Transcript Fundamentals of Neutronics : Reactivity Coefficients in

Fundamentals of Neutronics :
Reactivity Coefficients in Nuclear
Reactors
Paul Reuss
Emeritus Professor
at the Institut National des Sciences et Techniques
Nucléaires
Contents
A – Neutron balance
B – Temperature effects
C – Examples of design problems
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PART A
Neutron balance
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 Nordheim’s) equation
Uranium 235
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Inhour (or Nordheim’s) equation
Plutonium 239
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Neutron balance
The criticality is possible if the size is sufficient
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Fermi’s 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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Dancoff’s 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
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
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) > 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
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