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

 A time-independent  can be sustained in the reactor with no Q IV.1 CRITICALITY Objective solutions of the diffusion eq. in a finite homogeneous media exist without external sources 1st 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: criticality  A time-independent  can be sustained in the reactor with no Q

+ 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 (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)

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 J -K

and criticality for keff = 1 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

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

Solution in an  media: use of Fourier transform 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 (see chap.VII) 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

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

Examples of moderation kernels Reminder: convolution product in r  product of Fourier transforms in B Two-group diffusion Fast group:  Criticality eq.: G-group diffusion 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.

IV.2 REFLECTORS INTRODUCTION No bare reactor Thermal reactors 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

One-speed diffusion model 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

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 = ?

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 

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:

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?

fast flux thermal flux core reflector

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