1. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 1 CH.VI: REACTIVITY BALANCE AND REACTOR CONTROL REACTIVITY BALANCE OPERATION AND CONTROL CHARACTERISTIC TIMES INTRODUCTION TO PERTURBATION THEORY NEUTRON IMPORTANCE REACTIVITY COEFFICIENTS DEFINITION EXAMPLES LONG-TERM NEED FOR REACTIVITY CONTROL CONTEXT ISOTOPE CONCENTRATION EVOLUTION XENON EFFECT XENON POISONING XENON OSCILLATIONS MEANS TO ENSURE CONTROL EXTERNAL MEANS REACTIVITY EVOLUTION

2. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 2 VI.1 REACTIVITY BALANCE OPERATION AND CONTROL Variation of the reactor parameters  reactivity  Loss of the neutron cycle equilibrium  transient  Control Criticality to maintain/manage in all circumstances: power, shutdown, cold shutdown, new/used fuel, whatever q ty of fission products…  Reactivity margins:  available at any moment  same magnitude as and opposite sign to the reactivity change caused by any factor affecting   Characteristic time comparable to that on which  occurs

3. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 3 CHARACTERISTIC TIMES Some orders of magnitude Consumption of fissile matter1000 h Xenon effect (see below)10 h Delayed n10 s Circulation of coolant in the primary circuit10 s Transit of the coolant in the core1 s Heat transfer from the fuel element to the coolant 0.1 s Asymptotic period at the prompt-critical threshold for  =10 -8 s (small fast reactor) 10 -3 s Mean lifetime of the n10 -3 - 10 -8 s

4. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 4 INTRODUCTION TO PERTURBATION THEORY Necessity to be able to compute in all situations In practice, calculation of  rarely possible because  Actual reactor geometry  ideal geometry used in the computations  Presence of detectors in the core  Consummation and production of isotopes = non-uniform f(t) Simple way to estimate  : perturbation from a reference stationary state  Perturbed state:

5. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 5 Let  : arbitrary weight function  Static reactivity: If : solution of the adjoint reference problem Reference statePerturbed state JoJo J = J o +  J KoKo K = K o +  K oo  =  o +  0 ()() 1 st order

6. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 6 NEUTRON IMPORTANCE Physical meaning of the adjoint flux Introduction of 1 n at point with velocity in a critical reactor  secondary n and   Corresponding augmentation of  ?  The more important the added n, the larger the increase Consider a reaction rate with Importance H(P) of a n – entering a collision at P – for R?  Direct contribution due to a collision at point P: f(P)  Expected contribution due to the next collisions: (see chap.2)

7. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 7  Adjoint equation: H(P)   *(P) Expression of the reaction rate based on importance? n emitted by the source, then transported to a 1 st collision Adjoint transport problem in differential form + adjoint BC for a reactor in vacuum: no importance of the outgoing n through  One speed case: ifsolution of the direct problem on the volume V of the reactor with BC in vacuum, then solution of the adjoint problem with adjoint BC in vacuum

8. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 8 Adjoint diffusion problem with BC at the extrapolated boundary One speed diffusion  diffusion operator: self-adjoint   *   (at a c st ) Ex: impact of a cross section variation: Variation of  weighted by the flux squared  Application:  of a control rod more important at mid-height in the core

9. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 9 VI.2 REACTIVITY COEFFICIENTS DEFINITION Reactivity variations calculable by perturbation theory  Trace back the causes of the variations of J and K ?  Modification of the isotope density  Dilatation due to the  of t o  Production/destruction of isotopes  Void rate (BWR mainly)  Move of matter (expulsion of coolant outside the core)  Modification of microscopic cross sections  Doppler effect (see chap.VIII) NB: Effects due to variations of power, or of fuel or coolant t o

10. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 10 Variation  T c of the fuel t o One speed diffusion model: Let, with  c : mean t o and the spatial distribution of T c. If perturbation  T c only affects  c : < 0 (dilatation) > 0 (Doppler)  0  : reactivity coefficient

11. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 11 In general N independent parameters  i  N reactivity coef. s.t. EXAMPLES Power coefficient If  i fct of , hence of P: Doppler coefficient Two t o to account for: fuel T c and moderator T m < 0 for stability! (Doppler effect) and Both < 0 Fast variationsSlower variations

12. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 12 VI.3 LONG-TERM NEED FOR REACTIVITY CONTROL CONTEXT Time-dependent issues considered up to now (see chap.V) on time scales characteristic of prompt/delayed n generation Longer-term time-dependent effects to be considered in the neutron balance: consumption of fissile material, decay of fission products…  Interaction: material consumption/production dependent on , which in turn depends on the material composition of the reactor Reaction rates  (Boltzmann)

13. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 Time scales likely to be different, however  Usually:  flux calculations with N i constant at each time step  t of the irradiation history of the fuel (from ‘begin of cycle’ (BOC) till ‘end of cycle’ (EOC))  then N i evolution (via a depletion code) at the end of the time step with  constant  Burnup calculations 13 Irradiation history T BOC T EOC ∆t (possibility to do better than an explicit Euler scheme but calculations of  are time-consuming)

14. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 14 ISOTOPE CONCENTRATION EVOLUTION Source balance for isotope i Positive sources Isotope i as a fission fragment (fraction  ji of fissions with j) Isotope i as a result of a n capture by isotope ‘i-1’ Radioactive decay from parent isotopes Negative sources Isotope i absorbing (capture + fission) a n Radioactive decay to daughter isotopes (Bateman equations)

15. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 15 VI.4 XENON EFFECT XENON POISONING  a (Xe 135 ) = 2.7 10 6 barn at 2200 ms -1 (thermal) !!  Particular role among all fission products Production? Let X, I be the atomic densities of Xe 135 and I 135 (stable) Fission  I = 0.061  X = 0.003 < 0.5 min 6.7 h9.2 h2.6 10 6 ans (  a (I 135 ) neglected) ( I = 2.89 10 -5 s -1 ) ( X = 2.09 10 -5 s -1 ) Linked to the current  Linked to the  before (Bateman equations) (Q: other assumptions?)

16. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 16

17. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 17 In stationary regime with constant flux: Saturation in Xe for Let : ratio of the nb of n absorbed by Xe over the nb of fission n  Reactivity (1G diffusion) :  Positive reactivity margin to have in store! and (U 235 )

18. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 18 Reactor shutdown in asymptotic regime We have [Xe] increases due to disintegration of I 135 without destruction by the n flux ([Xe] maximum after  11h), then decreases If   starting from a stationary regime, [Xe]  first before  Negative reactivity following the maximum in [Xe] Other isotope (poison) with similar effects: Sm (U 235 )

19. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 19 XENON OSCILLATIONS Reactor of large size, i.e. R(radius)/L(diffusion length) >> 1  Sufficiently distant regions:  Both critical  Might be seen as +/- uncoupled TimingZone 1Zone 2 t = 0  starts to swing   Short tConc.Xe?Production? Mainly due to fission  10h earlier (see  I >>  X ) Destruction? Present fission  X   X X Reactivity?  <0  >0 Swing increased     Longer tConc.Xe?XX XX Reactivity?  >0  <0 Swing reversed   … Accurate calculation? Complex (no point kinetics!) Risks? Power peaks, but long characteristic time  Easily detected Mitigation? Differential insertion of the control rods

20. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 20 VI.5 MEANS TO ENSURE CONTROL EXTERNAL MEANS Control rods  Highly absorbing isotopes (e.g. Ag 80%, In 15%, Cd 5%)  Impenetrable for thermal n  Decreasing  in their neighborhood  Reactivity source > or < 0 in normal operation  Prompt anti-reactivity source if scram Chemical poisons Boric acid: uniformly distributed reactivity source  spatial power distribution unchanged

21. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 21 Consumable poisons (e.g. borate pyrex rods B 2 O 3 ) Isotopes with high , initially put inside the reactor and depleted because of the (n,  ) reaction   of  a and compensation for:  of  a due to fission products  of ( f -  a ) due to the depletion of the fissile matter REACTIVITY EVOLUTION  Cold reactor, P = 0, no poisons (Xe, Sm): k eff = 1.229  Reactor in power, poisons in a steady state: k eff = 1.126 cause   due to the  of both the moderator and fuel t o Criticality? Obtained by partly inserting the control rods (PWR with fresh fuel)