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5

1

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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2

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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3

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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4

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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5

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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6

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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7

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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9

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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10

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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11

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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12

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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13

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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14

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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15

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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16

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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17

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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18

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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19

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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20

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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21

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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22

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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23

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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24

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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25

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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26

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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27

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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28

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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29

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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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