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CH.IV : CRITICALITY CALCULATIONS IN DIFFUSION THEORY
ONE-SPEED DIFFUSION MODERATION KERNELS REFLECTORS INTRODUCTION REFLECTOR SAVINGS TWO-GROUP MODEL
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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
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+ associated eigenfunctions: orthogonal basis
A unique solution positive everywhere fundamental mode Flux ! Eigenvalue of the fundamental – two ways to express it: = geometric buckling = f(reactor geometry) = 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)
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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
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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
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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
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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
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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
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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.
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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
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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
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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 = ?
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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
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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:
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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?
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fast flux thermal flux core reflector
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CH.IV : CRITICALITY CALCULATIONS IN DIFFUSION THEORY
ONE-SPEED DIFFUSION MODERATION KERNELS REFLECTORS INTRODUCTION REFLECTOR SAVINGS TWO-GROUP MODEL
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