Fuel BurnupCalculations and Uncertainties Outline Review lattice physics methods Different approaches to burnup predictions Linkage to fuel safety criteria Sources of uncertainty Survey of available codes Page 2 1
Reactor Physics Challenge Go from here to here without losing too much information Page 3 Additional Complications Temperature (or doppler effects) Strong spatial discontinuities between materials Water next to Zr and UO2 Neutron scattering is non-linear in energy, angle and space Time dependence Neutron population Material properties and compositions Page 4 2
Reactor Physics Computational Strategy Circa 1980 K. Smith, Reactor Core Methods, M&C 2003 Page 5 Modern Approaches K. Smith, Reactor Core Methods, M&C 2003 Page 6 3
Lattice to Whole Core Analyses K. Smith, Reactor Core Methods, M&C 2003 Page 7 A Brute Force Approach not possible K. Smith, Reactor Core Methods, M&C 2003 Page 8 4
Different Approaches to Analyses Deterministic methods Collision probabilities Discrete ordinate methods Method of characteristics Stochastic methods Monte carlo The choice of method is dictated by computational resources and desired accuracy Note that this accuracy directly affects burnup calculations and error can compound with time Page 9 Collision Probabilities Integral method based on assumption that flux at a point is dependent on the probability of a neutron transiting a region in space Page 10 5
Discrete Ordinates Refers to treatment of angular variable Spatial variable treatment varies Finite difference type approach Characteristics based methods Page 11 Methods of Characteristics Lagragian method that explicitly treats both spatial and angular variables Scalar fluxes calculated by integrating along a series of rays transiting through the problem domain Page 12 6
The Burnup Problem Branching constants bi.j are known Decay constants λj are known Flux, φ, and cross section, σ, derived from lattice physics analyses Page 13 Modeling Approaches Once again, choice depends on desired accuracy Modern approaches predict burnup pin by pin Historical approaches perform calculations at the lattice level Page 14 7
Computational Strategy Historically core burnup calculations have been performed based on precalculated cross section libraries Account for all relevant physics Fuel temperature Moderator conditions Exposure Xe/Sm Control rods etc. Page 15 Lattice Physics Computational Flow http://scale.ornl.gov/index.shtml Page 16 8
Application to Reactor Problems Lattice physics calculation applied to fuel assembly Output reduced to library Whole core multi-group diffusion simulation accesses fuel specific library This allows whole core simulation to account for changes in state variable (Tf, Tm, Dm, etc.) K. Smith, Reactor Core Methods, M&C 2003 Page 17 Latest Developments Main difference is that lattice physics is embedded into core diffusion code Eliminates intermediate library Better captures real physics Page 18 9
Linkages to Safety Criteria Input to fuel mechanical code Predict reactivity coefficients Peaking factors Fuel burnup Reactor operations Page 19 Input to Fuel Mechanical Code Most fuel performance processes dependent upon power Typically, a limiting power profile is chosen Core physics calculations needed to ensure that reality is within assumed values Page 20 10
Reactivity Coefficients Needed to ensure compliance with safety standards Reactivity coefficients are burnup dependent Calculation needed to assure proper values throughout the entire operating cycle Page 21 Peaking Factors Directly linked to AOO, LOCA and RIA fuel safety criteria LHGR derived from AOO analysis typically constrains power operation Similarly, LOCA calculation assume peaking factors that constrain power operations RIA simulation imposes radial peaking limits to constrain rod worth Page 22 11
Fuel burnup Limits derived from fuel mechanical simulation Imposed by regulatory authority Simulation needed to demonstrate compliance Measurements difficult and uncertain Page 23 Reactor Operations Modern online monitoring systems are coupled to 3-D core simulator Use pre-calculated cross section libraries Use simplified nodalization schemes to allow for real time results Operator aid to assess plant performance Not used to actuate safety functions Page 24 12
Mechanical Data Calculational error Stochastic error Sources of Uncertainty Page 25 Mechanical Uncertainties Manufacturing processes are all conducted with design tolerances Fuel pellet radius diameter Cladding diameters Spacer pitch Channel thickness Material properties are never exact UO 2 density Cladding material specification Soluble poison specification Operational impacts CRUD Rod bow Page 26 13
Uncertainty in Data Cross section measurements not exact Early techniques fairly uncertain for some materials Some of these measurements still in ENDF database Fe neutron transmission Thermal expansion coefficients Branching constants Decay constants Neutron yields Half life Page 27 Calculation Errors All deterministic methods employ some type of discretization scheme Finite differences Angular quadrature Energy partitioning (i.e. multi-group assumption) Convergence errors caused by ill formed solution Nodalization too coarse Bad quadrature weights Inherent errors from numerical methods Event well converged solutions are not perfect Page 28 14
Stochastic Errors Generally refer to Monte Carlo methods Modern codes typically employ continuous energy treatment No multi-group errors Can exactly represent complex geometry No finite difference errors Stochastic errors relate to under sampling Not enough particle histories to have good statistics Problem domain not fully sampled Even for well sampled problems, uncertainty remains Relates to the convolution of various probability distribution functions Page 29 Treatment of Uncertainty Typically handled by sensitivity calculations Mechanical uncertainties addressed by biasing model to extreme of tolerance Calculational uncertainties derived from assessment and applied as a bias Stochastic uncertainty addressed by upper bound 95/95 limit All of these treatments manifest themselves as an increase in margin between operating and safety limits Page 30 15
Typical Uncertainties K. Smith, Reactor Core Methods, M&C 2003 Page 31 So what is a Regulator to do? Take time to understand the physics and manufacturing processes Ask good questions Page 32 16
Survey of the more Common Physics Codes Page 33 WIMS http://www.answerssoftwareservice.com/wims/ Page 34 17
CASMO http://www.studsvik.com/documents/product-sheets/updated%20product%20sheets%20ssp/c5_2013-01_usa_r1.pdf Page 35 HELIOS http://www.studsvik.com/documents/product-sheets/updated%20product%20sheets%20ssp/helios.a4_el.pdf Page 36 18
MCU http://mcuproject.ru/eabout.html Page 37 SCALE http://scale.ornl.gov/index.shtml Page 38 19