1. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 1 CH.IV : CRITICALITY CALCULATIONS IN DIFFUSION THEORY CRITICALITY ONE-SPEED DIFFUSION MODERATION KERNELS REFLECTORS INTRODUCTION REFLECTOR SAVINGS TWO-GROUP MODEL

2. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 2 Objective solutions of the diffusion eq. in a finite homogeneous media exist without external sources 1 st study case: bare homogeneous reactor (i.e. without reflector) ONE-SPEED DIFFUSION With fission !!  Helmholtz equation with and BC at the extrapolated boundary:   : solution of the corresponding eigenvalue problem countable set of eigenvalues: IV.1 CRITICALITY criticality  A time-independent  can be sustained in the reactor with no Q

3. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 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. = material buckling = f(materials) Criticality:  Core displaying a given composition (B m c st ): determination of the size (B g variable) making the reactor critical  Core displaying a given geometry (B g c st ): determination of the required enrichment (B m )

4. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 4

5. 5 Time-dependent problem Diffusion operator:  Spectrum of real eigenvalues: s.t.with o = max i 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 J-K

6. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 6 Unique possible solution of the criticality problem whatever the IC: Criticality and multiplication factor k eff : production / destruction ratio Close to criticality:   o = fundamental eigenfunction associated to the eigenvalue k eff of:  media: Finite media: Improvement: with and criticality for k eff = 1

7. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 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. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 8 MODERATION KERNELS Definitions = moderation kernel: proba density function that 1 n due to a fission in is slowed down below energy E in = moderation density: 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 Objective: improve the treatment of the dependence on E w.r.t. one-speed diffusion

9. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 9 Inverting the previous expression: solution of Solution in finite media Additional condition: B 2  {eigenvalues} of (-  ) with BC on the extrapolated boundary   Criticality condition: with solution of  : fast non-leakage proba

10. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 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. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 11 INTRODUCTION No bare reactor Thermal reactors 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 (U 238 ) but  leakage from neutronics standpoint  Not considered here IV.2 REFLECTORS

12. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 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. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 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. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 14 Criticality reached for a thickness 2a satisfying this condition For a bare reactor:  Reflector savings:  In the criticality condition: As B c  << 1 : If same material for both reflector and moderator, with a D little affected by the proportion of fuel  D  D R Criticality: possible calculation with bare reactor accounting for 

15. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 15 TWO-GROUP MODEL Core Reflector Planar geometry: solutions s.t. ? Solution iff determinant = 0  2 nd -degree eq. in B 2  (one positive and one negative roots) For each root:

16. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 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 is then given to a constant. Why?

17. PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014 17 core reflector fast flux thermal flux