Fundamentals of Neutronics : Reactivity Coefficients in Nuclear Reactors

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Fundamentals of Neutronics Reactivity
Coefficients in Nuclear Reactors

• Paul Reuss
• Emeritus Professor
• at the Institut National des Sciences et
  Techniques Nucléaires

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Contents
A Neutron balance B Temperature effects C
Examples of design problems
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PART A

• Neutron balance

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Fission chain reaction

• Fissions ? Neutrons ? Fissions ? Neutrons ?
  Fissions ? Neutrons ?Etc.
• Fission yields
• About 200 MeV of energy (heat)
• About 2.5 fast neutrons (about 2 MeV)
• 2 fission products
• The scattering slows down the neutrons
  (thermalized neutron about 1/40 eV)

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Reactor types

• Fast neutron reactors
• Avoid the slowing down
• Use a highly enriched fuel
• Good neutron balance (breeding possible)
• Thermal neutron reactors
• Slow down the neutrons thanks to a moderator
• Great cross-sections of the fissile nuclei in the
  thermal range
• Therefore possibility to use a low enriched fuel
• Breeding impossible in practice

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Kinetics

• N ? kN ? k2N? k3N? k4N ?
• Equivalently N(0) exp(wt)
• Criticality k 1 or r (k - 1)/k 0
• Otherwise see inhour equation

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Inhour (or Nordheims) equationUranium 235
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Inhour (or Nordheims) equationPlutonium 239
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Neutron balance
The criticality is possible if the size is
sufficient
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Fermis four factor formula
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Uranium 238 capture cross-section(zoom)
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Uranium 238 effective integral
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Dancoffs factor (C)
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Examples for PWR cases
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Proposed k-infinity analysis
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Examples for PWR cases
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Examples for GFR cases
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PART B

• Temperature effects

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Stability of a reactor
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Temperature effects

• Doppler effect
• Broadening of the resonances
• Mainly of uranium 238 capture
• Negative (stabilizing) prompt effect
• Thermal spectrum effect
• No-proportionality of the absorption
  cross-sections
• Small effect (on f and h) for the PWRs
• Water expansion effect
• p decreases, f increases if Tm increases
• Main moderator effect for the PWRs

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Doppler effect resonance broadening
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Example of cross-section in the thermal range
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PART C

• Examples
• of design problems

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Main parameters of the PWR design

• Radius of the fuel
• Mainly thermal criteria
• Moderation ratio
• If it increases, p improves and f decreases
• There is an optimum of moderation
• A under-moderated design is chosen
• Fuel enrichment
• Get the adequate length of cycle

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Choice of the moderation ratio
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Problem of the boron poisoning

• Condition for a negative temperature coefficient
  ln(1/p) gt 1 f
• If CB increases, f decreases and this condition
  may be non fulfilled
• Therefore a limit on the boron concentration
• If the need of boron is greater than the limit,
  burnable poisons are used

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Evolution of the multiplication factor
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Burnable poisons

• Solid no positive expansion effect
• Efficient reduce the excess of reactivity at
  the beginning of cycle
• Burnable no more antireactivity at the end of
  cycle
• Usual materials B, Gd, Eu

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Problem of plutonium recycling

• Standard uranium fuel about 1 of plutonium
  after irradiation ? recycling interesting
• No FBR available ? recycling in the water
  reactors
• Great neutron absorption of the plutonium fuels ?
  control less efficient ? mixed core ? zoning of
  the MOX assemblies

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Evolution of the main heavy nuclides (PWR)
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Order of magnitude of the concentrations
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Typical isotopic composition of first generation
plutonium
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Main cross-sections in the thermal range
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Typical thermal spectra
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Problem of U/Pu interfaces
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Example of MOX PWR assembly
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Conclusions

• Major concerns criticality and negative
  temperature coefficients
• Criticality ? adjust the content in fissile
  material
• Temperature coefficients ? constraints on the
  design and the choice of materials
• Strong interactions between neutronics,
  thermalhydraulics, sciences of materials, etc.
• The safety analyses defines the limits
• The margins must be as great as possible to
  anticipate the evolutions
• Weight of history