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
1
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
 f   a
 D (r )   a (r )   f  (r )
 (r )  B 2 (r )  0
D
and BC at the extrapolated boundary:  (rs  n de )  0
  : solution of the corresponding eigenvalue problem
countable set of eigenvalues:
0  Bo2  B12  B22 ...
2
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:
 o
2
1. Bg 
= geometric buckling
o
= f(reactor geometry)
2. B 
2
m
 f   a
D
= material buckling
= f(materials)
Criticality: Bg2  Bm2
 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)
3
4
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Time-dependent problem
Diffusion operator:
J
-K
( J  K ) (r )   f  (r )  D (r )   a (r )
 Spectrum of real eigenvalues: o  1  2  ...
s.t.
with Bi2   ()
i   f  a  DBi2
o = maxi i associated to Bo2 : min eigenvalue of (-)  o
associated to o: positive all over the reactor volume
1  (r , t )
 ( J  K ) (r , t )
Time-dependent diffusion:
v t
Eigenfunctions i: orthogonal basis   (r , t )   ci (t )i (r )
 (r , t )   ci (0)i (r )e
i vt
i
i
 o < 0 : subcritical state
 o > 0 : supercritical state
t 
 co (0)o (r )
 o = 0 : critical state with  (r , t ) 
5
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Unique possible solution of the criticality problem whatever the
2
2
IC:
      DB 2    DB  DB
i
f
a
o
i
m
g
Criticality and multiplication factor
keff : production / destruction ratio
Close to criticality:  (r )  o (r )
keff 
 f
 f
Jo
1


Ko  a  DB 2
 a 1  L2 B 2
 o = fundamental eigenfunction associated to the eigenvalue
1
keff of:
K  J
 media:
k 
 f
a
k
 f
f
Finite media:
keff 
Improvement:
keff  pf .Pth 
1 L B
2
2
 fPth
pf
1  L2 B 2
and criticality for keff = 1
with
Bm2 
pf  1
L2
6
Independent sources
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Eigenfunctions i : orthonormal basis
( K  J ) (r )  Q(r )   Qii (r )
i
 (r )  
i
Qi
i (r )
2
DBi   a  f
Subcritical case with sources: possible steady-state solution
Qo
Qo
 (r ) 
o (r ) 
o (r )
2
 o
DBo   a  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
7
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 , Eth )   P(ro  r , Eth ) f th (ro )dro
V
with  Dth (r )   ath (r )  q(r , Eth )
P(ro  r , E )  f (| r  ro |)
 media: translation invariance 
Finite media: no invariance  approximation
Solution in an  media: use of Fourier transform
( a  DB )ˆ( B)  (2 )3 / 2 Pˆ ( B, Eth ) f ˆ( B)
2
(2 )
3/ 2
Pˆ ( B, Eth )
 f
 a  DB 2
 1  Bm2
8
Inverting the previous expression:  (r )   A(u ).eiB u .r du
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
m
solution of  (r )  Bm2 (r )  0
Solution in finite media
Additional condition: B2  {eigenvalues} of (-) with BC on the
extrapolated boundary  B 2  Bo2  Bg2
 Criticality condition:
Bm2  Bg2
with Bm2 solution of (2 )3 / 2 Pˆ ( Bm , Eth )
f
1 L B
2
2
m
1
 (2 )3/ 2 Pˆ ( B, Eth )  P ( B, Eth ) : fast non-leakage proba
9
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Examples of moderation kernels
Two-group diffusion
e 1r

1
1
Fast group: P(r , Eth )   r1
 P ( B, Eth )  r1 2 2 
D1 1  B
1  L12 B 2
4D1r
f
1
.
1
 Criticality eq.:
2 2
2 2
1  L2 B 1  L1 B
G-group diffusion
G 1
P ( B, Eth )  
i 1
 Criticality eq.:
1
1
L B 1





G 1 2
1  L2i B 2
1  (i 1 Li ) B 2
2
i
f
1  (i 1 L ) B
G
2
i
2
2
1
Age-diffusion (see Chap.VII) P(r , E ) 
 Criticality eq.:
fe
B 2
1 L B
2
2
1
 r 2 / ( E )
e
B 2
3 / 2  P ( B, Eth )  e
(4 ( E ))
(E) = age of n at en. E emitted at the fission en.
 = age of thermal n emitted at the fission en.
10
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
11
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
REFLECTOR SAVINGS
One-speed diffusion model
 D (r )   a (r )   f  (r )
 In the core:
  (r )  B  (r )  0 with B 
2
c
2
c
 In the reflector:
  (r ) 
 f   a
D

f  1
L2
k  1

L2
 DR  (r )   aR (r )  0
1
 (r )  0
2
LR
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
12
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)  A cos Bc x
0 xa
 x 
 x 


 ( x)  C cosh    E sinh   a  x  a  b
 LR 
 LR 
Flux continuity + BC:
 ( x)  A cos Bc x
 ( x) 
0 xa
 a  b | x | 
cos Bc a
 a  x  a  b
A
sinh 
b
sinh LR
LR


Current continuity:
DBc tan Bc a 
DR
b
coth
 criticality eq.
LR
LR
Q: A = ?
13
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 

a
2 Bc
DBc
b
LR tanh
 In the criticality condition: tan Bc 
DR
LR
As Bc << 1 :

D
b
LR tanh
DR
LR
If same material for both reflector and moderator, with a D little
affected by the proportion of fuel  D  DR
b
  LR tanh
b  LR :   b
LR
b  LR :   LR
Criticality: possible calculation with bare reactor accounting for 
14
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
TWO-GROUP MODEL
Core
 D11 (r )   a11 (r )   s11 (r )   f 11 (r )  f 22 (r )
 D2 2 (r )   a 22 (r )   s11 (r )
Reflector
 DR11 (r )   R11 (r )  0
 DR 22 (r )   R 22 (r )   R11 (r )
Planar geometry: solutions s.t.  i  B 2i ?
 D1B 2   a1   s1  f 1


  s1

 1   0 
    
2
D2 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
D
B
 a2

2
1
For each root:

2
 s1
15
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Solution in the core for [-a, a]:
1 ( x)  A1 cos B1x  A2 cosh B2 x
2 ( x)  A1
 s1
 s1
cos
B
x

A
cosh B2 x
1
2
2
2
D2 B1   a 2
D2 B2   a 2
Solution in the reflector for a  x  a+b:
ab x
1 ( x)  A3 sinh
L1R
2 ( x)  A3
 R1 / DR 2
ab x
ab x
sinh

A
sinh
4
1
1

L
L2 R
1R
L2
L2
R2
R1
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?
16
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
fast flux
thermal flux
core
reflector
17