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CH.IV : CRITICALITY CALCULATIONS IN DIFFUSION THEORY. CRITICALITY ONE-SPEED DIFFUSION MODERATION KERNELS REFLECTORS INTRODUCTION REFLECTOR SAVINGS TWO-GROUP MODEL. IV.1 CRITICALITY. criticality. Objective solutions of the diffusion eq. in a finite homogeneous - PowerPoint PPT Presentation
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CH.IV : CRITICALITY CALCULATIONS IN DIFFUSION THEORY
CRITICALITY
• ONE-SPEED DIFFUSION• MODERATION KERNELS
REFLECTORS
• INTRODUCTION• REFLECTOR SAVINGS• TWO-GROUP MODEL
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Objective solutions of the diffusion eq. in a finite homogeneous media exist without external sources
1st study case: bare homogeneous reactor (i.e. without reflector)
ONE-SPEED DIFFUSION
With fission !!
Helmholtz equation
with
and BC at the extrapolated boundary:
: solution of the corresponding eigenvalue problemcountable set of eigenvalues:
IV.1 CRITICALITY
)()()( rrrD fa
DB af
2
0)()( 2 rBr
0)( es dnr
criticality
...0 22
21
2 BBBo
A time-independent can be sustained in the reactor with no Q
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+ associated eigenfunctions: orthogonal basis
A unique solution positive everywhere fundamental mode
Flux !
Eigenvalue of the fundamental – two ways to express it:
1. = geometric buckling
= f(reactor geometry)
2. = material buckling
= f(materials)
Criticality:
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)
o
ogB
2
DB af
m
2
22mg BB
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Time-dependent problem
Diffusion operator:
Spectrum of real eigenvalues:
s.t. with
o = maxi i associated to : min eigenvalue of (-) o associated to o: positive all over the reactor volume
Time-dependent diffusion:
Eigenfunctions i: orthogonal basis
o < 0 : subcritical state
o > 0 : supercritical state
o = 0 : critical state with
)()()()()( rrDrrKJ af
...21 o
2iafi DB )(2 iB
2oB
),()(),(1
trKJt
tr
v
)()(),( rtctr iii
vt
iii
ierctr )()0(),(
)()0(),( rctr oot
J -K
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Unique possible solution of the criticality problem whatever the IC:
Criticality and multiplication factor
keff : production / destruction ratio
Close to criticality:
o = fundamental eigenfunction associated to the eigenvalue keff of:
media:
Finite media:
Improvement: with
2iafi DB 22
gmo DBDB
)()( rr o
2DBK
Jk
a
f
o
oeff
Jk
K1
fka
f
221 BL
fkeff
thfP
221.
BL
pfPpfk theff
221
1
BLa
f
22 1
L
pfBm
and criticality for keff = 1
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Independent sources
Eigenfunctions i : orthonormal basis
Subcritical case with sources: possible steady-state solution
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
)()()( rQrJK )(rQ iii
)()(2
rDB
Qr i
fai
i
i
)()()(2
rDB
Qr
Qr o
fao
oo
o
o
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MODERATION KERNELS
Definitions
= moderation kernel: proba density function that 1 n due to a fission in is slowed down below energy E in
= moderation density: nb of n (/unit vol.time) slowed down below E in
with
media: translation invariance Finite media: no invariance approximation
Solution in an media: use of Fourier transform
),()()( ththath ErqrrD
),( ErrP o
),( Erq
oothftho
V
th rdrErrPErq )(),(),(
)(ˆ),(ˆ)2()(ˆ)( 2/32 BEBPBDB ftha 22
2/3 1),(ˆ)2( ma
fth B
DBEBP
or r
r
|)(|),( oo rrfErrP
Objective: improve the treatment of thedependence on E w.r.t. one-speed diffusion
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Inverting the previous expression:
solution of
Solution in finite media
Additional condition: B2 {eigenvalues} of (-) with BC on the extrapolated boundary
Criticality condition:
with solution of
: fast non-leakage proba
0)()( 2 rBr m
222go BBB
22gm BB
2mB 1
1),(ˆ)2(
222/3
mthm BL
fEBP
),(),(ˆ)2( 2/3thth EBPEBP
udeuAr ruiBm .).()(
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Examples of moderation kernels
Two-group diffusion
Fast group:
Criticality eq.:
G-group diffusion
Criticality eq.:
Age-diffusion (see Chap.VII)
Criticality eq.:
rD
eErP
r
rth1
1 4),(
1
22
122
11
1
1
11),(
BLBDEBP r
th
11
1.
1 221
222
BLBL
f
1)(1 22
1
BL
f
i
G
i
22
1
1 1
1),(
BLEBP
i
G
ith
221
1
1
)(1
122
BLi
G
i
BLi
2/3
)(/
))(4(),(
2
E
eErP
Er
2
),( Bth eEBP
11 22
2
BL
fe B (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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INTRODUCTIONNo 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
IV.2 REFLECTORS
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REFLECTOR SAVINGS
One-speed diffusion model
In the core:
with
In the reflector:
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
0)(1
)(2
rL
rR
)()()( rrrD fa
0)()( 2 rBr c 222 11
L
k
L
f
DB afc
0)()( rrD aRR
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Solution in planar geometry
Consider a core of thickness 2a and reflector of thickness b (extrapolated limit)
Problem symmetry
Flux continuity + BC:
Current continuity:
criticality eq.
Q: A = ?
baxaL
xbaaBAx
axxBAx
RLb
c
c
R
||
sinhsinh
cos)(
0cos)(
RR
Rcc L
b
L
DaBDB cothtan
baxaL
xE
L
xCx
axxBAx
RR
c
sinhcosh)(
0cos)(
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Criticality reached for a thickness 2a satisfying this condition
For a bare reactor:
Reflector savings:
In the criticality condition:
As Bc << 1 :
If same material for both reflector and moderator, with a D little affected by the proportion of fuel D DR
Criticality: possible calculation with bare reactor accounting for
aB
aac
o 2
c
o Ba
2
RR
R
cc L
bL
D
DBB tanhtan
RR
R L
bL
D
Dtanh
RR L
bL tanh
bLb R :
RR LLb :
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TWO-GROUP MODELCore
Reflector
Planar geometry: solutions s.t. ?
Solution iff determinant = 0
2nd-degree eq. in B2
(one positive and one negative roots)
For each root:
)()()()()( 2211111111 rrrrrD ffsa
)()()( 112222 rrrD sa
)()()( 112222 rrrD RRR
0)()( 1111 rrD RR
ii B 2
0
0
2
1
22
21
21112
1
as
ffsa
BD
BD
22,1B
1
22
2
2
1
s
aBD
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Solution in the core for [-a, a]:
Solution in the reflector for a x a+b:
4 constants + 4 continuity equations (flux and current in each group)
Homogeneous linear system Annulation of the determinant to obtain a solution
Criticality condition
xBAxBAx 22111 coshcos)(
xBBD
AxBBD
Axa
s
a
s2
2222
121
2212
112 coshcos)(
RL
xbaAx
131 sinh)(
RRLL
RR
L
xbaA
L
xbaDAx
RR2
41
11
2132 sinhsinh
/)(
21
22
Q: the flux is then given to a constant. Why?
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core reflector
fast fluxthermal flux
