PHYS-H406 – Nuclear Reactor Physics – Academic year 2016-2017
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
1
PHYS-H406 – Nuclear Reactor Physics – Academic year 2016-2017
III.1 ONE SPEED BOLTZMANN
EQUATION
ONE SPEED TRANSPORT EQUATION

. (r , v, )  t (r , v) (r , v, )     s (r , v' , '  v, ) (r , v' , ' )dv' d'
o
4


1

 (v)    f (r , v' ) (r , v' , ' )dv' d'Q(r , v, )
o
4
4
 Suppressing the dependence on v in the Boltzmann eq.:
. (r , )  t (r ) (r , )    s (r , '  ) (r , ' )d' 
4
Let
c(r ) 
and
 s ( r )   f ( r )
 t (r )
 f ( r )
 (r ,  ' )d 'Q(r ,  )

4 4
: expected nb of secundary n/interaction,
 f ( r )
1
1
f (r , . ' ) 
( s ( r ,  '   ) 
)
2
c ( r ) t ( r )
4
(why?)
: distribution of the
scattering angle

. (r , )  t (r ) (r , ) 
(why?)
ct (r )
f ( '.) (r , ' )d'  Q(r , )

2 4
2
PHYS-H406 – Nuclear Reactor Physics – Academic year 2016-2017
Development of the scattering angle distribution in Legendre
polynomials:
f ( )  
l
2l  1
f l Pl (  )
2
,   . '
(2l  2m)!
 l 2m
with Pl (  )   (1) l
2 m!(l  m)!(l  2m)!
m 0
l/2
m
2 mn
,  Pm (  ) Pn (  )d 
1
2n  1
1
1
and fl  1 Pl ( ) f ( )d
Weak anisotropy
 f (r ) ct (r )
 s (r ,  '   ) 

(1  3  o  .' )
4
4
with
 o   f1 (r )
. (r , )  t (r ) (r , ) 
ct (r )
(1  3  o  '.) (r , ' )d'  Q(r , )

4 4
3
PHYS-H406 – Nuclear Reactor Physics – Academic year 2016-2017
INTEGRAL FORM
Isotropic scattering and source  (r , v)  R3
(see chap.II)
 In the one speed case:  (r )  R
with
K (r , ro ) 
e  v ( r ,ro )
3
4 r  ro
2
e  v ( r ,ro )
4 r  ro
2
S (ro , v)dro
(ct (ro ) (ro )  Q(ro )) dro
e  v ( r ,ro )
4 r  ro
2
= transport kernel
= solution for a point source
1
Q(r ,  ) 
 (r  ro )
4
in a purely absorbing media
(Dimensions !!??)
4
PHYS-H406 – Nuclear Reactor Physics – Academic year 2016-2017
RECIPROCITY THEOREM AND COROLLARIES
 (r ,  | ro , o )   (ro , o | r ,  )
S
V
ct (r )
. (r , )  t (r ) (r , ) 
f ( '.) (r , ' )d'  Q(r , )
2 4
+BC in vacuum
with
Proof
.  (r ,  | ro , o )  t (r ) (r ,  | ro , o ) 
c ( r ) t ( r )
f (  '. ) (r ,  ' | ro , o )d '

2
4
  (r  ro ) (   o )
 .  (r ,  | r1 , 1 )   t (r ) (r ,  | r1 , 1 ) 
  (r ,  | r1 , 1 )
c ( r ) t ( r )
4 f ('.) (r ,'| r1 ,1 )d'
2
  (r  r1 ) (   1 )
-
. ( (r ,  | ro , o ). (r ,  | r1 , 1 )) 
V
(BC in vacuum!)
  (r ,  | ro , o )
c ( r ) t ( r )
4 f ( '. )[ (r ,  | r1 , 1 ) (r ,  '| ro , o )
2
  (r ,  | ro , o ) (r ,  ' | r1 , 1 )]d  '
dr
  (r ,  | r1 , 1 ) (r  ro ) (   o )   (r ,  | ro , o ) (r  r1 ) (   1 )
4

V
c ( r ) t ( r )
4 f ('.)[ (r , | r1 ,1 ) (r , '| ro , o )   (r ,  | ro , o ) (r ,'| r1 ,1 )]d' dr
2
  (ro ,  | r1 , 1 ) (   o )   (r1 ,  | ro , o ) (   1 )
d
5
PHYS-H406 – Nuclear Reactor Physics – Academic year 2016-2017
Corollary
Isotropic source in ro
 (r | ro )   (ro | r )
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
t
i j
P
Then
t
Vi Pi 
j
 tj

  (r
Vi V j
o
1
    tj (r | ro ) dr dro
Vi
Vi V j
| r )dr dro 
V j Pjti
 ti
Nb of n emitted
in dro about ro
(dimensions!!)
Reaction rate in dr about
r per n emitted at ro
t
t
tiVi Pi


V
P
j
tj j j i
Rem: applicable to the absorption (Paij) and 1st-flight
collision proba’s (P1tij)
6
PHYS-H406 – Nuclear Reactor Physics – Academic year 2016-2017
Escape probabilities
Homogeneous region V with surface S
Po : escape proba for 1 n appearing uniformly and isotropically
in V
1
Po       (rs ,  | ro , o )
dro do n .ddS
4V
S n .   0 V o
o : absorption proba for 1 n incident uniformly and isotropically
on S
1
o 
S V
Po 
1
4V
  
 
 a (r ,  | rs , s )n .s ds dSdr d
 S n . s 0
  (ro ,o | rs ,)dro do n.ddS 
S n .   0 V o
S 1
.
4V S
     (r , 
o
o
| rs , )dro do n .ddS
S n .  0V o
4V
o   a
Po
S
Rem: applicable to the collision and 1st-flight collision probas
7
PHYS-H406 – Nuclear Reactor Physics – Academic year 2016-2017
III.2 DIFFUSION APPROXIMATION
CONTINUITY EQUATION
Objective: eliminate the dependence on the angular direction
 Boltzmann eq. integrated on  (see weak anisotropy):


. (r , v, )  t (r , v) (r , v, )     s (r , v' , '  v, ) (r , v' , ' )dv' d'
o
4


1
4

 (v)    f (r , v' ) (r , v' , ' )dv' d'Q(r , v, )
o
4
4
d

div ( J (r , v))  t (r , v) (r , v)    s (r , v'  v) (r , v' )dv'
o

  (v)  f (r , v' ) (r , v' )dv'Q(r , v)
o
with
J ( r , v) 
  (r , v, )d
4
 Angular dependence still explicitly present in the expression
of the integrated current (i.e. not a self-contained eq. in  (r , v))
8
PHYS-H406 – Nuclear Reactor Physics – Academic year 2016-2017
DIFFUSION EQUATION
Continuity eq.: integrated flux  (r , v) everywhere except for J (r , v)
 Still 6 var. to consider!
Objective of the diffusion approximation: eliminate the two
angular variables to simplify the transport problem
Postulated Fick’s law:
J ( r , v )   D( r , v )   ( r , v )
with D(r , v) : diffusion coefficient
[dimensions?]
(comparison with other physical phenomena!)


 ( D(r , v) (r , v))  t (r , v) (r , v)    s (r , v'  v) (r , v' )dv'

o
  (v)  f (r , v' ) (r , v' )dv'Q(r , v)
o
9
PHYS-H406 – Nuclear Reactor Physics – Academic year 2016-2017
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):
 div ( J (r , v))dV  0
V
 Continuity of the normal comp. of the current:J n (rs , v)  n .J (rs , v)
 Discontinuity of the normal derivative of the flux
 (rs , v)
 (rs , v)
D
 D
n
n
But continuity of the flux because
 (rs  n , v)   (rs  n , v)  


 (rs  n , v)
0
d 
 0
n
 Continuity of the tangential derivative of the flux
10
PHYS-H406 – Nuclear Reactor Physics – Academic year 2016-2017
External boundary: partial ingoing current vanishes
J  

n . (rs , v,  )d  0
n . 0
Not directly deductible from Fick’s law
(why?)
Weak anisotropy  1st-order development of the flux in 
 ( r , v,  ) 
1
1
( (r , v)  3.J (r , v))
(o (r , v)  .1 (r , v)) 
4
4
Expression of the partial currents
1
1
1
1
n
.


(
r
,
v
,

)
d



(
r
,
v
)


(
r
,
v
)


(
r
,
v
)

Dn .  (r , v)
o
1n

4
6
4
2
n .  0
1
1
1
1
J     n . (r , v, )d  o (r , v)  1n (r , v)   (r , v)  Dn .  (r , v)
4
2
4
6
n .  0
J  
with 1n (r , v)  n .1 (r , v)
11
PHYS-H406 – Nuclear Reactor Physics – Academic year 2016-2017
Partial ingoing current vanishing at the boundary:
n ' (rs , v)
1


J  (rs , v)  0
 (rs , v) 2 D(rs , v)
Linear extrapolation of the flux outside the reactor
 Nullity of the flux in
d e  2 D(rs , v) : extrapolation distance
Simplification
Use of the BC  (re , v)  0 at the extrapoled boundary re  rs  d e n
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)
12
PHYS-H406 – Nuclear Reactor Physics – Academic year 2016-2017
P1 APPROXIMATION IN ONE SPEED DIFFUSION
(link between cross sections
and diffusion coefficient)
Anisotropy at 1st order (P1 approximation):
 (r ,  ) 
1
( (r )  3.J (r ))
4
In the one speed transport eq.
ct (r )
. (r , )  t (r ) (r , ) 
f ( '.) (r , ' )d'  Q(r , )

2 4
0-order angular momentum
div J (r )  (1  c) t (r ) (r )  Q(r )
(one speed continuity eq.)
1st-order momentum
Preliminary:
  i d   0 , i  x, y , z
4
 i  j d  
4
4
 ij
3
, i, j  x, y, z
    d  0
i
4
j
k
, i , j , k  x, y , z
13
PHYS-H406 – Nuclear Reactor Physics – Academic year 2016-2017
Consequently
1  ( r )
P1


.


(
r
,

)
d



  x
3 x
4
P1


(
r
)

(
r
,

)
d



t ( r ) J x ( r )
 x t

4


4
P1

f
(

.

'
).

(
r
,

'
)
d

d

'


??
 x
4
Reminder:
f ( )  
l
2l  1
f l Pl (  )
2
,   . '
Addition theorem for the Legendre polynomials:
Pl ( . ' )  Pl ( .n ).Pl (  '.n ) 
l
e
im
...
ml
m0


  x Pl (.' ). (r , ' )dd' 
4 4
Thus:

4
2
2
4
' x 1l (r , ' )d' 
1l J x (r )
3
3
1  (r )
 t (r )(1  cf1 (r )) J x (r )    xQ(r , )d
3 x
4
14
PHYS-H406 – Nuclear Reactor Physics – Academic year 2016-2017
In 3D:
1
  (r )  ( t (r )   s1 (r )) J (r )  Q1 (r )
3
with
Q1i (r ) 
  Q(r , )d
i
, i  x, y, z
4
and  s1 (r )  c(r )t (r ) f1 (r )  c(r )t (r )  o 
Homogeneous material + isotropic sources
 Fick’s law with
1
J (r )  
  (r )
3( t   s1 )
1
D
3( t   s1 )
Transport cross section:
tr  t   o   s
 Approximation of the diffusion coefficient:
(without fission)
1
D
3 tr
15
PHYS-H406 – Nuclear Reactor Physics – Academic year 2016-2017
ONE-SPEED SOLUTION OF THE DIFFUSION
EQUATION (WITHOUT FISSION)
Infinite media
Diffusion at cst v,  homogeneous media, point source in O
  ( D(r )   (r ))   t (r ) (r )   s (r ) (r )  Q(r )
 D (r )   a (r )  Q (r )
Define
a / D   2
  (r )   2 (r ) 
1
Fourier transform: ˆ(k )   
 2 
 Green function:
3/ 2
 ik . r
e
  (r )dr
e r
G (r ) 
4Dr
 For a general source:  (r ) 

R3
Q
 (r )
D
 1 
ˆ(k )   
 2 
3/ 2
Q
D ( 2  k 2 )
Comparison with transport ?
e  |r  rs |
Q(rs )drs
4D | r  rs |
16
PHYS-H406 – Nuclear Reactor Physics – Academic year 2016-2017
Particular cases (see exercises)
• Planar source Q(r )  Q( x, y, z )  Qo ( x  xo )
e  | x  xo |
 p ( x | xo ) 
2D
Q(r )  Q( R, ,  )  Qo ( R  Ro )
• Spherical source
Ro (e  |R  Ro |  e  ( R  Ro ) )
s ( R | Ro ) 
2DR
• Cylindrical source Q(r )  Q(r ,  , z )  Qo (r  ro )
2
 ( ) | r  ro |
ro
c (r | ro ) 
K

2D
o
( ( )) d with
o
As
K o ( | r  ro |) 
K o (u )  
K (r ) I n (ro ) if
in  n
e
 K (r )I (r ) if
n  
 n o n


r  ro
1
e  ut
t 1
2
dt
Kn(u), In(u):
modified
Bessel fcts
r  ro
ro K o (r ) I o (ro ) if
c (r | ro )  
D  K o (ro ) I o (r ) if
r  ro
r  ro
17
PHYS-H406 – Nuclear Reactor Physics – Academic year 2016-2017
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)
Q(r )  Qo ( x)
BC at the extrapolated boundary:
Virtual sources:
 (a )  0
Qv (r )  A ( x  a)  A ( x  a)
18
PHYS-H406 – Nuclear Reactor Physics – Academic year 2016-2017
Flux induced by the 3 sources:
 ( x) 
BC

1
(Qo e  | x|  Ae ( x  a )  Ae ( a  x ) ) , x  [a, a]
2D
sinh  (a  | x |)
 ( x)  Qo
2D cosh a
  ( x ) , x  [  a ,  a ]
Uniform source (slab of physical thickness 2a)
Solution in  media (source of constant intensity):
'
1



Diffusion BC:
 x a
de
Solution in finite media:
Q
 
a
 ( x)    A cosh x
cosh x
)
Accounting for the BC:  ( x)   (1 
cosh  (a  d e )
19
PHYS-H406 – Nuclear Reactor Physics – Academic year 2016-2017
Diffusion length
Let
L
1
: diffusion length

 tr  a 
1
D
We have L 


 a 3 tr  a
3
2
| x|
Planar source:
L L
 ( x) 
e
2D
 L = relaxation length
Point source: use of the migration area (mean square distance to absorption)


  (r )4r dr

r
e
dr


 6L

 re dr
 r2  
o

o

 r2
o
r 2 (r )4r 2 dr
2
3  r
2

o
 r
20
PHYS-H406 – Nuclear Reactor Physics – Academic year 2016-2017
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:
 g (r )  
E g 1
Eg
 (r , E )dE    (r , E )dE , g  1...G
g
Total cross section of group g:
1
tg (r ) 
t (r , E ) (r , E )dE

 g (r ) g
(reaction rate conserved)
Diffusion coefficient for group g AND direction x
Dgx (r ) 
1
 D( r , E )
 g ( r )
g
x
 (r , E )
dE
x
Isotropic case: J g (r )  Dg (r )g (r )
( possible loss of isotropy!)
21
PHYS-H406 – Nuclear Reactor Physics – Academic year 2016-2017
Transfer cross section between groups:
 sg ' g (r ) 
Fission in group g:
 fg (r ) 
1
g ' (r ) g
  (r , E '  E ) (r , E ' )dE ' dE
s
g'
1
 f (r , E ) (r , E )dE

 g (r ) g
External source:
 g    ( E )dE
g
Qg (r )   Q(r , E )dE
g
Multi-group diffusion equations
G
  ( Dg (r )   g (r ))   tg (r ) g (r )    sg ' g (r ) g ' (r )
g '1
G
  g   fg ' (r ) g ' (r )  Qg (r ) , g  1..G
g '1
Removal cross section:
  rg (r )  tg (r )   sgg (r )   ag (r )    sgg' (r )
g ' g
= proba / u.l. that
a n is removed
from group g 22
If thermal n only in group G  sg’g = 0 if g’ > g
PHYS-H406 – Nuclear Reactor Physics – Academic year 2016-2017
g 1
  ( Dg (r )   g (r ))   rg (r ) g (r )    sg ' g (r ) g ' (r )
g '1
G
  g   fg ' (r ) g ' (r )  Qg (r ) , g  1..G
g '1
SOLUTION METHOD
Characteristic quantities of a group = f() usually
 Multi-group equations = reformulation, not solution!
 Basis for numerical schemes however (see below)
23
PHYS-H406 – Nuclear Reactor Physics – Academic year 2016-2017
III.4 1st-FLIGHT COLLISION
PROBABILITIES METHODS
MULTI-GROUP APPROXIMATION
Integral form of the transport equation
 ( r , v,  )  
R3
r  ro
e  v ( r ,ro ) 



2

r  ro
r  ro

r  ro
e  v ( r ,ro ) 



2

r  ro
r  ro

 3
R



 4


o

Q(ro , v,  )dro


[ s (ro , v' ,  '  v,  ) 
 (v )
 f (ro , v' )]
4
  (ro , v' ,  ' )dv' d ' dro
Isotropic case with the energy variable:
e  v ( r ,ro )
 (r , E )  
R3
 3
R
e  v ( r ,ro )
r  ro
2
r  ro


o
2
Q(ro , E )dro
[ s (ro , E '  E )   ( E ) f (ro , E ' )]. (ro , E ' )dE ' dro
24
PHYS-H406 – Nuclear Reactor Physics – Academic year 2016-2017
Energy discretization
Optical distance in group g:
s
 vg (r , ro )   tg (ro  s' )ds'
o
( r  ro  s  )
Multi-group transport equations (isotropic case)
 g (r )  
R3
e
 vg ( r , ro )
4 r  ro
2
( sgg (ro ) g (ro )  S g (ro )) dro
, g  1...G
g 1
with source: S g (r )   g   fg ' (r )g ' (r )  Qg (r )    sg 'g (r )g ' (r )
g'
g '1
(compare with the integral form of the one speed Boltzmann eq.)
25
PHYS-H406 – Nuclear Reactor Physics – Academic year 2016-2017
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
26
PHYS-H406 – Nuclear Reactor Physics – Academic year 2016-2017
IMPLEMENTING THE FIRST-COLLISION
PROBABILITIES METHOD
Integral form of the one speed, isotropic transport equation
 (r )   K (r , ro )( s (ro ) (ro )  S (ro ))dro   3 K (r , ro )Qt (ro )dro
R
3
R
where S contains the various sources, and
K (r , ro ) 
e  v ( r ,ro )
4 r  ro
2
Partition of the reactor in small volumes Vi:
• homogeneous
• on which the flux is constant (hyp. of flat flux)
27
PHYS-H406 – Nuclear Reactor Physics – Academic year 2016-2017
Multiplying the Boltzmann eq. by t and integrating on Vi:
  (r ) (r )dr    Q (r )dr   (r )K (r , r )dr
t
t
j
Vi
o
o
Vj
t
o
Vi
Then, given the homogeneity of the volumes:
1
1
i    (r )dr , Qti   Qt (r )dr
Vi Vi
Vi Vi
Vi  tii   Pj1t iV j Qtj
 dr Q (r )   (r ) K (r , r )dr
avec
j
o
Pj1t i 
t
o
Vj
t
o
Vi
 Q ( r ) dr
t
o
o
Vj
Uniform source 
1t
j i
P

 P
1t
j i


Vj
1
V ti K (r , ro ) V j dr dro :
i
proba that 1 n unif. and isotr. emitted in Vi undergoes its 1st
collision in Vj
Vi  tii   Pj1t iV j ( sj j  S j )
j
(+ flat flux)
28
PHYS-H406 – Nuclear Reactor Physics – Academic year 2016-2017
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
1t
P
 i j  1
Infinite reactor:
j
Finite reactor in vacuum:
1t
P
 i j  Pio  1
j
with Pio: leakage proba outside the reactor without collision for
1 n appearing in Vi
1t
P
Finite reactor:  i  j  PiS  1
j
with PiS: leakage proba through the external surface S of the
reactor, without collision, for 1 n appearing in Vi
29
PHYS-H406 – Nuclear Reactor Physics – Academic year 2016-2017
For the ingoing n:

Sj
 SS  1
j
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
tiVi Pi1t j  tjV j Pj1t i
4Vi
Si   t
PiS
S
30
PHYS-H406 – Nuclear Reactor Physics – Academic year 2016-2017
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
 Collision
 Collision
 Collision
in the cell proper
in an adjacent cell
after crossing one cell
after crossing two cells, …
t
c
Pj1

P
i
j i  PjS .Si  PjS .SS .SI  PjS .SS .SS .Si  ...
1t
j i
P
P
c
j i

PjS .Si
1  SS
Second term: Dancoff effect (interaction between cells)
31
PHYS-H406 – Nuclear Reactor Physics – Academic year 2016-2017
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 
32