Lesson 3 Objectives

1
Lesson 3 Objectives

• Finish up MAGICMERV
• Transport theory overview for users
• Comparison of Deterministic vs Stochastic
• Discrete ordinates
• Monte Carlo

2
MAGICMERV

• Simple checklist of conditions that MIGHT result
  in an increase in k-eff.
• Mass
• Absorber loss
• Geometry
• Interaction
• Concentration
• Moderation
• Enrichment
• Reflection
• Volume

2
3

• Discrete ordinates overview

4
Neutron balance equation

• Scalar flux/current balance

• Used in shielding, reactor theory, crit. safety,
  kinetics
• Problem No sourceNo solution !

5
K-effective eigenvalue

• Changes n, the number of neutrons per fission

• Advantages
• Everybody uses it
• Guaranteed real solution
• Fairly intuitive (if you dont take it too
  seriously)
• Good measure of distance from criticality for
  reactors
• Disadvantages
• No physical basis
• Not a good measure of distance from criticality
  for CS

6
Buckling eigenvalue

• Adds extra leakage using Diffusion Theory
  buckling

• Advantages
• Physical basis
• Mildly intuitive (after you work with it awhile)
• Good measure of distance from criticality for
  reactors
• Disadvantages
• No guaranteed real solution
• Not intuitive for CS (?)

7
Time (a) eigenvalue

• Adds extra leakage using Diffusion Theory
  buckling

• Advantages
• Physical basis
• Mildly intuitive for kinetics work
• Disadvantages
• No guaranteed real solution
• Not intuitive for CS (?)

8
Material search eigenvalue

• Change atom density of some isotope(s) to achieve
  balance

• Advantages
• Ultimate physical basis
• Great for design
• Guaranteed real solution (fuel isotope)
• Disadvantages
• No guaranteed real solution (non-fuel isotope)
• Only useful for answering particular questions

9
Neutron transport

• Scalar vs. angular flux
• Boltzmann transport equation
• Neutron accounting balance
• General terms of a balance in energy, angle,
  space
• Deterministic
• Subdivide energy, angle, space and solve equation
• Get k-effective and flux
• Monte Carlo
• Numerical simulation of transport
• Get particular flux-related answers, not flux
  everywhere

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Deterministic grid solution

• Subdivide everything Energy, Space, Angle
• Eulerian grid Fixed in space
• Use balance condition to figure out each piece

11
Stochastic solution

• Continuous in Energy, Space, Angle
• Lagrangian grid Follows the particle
• Sample (poll) by following typical neutrons

Absorbed
Fissile
Fission
Particle track of two 10 MeV fission neutrons
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Discrete ordinates

• We will cover the mathematical details in 3 steps
• Numerical methods to determine the neutron flux
  field from source
• Numerical treatment of source
• Putting it together into a solution strategy
• Goal Give you just enough details for you to be
  an intelligent user
• Cocktail party knowledge

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Advantages and disadvantages of each
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I. Numerical treatment

• Subset of NE581
• We will use the easiest form 1D slab
• Discretization in all 3 dimensions space,
  energy, direction
• Space Relatively smooth, but equation has
  spatial derivativesFinite difference method
• Energy Extremely non-smooth (resonances), but no
  derivativesMultigroup treatment (complexity
  reduction)
• Direction Smooth, no derivativesQuadrature
  integration
• Named for the directional treatment

15
Directional treatment

• Mathematical basis Quadrature integration

• Lots of math research done in defining optimum
  (m,w) sets (quadratures)
• Most commonGaussian quadratures
• mn are roots of Pn(m)
• wn are chosen to perfectly integrate the first n
  moments

16
Application to our problem

• Solve for the flux only in particular directions
• or
• Scalar flux found with quadrature integration
• Bottom line for us
• More anglesmore accuracy
• CSAS1X defaults to 8 directions (S8)
• Available4, 8, 16, 32,

17
Energy treatment Multigroup

• Mathematical basis Complexity reduction using
  assumed flux spectra
• Define

• Find equation for by integrating the
  continuous equation

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Energy treatment, contd

• Weight cross sections with an assumed flux
  spectrum shape
• Cross sections are only as good as the assumed
  shapes
• Common assumption (Fission, 1/E, Maxwellian) for
  smooth cross sections spectrum from resonance
  treatments for resonance cross sections
• Bottom Line for us
• More groups are (theoretically) better
• In practice it depends on the group structure and
  the assumed spectra within the groups
• We choose the group structure by choosing from
  the choices given Hansen-Roach, 27 group, 44
  group, 238 group, etc.

19
Space treatment Cell centered finite difference

• Mathematical basis Subdivide the space into
  homogeneous cells, integrate equation over each
  cell

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Space treatment, contd

• One equation, three unknowns

• Incoming boundary flux (i.e., for
  mgt0, for mlt0) is known as the
  outgoing of neighbor
• Eliminates one unknown
• Still left with 1 equation, 2 unknowns, so we
  must assume another relationship among the three
  unknowns.
• Step
• Guaranteed positive flux, but not very accurate
• Diamond Difference
• Surprisingly accurate, but requires a negative
  flux fixup

21
Space treatment, contd

• Bottom Line for us
• More cells are better
• SCALE picks the spatial discretization
  automatically, but you can control it
• (Something of a kludge)

22
II. Source treatment anisotropic scattering

• Mathematical basis Approximation of functions
  with orthogonal basis functions
• Usual fit Low, odd-ordered Legendre polynomials

23
Source treatment, contd

• Bottom Line for us
• Cross-sections libraries are built with a maximum
  Legendre scattering order prescribed
• More is better, but more resource intensive
• Generally, for criticality
• is too small
• is too large
• (Need much higher to handle gamma ray transport)

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III. Iterative solution strategy

• Assume a scalar flux shape for each space
  cell and energy group
• Compute the spatial fission distribution and,
  from it, the fission neutron source
• For each energy group, g1,2,G (High energy to
  low)
• Find down-scatter from fluxes already caculated
  in this loop
• Find up-scatter (if any) from previous
  iterations fluxes
• Use best available angular fluxes to get
  within-group scattering terms
• For each angular direction, n
• Sweep over cells (following flow of neutrons),
  calculating the cell average angular fluxes,
• Contribute to the scalar flux in each cell,
• Repeat C-E until the group scalar fluxes converge
  (inner iteration)
• Repeat 2-4 until the fission spatial shape and
  k-effective eigenvalue converge

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Iterative solution, contd

• Bottom Line for us
• Flux solution is an iterative, bootstrap method
  with 2 levels of iteration
• Outer (or power iteration) consists of a sweep
  over all groups to get an improved spatial
  fission shape
• Inner iteration occurs within each group and
  consists of a sweep over each direction and cell
  in the group to get an improved within-group
  scattering source
• We get one k-effective eigenvalue per outer

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• Monte Carlo overview

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Monte Carlo

• We will cover the mathematical details in 3 steps
• General overview of MC approach
• Example walkthrough
• Special considerations for criticality
  calculations
• Goal Give you just enough details for you to be
  an intelligent user

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General Overview of MC

• Monte Carlo Stochastic approach
• Statistical simulation of individual particle
  histories
• Keep score of quantities you care about
• Most of mathematically interesting features come
  from variance reduction methods
• Gives results PLUS standard deviation
  statistical measure of how reliable the answer is

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Mathematical basis

• Statistical simulation driven by random number
  generator 0ltxlt1, with a uniform distribution
• p(x)dxprob. of picking x in (x, xdx)dx
• Score keeping driven by statistical formula

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Simple Walkthrough

• Six types of decisions to be made
• Where the particle is born
• Initial particle energy
• Initial particle direction
• Distance to next collision
• Type of collision
• Outcome of scattering collision (E, direction)
• How are these decisions made?

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Decision 1 Where particle is born

• 3 choices
• Set of fixed points (first generation only)
• User specifies how many chosen from each
• Uniformly distributed (first generation only)
• KENO picks point in geometry
• Rejected if not in fuel
• From previous collision sites
• After first generation, previous generations
  sites used to start new fissions

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Decision 2 Initial particle energy

• From a fission neutron energy spectrum
• Complicated algorithms based on advanced
  mathematical treatments
• Rejection method based on bounding box idea

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Decision 3 Initial particle direction

• Easy one for us because all fission is isotropic
• Must choose the longitude and latitude that
  the particle would cross a unit sphere centered
  on original location
• Mathematical results
• mcosine of angle from polar axis
• Fazimuthal angle (longitude)

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Decision 4 Distance to next collision

• Let sdistance traveled in medium with St
• Prob. of colliding in ds at distance s
• (Prob. of surviving to s)x(Prob. of colliding in
  dssurvived to s)
• (e- St s)x(Sts)
• Using methods from NE582, this results in

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Decision 5 Type of collision

• Most straight-forward of all because straight
  from cross sections
• Ss / St probability of scatter
• Sa / St probability of absorption
• Therefore, if xlt Ss / St , it is a scatter
• Otherwise particle is absorbed (lost)
• Weighting in lieu of absorptionfancy term for
  following the non-absorbing fraction of the
  particle

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Decision 6 Outcome of scattering collision

• In simplest case (isotropic scattering), a
  combination of Decisions 3 and 5
• Decision 5-like choice of new energy group

• Decision 3 gets new direction
• In more complicated (normal) case, must deal with
  the energy/direction couplings from elastic and
  inelastic scattering physics

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Special considerations for criticality safety

• Must deal w/ generationsouter iteration
• Fix a fission source spatial shape
• Find new fission source shape and eigenvalue
• User must specify of generations AND of
  histories per generation AND of generation to
  skip
• Skipped generations allow the original lousy
  spatial fission distribution to improve before we
  really start keeping score
• KENO defaults 103 generations of 300 histories
  per generation, skipping first 3