Fundamentals of Neutronics : Reactivity Coefficients in Nuclear Reactors Paul Reuss Emeritus Professor at the Institut National des Sciences et Techniques Nucléaires

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

PART A Neutron balance

4 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)

5 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

6 Kinetics N  kN  k 2 N  k 3 N  k 4 N  … Equivalently : N(0) exp(  t) Criticality : k = 1 or :  = (k - 1)/k = 0 Otherwise : see inhour equation

7 Inhour (or Nordheim’s) equation Uranium 235

8 Inhour (or Nordheim’s) equation Plutonium 239

9 Neutron balance The criticality is possible if the size is sufficient

10 Fermi’s four factor formula

11 Uranium 238 capture cross-section (zoom)

12 Uranium 238 effective integral

13 Dancoff’s factor (C)

14 Examples for PWR cases

15 Proposed k-infinity analysis

16 Examples for PWR cases

17 Examples for GFR cases

PART B Temperature effects

19 Stability of a reactor

20 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 T m increases –Main moderator effect for the PWRs

21 Doppler effect : resonance broadening

22 Example of cross-section in the thermal range

PART C Examples of design problems

24 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

25 Choice of the moderation ratio

26 Problem of the boron poisoning Condition for a negative temperature coefficient : ln(1/p) > 1 – f If C B 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

27 Evolution of the multiplication factor

28 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…

29 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

30 Evolution of the main heavy nuclides (PWR)

31 Order of magnitude of the concentrations

32 Typical isotopic composition of first generation plutonium

33 Main cross-sections in the thermal range

34 Typical thermal spectra

35 Problem of U/Pu interfaces

36 Example of MOX PWR assembly

37 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