Modeling of Spent Fuel Pools with Multi stage, Response function Transport (MRT) Methodologies

Transcription

1 Modeling of Spent Fuel Pools with Multi stage, Response function Transport (MRT) Methodologies Prof. Alireza Haghighat Virginia Tech Virginia Tech Transport Theory Group (VT 3 G) Director of Nuclear Engineering and Science Lab (NSEL) at Arlington Nuclear Engineering Program, Mechanical Engineering Department INMM CRC Modeling and Simulation for Nuclear Nonproliferation, Howard H. Baker Jr. Center for Public Policy University of Tennessee, Oct 20,

2 Objective Particle Transport Theory Determine the expected number of particles in a phase space (d 3 rded) at time t: n( r, 3 E, ˆ, t) d rded z d 3 r dω Ω de r y x Number density is used to determine angular flux/current, scalar flux and current density, partial currents, and reaction rates. 2

3 Simulation Approaches Deterministic Methods Solve the linear Boltzmann equation to obtain the expected flux in a phase space Statistical Monte Carlo Methods Perform particle transport experiments using random numbers (RN s) on a computer to estimate average properties of a particle in phase space 3

4 Deterministic Linear Boltzmann Equation Integro-differential form ) ˆ,, ( ) ˆ ',, ( ), ( ' ' 4 ) ( ) ˆ ',, ( ) ˆ ' ˆ, ', ( ' ' ) ˆ,, ( ), ( ) ˆ,, (. ˆ E r S E r E r d de E E r E E r d de E r E r E r f s Integral form ), ( 0 ) ', ( ' ) ˆ,, ( ') ( ' ) ˆ,, ( s E E r r s R r r e E r e r Q r r d E r streaming collision scattering fission Independent source 4

5 Integro-differential - Solution Method Angular variable: Discrete Ordinates (Sn) method: A discrete set of directions { ˆ m} and associated weights {w m } are selected ˆ m. ( r, E, ˆ m ) ( r, E) ( r, E, ˆ m ) q( r, E, ˆ m ) V V X Spatial variable Integrated over fine meshes using FD or FE methods Energy variable m, g, A V ijk d 3 r V m, g ijk ( r ) Integrate over energy intervals to prepare multigroup cross sections, σ g 1 1 5

6 Integral - Solution method Method of Characteristic (MOC): Model is partitioned into coarse meshes and transport equation is solved along the characteristic paths (k) (parallel to each discrete ordinate (n)), filling the mesh, and averaged Q ( ) (0)exp( ) (1 exp( )) gmi,, gmik,,, tmik,, gmik,,, gi, tmik,, gi, tmik,, gi, 6

7 Deterministic Issues/Challenges/Needs Robust numerical formulations (e.g., adaptive differencing strategy) Algorithms for improving efficiency (i.e., acceleration techniques synthetic formulations and pre conditioners) Use of advanced computing hardware & software environments Pre and post processing tools Multigroup cross section preparation Benchmarking 7

8 Monte Carlo Methods Perform an experiment on a computer; exact simulation of a physical process Pathlength r ln t Type of collision s t Scattering angle (isotropic scattering) Sample S (r, E, Ω) Tally (count) absorbed Issue: Precise expected values; i.e., small relative uncertainty, Variance Reduction techniques are needed for real world problems! R x x x 8

9 Deterministic vs. Monte Carlo Item Deterministic MC Geometry Discrete/ Exact Exact Energy treatment Discrete Exact cross section Direction Discrete/ Truncated series Exact Input preparation Difficult simple Computer memory Large Small Computer time Small Large Numerical issues Convergence Statistical uncertainty Amount of Large Limited information Parallel computing Complex Trivial 9

10 Why not MC only? Because of the difficulty in obtaining detail information with reliable statistical uncertainty in a reasonable time Example situations Real time simulations Obtaining energy dependent flux distributions, Time dependent simulations, Sensitivity analysis, Determination of uncertainties 10

11 Why not use advanced hardware? VT 3 G has developed vector and parallel algorithms, and developed two large codes: PENTRAN (1996) and TITAN (2004) Why not hybrid methods? Deterministic deterministic (differencing schemes, different numerical formulations, generation of multigroup cross sections, generation of angular quadratures, acceleration techniques) (VT 3 G has developed various algorithms; a few have been implemented in PENTRAN and TITAN) Monte Carlo deterministic (variance reduction with the of use deterministic adjoint) (VT 3 G has developed CADIS, A 3 MCNP in 1997; CADIS has become popular recently!) 11

12 V V X VT 3 G Milestones & Contributing Current/Former Students ( ) Vector computing of 1 D Sn spherical geometry algorithm Development an adjoint methodology for simulation TMI 2 reactor Vector and parallel processing of 2 D Sn algorithms Simulation of Reactor Pressure Vessel (RPV) Parallel processing of 2 D Sn algorithms & Acceleration methods Determination of uncertainties in the RPV transport calculations 3 D parallel Sn Cartesian algorithms Monte Carlo for Reactor Pressure Vessel (RPV) benchmark using Weight window generator; deterministic benchmarking of power reactors Directional Theta Weight (DTW) differencing formulation PENTRAN (Parallel Environment Neutral Particle TRANsport) code system CADIS (Consistent Adjoint Driven Importance Sampling) formulation for Monte Carlo Variance Reduction A 3 MCNP (Automated Adjoint Accelerate MCNP) Parallel Angular & Spatial Multigrid acceleration methods for Sn transport Hybrid algorithm for PGNNA device PENMSH & PENINP for mesh and input generation of PENTRAN Ordinate Splitting (OS) technique for modeling a x ray CT machine Simplified Sn Even Parity (SSn EP) algorithm for acceleration of the Sn method RAR (Regional Angular Refinement) formulation Pn Tn angular quadrature set FAST (Flux Acceleration Simplified Transport) PENXMSH, An AutoCad driven PENMSH with automated meshing and parallel decomposition CPXSD (Contributon Point wise cross section Driven) for generation of multigroup libraries TITAN hybrid parallel transport code system & a new version of PENMSH called PENMSHXP ADIES (Angular dependent Adjoint Driven Electron photon Importance Sampling) code system INSPCT S (Inspection of Nuclear Spent fuel Pool Calculation Tool ver. Spreadsheet), a MRT algorithm TITAN fictitious quadrature set and ray tracing for SPECT (Single Photon Emission Computed Tomography) FMBMC ICEU (Fission Matrix Based Monte Carlo with Initial source and Controlled Elements and Uncertainties) New WCOS (Weighted Circular Ordinated Splitting) Technique for the TITAN SPECT Formulation Adaptive Collision Source (ACS) for Sn transport AIMS (Active Interrogation for Monitoring Special nuclear materials), a MRT algorithm TITAN SDM includes Subgroup Decomposition Method for multigroup transport calculation TITAN IR TITAN with iterative image Reconstruction for SPECT RAPID Real time Analysis for spent fuel Pool in situ detection ADIES Prof. Haghighat Prof. R. Mattis, Pitt. Prof. B. Petrovic, GT Dr. M. Hunter, W Prof. B. Petrovic, GT Dr. G. Sjoden, DOD Dr. J. Wagner, ORNL Dr. B. Petrovic Dr. G. Sjoden, DOD Dr. J. Wagner, ORNL Dr. V. Kucukboyaci, W Dr. B. Petrovi, GT Prof. Haghighat Prof. Hgahighat Dr. G. Longonil, PNNL Dr. A. Patchimpattapong, IAEA Dr. A. Alpan, W Dr. C. Yi, GT Dr. B. Dionne, ANL Dr. W. Walters, Postdpc Dr. C. Yi, GT Dr. M. Wenner, W Dr. K. Royston, ORNL Dr. W. Walters, Postdoc. N. Roskoff, PhD student. Dr. K. Royston, ORNL. DR. W. Walters, Postdoc. 12 TITAN INSPCT S AIMS

13 Remarks Particle transport based methodologies are need for real time simulation Particle transport codes, even those parallel with hybrid algorithms, are slow because of large number of unknowns 13

14 Development of Transport Formulations for Real Time Applications Physics Based transport methodologies are needed: Developed Based on problem physics partition a problem into stages (subproblems), For each stage employ response method and/or adjoint function methodology Pre calculate response function or adjoint function using an accurate and fast transport code Solve a linear system of equations to couple all the stages 14

15 Examples for Nondestructive testing: Optimization of the Westinghouse s PGNNA active interrogation system for detection of RCRA (Resource Conversation and Recovery Act) (e.g., lead, mercury, cadmium) in waste drums (partial implementation of MRT; 1999) Nuclear Safeguards: Monitoring of spent fuel pools for detection of fuel diversion (2007) (funded by LLNL) Nuclear nonproliferation: Active interrogation of cargo containers for simulation of special nuclear materials (SNMs) (2013) (in collaboration with GaTech) Spent fuel safety and security: Real time simulation of spent fuel pools for determination of eigenvalue, subcritical multiplication, and material identification (partly funded by I 2 S project, led by GaTech) (Ongoing) Image reconstruction for SPECT (Single Photon Emission Computed Tomography): Real time simulation of an SPECT device for generation of project images using an MRT methodology and Maximum Likelihood Estimation Maximization (MLEM) (filed for a patent, June 2015) 15

16 Spent Fuel Pool Inspection (Development of a tool for safeguards, funded by LLNL) Atucha 1 Spent fuel pool MMCA and Computer B R I D G E PHWR Spent Fuels gap Fuel: UO2 Coolant: D2O Rods/assm: 36 Active length: 5.3 m Ave BU: 6-11GWD/TU Top view of the pond A front view of the pond that contains closely packed spent fuel assemblies Objective Identification of missing/moved assemblies for safeguards Approach Combination of measurement and on line computation to obtain trending curves 16

17 Approach Develop a fast and accurate computation tool which can estimate the detector response for various combinations of Burnup Cooling time Pool lattice arrangement Fuel type (enrichment) Perform on line calculation of detector responses (neutron and/or gamma) for comparison with the corresponding measurements; create normalized response curves 17

18 How do we calculate the detector response? Standard or forward approach R d dv de V d 0 4 d d ( r, E) ( r, E, ˆ ) Adjoint approach R S Where, S is particle source, and is adjoint ( importance ) function 18

19 Derivation of (R) formulation in terms of Forward transport equation is expressed by where H ˆ ( r, E) t H Adjoint transport equation is expressed by q de' 0 4 in V 0 on for nˆ ˆ 0 d' ( r, E' s E, ˆ ' ˆ ) where H q in V on for n ˆ ˆ 0 H ˆ ( r, E) t de' 0 4 d' ( r, E s E', ˆ ˆ ') 19

20 Application of Adjoint function Detector Response If we derive the commutation relation between the forward and adjoint transport equations, H H q q Then, we obtain the following equality q q If we consider Then q d R q 20

21 Example Calculation of detector Response: Detector Where, R Standard d H S Adjoint Methodology R Where, H S d

22 Multi stage response function Methodology R n S n n Multiplication Source by Assembly Stage 1 Source (S = S intrinsic + S subcritical Multiplication ) Intrinsic Source Spontaneous fission & (, n) from fuel burnup calculation (ORIGEN ARP) (Created a database) Multiplication Source Strength (#/source) (Fission Matrix) Assembly Position (x,y) S4 S5S6 S1 S2S Subcritical Multiplication Fission matrix (FM) method Use MCNP Monte Carlo to obtain a i,j for each pool type (Created a database for coef. a ij ) F i N i1 a i, j ( F j S int. j ) Stage 2 Adjoint function Use the deterministic parallel PENTRAN Sn transport code (Created a database for multigroup adjoint for different lattice sizes)

23 Derivation of Fission Matrix (FM) Formulation Eigenvalue formulation in operator form is expressed by Where,,,Ω Ω σ, Ω σ, E, 4 Ω ν, 23

24 , FM Derivation (cont) We may rewrite above equation as Where,,, & Ω ν, 24

25 Eigenvalue Fission Matrix (FM) Formulation, k is eigenvalue is fission source, is fixed source in cell j, is the number of fission neutrons produced in cell due to a fission neutron born in cell. Subcritical multiplication,,, ), is the number of fission neutrons produced in cell due to a source neutron born in cell. 25

26 For Atucha 1 Application Subcritical Multiplication formulation,, is fission source, is fixed source in cell j, is the number of fission neutrons produced in cell due to a fission neutron born in cell., is the number of fission neutrons produced in cell due to a source neutron born in cell. For this application, we have demonstrated that,, 26

27 Determination of FM elements ( It is demonstrated that for this application assemblies can be grouped into three categories: corner (i.e. 3 adjacent assemblies), edge (i.e. 5 adjacent assemblies) and interior (i.e. 8 adjacent assemblies). 6x9 array of assemblies Corner Assemblies Edge Assemblies Interior Assemblies 27

28 Determination of FM Elements Coupling is highly localized Consider two only rows of assemblies surrounding each assembly 6x9 array of assemblies Geometric similarity For example, green, red and blue arrows indicate the edge assemblies with identical coefficients Calculation yields a i,j for fixed i Three fixed source calculations for the three categories of assemblies to determine a i,j

29 Note Eigenvalue Formulation Similarly, we can write a FM formulation for the eigenvalue calculation, Where, K is the multiplication factor 29

30 Testing the Simplified FM Methodology Same a i,j coefficients used for every case Solve system of equations F i ( F Four test spent fuel scenarios 1. 2x6 array, uniform source 2. 9x6 array, uniform source 3. 9x6 array, 27 assemblies on the left with source strength 1, the rest with source strength x6 array, uniform source N i1 a i, j j S int. j ) MCNP calculation as a benchmark 4.

31 FM Results Excellent agreement with Monte Carlo (<1%) Assembly Arrangement Case M (MCNP) M (Fission Matrix) Difference MCNP Uncertainty 1-2x6, uniform % x6, uniform % x6, non-uniform % x6, uniform % Very fast <1s for Fission matrix method as compared to ~1hr for the standard Monte Carlo

32 Multi stage response function Methodology R n S n n Multiplication Source by Assembly Stage 1 Source (S = S intrinsic + S subcritical Multiplication ) Intrinsic Source Spontaneous fission & (, n) from fuel burnup calculation (ORIGEN ARP) (Created a database) Multiplication Source Strength (#/source) (Fission Matrix) Assembly Position (x,y) S4 S5S6 S1 S2S Subcritical Multiplication Fission matrix (FM) method Use MCNP Monte Carlo to obtain a i,j for each pool type (Created a database for coef. a ij ) F i N i1 a i, j ( F j S int. j ) Stage 2 Adjoint function Use the deterministic parallel PENTRAN Sn transport code (Created a database for multigroup adjoint for different lattice sizes)

33 Real time Tool: INSPCT S (Inspection of Nuclear Spent fuel Pool Computing Tool Spreadsheet) INSPCT S solves R n S n By interpolation, source and adjoint function are determined using databases of the decay neutrons, fission matrix coefficients, and adjoint distributions n (Use of Fission Matrix & Adjoint)

34 Design of passively safe and secure spent fuel pool (SFP) Considering a typical pool, e.g., AP1000 pool shown below 34

35 Background Standard approach Full Monte Carlo calculations face difficulties in this area Convergence is difficult due to low coupling between regions (due to absorbers) Convergence can also be difficult to detect Computation times are very long, especially to get detailed information Changing pool configuration requires complete recalculation Fission Matrix (FM) approach It can address the above issues Fission matrix coefficients are pre calculated using Monte Carlo Computation times are much shorter, with no convergence issues Detailed fission distributions are obtained at pin level Changing pool assembly configuration does not require new precalculations No additional Monte Carlo 35

36 Eigenvalue Fission Matrix (FM) Formulation, k is eigenvalue is fission source, is fixed source in cell j, is the number of fission neutrons produced in cell due to a fission neutron born in cell. Subcritical multiplication,,, ), is the number of fission neutrons produced in cell due to a source neutron born in cell. 36

37 Developed a Multi stage methodology for determination of FM coefficients As the computational size (for I 2 S reactor design) ,216 total fuel pins/ fission matrix cells Considering 24 axial segments per rod, then 653,184 9x9 array of assemblies in a pool Standard FM would require 653,184 separate fixedsource calculations to determine the coefficient matrix A matrix of size N x N = E+11 total coefficients (> 3.4 TB of memory is needed) The standard approach is clearly NOT feasible We have developed a multi stage approach to obtain detailed FM coefficients (in the process of filing for a patent) Assembly with 19x19 lattice; 25 positions are reserved for control rods 37

38 Remarks on the multi stage methodology Coefficients are calculated at different stages including: Pin wise (axially dependent) for one assembly for different burnups, cooling times, enrichments, and lattice structures For assemblies in the pool (pin wise or regional) Reduction in computation time and memory Computation time Geometric similarity Geometric symmetry Degree of coupling Sensitivity of the coefficients to different parameters Memory Taking advantage of a sparse matrix indexing 38

39 RAPID tool Developed the RAPID (Real time Analysis spent fuel Pool In situ Detection) tool for determination of Eigenvalue Subcritical multiplication Pin wise, axial fission density With application to Criticality safety Safeguards Nonproliferation and materials accountability 39

40 RAPID code system Structure Pre Calculation (one time): 1. Burnup Calculation to obtain material composition 2. Fission Matrix Coefficient Generation Real time Analysis: 1. Run Fission Matrix Code 2. Process Results 40

41 Test Problems (9x9 assemblies) 41

42 Case 3 Eigenfunction Comparison of RAPID with MC 1.80E E E 01 Reference Solution Fission Source 1.20E E E E E E E Assembly number MCNP FM 42

43 Case 11 Eigenfunction Comparison with RAPID with MC 1.80E E E 01 Reference Solution Fission Source 1.20E E E E E E E Assembly number MCNP FM 43

44 Case 4 Eigenfunction distribution Comparison with RAPID with MC Reference Solution 44

45 Comparison of calculated M RAPID vs. MCNP Case FM MCNP Error in M M Time (min) M Time (min) 1 σ Uncertainty (FM vs MCNP) Speedup (FM vs MCNP) 1x % x % x % x % 236 *Note that the FM technique also provides pin wise, axial dependent fission source or power. 45

46 Fission density plots (X Y integrated) Axial fission density for entire assembly 46

47 Fission density plots (Y Z integrated) x fission density for entire assembly 47

48 Fission density plots (Single Pin) Axial fission density at position (row=11, column=10, ) 48

49 Fission density in a segment of fuel rods Fission density for a segment of fuel rod at y=10 & z=72 along x axis 49

50 Developed a post processing tool for demonstration of Fission Density Y LEVEL ANIMATION Z LEVEL ANIMATION 50

51 Conclusions MRT methodology allows for development of real time tools for analysis of nuclear systems INSPCT s is capable of identification of material diversion; currently, we are exploring the possibility of its use for identification of fuel misload in pools and casks The current version of the RAPID can quickly and accurately calculate the eigenvalue and eigenfunction (fission density) for a spent fuel pool By pre calculating a database of FM coefficients for different conditions, pool simulation can be performed in real time allowing for changes in configurations (assembly shuffling, removal and addition) for various burnup, cooling times, lattice structure, and enrichment We are working on a novel capability for RAPID for identification of fuel burnup and isotopics 51

52 Thanks! Questions? 52