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CH.II: NEUTRON TRANSPORTINTRODUCTORY CONCEPTS• ASSUMPTIONS• NEUTRON DENSITY, FLUX, CURRENT• REACTION RATE• FLUENCE, POWER, BURNUP

TRANSPORT EQUATION • NEUTRON BALANCE• BOLTZMANN EQUATION • CONTINUITY AND BOUNDARY CONDITIONS • INTEGRAL FORMS • FORMAL SOLUTION USING NEUMANN SERIES

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II.1 INTRODUCTORY CONCEPTSASSUMPTIONS

1. Interactions between n – matter: quantum problemBut (n) << characteristic dimensions of the reactor E

2. Density of thermal n: ~ 109 n/cm3

Atomic density of solids: ~ 1022 atoms/cm3

Interactions n – n negligible Linear equation for the neutron balance

3. Statistical treatment of the n, but small fluctuations about the average value of their flux

n: classical particles, not interacting with each other, whose average value of their probability density of presence is accounted for

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NEUTRON DENSITY, FLUX, CURRENT

Variables

Position: 3

Speed: 3 or kinetic energy + direction

(Time: 1 t)

Angular neutron density

: nb of n in about with a speed in [v,v+dv] and a direction in about

Neutron density

Angular neutron density whatever the direction

3

r

rdd

vv

dtvrNtvrn ),,,(),,(4

(similar definition with variables (r,E,) or (r,v))(dimensions of N in both cases?)

,E

dvdrdtvrN ),,,( r

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Angular flux

s.t.

Total neutron flux Integrated flux

(Angular) current density

Nb of n flowing through a surface / u.t. (net current)

Isotropic distribution?

),,(.),,,(),,(4

tvrnvdtvrtvr

),,,(.),,,( tvrNvtvr [dim ?]

),,,(),,,(),,,( tvrNvtvrtvrJ

dvdSdntvrdvdSdntvrJoo

.),,,().,,,(

44

),(4

1),,( vrvr

0.),,(4

dSdnvr

),,,( tvr

Net current = 0 , hence flux spatially cst?

Wrong!!A reactor is anisotropic But weak anisotropy often ok (1st order)

dvtvrtro

),,(),(

[dim ?]

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REACTION RATE

Nb of interactions / (volume.time): R

Beam of incident n on a (sufficiently) thin target (internal nuclei not hidden):

R = N.(r,t).(r) = (r).(r,t)

Or: interaction frequency = v [s-1]

n density = n(r,t) [m-3]

R = n(r,t).v(r) = (r).(r,t)

General case: cross sections dependent on E ( v)

Rem: = f(relative v between target nucleus and n) while = f(absolute v of the n) implicit assumption (for the moment): heavy nuclei immobile

(see chap. VIII to release this assumption)

R = Nb nuclei

cm3

Nb n

cm2.s

Cross sectional area of a nucleus (cm2)

dvtvrvrdvtvrnvrvRoo

),,(),(),,(),( ***

x x

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Rem: differential cross sections

Scattering speed after a collision?

Conditional probability that 1 n with speed undergoing 1 collision at leaves it with a speed in [v’, v’+dv’] and a direction in about

with

Scattering kernel s.t.

Isotropic case:

),(

'')'.,',,(

),(

'')',',,(

vr

ddvvvr

vr

ddvvvr

s

s

s

s

)',',,().,()',',,( vvrKvrvvr ss

)',',,( vvrK

rv

''d

)',(4

1)',',,( vvrvvr ss

),('')',',,(4

vrddvvvr sso

[dim ?]

Why?

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FLUENCE, POWER, BURNUP

Fluence

Characteristics of the irradiation rate ([n.cm-2] or [n.kbarn-1])

Power

Linked to the nb of fission reactions

Burnup

Thermal energy extracted from one ton of heavy nuclei in fresh fuel

Fluence x <fission cross section> x energy per fission

')',(),( dttrtrt

o

rdtrrtPV f ),()()(

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II.2 TRANSPORT EQUATION

NEUTRON BALANCE

Variation of the nb of n (/unit speed) in volume V, in dv about v, in about

rdt

tvr

vrd

t

tvrNrdtvrN

t VVV

),,,(1),,,(

),,,(

)),,,((),,,(),,,(),,,(1

tvrJdivtvrtvrSt

tvr

v t

Sources Losses due toall interactions

Losses throughthe boundary

Gauss theorem

V

rdtvrSV

),,,( rdtvrvrtV

),,,(),( SdtvrJ ).,,,(

(n produced in about , dv about v, about )

rdd

r (n lost in about , dv about v, about )

rdd

r

d

dS

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Rem: general form of a conservation equation

BOLTZMANN EQUATION (transport)

Without delayed nSources?

Steady-state form

Sourcescurrentdivdensityt

)(

),,,( tvrS

'')',',(),',',(),,(),(),,(.4

ddvvrvvrvrvrvr sot

),,('')',',()',()(4

1

4

vrQddvvrvrv fo

Scattering External source

Fission

)(4

1v

''),',',(),',',(

4

ddvtvrvvrso

),,,( tvrQ

''),',',()',(4

ddvtvrvrfo

Total nb/(vol. x time) of n due to all fission at r

Fraction in dv about v, d about

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Compact notation

with

(destruction-scattering operator)

and

(production operator)

Non-stationary form

'')',',()',()(4

1),,)((

4

ddvvrvrvvrJ fo

'')',',(),',',(

),,(),(),,(.),,)((

4

ddvvrvvr

vrvrvrvrK

so

t

QJK

QKJtv

)(1

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With delayed n

Concentration Ci of the precursors of group i:

with i = (ln 2) / Ti

Def: production operator for the delayed n of group i, i = 1…6:

Total production operator:

Let

t

trCi ),(

'')',()(4

1)(

4

ddvvrvJ foii

4

)().,(),,,)()(1(),,,)((

6

1

vtrCtvrJtvrJ i

iii

o

4

)().,(),,(

vtrCtvrF i

ii

),( trCii

dvdtvrvrfo),,,(),(

4

prompt n

i

Fraction/(vol. x time) of n due to all fissions at r in group i

Radioactive decay

Fraction in dv about v, d about

Production of delayed n of group i / (vol. x time)

(precursors assumed to be a direct product of a fission)

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System of equations for the transport problem with delayed n

Stationary regime

Reduction to 1 equation:

Production operator: equivalent to having J Jo (prompt n) iff

Formalism equivalent, with or without delayed n, with a modified fission spectrum

0),,(),,(])1[(6

1

vrQvrJKJ iii

o

)()()1()(~)(6

1

vvvv iii

oo

6...1,),,)((),,(),,(

itvrJtvrF

t

tvrFiiii

i

),,,(),,(),,,(])1[(),,,(1 6

1

tvrQtvrFtvrKJt

tvr

v iii

o

(Why in stationary regime?)

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CONTINUITY AND BOUNDARY CONDITIONS

Nuclear reactors: juxtaposition of uniform media ( indep. of the position) How to combine solutions of

in the media?

Let : discontinuity border (without superficial source)

Integration on a distance [-,] about in the direction

continuity on Boundary condition (convex reactor surrounded

by an vacuum):

),,(),,(),(),,(. vrSvrvrvr t

0)(),,(),,(0

Ovrvr ss

dvrS

dvrvrdvr

s

ssts

),,(

),,(),(),,(.

0.,0),,( nrvr ss

sr

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INTEGRAL FORMS

If s = distance covered in the direction of the n:

Lagrange’s variation of constants:

Let

(interpretation ?)

: optical thickness (or distance) [1]

),,(),,(),(),,(

vsrSvsrvsrds

vsrdooot

o

'),,'(),,("),"(

' dsvsrSevsr o

s dsvsr

o

ot

s

s

dsvsrSevro

dsvsrts

o ),,(),,('),'(

'),'(),( dsvsrrr ot

s

oov

srr o

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Yet

If both scattering and independent source are isotropic

After integration to obtain the total flux:

Rem: S fct of !!

')(2 dsdrrrrd oo

''),),((),,( ))(,(

4

dsdvsrSevr oo

srr ov

oo

ooR

o

rr

rdrr

rrvrS

rr

e ov

),,(3 2

),(

ooRo

rr

rdvrSrr

evr

ov

),(4

),(3 2

),(

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Explicit form of the integral equation for the angular flux

We have

and

Thus

(interpretation ?)

),,('')',',(),',',(

'')',',()',()(4

1),,(

4

4

vrQddvvrvvr

ddvvrvrvvrS

so

fo

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

oo

ooR

o

rr

rdrr

rrvrS

rr

evr

ov

),,(),,(3 2

),(

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Transition kernel Transport process: proba distribution of the ingoing coordinates in the next collision, given the outgoing coordinates from the previous one

Collision kernel

Impact: entry in 1 collision exit

Compact notation :

)'().'(.'

'

')',(),,',','( 2

),'(

vvrr

rr

rr

evrvrvrT

rr

t

v

)'()','(

)]','(4)(

),',','([),,',','( rr

vr

vrv

vvrvrvrC

t

fs

)',','(';),,( vrPvrP

1)'( dPPPC

)(1)'( vedPPPT

Captures not considered

Fissions : 1

= 1 for an infinite reactor

(based on the negative exponential law)

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Collision densities

Ingoing density: = expected nb of n entering/u.t. in a collision with coordinates in dP about P

Outgoing density: = expected nb of n leaving/u.t a collision with coordinates in dP about P

Evolution equations

')'()'()( dPPPTPP

dPP)(

)()()( PPP t

dPP)(

')'()'()()( dPPPCPPQP

")"()"()(

"')'()'"()"(')'()'()(

dPPPKPPI

dPdPPPTPPCPdPPPTPQP

Interpretation ?

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Rem:

equ. of (P) = (equ. of (P)) x t(P)

Possible interpretation of n transport as a shock-by-shock process

FORMAL SOLUTION USING NEUMANN SERIES

Let

j(P): ingoing collision density in the jth collision

: solution of the transport equation

Not realistic: infinite summation… Basis for solution algorithms

)()( PIPo

...1,')'()'()( 1 jdPPPKPP jj

)()(0

PP jj

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CH.II: NEUTRON TRANSPORTINTRODUCTORY CONCEPTS• ASSUMPTIONS• NEUTRON DENSITY, FLUX, CURRENT• REACTION RATE• FLUENCE, POWER, BURNUP

TRANSPORT EQUATION • NEUTRON BALANCE• BOLTZMANN EQUATION • CONTINUITY AND BOUNDARY CONDITIONS • INTEGRAL FORMS • FORMAL SOLUTION USING NEUMANN SERIES