PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
CH.IV : CRITICALITY CALCULATIONS
IN DIFFUSION THEORY
CRITICALITY
• ONE-SPEED DIFFUSION
• MODERATION KERNELS
REFLECTORS
• INTRODUCTION
• REFLECTOR SAVINGS
• TWO-GROUP MODEL
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
IV.1 CRITICALITY
Objective
solutions of the diffusion eq. in a finite homogeneous
criticality
media exist without external sources
 A time-independent  can be sustained in the reactor with no Q
1st study case: bare homogeneous reactor (i.e. without reflector)
ONE-SPEED DIFFUSION
With fission !!
 Helmholtz equation
with
B 
2
 D   ( r )   a ( r )    f  ( r )
 (r )  B  (r )  0
2
 f   a
D
and BC at the extrapolated boundary:  ( rs  n d e )  0
  : solution of the corresponding eigenvalue problem
2
2
2
countable set of eigenvalues:
0  B o  B1  B 2 ...
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
+ associated eigenfunctions: orthogonal basis
 A unique solution positive everywhere  fundamental mode
 Flux !
Eigenvalue of the fundamental – two ways to express it:
1.
B 
2
g
 o
o
= geometric buckling
= f(reactor geometry)
2. B
2
m

 f   a
D
= material buckling
= f(materials)
2
2
Criticality: B g  B m
 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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PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Time-dependent problem
J
-K
( J  K ) ( r )    f  ( r )  D   ( r )   a  ( r )
Diffusion operator:
 Spectrum of real eigenvalues:  o  1   2  ...
s.t.
 i    f   a  DB i
2
Bi   (   )
2
with
o = maxi i associated to B o2 : min eigenvalue of (-)  o
associated to o: positive all over the reactor volume
Time-dependent diffusion:
1  (r , t )
v
t
 ( J  K ) ( r , t )
Eigenfunctions i: orthogonal basis   ( r , t )  
 (r , t ) 

c i ( 0 ) i ( r ) e
 i vt
c i ( t ) i ( r )
i
i
 o < 0 : subcritical state
 o > 0 : supercritical state
t
  c o ( 0 ) o ( r )
 o = 0 : critical state with  ( r , t )  
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Unique possible solution of the criticality problem whatever the
2
2
IC:
2
   DB  DB
       DB
i
f
a
o
i
m
g
Criticality and multiplication factor
keff : production / destruction ratio
Close to criticality:  ( r )   o ( r )
k eff 
J o
K o

 f
 a  DB
2

 f
1
a 1 L B
2
2
 o = fundamental eigenfunction associated to the eigenvalue
1
keff of:
K   J
 media:
k 
Finite media:
Improvement:
 f
a
k eff 
k
 f
f
1 L B
2
2
k eff   pf . Pth 
  fPth
 pf
1 L B
2
2
and criticality for keff = 1
with
Bm 
2
 pf  1
L
2
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Independent sources
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Eigenfunctions i : orthonormal basis
( K  J ) ( r )  Q ( r ) 
 Q  (r )
i
i
i
 (r ) 

i
Qi
DB
2
i
  a 
i (r )
f
Subcritical case with sources: possible steady-state solution
 (r ) 
Qo
 o
o (r ) 
Qo
DB
2
o
  a 
o (r )
f
 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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PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
MODERATION KERNELS
Objective: improve the treatment of the
dependence on E w.r.t. one-speed diffusion
Definitions
P ( ro  r , E ) = moderation kernel: proba density function that 1
n due to a fission in ro is slowed down below energy E in r
q ( r , E ) = moderation density: nb of n (/unit vol.time) slowed
down below E in r
q ( r , E th ) 

P ( ro  r , E th )  f  th ( ro ) d ro
V
with
 D   th ( r )   a th ( r )  q ( r , E th )
P ( ro  r , E ) 
 media: translation invariance 
Finite media: no invariance  approximation
f (| r  ro |)
Solution in an  media: use of Fourier transform
3/2
(  a  DB )ˆ ( B )  ( 2 ) Pˆ ( B , E th )  f ˆ ( B )
2
( 2 )
3/2
Pˆ ( B , E th )
 f
 a  DB
 1  Bm
2
2
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Inverting the previous expression:  ( r )  
solution of
A ( u ). e
iB m u . r
du
  ( r )  B m ( r )  0
2
Solution in finite media
Additional condition: B2  {eigenvalues} of (-) with BC on the
2
2
2
extrapolated boundary  B  B o  B g
 Criticality condition:
with

2
Bm
( 2 )
3/2
solution of
( 2 )
Bm  Bg
2
3/2
Pˆ ( B , E th )  P ( B , E th )
2
Pˆ ( B m , E th )
f
1 L B
2
2
m
1
: fast non-leakage proba
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Examples of moderation kernels
Two-group diffusion
Fast group:
e
P ( r , E th )   r 1
  1r
4  D1 r
f
 Criticality eq.:

1 L B
2
2
2
.
1
1 L B
2
1
2
 r1
P ( B , E th ) 
1
D1  1  B
2
2

1
1  L1 B
2
2
1
G-group diffusion
G 1
P ( B , E th ) 

i 1
 Criticality eq.:
1
2
1  Li B
2
 

2
f
1  (
G
i 1
2
i
L )B
2
 Criticality eq.:
 fe
2
i 1
2
Li ) B
2
 r / ( E )
2
2
1 L B
2
1  (
G 1
1
Age-diffusion (see Chap.VII) P ( r , E ) 
B
1
2
L i B  1
1
e
( 4 ( E ))
3/2
 P ( B , E th )  e
B
2
(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
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
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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PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
REFLECTOR SAVINGS
One-speed diffusion model
 In the core:
 D   ( r )   a ( r )    f  ( r )
   ( r )  B  ( r )  0 with
2
c

 f   a
D

f  1
L
2

k  1
L
2
 D R   ( r )   aR  ( r )  0
 In the reflector:
 (r ) 
B 
2
c
1
L
2
R
 (r )  0
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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PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Solution in planar geometry
Consider a core of thickness 2a and reflector of thickness b
(extrapolated limit)
Problem symmetry 
 ( x) 
 ( x) 
0 xa
A cos B c x
 x 
  E sinh
C cosh 

 LR 
 x 


L 
 R 
a  x ab
Flux continuity + BC:
 ( x) 
 ( x) 
A cos B c x
A
cos B c a
b
LR
sinh
 a  b | x | 

sinh 

LR


0 xa
a  x ab
Current continuity:
DB c tan B c a 
DR
LR
coth
b
LR
 criticality eq.
Q: A = ?
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Criticality reached for a thickness 2a satisfying this condition

2
a

For a bare reactor:
o
Bc
 Reflector savings:
  ao  a 
 In the criticality condition:
As Bc << 1 :
 
D
DR
tan B c  
L R tanh

a
2 Bc
DB c
DR
L R tanh
b
LR
b
LR
If same material for both reflector and moderator, with a D little
affected by the proportion of fuel  D  DR
  L R tanh
b  L R :   b
b
LR
b  L R :   L R
Criticality: possible calculation with bare reactor accounting for 
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
TWO-GROUP MODEL
Core
 D1  1 ( r )   a 11 ( r )   s 11 ( r )    f 11 ( r )    f 2 2 ( r )
 D 2   2 ( r )   a 2 2 ( r )   s11 ( r )
Reflector
 D R 1   1 ( r )   R 1 1 ( r )  0
 D R 2   2 ( r )   R 2 2 ( r )   R 1 1 ( r )
 i  B i
2
Planar geometry: solutions s.t.
 D1 B 2   a 1   s1   


  s1

f1
?
  1   0 
    
2
   
D 2 B   a 2    2   0 

f 2
Solution iff determinant = 0
 2nd-degree eq. in B2
2
B
 1 , 2 (one positive and one negative roots)
2
D2 B   a2
1
For each root:

2
 s1
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PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Solution in the core for [-a, a]:
 1 ( x )  A1 cos B1 x  A 2 cosh B 2 x
 2 ( x )  A1
 s1
D2 B   a2
2
1
cos B1 x  A 2
 s1
D2 B   a2
2
2
cosh B 2 x
Solution in the reflector for a  x  a+b:
abx
 1 ( x )  A3 sinh
 2 ( x )  A3
L1 R
 R1 / D R 2
1
2
LR 2

1
2
LR1
sinh
abx
L1 R
 A 4 sinh
abx
L2 R
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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PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
fast flux
thermal flux
core
reflector
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