fuel burnup calculations and uncertainties

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Fuel Burnup Calculations and Uncertainties

International Atomic Energy AgencyPage 2

Outline

� Review lattice physics methods

� Different approaches to burnup predictions

� Linkage to fuel safety criteria

� Sources of uncertainty

� Survey of available codes

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Reactor Physics Challenge

Go from here to � here without losing too much information


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

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Reactor Physics Computational Strategy

K. Smith, “Reactor Core Methods,” M&C 2003

Circa 1980


Modern Approaches


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Lattice to Whole Core Analyses



A Brute Force Approach … not possible


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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


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

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Discrete Ordinates

� Refers to treatment of angular variable

� Spatial variable treatment varies• Finite difference type approach

• Characteristics based methods


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

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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


Modeling Approaches

� Once again, choice

depends on desired

accuracy

� Modern approaches

predict burnup pin by

pin

� Historical approaches

perform calculations at

the lattice level

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Computational Strategy

� Historically core burnup

calculations have been

performed based on pre-

calculated cross section

libraries

• Account for all relevant

physics

� Fuel temperature

� Moderator conditions

� Exposure

� Xe/Sm

� Control rods

� etc.


Lattice Physics Computational Flow

http://scale.ornl.gov/index.shtml

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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.)



Latest Developments

� Main difference is that lattice physics is embedded

into core diffusion code• Eliminates intermediate library

• Better captures real physics

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Linkages to Safety Criteria

� Input to fuel mechanical code

� Predict reactivity coefficients

� Peaking factors

� Fuel burnup

� Reactor operations


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

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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


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

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Fuel burnup

� Limits derived from fuel mechanical simulation• Imposed by regulatory authority

� Simulation needed to demonstrate compliance• Measurements difficult and uncertain


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

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Sources of Uncertainty

� Mechanical

� Data

� Calculational error

� Stochastic error


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• UO2 density

• Cladding material specification

• Soluble poison specification

� Operational impacts• CRUD

• Rod bow

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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


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

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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


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

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Typical Uncertainties



So what is a Regulator to do?

� Take time to understand the physics and

manufacturing processes

� Ask good questions

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Survey of the more Common Physics Codes


WIMS

http://www.answerssoftwareservice.com/wims/

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CASMO

http://www.studsvik.com/Documents/Product-sheets/Updated%20product%20sheets%20SSP/C5_2013-01_USA_R1.pdf


HELIOS

http://www.studsvik.com/Documents/Product-sheets/Updated%20product%20sheets%20SSP/helios.a4_el.pdf

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MCU

http://mcuproject.ru/eabout.html


SCALE

http://scale.ornl.gov/index.shtml