The Advanced Fuel Cycle Initiative Status of Neutronics Modeling Won Sik Yang Argonne National...

The Advanced Fuel Cycle Initiative

Status of Neutronics Modeling

Won Sik Yang

Argonne National Laboratory

NEAMS Reactor Simulation Workshop

May 19, 2009

May 19, 2009 NEAMS Reactor Simulation Workshop 2

Within the current knowledge of physics, theory and governing equations are well known

– Boltzmann equation for neutron transport– Bateman equation for fuel composition evolution

The coefficients of these equations are determined by nuclear data, geometry, and composition

– Nuclear data are for the most part relatively well known for the most commonly used nuclides

• But still improved data are required to reduce design uncertainties– Geometry and composition have stochastic uncertainties and are affected

by thermal, mechanical, irradiation, and chemical phenomena• These coupled phenomena are not as well described, and they can

dominate the analysis errors The challenge in neutronics analysis is to determine the solution

efficiently by taking into account geometric complexity and complicated energy dependence of nuclear data

Status of Neutronics Analyses


Monte Carlo simulation with MCNP5 (INL)

– Reaction rate tally uncertainties < 1% C/E values for U-235 fission rate distribution in CIRANO-2A (Blanket) and

CIRANO-2B (Reflector) experiments

Reaction Rate Traverse Example

CIRANO-2A U-235 Fission

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

0 15 30 45 60Radial Distance (cm)

No

rmal

ized

Rea

ctio

n R

ate

0.94

0.95

0.96

0.97

0.98

0.99

1.00

1.01

1.02

C/E

Experiment

MCNP5 (ENDV/B-VII)

C/E (MCNP5)

CIRANO-2B U235 Fission

0.6

0.7

0.8

0.9

1.0

1.1

1.2

0 15 30 45 60

Radial Distance (cm)

No

rmal

ized

Rea

ctio

n R

ate

0.90

0.95

1.00

1.05

1.10

1.15

1.20

C/E

Experiment

MCNP5 (ENDV/B-VII)

C/E (MCNP5)


1989OctobreJanvier Février Mars Avril Mai Juin Juillet août Septembre Novembre Décembre

600

400

200

MW

4746

1989OctobreJanvier Février Mars Avril Mai Juin Juillet août Septembre Novembre Décembre

600

400

200

MW

4746

OctobreJanvier Février Mars Avril Mai Juin Juillet août Septembre Novembre Décembre

600

400

200

MW

4746

400

600

200

MW

1990 OctobreJanvier Février Mars Avril Mai Juin Juillet août Septembre Novembre Décembre

4948

400

600

200

MW


4948600

200

MW600

200

MW


4948


4948

OctobreJanvier Février Mars Avril Mai Juin Juillet août Septembre Novembre Décembre

4948

Negative Reactivity Transients of PHENIX

Four unexpected scrams occurred in 1989 - 1990 due to short negative reactivity transients (200 ms) with the same signal shape

Several potential explanations were given, but not satisfactory

Experiments are planned for PHENIX end-of-life tests for further investigation


Current and Target Uncertainties for sodium cooled fast reactors

Generation IV Target Uncertainties

Parameter

Current Uncertainty (SFR) Targeted

UncertaintyInput data origin

Modeling origin

Multiplication factor, Keff (k/k) 1% 0.5% 0.3%

Power peak 1% 3% 2%

Power distribution 1% 6% 3%

Conversion ratio (absolute value in %) 5% 2% 2%

Reactivity coefficients (component) 20% 20% 10%

Control rod worth (total) 5% 4% 2%

Burnup reactivity swing (k/k) 0.7% 0.5% 0.3%


The final objective is to produce an integrated, advanced neutronics code that allows the high fidelity description of a nuclear reactor and simplifies the multi-step design process

– Integration with thermal-hydraulics and structural mechanics analyses to account for reactivity feedbacks due to geometry deformation accurately

Required modeling capabilities

– Reactivity and power distribution (coupled neutron and gamma heating)

– Non-equilibrium and equilibrium fuel cycle analyses• Refueling, fuel shuffling, and ex-core models

– Perturbation and sensitivity analyses• Uncertainty analysis and optimization

– Transient analysis (coupled with T/H and T/M analyses)• Reactivity coefficients and kinetics parameters

– Shielding, decay heat, coolant activation and dose rate calculations, etc.

Objectives and Requirements


Utilize modern computing power and computational techniques

– Meshing, domain decomposition strategies, parallel linear solvers, new visualization techniques, etc

Allow uninterrupted applicability to core design work

– Phased approach for multi-group cross section generation• Simplified multi-step schemes• Online cross section generation

– Adaptive flux solution options from homogenized assembly geometries to fully explicit heterogeneous geometries in serial and parallel environments

• Allow the user to smoothly transition from the existing homogenization approaches to the explicit geometry approach

• Rapid turn-around time for scoping design calculations• Detailed models for design refinement and benchmarking calculations

Selected Approaches


Adaptive Flux Solution Options

Unified geometrical framework

– Unstructured finite element analysis for coupling with structural mechanics and thermal-hydraulics codes

Homogenized assembly

Homogenized assembly internals

Homogenized pin cells

Fully explicit assembly


PN2ND– Second-order, even-parity transport equation (CG solve)– 1-D, 2-D, 3-D Cartesian with general reflected and vacuum b.c.s– Spherical harmonics combined with Serendipity and Lagrangian FE

SN2ND– Second-order even-parity transport equation (CG solve)– 2-D & 3-D Cartesian with general reflected and vacuum b.c.s– Discrete ordinates combined with Serendipity and Lagrangian FE

MOCFE– First-order transport equation (long characteristics)– 3-D Cartesian with general reflected and vacuum b.c.s– Discrete ordinates combined with Serendipity and Lagrangian FE

NODAL: hybrid finite element method for structured geometries– Will replace nodal diffusion and VARIANT options in DIF3D– Use as an multi-grid preconditioner for other solvers

Flux Solvers Available in UNIC


Takeda Benchmark 4

Control Rod In Control Rod Half Control Rod Out

Reference 0.88001 ± 0.00038 0.98340 ± 0.00039 1.09515 ± 0.00040

PN2ND 0.87960 0.98365 1.09599

SN2ND 0.87877 0.98275 1.09494

MOCFE 0.87796 0.98164 1.09353


ABTR Whole-Core Calculations

AngularDirections

Spatial Mesh Approximation

78243 113873 461219 671219 785801

32 -241 -233 -69 -64 -59

50 -220 -210 -47 -40 -37

72 -225 -217 -51

98 -216 -207 -43

288 -216


ZPPR-15 Critical Experiments

Computational Mesh and Example Flux Solutions of ZPPR-15 Critical Experiment

Flux expansion order Scattering order EigenvalueP1 P1 0.99258P3 P3 0.99640P5 P3 0.99651

Monte Carlo (VIM) 0.99616±0.00010


2D OECD/NEA C5G7 Benchmark

Thermal Group Flux in CoreThermal Group Flux in Pin Cell

Reference 1.18655 ± 0.00010

MOCFE 1.18649


Parallel Implementation

The scalability to peta-scale computing resources has been demonstrated

– 163,840 cores of BlueGene/P (Argonne) – 131,072 cores of XT5 (ORNL) – Over 75% weak scalability

Cores4π

Angleskeff

Fission Iters. / Time

TotalTime(sec)

SourceUpdate(sec)

WeakScaling

32,768 32 0.96006 23 / 152 3493 2934 100%

49,152 48 0.96004 23 / 152 3510 2933 100%

65,536 64 0.96007 23 / 153 3526 2934 99%

73,728 72 0.96015 23 / 156 3593 2934 97%

131,072 128 0.96019 27 / 156 4209 3437 83%*

163,840 160 0.96019 27 / 173 4676 3436 75%*

Weak Scaling Study by Angle on BlueGene/P (PHENIX EOL test)


Parallel Implementation

Weak Scaling Study by Angle on XT5 (PHENIX EOL test)

Cores4π

Angleskeff

FissionIters. / Time

TotalTime(sec)

SourceUpdate(sec)

EffectiveWeak

Scaling

32,768 32 0.96017 25 / 63 1574 851 100%

49,152 48 0.96014 22 / 64 1399 746 99%

65,536 64 0.96017 22 / 64 1402 745 99%

98,304 96 0.96017 25 / 65 1623 847 97%

114,688 112 0.96017 26 / 65 1687 882 97%

131,072 128 0.96029 28 / 68 1902 948 93%


PHENIX End-of-Life Experiments

Participating in the PHENIX end-of-life experiments Whole-core geometry is required (no symmetry) using homogenized

fuel and explicit control rods Space/angle convergence study completed using over 4 billion DOF on

up to 163,840 cores of Blue Gene/P Energy discretization study is ongoing

0.4 MeVMax/Min=1.78

900 eVMax/Min=16.9

2 eVMax/Min=84.2 600 eV Flux and Radial Mesh


ZPR-6 Critical Experiments

Two ZPR-6 critical experiments are targeted for V&V in 2009 (Assemblies 6A and 7)

Explicit fuel plate representation allows direct comparison to legacy homogenization methods

Spatial mesh requirements are large; U-235 plates are 1/16th in thick

Preliminary studies performed on BG/P and Jaguar up to 130,000 processors indicate that over 10 billion DOF will be required to resolve the space-angle-energy mesh


ZPR-6 Critical Experiments

14 MeV Flux / Mesh

U-235 Plate Power


A modular version has been integrated into UNIC for on-line generation of multi-group cross sections of each spatial region with given material and temperature distribution

– Standalone code to generate ISOTXS datasets for legacy tools Ultrafine group (2082 groups) transport calculations

– Homogeneous mixture, and 1-D slab and cylindrical geometries

– Resolved resonance self-shielding with numerical integration of point-wise cross sections using the narrow resonance (NR) approximation

– Unresolved resonance self-shielding with the generalized resonance integral method

– Elastic scattering transfer matrices obtained with numerical integration of isotopic scattering kernel in ENDF/B data

Advanced Multi-group Cross Section Generation Code MC2-3

*

*1 1

( )

0

( ) ( ) ( )1( ) (2 1) ( ) ( )

(1 )

g g

g g

i u u i Nu ui i il s l ssl n n cu u

nlg i

u u e Pg g du du n f u P


1-D hyperfine group (~100,000) transport capability

– Consistent P1 transport calculation for entire resolved resonance energy range (< ~1 MeV) with anisotropic scattering sources

– Optionally used for accurate resolved resonance self-shielding and scattering transfer matrix generation

Efficient strategy to generate accurate multi-group cross sections for heterogeneous assembly or full-core calculations is being developed by combining various solution options

– 1-D hyperfine group cell calculation

– 1-D ultrafine group whole-core calculation (with homogenized regions)

– 2-D MOCFE calculation in several hundred groups

Advanced Multi-group Cross Section Generation Code MC2-3


MC2-3.0 and Coupling with UNIC

Smooth cross sections, unresolved and resolved

resonances, inelastic and (n,2n) distribution, fission spectrum,

Legendre data.

Pointwise cross sections(capture, fission, scattering, total)

Binary files, f(T)

MC2 Libraries

Self-shield unresolved resonance

Self-shield resolved resonance

Evaluate scattering matrices

UFG consistent P1 calculation

Group collapsing

Optionally, HFG P1 calculation

MC2-3

MC2-3 input

Import meshes

Multigroup cross sections

Neutronics Solution

PN2DN SN2ND MOCFE

T/H

CUBIT

NeK

UNIC

UNIC input

ETOE-2


Reconstructed Pointwise Cross Sections (ENDF/B-VII.0)


Hyper-Fine-Group Spectrum Calculation

Inner core composition of ZPR-6/6A

0.0

0.2

0.4

0.6

0.8

1.E+02 1.E+03 1.E+04 1.E+05 1.E+06

Energy (eV)

No

rma

lize

d F

lux

Ultra FG (NR flux)Hyper FG

0.0

0.2

0.4

0.6

0.8

1.E+05 1.E+06Energy (eV)

No

rmal

ized

Flu

x

Hyper FG

Ultra FG


Hyper-Fine-Group vs. Ultra-Fine-Group Spectra

0.0

0.2

0.4

0.6

0.8

1.E+02 1.E+03 1.E+04 1.E+05 1.E+06

Energy (eV)

No

rma

lize

d F

lux

Ultra FG (NR flux)

Hyper FG

Hyper FG (No Anisotropic Source)

0.0

0.2

0.4

0.6

0.8

1.E+05 1.E+06

Energy (eV)

No

rma

lize

d F

lux

0.0

0.2

0.4

0.6

0.8

1.E+04 1.E+05

Energy (eV)

No

rma

lize

d F

lux

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

184.6 186.2 187.7 189.3 190.9

Energy (eV)

Tota

l XS (bar

n)

0

1

2

3

Norm

aliz

ed F

lux

U-238 total XS

UFG flux

HFG flux


LANL Criticality Assembly Benchmarks (UFG Calculation)

0

1

2

3

4

1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08

Energy (eV)E

* Flu

x(E)

.

Jezebel

Big-10

Godiva

ZPR-67

Flattop

Multiplication factors are in an excellent agreement within 0.15% ∆ρ by taking into account the anisotropy of inelastic scattering

-500

0

500

1000

k

(pcm

)

Isotropic inelastic scattering

Anisotropic inelastic scattering


MC2-3 vs. VIM for ZPR-6/7 (Standalone UFG Calculation)

0.0

2.5

5.0

1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Energy (eV)

Norm

alize

d Ne

utro

n Fl

ux

Inner Core(MC2-3)

Inner Core (VIM)

Radial Blanket (MC2-3)Radial Blanket (VIM)

Region VIMMC2-3

(k pcm)

Inner Core1.22945

±0.0003810

Outer Core1.22482

±0.00048-36

Radial Blanket0.33513

±0.00043485

Axial Blanket0.33215

±0.00048440

-50

-25

0

25

50

1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Energy (eV)

% D

iff

of

Tota

l X

S

U-238

U-235

Pu-239

Fe-56



PSR Rod

Detector

Reflector

Blanket

Outer core

Inner core

Sodium

Pu-U

-Mo F

uel

Matrix Tube

Depleted U

Stainless S

teel

Stainless S

teel

Sodium

Sodium

Sodium

Depleted U

Drawer

x (width)

y (height)

z (length)


A Realistic View of ZPPR-15 Double Fuel Column Drawer

0 10 20 30 40 50 60 70 80 90

0.0

1.0

2.0

3.0

4.0

5.0

STAINLESS STEEL

PU-U-MO FUELDEPLETED URANIUM

DEPLETED URANIUM

DEPLETED URANIUM

STAINLESS STEEL

DEPLETED URANIUM

SODIUM

DEPLETED URANIUM

DEPLETED URANIUM

SODIUM

STEEL BLOCK

SODIUM

STAINLESS STEEL

SODIUM

PU-U-MO FUEL

STAINLESS STEEL

DEPLETED URANIUM

STAINLESS STEEL

Matrix tube Drawer

SODIUMSODIUM

SODIUMSODIUM

SODIUM

SODIUM

VoidZ

X



Three loading configurations of ZPPR-15 Phase A were analyzed– Loading 15: initial criticality– Loading 16: reference configuration for sodium void worth measurement– Loading 20: configuration with an 18” sodium void in part of inner core

VIM - Exp DIF3D - Exp

Data Configuration Experiment VIM ∆k, pcm DIF3D Sn ∆k, pcm

ENDF/B-V.2

L15 1.00046 0.99647 -399 0.99525 -521

L16 0.99627 0.99200 -427 0.99104 -523

L20 0.99853 0.99529 -324 0.99428 -425

Void Worth (pcm) 226 329 324

ENDF/B-VII.0

L15 1.00046 0.99985 -61 0.99905 -141

L16 0.99627 0.99571 -56 0.99489 -138

L20 0.99853 0.99742 -111 0.99741 -112

Void Worth (pcm) 226 171 252

* Standard deviations of Experiment and VIM ≤ 0.00021


Summary

An initial version of new multi-group cross section generation code MC2-3 has been developed

– Preliminary tests showed significantly improved performance relative to MC2-2

– Integrated with UNIC for online cross section generation• Consistent thermal feedbacks• Account for spectral transition effects

Second order solvers PN2ND and SN2ND have been improved

– SN2ND demonstrated good scalability to >100,000 processors

– Working on enhancing the anisotropic scattering iteration

– Fixing the load imbalance for reflected boundary conditions

– Starting next phase of pre-conditioner development• p-refinement multi-grid and • Algebraic multi-grid beyond that or possibly h-refinement


Summary

First order solver MOCFE

– Improving parallel performance with Krylov Method

– Added more elements to ray tracing capabilities

– Adding back projection for parallel Started NODAL

– Implement Krylov solution technique to fix some convergence problems

– Eliminate memory problems and 1970s architecture

– Will investigate energy parallelization on multi-core machines (8-32 cores)


Backup Slides


The MCNP perturbation option was used to determine the difference in net neutron production in every fuel assembly as a resulting of reducing

– Fuel density by 2%, cladding density by 5%, and coolant density by 50% While the fuel density reduction showed reasonable results, the clad and

coolant density effects still showed significant statistical variations

– Observed statistical errors are less than 2% for the fuel density perturbation

– However, as large as 41% for the cladding density perturbation and 100% for the coolant density perturbation

Direct perturbation calculations showed even worse results

– Relative statistical uncertainties of the re-converged production rates are often above 50%, and in some cases reach 100%

– The re-converged calculation ran 50,000 histories per cycle for 160 active cycles, each of which took 1000 minutes on a 2.7-GHz Opteron processor

Perturbation Evaluation with MCNP (LANL)


NGNP with 60-degree periodic symmetry Core multiplication factor converges relatively quickly Power distribution converges very slowly

Number of neutron histories

100M 20M 5M

Eigenvalue1.45598

0.0001

1.45599

0.0002

1.45607

0.0003

CPU time, hr 1765 360 145

Variation, %

RMS 0.3 1.6 2.4

Max 0.6 3.2 5.4

C

C

C

C

C

C

0.980.970.96

0.990.980.98

1.021.010.99

1.031.010.99

0.930.910.88

1.061.051.03

C

C

C

1.051.051.03

0.930.930.92

0.860.860.85

0.880.870.87

0.820.810.79

0.890.880.85

0.860.850.83

0.930.920.90

0.980.970.95

C

0.930.930.92

0.860.860.85

1.051.041.03

1.231.221.20

1.141.131.11

1.161.141.11

1.241.211.19

1.051.031.00

0.860.850.84

1.000.990.97

C

C

1.021.021.00

0.890.890.88

1.231.241.21

1.221.211.20

0.880.870.86

1.031.021.00

C

C

1.021.031.01

0.820.830.81

1.151.161.13

1.141.121.11

0.820.810.81

1.031.011.00

C

0.991.000.98

0.880.890.86

1.131.141.11

1.151.141.13

0.890.880.87

0.920.910.91

C

C

0.970.990.97

0.860.870.86

1.221.231.22

1.241.231.22

0.860.860.86

1.051.041.03

C

0.920.940.92

1.041.061.05

1.051.041.05

0.920.920.92

C

1.051.061.03

0.860.870.86

1.231.251.23

1.221.221.24

0.860.860.87

0.980.970.98

C

C

0.920.940.92

0.890.900.89

1.151.161.16

1.131.131.16

0.880.880.90

0.990.991.01

C

1.021.041.02

0.820.830.82

1.131.151.13

1.151.151.19

0.820.820.85

1.021.021.05

C

C

1.021.031.02

0.880.890.88

1.221.231.21

1.231.231.29

0.890.890.93

1.021.021.06

C

C

0.991.000.99

0.860.860.87

1.041.051.04

1.231.231.25

1.151.151.18

1.131.131.16

1.221.221.27

1.041.041.09

0.860.860.90

0.920.920.97

C

0.970.980.98

0.920.930.93

0.860.860.87

0.890.890.91

0.820.820.84

0.880.880.91

0.860.860.90

0.920.930.96

1.051.051.10

C

C

C

1.051.051.06

0.920.920.94

1.021.021.04

1.021.031.06

0.991.001.03

0.970.981.03

C

C

C

C

C

C

100 M20 M 5 M

– Asymmetric assembly power distribution is observed

– Extremely large number of histories would be required for converged pin power distribution

Convergence of Assembly Power Distribution


Comparison of whole core depletions performed by GA, BNL, and ANL– MONTEBURNS (MCNP5+ORIGEN2)

– Simple cubic lattice model

– CPU time: ~40 hours for 50K and ~100 hours for 100K histories

Much larger number of histories are required for converged flux solutions

0.88

0.92

0.96

1.00

1.04

1.08

0 540 1080 1620 2160 2700 3240 3780

Burnup, EFPD

K-e

ff

GA BNL ANL-50K ANL-100K

DB-MHR benchmark

– Cycle length = 540 EFPD

– Total 7 cycles

– 6 burn steps per cycle (90 days interval)

– 50K and 100K neutron histories per burn step

Note that there are ~3 billion fuel particles

Depletion with Monte Carlo Method


APPLO2:172-group CP and 28-group MOC calculation

CRONOS2: 8-group diffusion calculation (finite element method)

% difference in fission rate distributions from MCNP4C (3D core)

Control Rod

Position

Control Rod Worth

TRIPOLI4 ±38 pcm

NEPHTIS, % Diff.

Homogeneous Heterogeneous

ARI 18,341 0.90 -1.05

ORI 7,083 0.98 0.64

SRI 5,676 -3.87 -3.35

Homogenous Element

Heterogeneous Element

CEA: NEPHTIS Verification Results


Power Distribution of Fuel Block (CR Inserted)

0.297-3.60

0.353-2.55

0.365-2.99

0.473-1.69

0.427-1.00

0.471-0.69

0.442-1.43

0.6432.05

0.552-1.11

0.545-0.60

0.531-0.42

0.5240.14

0.5951.34

0.6931.22

0.606-0.74

0.614-0.28

0.6290.94

0.6710.86

0.8230.59

0.9090.45

1.0620.38

1.1200.27

0.668-0.94

0.6920.08

0.7170..04

0.7950.11

0.8530.12

0.9920.56

1.0580.57

1.1700.18

1.220-0.08

0.742-0.23

0.7580.13

0.8050.16

0.846-0.08

0.9430.60

1.0060.01

1.1210.22

1.174-0.07

1.2630.04

1.303-0.01

0.8320.54

0.8620.16

0.934-0.18

0.9690.78

1.0760.23

1.129-0.03

1.228-0.36

1.269-0.19

1.343-0.16

1.3700.32

0.8890.24

0.9390.02

0.971-0.05

1.0460.33

1.093-0.02

1.192-0.54

1.230-0.03

1.312-0.26

1.346-0.15

1.410-0.15

0.9650.29

0.9860.10

1.0380.13

1.074-0.20

1.1530.00

1.198-0.33

1.279-0.16

1.320-0.45

1.382-0.04

1.413-0.08

1.0480.18

1.073-0.12

1.1310.01

1.1640.13

1.2430.21

1.287-0.23

1.360-0.12

1.384-0.39

1.0940.02

1.1200.28

1.1480.03

1.2150.17

1.2500.28

1.335-0.29

1.367-0.06

1.432-0.05

1.1371.03

1.1930.05

1.2260.19

1.302-0.15

1.343-0.16

1.416-0.08

MCNP5% diff

k∞ = 0.58326±0.00035 (MCNP5) 0.58375 (DeCART)

RMS = 0.76 %Max. = 3.60 %


Effective Multiplication Factors for 2D and 3D VHTRs with Heterogeneous Fuel Compact

Geometry Control Rod PositionMCNP5±20 pcm

DeCART , ∆ pcm

190 Group 47 Groups

2DARO- Standard block 1.46245 187 573

ARI 1.09752 14 788

3DARO- Standard block 1.46379 439

ARO 1.45791 123

All Rods Out (ARO) All Rods In (ARI) Operating Rods In (ORI)


2D Power Distributions

ARO ARI ORI


2D Block Power Comparison with MCNP5

RMS 0.5%, Max 0.92%

Heterogeneous Fuel, ARO

1.62-0.29

0.950.68

1.70-0.31

1.010.32

0.69-0.03

1.60-0.20

1.68-0.08

1.330.13

0.900.51

0.55-0.30

0.950.68

1.010.46

0.920.54

0.710.04

0.53-0.68

0.65-0.27

0.66-0.24

0.52-0.39

RMS 0.38%, Max 0.68%Homogeneous Fuel, ORI

0.80-1.88

0.54-0.29

1.72-0.51

0.78-0.14

0.62-1.12

1.15-0.40

2.130.69

1.830.53

1.060.68

0.39-0.74

0.54-0.29

1.150.97

1.260.99

0.900.14

0.58-1.45

0.71-0.47

0.880.13

0.49-0.91

RMS 0.82%, Max -1.88%Homogeneous Fuel, ARI

ARO ARI ORI


3D Flux Distribution for All Rods Out (ARO) Case

0.13 eV1 eV7 eV1 MeV


3D Flux Distribution for Operating Control Rods In (ORI)

0.13 eV1 eV7 eV1 MeV

The Advanced Fuel Cycle Initiative Status of Neutronics Modeling Won Sik Yang Argonne National...

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