NE 455555 Nuclear Reactor Analysis II

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NE 455/555Nuclear Reactor Analysis II

• Lecture 5Structure of , power
  iteration

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Remarks about the derivation of the multigroup
equations

• Remark 1
• We can use our multigroup derivation procedure
  (Lecture 4) to perform a group collapse.
• Start from the multigroup equations with a large
  number (G1) groups.
• Result is a similar system with fewer (G2)
  groups.
• Need a spectral function defined on the
  fine group structure.
• Trade-off loss of accuracy, but less expensive
  computationally.
• Remark 2
• Our multigroup derivation will be exact if
• and
  are constants for E and
  E in each energy group, OR

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What is the structure of our scattering transfer
matrix?

• In scattering processes, high-energy neutrons
  lose energy and thermal neutrons gain or lose
  energy.
• If we have only one thermal energy group, then
  sincethen

Upscattering
Downscattering
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It is also possible to have direct coupling of
our energy groups.

• Or, neutrons can scatter downward at most one
  group at a time.

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We now assume that there is no upscatter.

• We can define the removal cross-section
• Then, with no upscatter, our multigroup equations
  become
• These are the multigroup diffusion equations with
  fission and downscattering (but no upscattering).

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How do we solve these equations numerically?

• If we write out the steady-state, fixed source
  multigroup equations we getwhere

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This suggests the following iteration scheme

• and
• The boundary conditions are vacuum

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When do we stop iterating?

• A couple of remarks are in order
• At each step in the scheme, we only need to solve
  a one-group diffusion equation.
• In one-dimensional problems, each 1-group
  diffusion problem can be solved directly
  (Gaussian elimination).
• 2- or 3-d problems require an iterative solver in
  each group. These are the inner iterations.
• A single sweep through all the groups is called
  an outer iteration.

where typically e 10-5