CH.VI: REACTIVITY BALANCE AND REACTOR CONTROL

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. 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 qty 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. CHARACTERISTIC TIMES Some orders of magnitude

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. Let : arbitrary weight function • Static reactivity: If : solution of the adjoint reference problem 1st order

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 ImportanceH(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. Adjoint equation: H(P)  *(P) Expression of the reaction rate based on importance? n emitted by the source, then transported to a 1st collision Adjoint transport problem in differential form + adjoint BC for a reactor in vacuum: no importance of the outgoing n through  One speed case: if solution 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. Adjoint diffusion problem with BC at the extrapolated boundary One speed diffusion  diffusion operator: self-adjoint  *   (at a cst) 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. 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 to • 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 to

10. Variation Tc of the fuel to One speed diffusion model: Let , with c: mean to and the spatial distribution of Tc. If perturbation Tc only affects c: > 0 (Doppler) < 0 (dilatation)  0  : reactivity coefficient

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 to to account for: fuel Tc and moderator Tm < 0 for stability! (Doppler effect) and Both < 0 Fast variations Slower variations

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. Time scales likely to be different, however • Usually: • flux calculations with Ni constant at each time step t of the irradiation history of the fuel (from ‘begin of cycle’ (BOC) till ‘end of cycle’ (EOC)) • then Ni evolution (via a depletion code) at the end of the time step with  constant • Burnupcalculations (possibility to do betterthan an explicit Euler scheme but calculations of  are time-consuming) Irradiation history TBOC TEOC ∆t

14. ISOTOPE CONCENTRATION EVOLUTION Source balance for isotope i Positive sources • Isotope i as a fission fragment (fraction jiof fissions with j) • Isotope i as a result of a n capture by isotope ‘i-1’ • Radioactive decay from parent isotopes Negative sources • Isotope iabsorbing (capture + fission) a n • Radioactive decay to daughter isotopes (Bateman equations)

15. VI.4 XENON EFFECT XENON POISONING a(Xe135) = 2.7 106 barn at 2200 ms-1 (thermal) !! • Particular role among all fission products Production? Let X, I be the atomic densities of Xe135 and I135 Fission I = 0.061 X = 0.003 (stable) < 0.5 min 6.7 h 9.2 h 2.6 106 ans (Bateman equations) (a(I135) neglected) (I = 2.89 10-5 s-1) (X = 2.09 10-5 s-1) Linked to the current  Linked to the  before (Q: other assumptions?)

17. and 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! (U235)

18. Reactor shutdown in asymptotic regime We have [Xe] increases due to disintegration of I135 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 (U235)

19. Accurate calculation? Complex (no point kinetics!) XENON OSCILLATIONS Reactor of large size, i.e. R(radius)/L(diffusion length) >> 1 • Sufficiently distant regions: • Both critical • Might be seen as +/- uncoupled Risks? Power peaks, but long characteristic time  Easily detected Mitigation? Differential insertion of the control rods

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. Consumable poisons (e.g. borate pyrex rods B2O3) Isotopes with high , initially put inside the reactor and depleted because of the (n,) reaction   of aand 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): keff = 1.229 • Reactor in power, poisons in a steady state: keff = 1.126 cause   due to the  of both the moderator and fuel to Criticality? Obtained by partly inserting the control rods (PWR with fresh fuel)