CH.IV

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CH.IV  CRITICALITY CALCULATIONS IN DIFFUSION
THEORY

• CRITICALITY
• ONE-SPEED DIFFUSION
• MODERATION KERNELS
• REFLECTORS
• INTRODUCTION
• REFLECTOR SAVINGS
• TWO-GROUP MODEL

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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

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
• 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)

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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 lt 0 subcritical state
• ?o gt 0 supercritical state
• ?o 0 critical state with

J
-K
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• 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(lt0) ? 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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• 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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• 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
• 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.
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IV.2 REFLECTORS

• 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 (U238) but ? leakage from
  neutronics standpoint
• ? Not considered here

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• 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? ltlt 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 ? ab
• 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
reflector
core
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CH.IV  CRITICALITY CALCULATIONS IN DIFFUSION
THEORY

• CRITICALITY
• ONE-SPEED DIFFUSION
• MODERATION KERNELS
• REFLECTORS
• INTRODUCTION
• REFLECTOR SAVINGS
• TWO-GROUP MODEL

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