Reactor Core Methods. Kord Smith Studsvik Scandpower

Transcription

1 Kord Smith Studsvik Scandpower

2 Presentation Outline 1. Background for LWR Core Analysis 2. Modern LWR Design Requirements 3. Factorization of the Core Analysis Space 4. Early Analysis Methods 5. Lattice Physics Applications 6. Prerequisites For Advanced Nodal Models 7. Lattice Physics Models 8. Advanced Nodal Methods 9. Assembly Homogenization 10. Fuel Depletion Modeling 11. Pin Power Recovery 12. Nodal Method Verification 13. Refinements/Applications 14. Looking to the 21 st Century 2of 56

3 1. Applications of Reactor Physics Chicago Pile (CP-1, December 2, 1942) 3of 56

4 1. Computational Requirements One Portable Super Computer: Enrico Fermi 4of 56

5 1. Simple Core Models Four- and Six-Factor Formulas: k ( f p) L L eff th th fast where, th fuel f, fuel a fuel a fuel clad mod erator a a a Fuel thermal eta f... Thermal utilization factor p Fast fission factor Resonance escape probability Lth Thermal non-leakage probability (geometry) Lfast Fast non-leakage probability (geometry) 5of 56

6 1. Early Design of Reactors Built special experiments/fit parameters/use simple models Measure eta, thermal utilization, fast fission, etc. Fit data to assumed functional form (e.g., fuel/coolant ratio, pin diameter, etc.) Geometrical approximations (thermal diffusion lengths, buckling, etc.) Pencil and paper designs Built exact mockup criticals Deduce few-group cross sections from criticals/integral measurements Simple computational models (i.e., 1-D, 2-D homogeneous diffusion theory) Extensive use of good Engineering Judgment 6of 56

7 1. Focus of the 1950 s Hundred of reactors/criticals built of many designs Analysis Progression: Integral experiments/simple analytical methods Integral experiments to deduce parameters/simple computational models Differential cross sections measurements/complex computational methods/ criticals for testing/verification Methods driven by Naval Reactors needs, (STR, Nautilus) Shippingport Nuclear Power Station, critical in of 56

8 1. Analytical Concepts of the 1950 s Physical insight leads to simple mathematical models Resonance integrals NR and NRIM Approximations Equivalence theory Dancoff factors Resonance escape Slowing down kernels Flux disadvantage factors Fermi age theory Migration area Thermal utilization factors Thermal diffusion lengths Critical buckling Reflector savings 8of 56

9 1. Extensive Model Improvements see ANL-5800 (1963) Section 3: Constants for thermal homogeneous systems Thermal neutron spectrum Effective cross sections Thermal group diffusion parameters Slowing down parameters Non-thermal parameters Infinite multiplication Section 4: Constants for thermal heterogeneous systems Thermal utilization Resonance escape probability Fast effect Neutron diffusion in lattices Integral data 9of 56

10 2. Cross Section Measurements Full energy range (0-20 MeV) measurements needed Data is independent of reactor design Requires reasonably complex computational models 10 of 56

11 The Sexy Years of Nuclear Engineering This slide has been intentionally removed This presentation originally contained a slide which attempted to break the monotony and add levity to the presentation. I am guilty of having given insufficient attention to the possible negative implications of this slide, and I would like to apologize to all those who have been injured as a result. Rest assured that I am now much more sensitive to such issues. I hope that you can forgive me for this lapse of judgment. I would like to thank those who have had the courage to bring this to my attention. Kord Smith 11 of 56

12 2. Inexorable Link Between Digital Computing/Reactor Analysis ENIAC 12 of 56

13 2. Modern LWR Core Design Fuel procurement analysis: Enrichment specification Burnable absorber design Economics analysis Reload Core Design: Selection of optimum fuel loading pattern Selection of coolant flow and control rod strategy (BWR) Computations of margins to design safety limits Static Safety Analysis: Calculations of nominal and off-nominal power shapes ( fly spec analysis) Calculations of rod worths, shutdown margins, reactivity coefficients DNBR analysis 13 of 56

14 2. Modern Design Requirements Transient Safety Analysis: Reactivity insertion accidents Loss of coolant accidents Loss of off-site power Operational Support: Pre-calculations of core monitoring data Calculations of startup sequences Computation of parameters needed for setting of operating limits Core monitoring: On-line 3-D computation of margins (MCPR, MLHGR, etc) Bottom Line: 10,000 s of core calculations required per cycle of operation 14 of 56

15 2. Deterministic Transport Scale of problem: Number of fuel Assemblies 200 Number of axial planes 100 Number of pins per assembly 300 Number of depletion regions per pin 10 Number of angular directions 100 Number of neutron energy groups 100 Total unknowns 600 Billion At 100 FLOPS/unknown on 1 gigaflop machine = 16 CPU hours Not yet (or even soon) tractable for routine analysis 15 of 56

16 2. Direct Monte Carlo? Scale of problem: Number of fuel Assemblies 200 Number of axial planes 100 Number of pins per assembly 300 Number of depletion regions per pin 10 Number of isotopes to be tracked 100 Total unknowns 6 billion tallies Further complicating factors LWRs need ~1% statistics on assembly-wise peak pin power 10 6 histories yields 1.% statistics for one assembly (dominance ratio ~0.75) 10 6 x 200 x 100 =20 billion histories (~ 5000 hr on 2.0 GHz PC) Source distribution if far more difficult to converge for a full-core (dominance ratio > 0.995) (50 times harder to converge than single assembly) If Moore s Law holds (factor of 2 every 18 months), LWR Monte Carlo core calculations will be reduced to 1 hr (single CPU) in the year 2030! 16 of 56

17 2. Core Analysis Limitations Cross Section Knowledge Extremely small but asymptotic Engineering Limitations Not important significant and asymptotic Computer Resources None better, not asymptotic Modeling Approximations Many fewer, not asymptotic Year of 56

18 3. Factorization in the s 1-D pin-cell with great detail: Resonance treatment by equivalence theory Multigroup energy treatment with ~100 groups Few region cylindrical transport with collision probability methods 2-D assembly calculation with intermediate detail: Homogenize cross sections over square pin-cell regions Collapse pin-cell cross section to few groups (e.g., 2-4) 2-D finite-difference diffusion calculations ~3-D core calculations: Assembly homogenized cross sections Few groups (e.g., 1-2) Radial (1-D or 2-D) / axial (1-D) 18 of 56

19 3. PWR Analysis in the s 1-D pin-cell with great detail: Resonance treatment by equivalence theory Multigroup energy treatment with ~100 groups Few region cylindrical transport with collision probability methods (i.e., LEOPARD code) 2-D core calculations with intermediate detail: Homogenize cross sections over square pin-cell regions Collapse pin-cell cross section to few groups (e.g., 2-4) 2-D finite-difference diffusion calculations (i.e., PDQ/HARMONY) 3-D flux synthesis fine-mesh radial and 1-D axial (KAPL and BAPL) 3-D homogenized core calculations: Homogenized cross sections Few groups (e.g., 1-4) 2-D radial / 1-D axial factorization ( poor man s flux synthesis) 19 of 56

20 3. BWR Factorizations 1-D pin-cell with great detail: Resonance treatment by equivalence theory Multigroup energy treatment with ~100 groups Few region cylindrical transport with collision probability methods 2-D assembly calculation with intermediate detail: Homogenize cross sections over square pin-cell regions Collapse pin-cell cross section to few groups (e.g., 2-4) 2-D finite-difference diffusion calculations 3-D core calculations: Assembly homogenized cross sections One neutron energy group Full 3-D representation (one node per assembly radial) Thermal-hydraulic feedback required 20 of 56

21 4. Early BWR Nodal Models Coarse Mesh Finite-Difference (CMFD) very inaccurate on assembly-size mesh FLARE (1964) k S ( w S w S ), w p 6 p p q pp qp q1 pp 1 6 q1 w pq where w g M h g M h pq (1 )( p / 2 ) ( p / ) 21 of 56

22 4. Improved Nodal Models TRILUX PRESTO POLCA SIMULATE PANACEA NODE-B Common Features: One unknown per assembly One or one-and-a-half groups (fast/thermal leakage corrections) Some tunable parameters Albedo reflector models Shortcomings: Accuracy, 5-10% on assembly-averaged powers, dependent on core loadings Memory requirements 20 Kbytes; CPU times ~ minutes per statepoint Inconsistent (don t satisfy diffusion equation in fine-mesh limit) 22 of 56

23 5. Early Lattice Physics BWR bundle design requires 2-D lattice analysis: Large water gaps require enrichment distributions to control local peaking Internal water rods used to enhance moderation at high void Gadolinium used as a burnable absorber Control blades are very localized absorbers Early lattice codes simply used 2-D diffusion computations to capture spatial effects. Corrections used to treat finite-mesh (e.g., g-factors) Corrections used to treat transport effects (e.g., blackness theory) Depletion is performed for each pin 23 of 56

24 5. WIMS: first true lattice code WIMS pioneered the concept of modularity 69 group UKNDL library Numerous resonance models Pin-cell model Numerous 2-D models: Diffusion theory Collision probability Discrete ordinates Method of Characteristics (much later) Depletion capabilities Parameter edits for many types of downstream tools: Fine-mesh diffusion theory Fine-mesh transport theory Assembly-homogenized data for nodal codes Applications in gas reactors, fast reactors, HWRs, and LWRs 24 of 56

25 5. LWR lattice codes WIMS PHEONIX CPM CASMO HELIOS DIT APOLLO-2 MULTI-MEDIUM TGBLA DRAGON (UKAEA) (ASEA ABB Westinghouse BNFL) (Studsvik/EPRI) (Studsvik Scandpower) (Studsvik Scandpower) (C-E ABB Westinghouse BNFL) (CEA/Framatome/EDF) (KWU Siemens Siemens/Framatome) (Toshiba/G-E) (Ecole Polytechnique Montreal) 25 of 56

26 5. Data For 2-D 2 D Cartesian Model Physical Geometry 1-D Cylindrical 2-D Homogenized (white b.c.) Geometry Problems: 1-D approximate b.c. Preserving reaction rates in x-y geometry x-y mesh effects Transport-to-diffusion effects 26 of 56

27 5. Fine-mesh Diffusion Models Use Lattice calculation directly to produce x-y data Select characteristic pin-types: Edge pins Water holes Pins next to water holes Burnable absorbers Compute SPH homogenization to approximately preserve reaction rates Iteratively compute g-factors to preserve average reaction rates Extend lattice calculations to four ¼ bundles (colorset) Better estimates of edge pin reaction rates, flux gradients 27 of 56

28 6. Advanced Nodal Models Propositions: If one could solve accurately assembly-homogenized nodal diffusion problems, one might be able to produce 3-D reactor solutions 100 times faster than using 2-D pin-by-pin methods. By using lattice data directly, many of the difficulties of making pin-cell homogenized diffusion models match lattice results could be avoided. Fast accurate nodal methods could permit transient analysis to be performed with much higher accuracy than obtained with existing methods Accurate nodal methods can be used for both PWRs and BWRs Required steps: Efficient assembly lattice physics tools Accurately solve 3-D diffusion equations Define assembly-homogenized parameters directly from lattice calculations Reconstruct pin-wise powers and reaction rates Treat depletion effects 28 of 56

29 M&C Solution to Methods Disagreement This slide has been intentionally removed This presentation originally contained a slide which attempted to break the monotony and add levity to the presentation. I am guilty of having given insufficient attention to the possible negative implications of this slide, and I would like to apologize to all those who have been injured as a result. Rest assured that I am now much more sensitive to such issues. I hope that you can forgive me for this lapse of judgment. I would like to thank those who have had the courage to bring this to my attention. Kord Smith 29 of 56

30 7. Lattice Calculations 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 ( )] = 1740 total state points 30 of 56

31 7. Lattice Physics Models Discrete ordinates in homogenized Cartesian geometry Collision Probability Methods (CP) Current Coupling Collision Probability Methods (CCCP) Method of Characteristics (MOC) Monte Carlo Methods 31 of 56

32 7. CCCP Spatial/Angular Coupling MOX Pincell k-eff vs. angular representation 2 surface segments 4 surface segments 8 surface segments of 56

33 7. Long Characteristics (MOC) Modeling Approximations: Cyclic azimuthal tracking Exact boundary conditions Product quadrature (azimuthal x polar) Flat Source (Step Characteristics) Programming Considerations: Efficient ray tracing Minimize operations Minimize storage Minimize stride 1700 Statepoints requires about 1 CPU hr on 2.0 GHz PC m m /cos Q m m m gk j g gk /cosj gi,, jk, gi,, jk, e (1 e ) m 4 g 33 of 56

34 8. Advanced Nodal Methods Higher-order difference equations QUABOX/CUBBOX Classical finite-element methods Many unknowns with 4-th or 5-th order expansions Iterative solutions are costly because of tight coupling Response matrix methods High-order surface spatial representations needed Intra-assembly heterogeneity and depletion difficult to model Transverse integrated nodal methods Most successful advanced nodal methods (as of 1980) Most widely used for production analysis today 34 of 56

35 8. Transverse Integration Dg g( xyz,, ) Dg g( xyz,, ) Dg g( xyz,, ) agg( xyz,, ) Qg( xyz,, ) x x y y z z G G 1 Qg( x, y, z) g fg' g' ( x, y, z) gg' g' ( x, y, z) k eff g' 1 g' Dg gx( x) aggx( x) Qgx( x) Lgy( x) Lgz( x) x x y z 1 1 gx( x) dy dzg ( x, y, z) y z and 1 Lgy( x) dz Dg g ( x, y, z) z dy 1 Lgz( x) dy Dg g ( x, y, z) y dz y y z z Transverse Leakage Fit to Quadratic Polynomial 35 of 56

36 8. Polynomial Approximations 0 gx gxn n 0 where, f 4 ( x) a f ( x) ( x) 1 x f1( x) x 2 1 f2( x) f3( x) ( )( ) f x ( ) ( )( )( ) 36 of 56

37 8. Popular Nodal Methods Nodal Expansion Method (NEM, ) Polynomial 1-D flux expansions Quadratic transverse leakage fit Partial current inner iterations Analytic Nodal Method (ANM, ) Analytic solution to 1-D coupling equations Buckling, flat, and quadratic polynomial transverse leakages Node-averaged fluxes iteration NGFM, DIF3-D Nodal, ILLICO, NESTLE,.. 37 of 56

38 8. Non-linear acceleration methods Non-linear Iterative Acceleration (1983) Applicable to most nodal kernels (NEM, ANM, etc.) All iterations performed with 7-point (3-D) stencil Minimized computer storage and CPU requirements i1 i i1 i g g g g Jg Dg Dg x x Accuracy in solving 3-D homogenized diffusion equation ~1.0% on nodal powers 3-D PWR/BWR statepoints about 5 CPU seconds on 2.GHz PC T-H, cross section evaluation, boron searches, Xe search 38 of 56

39 9. Homogenization Equations Known Reference Heterogeneous Solution: G G 1 Jˆ ˆ ˆ ˆ ˆ ˆ ˆ g() r ag() r g() r ˆ g fg' () r g' () r gg' () r g' () r keff g' 1 g' 1 Homogenized Equations: G G 1 J () r () r () r () r () r () r () r g ag g g fg' g' gg' g' keff g' 1 g' 1 Homogenized Constraints: S V i i and ˆ () r ˆ () r dr () r dr g g g g Jˆ () r ds J () r ds g S i V i g 39 of 56

40 9. Homogenization Paradox Homogenized Parameters: i g and D i g V i ˆ () r ˆ () r dr i () r dr Jˆ () r ds Si () r ds S i g V g g g g Which Surface? 40 of 56

41 9. Koebke s s Heterogeneity Factors HF+ HF- HF- HF+ Iterate on diffusion coefficients until HF+ and HF- are the same Continuity (discontinuity) condition: HF HF i i i 1 i 1 41 of 56

42 9. Discontinuity Factors Let + - heterogeneity factors be different (Discontinuity Factors) Approximate DF s from single-assembly lattice calculation (ADFs) Het Hom ADF+ I ADF- I+1 Het Hom 42 of 56

43 9. Applications of ADFs Use of ADFs reduces typical homogenization errors by about a factor of three: PWRs 3-5% errors reduced to ~ 1.0% BWRs 10% errors reduced to ~ % Little computational burden: Available as edits from lattice calculation Treat as additional homogenization parameters DFs very useful in treating PWR baffle/reflector as explicit nodes 1-D fuel/baffle/reflector problem used to generate DFs Accounts for transport/diffusion effects Accounts for inherent spatial/spectral approximations in nodal model. 43 of 56

44 10. Intra-assembly assembly Depletion Effects First developed by Wagner and Koebke at KWU Intra-assembly depletion (spatial) effects treated with space dependent cross sections (homogenized) 1 1 Dg( r) gx( x) ag( r) gx( x) Qgx( x) Lgy( x) Lgz( x) x x y z G G 1 Q ( x, y, z) ( r) ( x, y, z) ( r) ( x, y, z) g g fg' g' gg' g' keff g' 1 g' 1 Track assembly-surface exposures and assume quadratic profiles of exposure Treat spatially varying cross section contributions as addition non-linear sources like transverse leakages. 44 of 56

45 10. Assembly Spectral Interactions Interface instantaneous (spectral) effects i o o 2 21 ( ) 1 a2 a2 Interface depletion (spectral) effects a 21 h o o b E E 1 2() e 1 21() e ( de de) E () e E () e a2 E 1 21() E 0 a2 e de () e Important in 2 groups, reduced in importance as more groups are used 45 of 56

46 11. Pin Power Recovery After nodal solution, pin powers must be recovered, as pin-wise limits are used in safety/licensing Response matrix methods (Henry, MIT) indirectly yield pin powers Large amount of data required Accuracy limited by surface spatial expansions Imbedded local calculations: ROCS/MC Perform assembly 2-D pin-by-pin diffusion with b.c. from 3-D nodal Use axial shapes from 3-D nodal Reasonably computationally intensive SIMULA/SIMTRAN (Aragones and Ahnert) Non-linear iteration methods used with coarse mesh 3-D LD F-D Multiple planes of 2-D pin-by-pin diffusion Direct pin power reconstruction by superposition of nodal and lattice powers Pioneered by Wagner and Koebke at KWU 46 of 56

47 11. Pin Power Reconstruction Assume separability of pin-wise powers from lattice code and the homogenized power shape from nodal code. 1. Iteratively determine flux shapes along the edges of the nodes: Assume quadratic flux variation along an edge Used edge-averaged fluxes, and continuity of flux and derivatives at corner points as constraints 2. Assume a non-separable form for the radial flux expansion within a node 3. Use node-average fluxes, surface-averaged fluxes, and surface-averaged fluxes, and corner point fluxes/derivatives as expansion constraints 4. Use surface-integrated and node-average exposures to approximate the intra-nodal shape of fission cross sections 5. Integrate over pin-cell regions to get homogenized pin powers 6. Multiply homogenized powers by lattice pin powers (peaking factors) 47 of 56

48 12. Direct Nodal Method Verification 48 of 56

49 12. Nodal Method Accuracy Operating Reactors PWRs Axially-integrated reaction rates ~ 1.0% rms 3-D reaction rates ~ 3.0% rms BWRs Axially-integrated reaction rates ~ 1.5% rms 3-D reaction rates ~ % rms Pin powers vs. BOL criticals Axially-integrated pin powers ~1.0% rms Numerical tests vs. 2-D full core lattice depletion calculations PWRs Assembly powers ~1.0% rms Pin powers ~1.5% max MOX pin powers ~2.5% max 49 of 56

50 13. Nodal Refinements Hexagonal Geometry KWU, ANL Conformal Mapping (Chou) MOX applications: Analytic expansion functions Form function refinements Transport effects More energy groups Microscopic isotropic tracking Elimination of nodal/reconstruction inconsistencies: Finite-element like non-separable flux expansions (AFEN) Iterative solution improvements re-homogenization enhancements Nodal methods (VARIANT code at ANL) Direct treatment of cross sections heterogeneity High-order heterogeneous flux expansions Direct treatment of transport effects 50 of 56

51 13. Extended Applications Formal Core Loading Optimization: Stochastic optimization Simulated annealing (FORMOSA, SIMAN) Genetic Algorithms Direct Searches 10,000 to 100,000 of patterns are depleted to determine a core design 2-D initially and 3-D is presently feasible On-line Core monitoring Direct 3-D core calculations on-line Automatic predictions of future reactor state On-line computation of refueling shutdown margins 51 of 56

52 13. Expanding Transient Applications Growing application of 3-D transient methods New physics testing procedures Dynamic rod worth measurements Eliminate traditional licensing approximations Limits for PWR peak enthalpies for ejected rod accidents Linking to systems thermal-hydraulic codes Elimination of point and 1-D approximations Virtually unlimited applications for systems analysis Full scope training simulator core models 4-10 Hz executions with core design nodalization Realistic cycle-specific core models (INPO 96-02) Just-in-time training 52 of 56

53 13. BWR transient applications Direct 3-D evaluations of decay ratios On-line BWR stability analysis On-line BWR stability predictions for proposed maneuvers In Phase Out of Phase 53 of 56

54 14. New Factorization Boundaries Direct 3-D pin-by-pin models (see PHYSOR 2002, Seoul, Korea) Diffusion and transport Pin-cell homogenization approximations? Data explosion with detailed isotopics? New Synthesis methods (see PHYSOR 2002, Seoul, Korea) Direct use of full-core 2-D lattice calculations Simplified axial transport coupling (very fine radial mesh) Expanded Monte Carlo Applications Lattice physics applications? Steady-state core depletions? 54 of 56

55 14. Accuracy Limitations Limits to accuracy improvements Mechanical knowledge Assembly mechanics (e.g., BWR channel bowing) Crud buildup (e.g., axial offset anomaly) Manufacturing uncertainties (e.g., IFBA coatings) Fuel cycling history (e.g., fission gas migration) Feedback modeling Where is the water? Local hydraulic information Pin-wise fuel temperatures Cross section uncertainties Availability of refined ENDF sets Unresolved resonance models Thermal scattering models 55 of 56

56 14. Concerns for the Future Knowledge retention: Who under the age of 40 understands resonance theory? What is crystalline binding? What is reactivity? Too much reliance on the black boxes? When have we exceeded the applicability of the methods? How do we establish analysis uncertainties? Are we capable of building new reactor types? How many people understand existing safety/licensing? Is DOE capable of building a new generation reactor? When will utilities be ready to invest in the next generation? 56 of 56