CH.IV : CRITICALITY CALCULATIONS IN DIFFUSION THEORY

1. CH.IV : CRITICALITY CALCULATIONS IN DIFFUSION THEORY CRITICALITY • ONE-SPEED DIFFUSION • MODERATION KERNELS REFLECTORS • INTRODUCTION • REFLECTOR SAVINGS • TWO-GROUP MODEL

2. IV.1 CRITICALITY criticality Objective solutions of the diffusion eq. in a finite homogeneous media existwithoutexternal sources 1ststudy case: barehomogeneousreactor(i.e. withoutreflector) ONE-SPEED DIFFUSION With fission !! • Helmholtz equation with and BC at the extrapolatedboundary:   : solution of the correspondingeigenvalueproblem countable set of eigenvalues:  A time-independent  can be sustained in the reactor with no Q

3. + associated eigenfunctions: orthogonal basis • A unique solution positive everywhere  fundamental mode • Flux ! Eigenvalue of the fundamental – two ways to express it: 1. = geometric buckling = f(reactor geometry) 2. = materialbuckling = f(materials) Criticality: • Core displaying a given composition (Bm cst): determination of the size (Bg variable) making the reactor critical • Core displaying a given geometry (Bg cst): determination of the required enrichment (Bm)

5. J -K Time-dependent problem Diffusion operator:  Spectrum of real eigenvalues: s.t. with o = maxi i associated to : min eigenvalue of (-) o associated to o: positive all over the reactor volume Time-dependent diffusion: Eigenfunctions i: orthogonal basis  • o < 0 : subcritical state • o > 0 : supercritical state • o = 0 : critical state with

6. Unique possible solution of the criticality problem whatever the IC: Criticality and multiplication factor keff: production / destruction ratio Close to criticality: • o = fundamental eigenfunction associated to the eigenvalue keff of:  media: Finite media: Improvement: with and criticality for keff = 1

7. Independent sources Eigenfunctions i : orthonormal basis Subcritical case with sources: possible steady-state solution • Weak dependence on the expression of Q, mainly if o(<0)  0 • Subcritical reactor: amplifier of the fundamental mode of Q  Same flux obtainable with a slightly subcritical reactor + source as with a critical reactor without source

8. Objective: improve the treatment of the dependence on E w.r.t. one-speed diffusion MODERATION KERNELS Definitions = moderation kernel: proba density function that 1 n due to a fission in is slowed down below energyE in = moderationdensity: nb of n (/unit vol.time) slowed down below E in with  media: translation invariance  Finite media: no invariance  approximation Solution in an  media: use of Fourier transform

9. Inverting the previous expression: solution of Solution in finite media Additional condition: B2  {eigenvalues} of (-) with BC on the extrapolated boundary  • Criticalitycondition: with solution of  : fast non-leakage proba

10. Examples of moderation kernels Two-group diffusion Fast group:  • Criticality eq.: G-group diffusion • Criticality eq.: Age-diffusion(see Chap.VII)  Criticality eq.: (E) = age of n at en. E emitted at the fission en.  = age of thermal n emitted at the fission en.

11. IV.2 REFLECTORS INTRODUCTION No bare reactor Thermalreactors Reflector • backscatters n into the core • Slows down fast n (composition similar to the moderator)  Reduction of the quantity of fissile material necessary to reach criticality  reflector savings Fast reactors n backscattered into the core? Degraded spectrum in E  Fertile blanket (U238) but  leakage from neutronics standpoint  Not considered here

12. REFLECTOR SAVINGS One-speed diffusion model • In the core:  with • In the reflector:  Solution of the diffusion eq. in each of the m zones  solution depending on 2.m constants to be determined • Use of continuity relations, boundary conditions, symmetry constraints… to obtain 2.m constraints on these constants • Homogeneous system of algebraic equations: non-trivial solution iff the determinant vanishes • Criticality condition

13. Solution in planar geometry Consider a core of thickness 2a and reflector of thickness b (extrapolated limit) Problem symmetry Flux continuity +BC: Current continuity:  criticality eq. Q: A = ?

14. Criticality reached for a thickness 2a satisfying this condition For a bare reactor:  Reflector savings: • In the criticality condition: As Bc << 1 : If same material for both reflector and moderator, with a D little affected by the proportion of fuel  D  DR Criticality: possible calculation with bare reactor accounting for 

15. TWO-GROUP MODEL Core Reflector Planar geometry: solutions s.t. ? Solution iff determinant = 0  2nd-degree eq. in B2  (one positive and one negative roots) For each root:

16. Solution in the core for [-a, a]: Solution in the reflector for a  x  a+b: 4 constants + 4 continuity equations (flux and current in each group) • Homogeneous linear system • Annulation of the determinant to obtain a solution • Criticality condition Q: the flux isthengiven to a constant. Why?

17. fast flux thermal flux reflector core