t − m ˙ b p , m ˙ t = m ˙ s , 0 + G ( P − P s e t ) + K i ∫ ( P − P s e t ) d t
The power-flow map is obtained by solving ρ = 0 for q at fixed flow and rod position, using αeq with the initial pressure: ρrod + αv (ᾱeq (q,W) − ᾱ0 ) + αD ΔTf (q − q0 ) = 0.
Engineering interpretation The equilibrium power for a given flow is where the negative power feedback balances the positive void reactivity from the extra flow. Because the model has the same feedback for a flow increase and a rod withdrawal, both give a stable new power; the map is monotone. Assumptions and limitations Uniform axial heating; quality-based void with a constant slip ratio; constant inlet subcooling model without transport delay; steam pressure equal in core and dome; no carry-under.
4. Inputs and Operating Conditions
Initial power and flow (a critical steady state is constructed: the rod reactivity is taken as the zero of the scale), a schedule of up to 12 manoeuvres (rod reactivity change in pcm, core-flow change in % of rated, bypass steam demand in % of rated steam flow, pressure set-point change in kPa, each ramped in linearly over the ramp time), the plant and thermal-hydraulic data, the kinetics and feedback constants and the pressure-regulator constants. Reactivity above about 275 pcm (half of βeff = 550 pcm) is outside the prompt-jump approximation and is flagged. Times, ramps and steps in the schedule are all in seconds.
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
Power (neutron and heat), core flow, reactivity and its components, void and exit quality, pressure and the steam balance as functions of time; the power-flow map; the void and exit quality versus power. At rated conditions the model gives a steam flow of 1782.9 kg/s, a core enthalpy rise of 264 kJ/kg, an inlet subcooling of 49.3 kJ/kg, a boiling boundary at 18.7 % of the height, an exit quality of 14.3 %, a core-average void of 38.6 % and an exit void of 70.6 %; the saturation data at 7000 kPa are hf = 1266.49, hfg = 1505.4, hg = 2771.9 kJ/kg, ρf = 740.1 and ρg = 36.57 kg/m³. The Doppler defect at rated power is -1400 pcm. Along the reference rod line the equilibrium power is 78.94 % at 70 % flow, 52.47 % at 40 % flow and 106.2 % at 110 % flow.
Table 5.1 — Equilibrium power-flow map (rated initial state; generated from the code).
Core flow [%] Power, rods −100 pcm [%] Power, reference rods [%] Power, rods +100 pcm [%] Core-average void [%] 40 50.84 52.47 54.14 43.72 55 64.61 66.56 68.55 42.21 70 76.74 78.94 81.18 40.87 85 87.59 90 92.44 39.68 100 97.42 100 102.6 38.6 110 103.5 106.2 108.9 37.94
6. Numerical Method
The 10 state equations (six precursors, heat flux, void, pressure and the integral of the pressure error) are integrated with the classical fourth-order Runge–Kutta method with a fixed step of 0.02 s; the neutron power follows algebraically from the prompt-jump relation at each stage. The manoeuvre schedule is evaluated as continuous ramps. The equilibrium solver is a bisection on the reactivity balance. The default case takes 63 ms.
7. Verification and Numerical Accuracy
Verification shows that the equations are solved correctly; it is not validation. Findings during development: (1) the first version of the pressure regulator took the rated steam flow as the turbine flow at the start, which produced a pressure transient at t = 0 whenever the initial power was not 100 % (Exercise 2 showed a pressure loss of about 30 kPa before the manoeuvre); the turbine flow now starts at the initial steam production and the steady-state test T1 was extended to a 70 % power / 65 % flow state; (2) the first step-convergence test compared output arrays recorded at different times for the coarse step (13 steps of 0.04 s = 0.52 s instead of 0.5 s) and reported a 2.5 % “error”; the coarse step was changed to 0.05 s, which divides the output interval, and the ratio of errors is now 42 for a 2.5 times larger step; (3) the initial slip ratio of 2.0 gave a core-average void of only 32 %; it was reduced to 1.4 (39 %); (4) the rod manoeuvre of Exercise 3 was first +40 pcm, which changes the power by only 1 % because of the strong power feedback (about 38.2 pcm per % power), so +100 pcm is used. The suite runs in the application and in Node.
Table 7.1 — Release verification suite (13 of 13 passed; generated from an actual run).
ID Test Criterion Result Measured T1 Initial condition is a steady state: power, void and pressure stay unchanged for 120 s with no manoeuvre, at rated conditions and at 70 % power / 65 % flow deviation ≤ 1e-9 (% or relative) PASS 0 T2 Heat and steam balance at rated conditions: steam flow = Q/(h_g − h_fw), core enthalpy rise = Q/W, inlet subcooling from the feedwater mixing, exit quality from the energy balance relative difference ≤ 1e-12 PASS steam 1782.9 kg/s; exit quality 14.26 %; core-average void 38.60 % T3 Void-quality relation with slip: closed-form ∫α dx vs. Simpson quadrature (four exit qualities); homogeneous limit (S = 1); α(0) = 0 and α(1) = 1 ≤ 1e-9; ≤ 1e-12 PASS 2.94e-13 ; 0 T4 Equilibrium (rod-line) solver: reproduces the initial power at the initial flow and rods, zero net reactivity at the solution, and power increases with flow and with rod reactivity |q − 1| ≤ 1e-9; residual ≤ 1e-6 pcm; monotone PASS q(100 %, 0) = 1.000000000000; residual 4.55e-13 pcm; power at 55 % flow 66.56 % T5 Transient vs. algebraic equilibrium: after a flow decrease, a rod withdrawal and a pressure set-point increase (900 s) the simulated power equals the equilibrium power of the equilibrium solver relative difference ≤ 1e-4 PASS 1.84e-8 T6 Prompt-jump kinetics vs. the full six-group point-kinetics equations solved by matrix exponential, +10 pcm step, feedback off, at 5, 15 and 30 s relative difference ≤ 1e-3 PASS 8.48e-5 T7 Steam mass balance: ∫(ṁ_prod − ṁ_turbine − ṁ_bypass) dt equals C_s·ΔP; the regulator returns the pressure to its set-point after a bypass opening closure ≤ 1e-3 of the largest flow-time; |P_end − P_set| ≤ 0.5 kPa PASS closure 5.09e-8 kg; min P 6990.49 kPa; final 7000.000 kPa T8 Signs of the feedbacks: more flow → less void → positive void reactivity and higher power; higher power → negative Doppler reactivity; the Doppler defect at rated power equals α_D·ΔT_f signs as stated; defect −1400 pcm PASS Doppler defect -1400 pcm; peak 125.45 % T9 Step-size convergence of the transient: production step 0.02 s vs 0.01 s (and 0.05 s vs 0.01 s for the error ratio) difference ≤ 1e-6 at 0.02 s; ratio ≥ 8 PASS 4.84e-11 ; ratio 42.0 T10 IC round trip restores every input including the manoeuvre schedule; 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 T11 All laboratory exercise set-ups are valid and produce finite results 6 of 6 PASS slowest 106 ms T12 Default-case regression values (final and minimum power, equilibrium power, core-average void) max relative difference ≤ 1e-9 to the stored values PASS 0 (final 81.177 %) T13 Invalid inputs are refused: slip below 1, negative ramp time, exit quality above 60 %, pressure outside the table range 4 of 4 rejected PASS 4 of 4
8. Model Parameters and Constants
Evidence classes: externally sourced derived assumption not verified . There is no earlier release. The steam-pressure coefficient of reactivity that follows from the model is 0.576 pcm per kPa (the void changes by -0.00443 % per kPa); it is a result, not an input.
Parameter Default Unit Class Basis and assessment Editable
hf , hfg , ρf , ρg (P) rows at 280 °C (6412 kPa) and 290 °C (7436 kPa) kJ/kg, kg/m³ externally sourced Saturated-water rows of the Keenan–Keyes steam tables (hf = 1236.0 / 1289.1, hfg = 1543.6 / 1477.1 kJ/kg, vf = 1.3321 / 1.3656 L/kg, vg = 0.03017 / 0.02557 m³/kg) as found in a secondary web copy; linear interpolation in pressure between the rows (valid 6412–7436 kPa). no hfw 921 kJ/kg not verified Enthalpy of feedwater at about 215 °C, recalled from steam-table values (not looked up for this temperature). yes Qrated , Wrated 3300 MWth assumption Generic 3300 MWth plant with a core flow of 12 500 kg/s (order of a large BWR); an assumption, not a specific plant. yes S 1.4 – assumption Slip ratio of the void-quality relation, chosen to give a core-average void of about 40 % at rated conditions. yes βeff , Λ 0.0055 – assumption Effective delayed-neutron fraction and prompt neutron generation time (4·10⁻⁵ s) typical of a BWR core; group fractions and decay constants of the Keepin-type U-235 set (secondary source) scaled to βeff . yes αv -130 pcm/% assumption Void coefficient (per percent void); of the order of the value found in AntraSIM-0140 (−164 pcm/% at zero void) and adjusted so that the power–flow slope is realistic; constant in this model. yes αD , ΔTf -2.8 pcm/K, K derived Doppler coefficient and fuel-temperature rise at rated power taken from the default case of AntraSIM-0140 (−2.88 pcm/K at 559 K, 500 K rise); the two simulators are consistent, both are illustrative. yes τf , τv 6 s assumption Fuel-rod thermal time constant (several seconds) and void response time (order of the coolant transit time); assumptions. yes Cs , G, Ki 6 kg/kPa, kg/s per kPa assumption Effective steam-volume capacity and PI pressure-regulator constants; chosen for a well-damped regulator; assumptions. yes
9. Model Scope and Limitations
Represented Prompt-jump six-group kinetics; constant void, Doppler and rod reactivity; heat balance with inlet subcooling from feedwater mixing; void-quality relation with slip and uniform heating; a fuel lag and a void lag; a steam volume with a PI pressure regulator and a bypass valve; the power-flow map. Not represented Density-wave and core-wide stability oscillations (the model has one void node and cannot show them), xenon, axial and radial power shape changes and thermal limits, void-coefficient dependence on void and burnup, recirculation-pump and jet-pump hydraulics (flow is an input), carry-under and separator dynamics, feedwater and level control, steam-line pressure drop and transport delay, reactor scram and the pressure limits of the vessel. Manoeuvres with very fast, large pressure changes (a turbine trip without bypass) cause reactivity beyond the prompt-jump limit and must not be simulated. The results are qualitative in shape and of the correct order of size only.
10. Data Provenance and Open Points
What was found Saturated-water rows at 280 and 290 °C from the Keenan–Keyes steam tables in a secondary web copy. What was not found The feedwater enthalpy at the chosen temperature (recalled), plant data for a specific BWR, a measured power-flow map, and recirculation-system data. Unverified items Feedwater enthalpy, slip ratio, βeff and Λ, void coefficient, fuel and void time constants, regulator constants, steam-volume capacity.
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, section 292008). Consulted at excerpt level (project copy).[2] J. H. Keenan, F. G. Keyes, P. G. Hill and J. G. Moore, Steam Tables , Wiley, 1969. Consulted at excerpt level through a secondary web reproduction of the saturated-water temperature table.[3] J. R. Lamarsh and A. J. Baratta, Introduction to Nuclear Engineering , 3rd ed., Prentice Hall, 2001. Further reading — not consulted (point kinetics, feedback).[4] J. G. Collier and J. R. Thome, Convective Boiling and Condensation , 3rd ed., Oxford University Press, 1994. Further reading — not consulted (void-quality relation with slip).[5] U.S. DOE, DOE Fundamentals Handbook — Nuclear Physics and Reactor Theory , DOE-HDBK-1019/2-93. Further reading — not consulted .
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 — Power-flow map and the rod line
Objective: see how power follows flow at fixed rods.
Learning outcomes. Read the power-flow map; explain why power falls when flow falls; identify the reactivity balance at the new equilibrium.
Load set-up flow from 100 % to 70 % over 60 s, rods fixed.
Follow the red path on the map. Read the final power and compare with the 70 % flow point of the reference line (Table 5.1). Read the void and the reactivity components at the end. Expected results. Power falls from 100 % to 78.942 % (equilibrium 78.938 %, Table 5.1: 78.94 %); the path passes a minimum of 77.367 % before it settles, reaching 90 % of the change after 60.5 s. The core-average void rises from 38.6 % to 40.87 %, giving -295 pcm of void reactivity, balanced by 294.8 pcm of Doppler reactivity (the fuel is colder). Largest reactivity during the manoeuvre: 28.3 pcm.
Discussion. Why does the power settle where the two feedbacks cancel?
Exercise 2 — Flow increase
Objective: raise power with flow from a low-power state.
Learning outcomes. Explain the void collapse; describe the power rise and the small overshoot; use the map to predict the final power.
Load set-up 70 % power at 65 % flow; flow +35 % over 60 s.
Read the power at the end and the maximum power; predict the final power from the map. Read the void before and after. Expected results. Power rises from 70 % to 93.923 % (equilibrium 93.927 %); maximum 95.388 % at 70 s (end of the ramp, then a slow decay). The void falls from 40 % to 37.42 %. The largest reactivity is 23.8 pcm, far below the prompt-jump limit of 275 pcm.
Discussion. Why is a slow flow ramp preferred for power increases?
Exercise 3 — Rod withdrawal at constant flow
Objective: quantify the power response to rod reactivity.
Learning outcomes. Explain how power feedback limits the power rise; estimate the power coefficient.
Load set-up +100 pcm over 20 s at rated conditions.
Read the final power, the peak and the void and Doppler reactivities at the end. Estimate the power coefficient from the rod worth and the power change. Expected results. The power settles at 102.62 % (peak 103.35 % at 30 s; 101.4 % at 15 s and 102.6 % at 40 s). At the end the void has fallen to 39.09 % (-63.34 pcm) and the Doppler reactivity is -36.65 pcm; together they cancel the 100 pcm of the rods. The power coefficient is about 38.2 pcm per % power.
Discussion. Why do BWR operators use flow rather than rods for large power changes?
Exercise 4 — Speed of the response: step versus ramp
Objective: show that a fast manoeuvre overshoots.
Learning outcomes. Compare the power response of a fast flow increase with that of a slow one; explain the overshoot with the time constants.
Load set-up 70 % power at 65 % flow; flow +20 % over 5 s.
Read the peak power and its time, and the final power. Compare with Exercise 2. Expected results. The power peaks at 97.935 % at 15 s and settles at 84.34 %; the largest reactivity is 105 pcm. The overshoot is 97.935 − 84.34 percentage points, because the void collapses at once while the Doppler feedback builds with the fuel time constant. (The slow +35 % ramp of Exercise 2 overshoots only to 95.388 % from a final 93.923 %.)
Discussion. Which time constants decide the size of the overshoot?
Exercise 5 — Bypass valve opening (pressure disturbance)
Objective: follow a steam-demand disturbance through pressure, void and power.
Learning outcomes. Explain the sequence: more steam demand → pressure falls → void rises → power falls; describe the recovery by the regulator.
Load set-up bypass steam demand +10 % of rated flow over 2 s at 10 s.
Read the lowest pressure and the lowest power and their times; read the steam balance at the end. Expected results. Pressure falls by 9.514 kPa to 6990.49 kPa; the void rises from 38.604 % to at most 38.628 % and the power dips to 99.544 % at 12.5 s. The regulator closes the turbine valves to 90 % of the steam flow, so that the pressure returns to 7000 kPa and the power to 100 %.
Discussion. Why is the power effect of a small bypass opening so small compared with a large pressure change?
Exercise 6 — Pressure set-point increase
Objective: quantify the pressure coefficient of reactivity.
Learning outcomes. Explain why a higher pressure gives a power increase; estimate the pressure coefficient.
Load set-up set-point +50 kPa over 5 s at 10 s.
Read the highest pressure, the maximum power and the void minimum. Expected results. Pressure rises to 7052.74 kPa; the void falls from 38.604 % to 38.5 % and the power peaks at 101.56 % at 15.5 s, ending at 100.75 % (the new equilibrium with a slightly lower void). The model’s pressure coefficient is 0.576 pcm per kPa.
Discussion. What is the risk of a rapid pressure increase in a BWR, and which systems limit it?