Fundamentals of Neutronics :Reactivity Coefficients in Nuclear

Reactors

Paul Reuss

Emeritus Professorat the Institut National des Sciences et Techniques

Nucléaires

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Contents

A – Neutron balance

B – Temperature effects

C – Examples of design problems

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(t)

• Criticality : k = 1 or : = (k - 1)/k = 0

• Otherwise : see inhour equation

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Inhour (or Nordheim’s) equationUranium 235

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Inhour (or Nordheim’s) equationPlutonium 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

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 ) 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

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) > 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