PHYS-H406 – Nuclear Reactor Physics – Academic year 2014-2015 1 ONE SPEED BOLTZMANN EQUATION ONE...
Transcript of PHYS-H406 – Nuclear Reactor Physics – Academic year 2014-2015 1 ONE SPEED BOLTZMANN EQUATION ONE...
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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
CH.III : APPROXIMATIONS OF THE TRANSPORT EQUATION
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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
')',()',(),()(),(.4
drrrrr st
III.1 ONE SPEED BOLTZMANN EQUATION
),(')',(4
)(
4
rQdrrf
)(
)()()(
r
rrrc
t
fs
)4
)()',((
)()(
1)'.,(
2
1
r
rrrc
rf fs
t
),(')',()'.(2
)(),()(),(.
4
rQdrfrc
rrr tt
'')',',(),',',(),,(),(),,(.4
ddvvrvvrvrvrvr sot
),,('')',',()',()(4
1
4
vrQddvvrvrv fo
(why?)
(why?)
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Development of the scattering angle distribution in Legendre polynomials:
with
and
Weak anisotropy
with
12
2)()(,
)!2()!(!2
)!22()1()(
1
1
22/
0
ndPP
mlmlm
mlP mn
nmml
lm
l
ml
'.,)(2
12)(
ll
l
Pfl
f
1
1)()( dfPf ll
)(1 rfo
)'.31(4
)(
4
)()',(
o
tfs
rcrr
),(')',()'.31(4
)(),()(),(.
4
rQdrrc
rrr ot
t
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INTEGRAL FORM
Isotropic scattering and source(see chap.II)
In the one speed case:
with
= transport kernel
= solution for a point source
in a purely absorbing media
(Dimensions !!??)
ooootRo
rr
rdrQrrcrr
er
ov
))()()((4
)(3 2
),(
2
),(
4),(
o
rr
orr
errK
ov
)(4
1),( orrrQ
ooRo
rr
rdvrSrr
evr
ov
),(4
),(3 2
),(
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RECIPROCITY THEOREM AND COROLLARIES
with
Proof
),|,(),|,( rrrr oooo
),(')',()'.(2
)(),()(),(.
4
rQdrfrc
rrr tt
)()(
'),|',()'.(2
)()(),|,()(),|,(.
4
oo
oot
ootoo
rr
drrfrrc
rrrrr
)()(
'),|',()'.(2
)()(),|,()(),|,(.
11
11
4
1111
rr
drrfrrc
rrrrr tt
)()(),|,()()(),|,(
')],|',(),|,(
),|',(),|,()['.(2
)()()),|,().,|,((.
1111
11
11
4
11
rrrrrrrr
drrrr
rrrrfrrc
rrrr
oooo
oo
oot
oo
)(),|,()(),|,(
')],|',(),|,(),|',(),|,()['.(2
)()(
1111
1111
4
oooo
oooot
V
rrrr
rddrrrrrrrrfrrc
),|,( 11 rr
),|,( oorr
-
Vdr
4 d
VS
+BC in vacuum
(BC in vacuum!)
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Corollary
Isotropic source in
Collision probabilities
Set of homogeneous zones Vi
Ptij : proba that 1 n appearing uniformly and isotropically in Vi will make a next collision in Vj
Then
Rem: applicable to the absorption (Paij) and 1st-flight
collision proba’s (P1tij)
or )|()|( rrrr oo
oi
o
V
tj
V
tji rdrd
VrrP
ji
1)|(
oo
VVtj
tjii rdrdrr
PV
ji
)|(
tijjtj
tjiiti PVPV
(dimensions!!)
ti
tijjPV
Reaction rate in dr about r per n emitted at ro
Nb of n emitted in dro about ro
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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
dSdndrdrrSV
Soosoo
VnS o
.),|,(1.
4 0.
dSdndrdV
rrP oooos
VnS
o
o
.4
1),|,(
0.
oao PS
V4
dSdndrdrrV
P oosoo
VnS
o
o
.),|,(4
1
0.
drdSddnrrS ssssa
nSV
o
s
.),|,(1
0.
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5 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
1
4
vrQddvvrvrv fo
'')',',(),',',(),,(),(),,(.4
ddvvrvvrvrvrvr sot
8
')',()',(),(),()),(( dvvrvvrvrvrvrJdiv sot
III.2 DIFFUSION APPROXIMATION
),(')',()',()( vrQdvvrvrv fo
dvrvrJ ),,(),(4
4 d
),( vr
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DIFFUSION EQUATION
Continuity eq.: integrated flux everywhere except forStill 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!)
),(),(),( vrvrDvrJ
),(')',()',()( vrQdvvrvrv fo
')',()',(),(),()),(),(( dvvrvvrvrvrvrvrD sot
),( vrD
),( vrJ),( vr
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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
0)),(( dVvrJdivV
),(.),( vrJnvrJ ssn
n
vrD
n
vrD ss
),(),(
0),(
),(),( 0
d
n
vnrvnrvnr s
ss
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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
0),,(.0.
dvrnJ s
n
)),(.),((4
1),,( 1 vrvrvr o
)),(.3),((
4
1vrJvr
),(6
1),(
4
1),,(. 1
0.
vrvrdvrnJ no
n
),(6
1),(
4
1),,(. 1
0.
vrvrdvrnJ no
n
),(.),( 11 vrnvrn
),(.2
1),(
4
1vrnDvr
),(.2
1),(
4
1vrnDvr
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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)
0),( vrJ s ),(2
1
),(
),('
vrDvr
vr
ss
sn
),(2 vrDd se
ndrr ese 0),( vre
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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:
))(.3)((4
1),( rJrr
),(')',()'.(2
)(),()(),(.
4
rQdrfrc
rrr tt
zyxidi ,,,04
zyxjid ijji ,,,,3
4
4
zyxkjidkji ,,,,,04
)()()()1()( rQrrcrJdiv t
(link between cross sectionsand diffusion coefficient)
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Consequently
Reminder:
Addition theorem for the Legendre polynomials:
Thus:
drQrJrcfr
x
rxxt ),()())(1)((
)(
3
1
4
1
'.,)(2
12)(
ll
l
Pfl
f
x
rdr P
x
)(
3
1),(. 1
4
)()(),()( 1
4
rJrdrr xtP
tx
??')',().'.( 1
44
Px ddrf
)(3
4')',('
3
22')',().'.( 11
444
rJdrddrP xllxlx
...)'.()..()'.(
0
iml
m
lmlll enPnPP
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In 3D:
with
and
Homogeneous material + isotropic sources
Fick’s law with
Transport cross section:
Approximation of the diffusion coefficient:
)()())()(()(3
111 rQrJrrr st
zyxidrQrQ ii ,,,),()(4
1
otts rrcrfrrcr )()()()()()( 11
)()(3
1)(
1
rrJst
)(3
1
1st
D
sottr
tr
D
3
1
(without fission)
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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:
)()()( rQrrD a
)()()()()())()(( rQrrrrrrD st
)()()( 2 rD
Qrr 2/ Da
rdrek rki )(2
1)(ˆ .
2/3
)(2
1)(ˆ
22
2/3
kD
Qk
Dr
er
r
G
4)(
sss
rr
R
rdrQrrD
er
s
)(||4
)(||
3
Comparison with transport ?
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Particular cases (see exercises)
• Planar source
• Spherical source
• Cylindrical source
As
)(),,()( oo xxQzyxQrQ
D
exx
oxx
op
2)|(
||
)(),,()( oo RRQRQrQ
DR
eeRRR
oo RRRRo
os
2
)()|(
)(||
)(),,()( oo rrQzrQrQ
dKD
rrr o
o
ooc ))((
2)|(
2
dt
t
euK
ut
o1
)(2
1
||)( orr
oooo
ooooooc rrifrIrK
rrifrIrK
D
rrr
)()(
)()()|(
onon
oonnin
noo rrifrIrK
rrifrIrKerrK
)()(
)()(|)|(
with Kn(u), In(u):modified Bessel fcts
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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:
)()( xQrQ o
0)( a
)()()( axAaxArQv
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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:
],[,)(2
1)( )()(|| aaxAeAeeQ
Dx xaaxx
o
],[,)(cosh2
|)|(sinh)( aaxx
aD
xaQx o
a
Q
eaxd
1'
xAx cosh)(
))(cosh
cosh1()(
eda
xx
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Diffusion length
Let : diffusion length
We have
Planar source:
L = relaxation length
Point source: use of the migration area (mean square distance to absorption)
1
L
L
x
eD
Lx
||
2)(
drrr
drrrrr
o
o
2
22
2
4)(
4)(
2
3
2 6Ldrre
drerr
r
o
r
o
a
DL
2
33
1
atr
atr
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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:
dEErErr
r t
ggtg ),(),(
)(
1)(
III.3 MULTI-GROUP APPROXIMATION
GgdEErdEErrg
E
Eg
g
g
...1,),(),()(1
dEx
ErErDrD
gx
rgx g
),(),(
1)( )(
)()()( rrDrJ ggg
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Transfer cross section between groups:
Fission in group g:
External source:
Multi-group diffusion equations
Removal cross section:
dEdEErEErr
r s
ggggsg ')',()',(
)(
1)(
'''
dEErErr
r f
ggfg ),(),(
)(
1)(
dEE
g
g )(
dEErQrQg
g ),()(
GgrQrr
rrrrrrD
ggfg
G
gg
ggsg
G
ggtggg
..1,)()()(
)()()()())()((
''1'
''1'
)()()()()( ''
rrrrr sgggg
agsggtgrg
= proba / u.l. thata n is removedfrom group g
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If thermal n only in group G sg’g = 0 if g’ > g
SOLUTION METHODCharacteristic quantities of a group = f() usually
Multi-group equations = reformulation, not solution! Basis for numerical schemes however (see below)
GgrQrr
rrrrrrD
ggfg
G
gg
ggsg
g
ggrggg
..1,)()()(
)()()()())()((
''1'
''
1
1'
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MULTI-GROUP APPROXIMATION
Integral form of the transport equation
Isotropic case with the energy variable:oo
ofosoo
o
o
rr
R
ooo
o
o
rr
R
rdddvvr
vrv
vvrrr
rr
rr
e
rdvrQrr
rr
rr
evr
ov
ov
'')',',(
)]',(4
)(),',',([
),,(),,(
42
),(
2
),(
3
3
III.4 1st-FLIGHT COLLISION PROBABILITIES METHODS
ooofosoo
rr
R
oo
o
rr
R
rddEErErEEErrr
e
rdErQrr
eEr
ov
ov
')',()].',()()',([
),(),(
2
),(
2
),(
3
3
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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.)
')'(),( dssrrr otg
s
oovg
srr o( )
GgrdrSrrrr
er oogogosggR
o
rr
g
ovg
...1,))()()((4
)(3 2
),(
)()()()()()( ''
1
1'''
'
rrrQrrrS ggsg
g
gggfg
ggg
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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
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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)
2
),(
4),(
o
rr
orr
errK
ov
oooosR o rdrSrrrrKr ))()()((),()(3
ootR o rdrQrrK )(),(
3
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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
oj
o
V
ti
V
tij
tij rdrd
VrrKPP
ij
1),(11
tjjtij
jitii QVPV 1
rdrrKrrdrQrdrr ot
V
oot
Vjt
V iji
),()()()()(
oot
V
ot
V
oto
Vtij
rdrQ
rdrrKrrQrd
P
j
ij
)(
),()()(1
,)(1
rdrV
iVii
)(1jjsjj
tij
jitii SVPV
rdrQV
Q t
Viti
i
)(1
avec
(+ flat flux)
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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
aboveReducing 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
11 tji
j
P
11 iotji
j
PP
11 iStji
j
PP
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– A
cad
em
ic y
ea
r 2
01
4-2
01
5
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
1 SSSjj
tijjtj
tjiiti PVPV 11
iSi
tSi PS
V4
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
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 properCollision in an adjacent cellCollision after crossing one cellCollision after crossing two cells, …
Second term: Dancoff effect (interaction between cells)
.........1 SiSSSSjSSISSjSSijScij
tij PPPPP
SS
SijScij
tij
PPP
1
.1
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
32
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
CH.III : APPROXIMATIONS OF THE TRANSPORT EQUATION