Using the Application Builder for Neutron Transport in Discrete OrdinatesC.J. HurtUniversity of TennesseeNuclear Engineering Department(This material is based upon work supported under a Department of Energy Nuclear Energy University Programs Graduate Fellowship.)

James D. FreelsOak Ridge National Laboratory, Research Reactors Division

Presented at:

ORNL is managed by UT-Battelle for the US Department of Energy

2 COMSOL Conference 2015 Boston, MA, October 7-9,2015

Presentation Outline

• Brief review of neutron transport

• Discrete ordinates methodology

• (Mis-)Use of the App. Builder• Transport Benchmarks

– Kobayashi benchmarks– Simple eigenvalue problems

• Results• Conclusions

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Neutron Transport

• Solve 7 independent variables:– Energy (1) – E – Angle (2) – Ω (ξ,ω)– Space (3) – r (x,y,z)– Time (1) – t

• Two primary methods:– Stochastic (MCNP5,

SCALE’s KENO, Serpent)– Deterministic (PARTISN,

DORT, Attila)

• Nuclear Datay

x

z

i

jr

Ωk

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Discrete Ordinates Method

• Gauss-Legendre quadrature for numerical integration over the angular domain:

• Spatial variables – finite difference, nodal, FEM

• Energy variables –multigroup method, split problem into energy groups

• Angular variables – discrete directions and quadrature approximation

𝜙𝜙𝑙𝑙𝑚𝑚(𝑟𝑟,𝐸𝐸) = 4𝜋𝜋𝑌𝑌𝑙𝑙𝑚𝑚 Ω′ 𝜓𝜓 𝑟𝑟,𝐸𝐸, Ω′ 𝑑𝑑Ω′ =

𝑛𝑛=1

𝑁𝑁

𝑌𝑌𝑙𝑙𝑚𝑚 Ω𝑛𝑛 ψ(𝑟𝑟,𝐸𝐸, Ω𝑛𝑛)

Flux moments

5 COMSOL Conference 2015 Boston, MA, October 7-9,2015

Weak form application mode.Multi-Group Neutron Transport Equation for Discrete Ordinates

Scattering Moment

cosine of scattering angle, Ω Ω′

Legendre polynomials

scattering cross section g→g’ (1/m)

Ω𝑛𝑛 𝛻𝛻 + Σ𝑡𝑡𝑔𝑔(𝑟𝑟) 𝜓𝜓𝑛𝑛

𝑔𝑔 𝑟𝑟, Ω𝑛𝑛 = 𝑞𝑞𝑒𝑒𝑒𝑒𝑡𝑡𝑔𝑔 𝑟𝑟

+ 𝜒𝜒𝑔𝑔 𝑔𝑔′=1

𝐺𝐺

𝜈𝜈Σ𝑓𝑓𝑔𝑔′(𝑟𝑟)𝜙𝜙𝑔𝑔′ 𝑟𝑟 +

𝑙𝑙=0

𝑆𝑆𝑆𝑆𝑆𝑆

𝑚𝑚=−𝑙𝑙

𝑙𝑙

𝑌𝑌𝑙𝑙𝑚𝑚∗ (Ω𝑛𝑛) 𝑔𝑔′=1

𝐺𝐺

Σ𝑠𝑠𝑙𝑙𝑔𝑔′→𝑔𝑔(𝑟𝑟)𝜙𝜙𝑙𝑙

𝑔𝑔′(𝑟𝑟)

𝜓𝜓𝑛𝑛𝑔𝑔

𝜙𝜙𝑙𝑙𝑔𝑔

Ω𝑛𝑛

r

Σ𝑡𝑡𝑔𝑔

𝜒𝜒𝑔𝑔

𝜈𝜈

Σ𝑓𝑓𝑔𝑔

Σ𝑠𝑠𝑙𝑙𝑔𝑔→𝑔𝑔′

𝑌𝑌𝑙𝑙𝑚𝑚∗

angular neutron flux(neutrons/m2-s)

neutron flux moment (neutrons/m2-s)

discrete ordinate direction, n

spatial coordinates

total cross section (1/m)

probability of neutron born in group g

average # of neutrons emitted per fission

fission cross section (1/m)

scattering moment cross sec. g→g’ (1/m)

conjugate spherical harmonic functions

Σ𝑠𝑠𝑙𝑙𝑔𝑔→𝑔𝑔′ =

−1

1𝑑𝑑𝜇𝜇0Σ𝑠𝑠

𝑔𝑔→𝑔𝑔′ 𝜇𝜇0 𝑃𝑃𝑙𝑙 𝜇𝜇0

𝜇𝜇0𝑃𝑃𝑙𝑙

Σ𝑠𝑠𝑔𝑔→𝑔𝑔′

external scattering source (neutrons/m3-s)

discrete ordinate index

energy group index

scattering order index

maximum scattering order (0 = isotropic)

𝑞𝑞𝑒𝑒𝑒𝑒𝑡𝑡𝑔𝑔

𝑛𝑛

𝑔𝑔

𝑙𝑙,𝑚𝑚

𝑆𝑆𝑆𝑆𝑆𝑆

6 COMSOL Conference 2015 Boston, MA, October 7-9,2015

• Neutron transport equation (simplified):

• “Streaming” terms:

Neutron Transport Equation – Streaming

Dimension Spatial Var. Angular Var. Ω 𝛁𝛁𝜓𝜓1D Cartesian

x 𝜇𝜇 𝜇𝜇𝜕𝜕𝜕𝜕𝜕𝜕

2D Cartesian

x,y 𝜇𝜇, 𝜂𝜂 𝜇𝜇𝜕𝜕𝜕𝜕𝜕𝜕 + 𝜂𝜂

𝜕𝜕𝜕𝜕𝑦𝑦

3D Cartesian

x,y,z 𝜇𝜇, 𝜂𝜂, 𝜉𝜉 𝜇𝜇𝜕𝜕𝜕𝜕𝜕𝜕 + 𝜂𝜂

𝜕𝜕𝜕𝜕𝑦𝑦 + 𝜉𝜉

𝜕𝜕𝜕𝜕𝜕𝜕

2DCylindrical

𝜌𝜌,z 𝜇𝜇, 𝜉𝜉 𝜇𝜇𝜌𝜌𝜕𝜕𝜕𝜕𝜌𝜌 𝜌𝜌 −

1𝜌𝜌𝜕𝜕𝜕𝜕𝜕𝜕 𝜂𝜂 + 𝜉𝜉

𝜕𝜕𝜕𝜕𝜕𝜕

Ω 𝛻𝛻𝜓𝜓 + 𝜎𝜎𝜓𝜓 = 𝑞𝑞

• Curvilinear geometry: The variation of the angular coordinate system with position introduces a troublesome angular derivative or “directional transfer” term

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Use of Application Builder to “streamline” physics building of G-by-N interfaces

Working Xsdirectory text file

COMSOL Application Builder1. Create model equations, BC’s,

variables2. Mix Xs3. Save Intermediary App. File

COMSOL Model:Run simulations

COMSOL App:1. Develop geometry,

solver, parameters, etc.2. Run edits3. Save as model file

SCALE:Material Processing

8 COMSOL Conference 2015 Boston, MA, October 7-9,2015

Kobayashi Benchmark Problems 1 – 3

Vacuum

Source

Vacuum

Source

• Difficult problems to assess:– “…the accuracy of the flux distribution for systems which

have void regions in a highly absorbing medium.”

• 3 problems in two cases:i. Pure absorberii. 50% scattering

• Expected “ray effects” allowed unmitigated

(1) (2)(3)

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Meshing, Solver and Problem Parameters• Swept, mapped mesh everywhere

– 2 cm mesh intervals, uniform

– Mesh control domains utilized

• Each model set up with ~45-125k elements– ~14-28 million DOF → 1-2 hr solution time

– 1+ compute nodes, dual quad core processors, 128 GB RAM

• PARDISO for single node batch runs, MUMPS for batch runs across multiple nodes

• Fully-coupled or segregated solvers both used– Segregated slower convergence, offers “lower limit” feature and

minimal memory savings

• Sn16 Level-Symmetric Quadrature

• Pn0 Scattering Order

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1.0E-08

1.0E-07

1.0E-06

1.0E-05

1.0E-04

1.0E-03

0 10 20 30 40 50 60

Flux

(n/c

m^2

./s)

X (cm), Y=95 cm, Z=35 cm

AnalyticCOMSOLPARTISN

Kobayashi Benchmark 3 Fluxes (w/o scattering)

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

0 20 40 60 80 100

Flux

(n/c

m^2

./s)

Y (cm), X,Z = 5 cm

Analytic

COMSOL

PARTISN

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

0 10 20 30 40 50 60

Flux

(n/c

m^2

./s)

X (cm), Y=55 cm, Z=5 cm

Analytic

COMSOL

PARTISN

Log plot of flux

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Kobayashi Benchmark 3 Fluxes (w/o scattering)

Log plot of flux

0.0E+002.0E-014.0E-016.0E-018.0E-011.0E+001.2E+001.4E+001.6E+001.8E+002.0E+00

0 20 40 60 80 100

Rat

io to

Ana

lytic

al S

olut

ion

Y (cm), X,Z = 5 cm

COMSOL

PARTISN

Analytic

0.0E+002.0E-014.0E-016.0E-018.0E-011.0E+001.2E+001.4E+001.6E+001.8E+00

0 10 20 30 40 50 60

Rat

io to

Ana

lytic

al S

olut

ion

X (cm), Y=55 cm, Z=5 cm

COMSOL

PARTISN

Analytic

-5.0E-010.0E+005.0E-011.0E+001.5E+002.0E+002.5E+003.0E+003.5E+004.0E+00

0 10 20 30 40 50 60

Rat

io to

Ana

lytic

al S

olut

ion

X (cm), Y=95 cm, Z=35 cm

COMSOLPARTISNAnalytic

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Simple Eigenvalue Problems

0.00E+00

1.00E-06

2.00E-06

3.00E-06

4.00E-06

5.00E-06

6.00E-06

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

Gro

up 1

Flu

x (n

/cm

2 -s)

Radial Position (m)

COMSOL

Keno-Va

HEU metal and water configurations, Sn2-4

keff

KENOVa MG 0.2834 --

COMSOL Sn2 0.2600 8.24%

COMSOL Sn4 0.2705 4.56%

COMSOL Sn6 0.2713 4.27%

MCNP CE 0.2748 --

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Summary and Conclusions• COMSOL-based neutron transport physics were

developed for the discrete ordinates method– Variable dimension (1-D, 2-D, 2-D Axisymmetric, 3-D)– Variable energy group, angular quadrature– Neutron cross sections from SCALE, mixed in COMSOL

• New use of the COMSOL Application Builder for streamlining physics and model building in the weak form interface

• Results from compared benchmarks promising, primary limitation is computation time

• Future HFIR applications for neutron transport approaches in COMSOL has a variety of avenues