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
ENERGY GROUPS
SOLUTION METHOD
1st–FLIGHT COLLISION PROBABILITIES METHODS

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

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

5. V 4 dr RECIPROCITY THEOREM AND COROLLARIES Proof d with S V
+BC in vacuum - V dr
(BC in vacuum!)
4 d

6. Collision probabilities
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 proba’s (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 Fick’s 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 Fick’s law (why?)
Weak anisotropy  1st-order development of the flux in
Expression of the partial currents
with

12. VALIDITY CONDITIONS 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. < 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. Homogeneous material + isotropic sources
In 3D:
with
and
Homogeneous material + isotropic sources
Fick’s law with
Transport cross section:
 Approximation of the diffusion coefficient:
(without fission)

16. Comparison with transport ?
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 (<0 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:
Virtual sources:

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 < … < Eg < … < 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. Multi-group diffusion equations
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. SOLUTION METHOD If thermal n only in group G  sg’g = 0 if g’ > g
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
1st–flight 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
ENERGY GROUPS
SOLUTION METHOD
1st–FLIGHT COLLISION PROBABILITIES METHODS    