ONE SPEED BOLTZMANN EQUATION

1
CH.III  APPROXIMATIONS OF THE TRANSPORT EQUATION

• ONE SPEED BOLTZMANN EQUATION
• ONE SPEED TRANSPORT EQUATION
• INTEGRAL FORM
• RECIPROCITY THEOREM AND COROLLARIES
• DIFFUSION APPROXIMATION
• CONTINUITY EQUATION
• DIFFUSION EQUATION
• BOUNDARY CONDITIONS
• VALIDITY CONDITIONS
• P1 APPROXIMATION IN ONE SPEED DIFFUSION
• ONE SPEED SOLUTION OF THE DIFFUSION EQUATION
• MULTI-GROUP APPROXIMATION

2
III.1 ONE SPEED BOLTZMANN EQUATION

• ONE SPEED TRANSPORT EQUATION
• ? Suppressing the dependence on v in the
  Boltzmann eq.
• Let expected nb of secundary n/interaction,
• and distribution of the
• scattering angle
• ?

(why?)
(why?)
3

• Development of the scattering angle distribution
  in Legendre polynomials
• with
• and
• Weak anisotropy
• with

4

• INTEGRAL FORM
• Isotropic scattering and source
• (see chap.II)
• ? In the one speed case
• with
• transport kernel
• solution for a point source
• in a purely absorbing media
• (Dimensions !!??)

5

• RECIPROCITY THEOREM AND COROLLARIES
• with
• Proof

S
V
BC in vacuum
-
?V
dr
(BC in vacuum!)
?4?
d?
6

• Corollary
• Isotropic source in
• Collision probabilities
• Set of homogeneous zones Vi
• Pti?j proba that 1 n appearing uniformly and
  isotropically in Vi will make a next collision in
  Vj
• Then
• Rem applicable to the absorption (Pai?j) and
  1st-flight collision probas (P1ti?j)

Nb of n emitted in dro about ro
(dimensions!!)
Reaction rate in dr about r per n emitted at ro
7

• Escape probabilities
• Homogeneous region V with surface S
• Po escape proba for 1 n appearing uniformly and
  isotropically in V
• ?o absorption proba for 1 n incident uniformly
  and isotropically on S
• Rem applicable to the collision and 1st-flight
  collision probas

8
III.2 DIFFUSION APPROXIMATION

• CONTINUITY EQUATION
• Objective eliminate the dependence on the
  angular direction ? Boltzmann eq. integrated on
  (see weak anisotropy)
• with
• ? Angular dependence still explicitly present in
  the expression of the integrated current (i.e.
  not a self-contained eq. in )

?4?
d?
9

• DIFFUSION EQUATION
• Continuity eq. integrated flux everywhere
  except for
• Still 6 var. to consider!
• Objective of the diffusion approximation
  eliminate the two angular variables to simplify
  the transport problem
• Postulated Ficks law
• with diffusion coefficient dimensions?
• (comparison with other physical phenomena!)
• ?

10

• BOUNDARY CONDITIONS
• Reminder BC in vacuum ? angular dependence
• ? not applicable in diffusion
• Integration of the continuity eq. on a small
  volume around a discontinuity (without
  superficial source)
• Continuity of the normal comp. of the current
• Discontinuity of the normal derivative of the
  flux
• But continuity of the flux because
• ? Continuity of the tangential derivative of the
  flux

11

• External boundary partial ingoing current
  vanishes
• Not directly deductible from Ficks law
  (why?)
• Weak anisotropy ? 1st-order development of the
  flux in
• Expression of the partial currents
• with

12

• Partial ingoing current vanishing at the
  boundary
• Linear extrapolation of the flux outside the
  reactor
• Nullity of the flux in extrapolation
  distance
• Simplification
• Use of the BC at the extrapoled boundary
• VALIDITY CONDITIONS
• Implicit assumption D material coefficient
• m.f.p. lt dimensions of the media ? last collision
  occurred in the media considered ? D fct of
  this media only
• Diffusion approximation questionable close to the
  boundaries
• BC in vacuum!
• Possible improvements (see below)

13

• P1 APPROXIMATION IN ONE SPEED DIFFUSION
• Anisotropy at 1st order (P1 approximation)
• In the one speed transport eq.
• 0-order angular momentum
• (one speed continuity eq.)
• 1st-order momentum
• Preliminary

(link between cross sections and diffusion
coefficient)
14

• Consequently
• Reminder
• Addition theorem for the Legendre polynomials
• ?
• Thus

15

• In 3D
• with
• and
• Homogeneous material isotropic sources
• Ficks law with
• Transport cross section

(without fission)
16

• ONE-SPEED SOLUTION OF THE DIFFUSION EQUATION
  (WITHOUT FISSION)
• Infinite media
• Diffusion at cst v, ? homogeneous media, point
  source in O
• Define
• Fourier transform
• ? Green function
• ? For a general source

Comparison with transport ?
17

• Particular cases (see exercises)
• Planar source
• Spherical source
• Cylindrical source
• As

with
Kn(u), In(u) modified Bessel fcts
18

• Finite media
• Allowance to be given to the BC!
• Virtual sources method
• Virtual superficial sources at the boundary (lt0
  to embody the leakages) ? no modification of the
  actual problem
• Media artificially extended till ?
• Intensity of the virtual sources s.t. BC
  satisfied
• Physical solution limited to the finite media
• Examples on an infinite slab
• Centered planar source (slab of extrapolated
  thickness 2a)
• BC at the extrapolated boundary

19

• Flux induced by the 3 sources
• BC ?
• Uniform source (slab of physical thickness 2a)
• Solution in ? media (source of constant
  intensity)
• Diffusion BC
• Solution in finite media
• Accounting for the BC

20

• Diffusion length
• Let diffusion length
• We have
• Planar source
• L relaxation length
• Point source use of the migration area (mean
  square distance to absorption)

21
III.3 MULTI-GROUP APPROXIMATION

• ENERGY GROUPS
• One speed simplification not realistic (E ?
  10-2,106 eV)
• Discretization of the energy range in G groups
• EG lt lt Eg lt lt Eo
• (Eo fast n EG thermal n)
• transport or diffusion eq. integrated on a group
• Flux in group g
• Total cross section of group g
• (reaction rate conserved)
• Diffusion coefficient for group g AND direction x
• (? possible loss of isotropy!)
• Isotropic case

22

• Transfer cross section between groups
• Fission in group g
• External source
• Multi-group diffusion equations
• Removal cross section
• ?

proba / u.l. that a n is removed from group g
23

• If thermal n only in group G ? ?sgg 0 if g gt
  g
• SOLUTION METHOD
• Characteristic quantities of a group f(?)
  usually
• Multi-group equations reformulation, not
  solution!
• Basis for numerical schemes however (see below)

24
III.4 1st-FLIGHT COLLISION PROBABILITIES METHODS

• MULTI-GROUP APPROXIMATION
• Integral form of the transport equation
• Isotropic case with the energy variable

25

• Energy discretization
• Optical distance in group g
• Multi-group transport equations (isotropic case)
• with source
• (compare with the integral form of the one speed
  Boltzmann eq.)

(
)
26

• Multi-group approximation
• ? Solve in each energy group a one speed
  Boltzmann equation with sources modified by
  scatterings coming from the previous groups (see
  convention in numbering the groups)
• Within a group, problem amounts to studying 1st
  collisions
• Iterative process to account for the other groups
• Remark
• Characteristics of each group f(?) !!!
• ? 2nd (external) loop of iterations necessary to
  evaluate the neutronics parameters in each group

27

• IMPLEMENTING THE FIRST-COLLISION PROBABILITIES
  METHOD
• Integral form of the one speed, isotropic
  transport equation
• where S contains the various sources, and
• Partition of the reactor in small volumes Vi
• homogeneous
• on which the flux is constant (hyp. of flat flux)

28

• Multiplying the Boltzmann eq. by ?t and
  integrating on Vi
• Then, given the homogeneity of the volumes
• Uniform source ?
• proba that 1 n unif. and isotr. emitted in Vi
  undergoes its 1st collision in Vj

avec
( flat flux)
29

• How to apply the method?
• Calculation of the 1st-flight collision probas
  (fct of the chosen partition geometry)
• Evaluation of the average fluxes by solving the
  linear system above
• Reducing the nb of 1st-flight collision probas to
  estimate
• Conservation of probabilities
• Infinite reactor
• Finite reactor in vacuum
• with Pio leakage proba outside the reactor
  without collision for 1 n appearing in Vi
• Finite reactor
• with PiS leakage proba through the external
  surface S of the reactor, without collision, for
  1 n appearing in Vi

30

• For the ingoing n
• with
• ?Sj proba that 1 n appearing uniformly and
  isotropically across surface S undergoes its 1st
  collision in Vj
• ?SS proba that 1 n appearing uniformly and
  isotropically across surface S in the reactor
  escapes it without collision across S
• Reciprocity 1
• Reciprocity 2

31

• Partition of a reactor in an infinite and regular
  network of identical cells
• Division of each cell in sub-volumes
• 1stflight collision proba from volume Vi to
  volume Vj
• Collision in the cell proper
• Collision in an adjacent cell
• Collision after crossing one cell
• Collision after crossing two cells,
• Second term Dancoff effect (interaction between
  cells)

32
CH.III  APPROXIMATIONS OF THE TRANSPORT EQUATION

• ONE SPEED BOLTZMANN EQUATION
• ONE SPEED TRANSPORT EQUATION
• INTEGRAL FORM
• RECIPROCITY THEOREM AND COROLLARIES
• DIFFUSION APPROXIMATION
• CONTINUITY EQUATION
• DIFFUSION EQUATION
• BOUNDARY CONDITIONS
• VALIDITY CONDITIONS
• P1 APPROXIMATION IN ONE SPEED DIFFUSION
• ONE SPEED SOLUTION OF THE DIFFUSION EQUATION
• MULTI-GROUP APPROXIMATION

?
?
?
?