Using the Application Builder for Neutron Transport in Discrete Ordinates

Transcription

1 Using the Application Builder for Neutron Transport in Discrete Ordinates C.J. Hurt University of Tennessee Nuclear Engineering Department (This material is based upon work supported under a Department of Energy Nuclear Energy University Programs Graduate Fellowship.) James D. Freels Oak Ridge National Laboratory, Research Reactors Division Presented at: ORNL is managed by UT-Battelle for the US Department of Energy

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

3 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 Data x z rˆ î y kˆ Ωˆ ĵ 3 Conference 2015 Boston, MA, October 7-9,2015

4 Discrete Ordinates Method Spatial variables finite difference, nodal, FEM Energy variables multigroup method, split problem into energy groups Angular variables discrete directions and quadrature approximation Gauss-Legendre quadrature for numerical integration over the angular domain: φφ ll mm (rr, EE) = 4ππ YY llll 4 Conference 2015 Boston, MA, October 7-9,2015 NN Ω ψψ rr, EE, Ω ddω = YY llll Ω nn ψ(rr, EE, Ω nn ) Flux moments nn=1

5 Weak form application mode. Multi-Group Neutron Transport Equation for Discrete Ordinates ψψ nn φφ ll Ω nn r Σ tt χχ νν Σ ff Σ ssss YY llll Ω nn + Σ tt (rr) ψψnn rr, Ω nn GG + χχ ννσ ff (rr)φφ 5 Conference 2015 Boston, MA, October 7-9,2015 = qq eeeeee SSSSSS rr + =1 ll=0 mm= ll angular neutron flux(neutrons/m 2 -s) neutron flux moment (neutrons/m 2 -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 ll rr GG YY llll ( Ω nn ) qq eeeeee nn ll, mm SSSSSS =1 Scattering Moment Σ ssss μμ 0 = Σ ssss (rr)φφll (rr) external scattering source (neutrons/m 3 -s) discrete ordinate index energy group index scattering order index maximum scattering order (0 = isotropic) 1 1 ddμμ 0 Σ ss μμ 0 PP ll μμ 0 cosine of scattering angle, Ω Ω PP ll Legendre polynomials Σ ss scattering cross section g g (1/m)

6 Neutron Transport Equation Streaming Neutron transport equation (simplified): Streaming terms: Dimension Spatial Var. Angular Var. Ω ψψ 1D Cartesian 2D Cartesian 3D Cartesian 2D Cylindrical 6 Conference 2015 Boston, MA, October 7-9,2015 x x,y x,y,z Ω + σσσσ = qq μμ μμ, ηη μμ, ηη, ξξ ρρ,z μμ, ξξ μμ ρρ μμ μμ xx + ηη yy μμ xx + ηη yy + ξξ ρρ 1 ρρ Curvilinear geometry: The variation of the angular coordinate system with position introduces a troublesome angular derivative or directional transfer term ηη + ξξ

7 Use of Application Builder to streamline physics building of G-by-N interfaces SCALE: Material Processing Application Builder 1. Create model equations, BC s, variables 2. Mix Xs 3. Save Intermediary App. File Working Xs directory text file App: 1. Develop geometry, solver, parameters, etc. 2. Run edits 3. Save as model file Model: Run simulations 7 Conference 2015 Boston, MA, October 7-9,2015

8 Kobayashi Benchmark Problems 1 3 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 absorber ii. 50% scattering Expected ray effects allowed unmitigated (1) (2) Vacuum Vacuum (3) Source Source 8 Conference 2015 Boston, MA, October 7-9,2015

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

10 Kobayashi Benchmark 3 Fluxes (w/o scattering) 1.0E-03 Flux (n/cm^2./s) 1.0E E E E E-03 Analytic PARTISN Flux (n/cm^2./s) 1.0E E E E E-08 Analytic PARTISN X (cm), Y=95 cm, Z=35 cm 1.0E Y (cm), X,Z = 5 cm 1.0E+00 Flux (n/cm^2./s) 1.0E E E-03 Analytic 1.0E-04 PARTISN 1.0E X (cm), Y=55 cm, Z=5 cm 10 Conference 2015 Boston, MA, October 7-9,2015 Log plot of flux

11 Kobayashi Benchmark 3 Fluxes (w/o scattering) 4.0E+00 Ratio to Analytical Solution 2.0E E E E E E E E E E E+00 PARTISN Analytic Y (cm), X,Z = 5 cm Ratio to Analytical Solution 3.5E E E E E E E E E-01 PARTISN Analytic X (cm), Y=95 cm, Z=35 cm Ratio to Analytical Solution 1.8E E E E E E E E E E+00 PARTISN Analytic X (cm), Y=55 cm, Z=5 cm 11 Conference 2015 Boston, MA, October 7-9,2015 Log plot of flux

12 Simple Eigenvalue Problems Group 1 Flux (n/cm 2 -s) 6.00E E E E E E-06 Keno-Va HEU metal and water configurations, Sn2-4 k eff KENOVa MG Sn % Sn % 0.00E Radial Position (m) Sn % MCNP CE Conference 2015 Boston, MA, October 7-9,2015

13 Summary and Conclusions -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 New use of the 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 has a variety of avenues 13 Conference 2015 Boston, MA, October 7-9,2015