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Overview of reactor core neutron transport codes
Jean Ragusa
Dept. of Nuclear EngineeringTexas A&M University College Station, TX
J. Ragusa, Texas A&M DOE-NPRCAFC meeting, August 11, 2006 1
Outline
The big pictureGeometry is challengingThe physics is challenging
Basic nuclear dataFirst order form of the integro-differential NTEMultigroup approximationDivide and conquer approach:
Lattice levelCore level
XS Multi-parameterization Current and future target accuraciesShort discussion on Monte Carlo
J. Ragusa, Texas A&M DOE-NPRCAFC meeting, August 11, 2006 2
The big picture: example of a PWR core
J. Ragusa, Texas A&M DOE-NPRCAFC meeting, August 11, 2006 3
Geometry is challenging
~ 20 cm ~ 3 m~ 1.2 cm
Cell
Core
Fuel Assembly
moderator/coolant
cladFuel pin
# meshes per cell6 to 48
# cells per assembly:264+25
# fuel assemblies in a LWR: 157 to 215
about 50,000 fuel pins (and actually 30 millions of fuel pellets) about 10 to 100 millions of 3-D spatial meshes excluding any intricacies related to radial reflector, upper/lower plenageometrical modeling
~ 4 m
30 to 40 axial planes
J. Ragusa, Texas A&M DOE-NPRCAFC meeting, August 11, 2006 4
Geometry is challenging
EFR:
271 pins per FA~400 FAs in core rings 1/2/3Blanket FAs: 0 to 164
Pin cell geometry
J. Ragusa, Texas A&M DOE-NPRCAFC meeting, August 11, 2006 5
Nature is challenging: simplified neutron life cycles
Thermal reactorfast neutron
Leaks
Doesnt leak
Slows down to thermal Absorbed fast
Leaks Absorbed In junkIn fuel
Causes fission CapturedIn junk In fuel
Captured Causes fission
J. Ragusa, Texas A&M DOE-NPRCAFC meeting, August 11, 2006 6
Nature is challenging: simplified neutron life cycles
Fast reactorfast neutron
Leaks
Doesnt leak
Scatters Absorbed
In junkIn fuel
Causes fission Captured
J. Ragusa, Texas A&M DOE-NPRCAFC meeting, August 11, 2006 7
Nature is challenging
Pu-239,fis
Na23,el
U238,fis
U238,el
J. Ragusa, Texas A&M DOE-NPRCAFC meeting, August 11, 2006 8
Nature is challenging : Typical spectrum per reactor type
0.00
0.10
0.20
0.30
0.40
0.50
1.00E-03 1.00E-01 1.00E+01 1.00E+03 1.00E+05 1.00E+07
Energy, eV
Nor
mal
ized
Flu
x/Le
thar
gy
PWRVHTRSCWRSFRGFRLFR
J. Ragusa, Texas A&M DOE-NPRCAFC meeting, August 11, 2006 9
Nature is challenging : coupled problem
XS depends on T and atom densities (which change because of TH, TM, but also because fuel depletes)
J. Ragusa, Texas A&M DOE-NPRCAFC meeting, August 11, 2006 10
Basic nuclear data
Activities related to nuclear data:Problem dependant data are often given in a multigroup format where an appropriateweighting function has been used to collapsed the data in energy group
Examples of evaluated nuclear data libraries
cross section data is often applied to wide range of specific data at the different stages of data processing and reactor applications!
J. Ragusa, Texas A&M DOE-NPRCAFC meeting, August 11, 2006 11
Basic nuclear data
Microscopic cross sections (XS):Differential microscopic XS:
Interaction in a medium is given by the macroscopic XS [probability of interaction per unit particle track length]
( )xyz E
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )( , ) ( ', ') ', '
'1( , ) ( ', ') '4 4
s s s
f s f s
E E E f E E
EE E E f E E E
=
=
i
,1
( ) ( )NISO
xyz iso xyz isoiso
E N E=
=
J. Ragusa, Texas A&M DOE-NPRCAFC meeting, August 11, 2006 12
(steady state) Integro-differential Neutron transport equation
First order form of the integro-differential neutron transport equation (NTE)
where the angular (energy-dependent) neutron flux is:
The phase-space for the solution of the NTE possesses 6 dimensions (r, v or r, and v = |v|)
Spatial discretization: FV, FEM (based either on the 1st form of the NTE or the 2nd order form).
Angular discretization: collocation of the angular flux on chosen directions m(short and long characteristics methods) or development of the angular on spherical harmonics basis.Energy discretization: multigroup (averaging the NTE on an energy bin)
(4 )
(4 )
( , ( , ' ' , ' ' ( , ' '
( ) ' ' ' ( , ' '4
s
feff
E E E dE d E E E
E dE v E d E
r r r r r
r r
i0
0
, ) + ( , ) , ) = ( , ) , )
1+ ( , ) , )
( , v ( ,E E n E r r, ) = ( ) , )
Recall, about 10 to 100 millions of 3-D spatial meshes Multiply by G energy groups (~100s < G < 10,000s) Multiply by directions (100s to low 1,000s) or angular moments (10s to 100s) a computational feat is needed to abide by first-principles
J. Ragusa, Texas A&M DOE-NPRCAFC meeting, August 11, 2006 13
Integro-differential Neutron transport equationFirst order NTE (multigroup and within group)
Even parity (second order formulation, within group)
(4 )
NG' '
1 (4 )g' g
( , ( , ' ' ( , '
' ' ( , ' for 1 g G
g g g g g gs
g g gs
g'=
d
d
r r r r, r
r, r
i
i
) + ( ) ) = ( ) )
+ ( ) )
NG' '
1 (4 )
' ( , '4
gg gf
g'=eff
v d
r r1
+ ( ) )
(4 )
( , ( , ' ' ( , ' ( ,sd Q
r r r r, r ri ) + ( ) ) = ( ) ) + )
( )
( ) ( )
s0 (4 )
1 (4 )
1 ( , ( , ( ,
' ' ( , '
+ ' ' ' ( , '
+cut
-cut
tt
L
even
L
s
odd
Q
d P
d P
+ + ++
+
=
+
=
+
r r r rr
r r
r r
i
i
) + ( ) ) = )( )
( ) )
( ) )
[ ] ( )
[ ] ( )
1s s
s1 (4 )
s1 (4 )
=
( , ' ' ( , '
( , ' ' ( , '
max
+cut
max
-cut
even
odd
tL
tL
L
tL
d P
d P
+ ++
= +
= +
r r r r
r r r r r
r r r r r
i
i
( ) ( ) ( ) ( )
( ) ( ) ) = ( ) )
( ) ( ) ) = ( ) )
where [ ]1 ( , = ( , ( ,2
r r r) ) )
J. Ragusa, Texas A&M DOE-NPRCAFC meeting, August 11, 2006 14
A few words on angular discretizations
Collocation (discrete ordinates aka SN)
Requires a set of directions d and weights d such that:
Iterations on the within source scatter
Process needs to be accelerated when s,0 / 1
Decouple angular directions from spatial discretization (in Cartesian geometries)
Short characteristics method: mesh sweeping done mesh cell by mesh cellLong characteristics method: solve the transport equation for a trajectory along d crossing the domain from boundary to boundary.
1(4 )
( ' ( , ' ( ,NDir
d dd
d
=
r r r) = ) )
1 1 *, , ,
, , ,1
( , ( , ( ( ( ,
( ( ( , ( ( ,
Li i i
d d d s m d m dm
Diri i i
m m d d d m d dd
Y Q
Y Y
+ +
=
=
r r r r r r
r r r
=1
(4 )
) + ( ) ) = ( ) ) ) + )
) = ) ) ) )
J. Ragusa, Texas A&M DOE-NPRCAFC meeting, August 11, 2006 15
A few words on angular discretizations
PN method (spherical harmonics)The angular is decomposed on the spherical harmonic basis
Take the angular moment of the NTE
spatial equations coupling several flux angular moments
*, ,
, ,
( , ( (
where ( ( ( ,
Nharm
m mm
m m
Y
d Y
=
r r
r r
=1
(4 )
) = ) )
) = ) )
( ),(4 )
( NTE md Y
)
J. Ragusa, Texas A&M DOE-NPRCAFC meeting, August 11, 2006 16
Multigroup approximation
A key idea is to preserve the reaction rates (physically observable quantities)
with
Is this a tour de force where the unknown Psi is needed to compute the multigroup cross sections???
1
1
1
(
( ( or ( ,
g
g
g
g
g
g
E
EEg g g
EE
E
dE E E
dE E EdE E
r r
r r r r rr
( , ) , )
( ) ) = ( , ) , ) ( ) =
)
( )1
NTE g
g
E
E
dE
(4 )
NG' '
1 (4 )g' g
( , ( , ' ' ( , '
' ' ( , ' for 1 g G
g g g g g gs
g g gs
g'=
d
d
r r r r, r
r, r
i
i
) + ( ) ) = ( ) )
+ ( ) )
NG' '
1 (4 )
' ( , '4
gg gf
g'=eff
v d
r r1
+ ( ) )
(4 )
( , ' ( , ',E d E
r r) = )
J. Ragusa, Texas A&M DOE-NPRCAFC meeting, August 11, 2006 17
Multigroup approximation
An a priori knowledge of the reactor spectrum (weighting function w below) is required to prepare the multigroup cross section set:
E.g., for LWR, w is: fission spectrum for E>1.3 MeV, 1/E slowing down spectrumMaxwellian distribution in the thermal range
Decent approximation when compared to a 172-g 2D UOX lattice calculation
1
1
(
(
g
g
g
g
E
EgE
E
dE E w E
dE w E
( ) )
=
)
0.00E+00
1.00E+13
2.00E+13
3.00E+13
4.00E+13
5.00E+13
6.00E+13
7.00E+13
1.E-09 1.E-08 1.E-07 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02
Energy (MeV)
Flux
Flux Bu = 0 MWd/t
Flux Bu = 50 000 MWd/t
J. Ragusa, Texas A&M DOE-NPRCAFC meeting, August 11, 2006 18
Multigroup approximation: self-shielding
Energy (eV)
Issue: The spectrum or weighting function w doesnt account for flux dip in resonances
Multigroup cross sections should be modified (self-shielded)
Multigroup cross sections are prepared/tabulated with respect to
Temperature (Doppler broadening)Dilution section b
In subsequent lattice calculations, the cross section values will interpolated in these tables.
Self-shielding factors
Group #
J. Ragusa, Texas A&M DOE-NPRCAFC meeting, August 11, 2006 19
Divide and conquer
In order to compute a reactor core, we currently rely on: 1. Appropriate multigroup cross section libraries with 100s of energy groups:
~200 g for thermal reactors, ~2000 g for fast reactors
2. A divide and conquer type of approach:
Fuel assembly level computations are performed with high accuracy (neutron transport calculations with 100s of groups, fine spatial discretization). Pin power form functions are stored for subsequent pin power reconstruction.
Equivalence step: spatial homogenization and group collapse of the assembly level results in order to generate few group cross section libraries for the core level calculation (few groups = 2 for thermal reactors, 232 for fast reactors). For LWRs, this step generated ADFs (assembly discontinuity factors)
Full core level computations during which the keff, reactivity, assembly averaged power distribution, etc , are obtained.
Pin power reconstruction step where the pin power is contained by combining the assembly level pin power form functions and the core level power distribution.
J. Ragusa, Texas A&M DOE-NPRCAFC meeting, August 11, 2006 20
Divide and conquer
gi
NTE
gi
Self-shieldingg
i,isot
Evaluation( )Eisot
Assembly level
Core level
Low-order few group Transport
pin pow. fun., ADFG
gisot
Library
Equivalence
Keff, power, etc
Pin power reconstruction
J. Ragusa, Texas A&M DOE-NPRCAFC meeting, August 11, 2006 21
Typical number of assembly level computations
Complete set of lattice calculations for a BWR includes:
Depletion calculations: Each depletion has about 50 burnup points Depletions for 3 different voids (0, 40, 80%) both with/without control rods
Branches from each depletion, for all independent variable, at 20 points: Void (3 points) Fuel temperature (3 points) Control rod (each type) Bypass void (3 points) Spacer type, detector type
Complete (HFP at least) set of calculations includes:
[50 x 3 x 2] + [20 x 3 x 2 x (3 + 3 + 1 + 3 + 2)] = 1740 total state points
These multi-parameterized cross section libraries are subsequently used in core cycle depletion calculations
J. Ragusa, Texas A&M DOE-NPRCAFC meeting, August 11, 2006 22
Accuracies
Current Operating Reactors (some due to K. Smith) Future ?Keff ~0.5% 0.1%PWRs
Axially-integrated reaction rates ~ 1.0% rms3-D reaction rates ~ 3.0% rms 1%
BWRsAxially-integrated reaction rates ~ 1.5% rms3-D reaction rates ~ 3.0-6.0% rms 1%
Rod worth ~2-8% 1%Local nuclide densities
Main elements ~5% 1%Others ~5-20% 1-5%
J. Ragusa, Texas A&M DOE-NPRCAFC meeting, August 11, 2006 23
MOX loaded PWR: pin-by-pin
MOX loaded PWR: 4 zones per FA
C5C7 OECD bench
J. Ragusa, Texas A&M DOE-NPRCAFC meeting, August 11, 2006 24
Recent trends in methodology
Eliminate or reduce the need for spatial homogenization of the fuel assemblyHomogenized Pin-by-pin core level computationPin-by-pin core level computation with limited homogenization (pin made of 2 regions: (1) coolant, (2) fuel+clad)
Increase the number of energy groupsAt the assembly level (10,000)At the core level (e.g., fast reactors can now be computed with 2000 g)
Avoid using diffusion theory at the core level SN or PN transport (with N being limited by the computer)Simplified PN transport
J. Ragusa, Texas A&M DOE-NPRCAFC meeting, August 11, 2006 25
Monte Carlo simulations
Def.:Perform particle transport experiments in a random manner on a computer to estimate average behavior of a particle in phase space
Advantages:Precise geometric modelingEnergy continuous cross sections
MC R&D needs:DepletionTemperature effects/couplingSource convergence
MC is primarily a V&V tool at BOL
Scale of problem:Number of fuel Assemblies 200Number of axial planes 100Number of pins per assembly 300Number of depletion regions per pin 10Number of isotopes to be tracked 100Total unknowns 6 billion talliesSource distribution if far more difficult to converge for a full-core(dominance ratio > 0.995)
J. Ragusa, Texas A&M DOE-NPRCAFC meeting, August 11, 2006 26
Summary and trends
Moving away from the classical divide & conquer approach for full blown application of first principles
Energy groups:10,000s range to simplify or remove self shielding formalism
Space-angle:Even parity methods:
Nowadays exists in 3-D (Variant-ANL)Intimate coupling of space and angleLeads to the assembly and storing of use matricesWell adapted for massively parallel libraries such as PETSc
SN methods:Short characteristics:
Only SN mesh in 3-D nowadays (Attila Transpire, IDT/Cronos-CEA)Mesh is swept cell by cell (on regular grid, there exist efficient parallel sweeping schemes; on unstructured or curvilinear meshes: a topic of R&D)
Long characteristics:Each trajectory is independent (natural parallelism)Hugh quantity of information to be stored for 3-D (topic of R&D)
J. Ragusa, Texas A&M DOE-NPRCAFC meeting, August 11, 2006 27
Discontinuity Factors
Let + - heterogeneity factors be different (Discontinuity Factors)Approximate DFs from single-assembly lattice calculation (ADFs)
HetHom
Het
Hom
ADF+ I
ADF- I+1
J. Ragusa, Texas A&M DOE-NPRCAFC meeting, August 11, 2006 28
Overview of reactor core neutron transport codesOutlineThe big picture: example of a PWR coreGeometry is challengingGeometry is challengingNature is challenging: simplified neutron life cyclesNature is challenging: simplified neutron life cyclesNature is challengingNature is challenging : Typical spectrum per reactor typeNature is challenging : coupled problemBasic nuclear dataBasic nuclear data(steady state) Integro-differential Neutron transport equationIntegro-differential Neutron transport equationA few words on angular discretizationsA few words on angular discretizationsMultigroup approximationMultigroup approximationMultigroup approximation: self-shieldingDivide and conquerDivide and conquerTypical number of assembly level computationsAccuraciesRecent trends in methodologyMonte Carlo simulationsSummary and trends