Multigroup diffusion equations

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NE 455/555Nuclear Reactor Analysis II

• Lecture 4 1/19/00
• Multigroup diffusion equations

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Our current model of neutrons in a reactor has a
number of limitations

• All neutrons are characterized by a single speed
  or energy
• Neutrons in a reactor have energies that span the
  range from 10 MeV to 0.01 eV - 9 orders of
  magnitude.
• Cross-sections depend sensitively on neutron
  energy.
• The distribution of neutrons, then, will also
  have a sensitive dependence on energy.
• We will not treat the energy as a continuous
  variable. We discretize it into energy intervals
  or groups.

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We will need to define multigroup fluxes and
equations to solve for them.

• We choose to define the group flux as the
  integral of the flux over the energy group.
• The equations will be coupled together.
• Example fission neutrons are born in the highest
  energy groups and cascade downward in energy as
  they are moderated.
• How many groups do we need?
• Many groups for fast reactors or fast neutron
  problems
• Few groups are good for thermal reactors
• We will need to calculate group-averaged
  cross-sections or multigroup constants.

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A heuristic derivation of the multigroup equations

• Consider a typical energy group g
• Lets balance the ways in which neutrons can
  enter or leave the group

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We need some definitions...

• First the scattering
• Absorption
• Source term
• Diffusion term define such that leakage
  from group g can be approximated as

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If we combine all the terms we get

• If we separate out the contribution of the source
  due to fissions, we obtainhere is the
  probability that a fission neutron is born with
  an energy in group g.

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We would like to be able to derive the multigroup
diffusion equations

• First, recall the energy dependent diffusion
  equationwhich we will assume holds for
  all points in spatial domain D, for all times
  tgt0, and for energies

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Multigroup derivation, cont

• Now we can approximate the integrals that we have
  over
• O.k. Lets integrate our energy dependent
  diffusion equation over the g-th energy group

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Multigroup derivation, cont

• At this point, we make the approximation that the
  flux is separablewhere is a precomputed
  spectral function.
• Now we make the definition of the group flux

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Multigroup derivation, cont

• Plug this into our previous equation, term by
  term(1)

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Multigroup derivation, cont

• (2)

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Multigroup derivation, cont

• (3)

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Multigroup derivation, cont

• (4)(5)

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Multigroup derivation, cont

• (6)
• (7)

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Combining these results gives us the multigroup
diffusion equation.

• Summary
• Determine the spectral function
• Determine the group structure
• Determine the multigroup constants
• Solve the multigroup diffusion equations