diffusion of neutrons overview basic physical assumptions generic transport equation diffusion...

DIFFUSION OF NEUTRONS

OVERVIEW

• Basic Physical Assumptions• Generic Transport Equation• Diffusion Equation• Fermi’s Age Equation• Solutions to Reactor Equation

HT2005: Reactor Physics T10: Diffusion of Neutrons 2

Basic Physical Assumptions

• Neutrons are dimensionless points

• Neutron – neutron interactions are neglected

• Neutrons travel in straight lines

• Collisions are instantaneous

• Background material properties are isotropic

• Properties of background material are known and time-independent


Ep E

E

10

9

-7

4.55 10cm; is in eV

0.01eV 4.55 10 cm

D(H) 10 cm

(a) (b)

Physical Model


θ

rm b

rc

rc

χm

v

v´

Collision Model


Initial Definitions

3 3

3 3 3 3

;

( , , ) Expected number of neutrons in within

x y zd dxdydz d dv dv dv

N t d d d d

r v

r v r v r v

2

1;

2( , )

vvmv

E

Ω v v Ω

Ω

r

ex

x

y

z

ey

v

ey


3

22

0 0 0 0 4

( , ) ( , , ) ( , , )

( , , , ) sin ( , , , )

sin

x y zn t N t dv dv dv N t d

N v t v dvd d N E t d dE

d d d

r r v r v v

r Ω r Ω Ω

Ω

2

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

( , , , ) ( , , )

1( , , , ) ( , , , )

vN E t N t N E t d dE N t d

mN v t v N t

N E t N v tmv

r Ω r v r Ω Ω r v v

r Ω r v

r Ω r Ω

Neutron Density


Angular Flux and Current Density

( , , , )

( , , ) ( , , )

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

( , , , ) ( , , , )

( , , , ) ( , , , )

Et

t N t

E t N E t vN E t

E t vN E t

E t E t

r Ω

J r v v r v

J r Ω v r Ω Ω r Ω

r Ω r Ω

J r Ω Ω r ΩdS

J

number of neutrons

crossing per 1 second

d

d

J S

S


Generic Transport Equation

.

changedue changedue changeduetimerate

of change to leakage to tomacro sources

collisions forcesof N throughS

We ignore macroscopic forces

Arbitrary volume V

3 3 3( , , ) ( , , ) ( , , )collV S V V

NN t d t d d Q t d

t t r v r J r v S r r v r


3 3 3( , , ) ( , , ) ( , , ) ( , , )S V V V

t d t d N t d N t d J r v S J r v r v r v r v r v r

x y zx y z

e e er

Generic Transport Equation

0collV

N NN Q d

t t v r

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

coll

N t NN t Q t

t t

r vv r v r v

Gauss Theorem:


Substantial Derivative

r

xy

z

Leonhard Euler's (1707-1783) description:

We fix a small volumeNt

We let a small volume move

Joseph Lagrange's (1736-1813) description

dNdt

( , , )dN N N N

N N tdt t t t

N N Nt m

r vr v

r vF

vr v

( , , )coll

dN NQ t

dt t

r v


( , , )coll

dN NQ t

dt t

r v

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

coll

N t NN t Q t

t t

r vv r v r v

coll

N N N NQ

t m t

Fv

r v

Transport (Boltzmann) Equation


Collision Term ,E Ω

r

x

y

z

,EΩ

2cm

, ,sterad eVs E E

Ω Ω

, , , ,s s BE E E E N Ω Ω Ω Ω

0 4

( )

( , , , ) ( , , , ) ( , ) ( , , , )tcoll Total absorption

Scatteringtothecurrent directionandenergy

NE E vN E t d dE E vN E t

t

r Ω Ω r Ω Ω r r Ω


Neutron Transport Equation

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

coll

N t NN t Q t

t t

r vv r v r v

0 4

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

E tE E E t d dE Q

v t

r Ω

Ω r Ω Ω r Ω Ω

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

( , , , ) 0 0 : ( )s s

E E initial condition

E t boundary freesurface condition

r Ω r Ω

R Ω Ω n

( , , , ) ( , , , )E t vN E t r Ω r Ω


Boundary Condition

( , , , ) 0 when 0sSE t

rr Ω n Ω

Ω

ns

xy

z

V

Volume V

Surface S

Outgoing direction

Outward normal

r

Ω

Incoming direction

Rs


Difficulties• Mathematical structure is too involved• Mixed type equation (integro-differential), no way

to reduce it to a differential equation• Boundary conditions are given only for a halve of

the values• Too many variables (7 in general)• Angular variable

( , ) ( , , , ) ; ( , ) ( , , , )n t N E t d dE t E t d dE r r Ω Ω r r Ω Ω


Angular Measures

180 Solar disks


φ

RCR

Plane Anglesd dss n

nr

cos r ddsd

r r

e s

re


2

AR

Solid Angles

d dAA n

nr e Ω

r

2 2

cosdA dd

r r

Ω A

Ω


• Infinite homogeneous and isotropic medium• Neutron scattering is isotropic in Lab-system

• Weak absorption Σa << Σs

• All neutrons have the same velosity v. (One-Speed Approximation)

• The neutron flux is slowly varying function of position

One-Group Diffusion Model


x

y

z ZJ J J

Derivation

J

Number of collisions s dV

2

cos

4 4Number of neutrons scattered within

Ω

Ω

s s

d dAdV dV

rd

dA

2

cos

4Number of neutrons reaching

srs

dAdV e

rdA

2 2

0 0 0

( ) cos sin4

srs

r

J r e d drd

Isotropic scattering

r = 0 is most important


Taylor’s series at the origin: 00 00

...x y zx y z

sin cos ; sin sin ; cosx r y r z r

00 00

( ) sin cos sin sin cosr r rx y z

r

2 2

000 0 0

cos sin4

srs

r

J r e d drdz

Derivation II


0

0

0

0

0

1

4 6

1

4 6

1

3

s

s

zs

Jz

Jz

Jz

Derivation III

0 0 0

1 1 1; ;

3 3 3z x ys s s

J J Jz x y

1

3x x y y z z x y zs

J J Jx y z

J e e e e e e


Fick’s Law

1( ) ( ); ( )

3 x y zs x y z

J r r r e e e

1( ) ( );

3 3s

s

D D

J r r

CM-System → Lab-System:1

(1 );tr s trtr

1( ) ( );

3 3tr

tr

D D

J r r


scos scos

tr s s s s s

n cos cos cos . . . . . cos

2 3

Transport Mean Free Path

s

tr

; 1 cos ; 1 cos1 cos

str tr s tr s

Transport correction =

A number of anisotropic collisions is replaced by one isotropic

Information about the original direction is lost


Diffusion Equation

Change rate Production Leakage Absorption

of rate rate raten

3

Production( , ) ( , ) ( , )

rate f

nt t Q t

cms

r r r

3

Absorption( , ) ( , )

rate a

nt t

cms

r r


Leakage Rate

(x,y,z)

x

y

z

dx

dy

dz

Jz2

2

( , , ) ( , , )z z z

z dz z

L J x y z dz dxdy J x y z dxdy

D dxdy D dxdydzz z z

2

2

2

2

2

2

x

y

z

L D dxdydzx

L D dxdydzy

L D dxdydzz

2 2 22

2 2 2Leakage from a unit volume D D

x y z


Change rate Production Leakage Absorption

of rate rate raten

21;a ext fD Q Q Q

v t

Time-dependent:

Time-independent: 2 0aD Q

Time-independent from a steady source

2

2 22

0

1 10;

3 3

a

a s a tr

a

D Q

DQ L

L D

Diffusion Equation


2 2 22

2 2 2

2 2

2 2 2

22

2 2 2 2 2

1 1

1 1 1sin

sin sin

x y z

rr r r r z

rr r r r r

Cartesian geometry

Cylindrical geometry

Spherical geometry

Laplace’s Operator


Symmetries

2 1 nn

d dr

r dr dr

n = 0 for slab

n = 1 for cylindrical

n = 2 for spherical

x

y

z

Slab geometry

22

2x

r

Spherical geometry

2 22

1r

r r r

r

Cylindrical geometry

2 1r

r r r

z


General Properties

• Flux is finite and non-negative

• Flux preserves the symmetry

• No return from a free surface

• Flux and current are continues

• Diffusion equation describes the balance of neutrons


0 0

0 0

1 1,

4 6 4 6s s

J Jz z

for +z - direction: A trA A B trB B

z z4 6 4 6

A trA A B trB B

z z4 6 4 6

for -z - direction:

BA

Dz

DzA

AB

B

A B

z

AB

Interface Conditions


00

0

00

0; 04 6

3

2

tr

tr

Jx

x

Straight line extrapolation from x = 0 towards vacuum: 0 0

3( )

2 tr

x x

2( ) 0 ( 0.71)

3 trx for x exact

extrapolation length = 0.71 tr

Free surface

Diffusion eq.

Transport equation

0.66 tr 0.71 tr

0 00 0

1 1

0.66 0.71tr trx x

Boundary Condition


x = 0

( )x

2

2

20

2 2

( ) ( ) 0

( )1 ( )( )

a

dD x Q xdx

Q xd Q xx

dx L D D

2

2 2

0 0

0

1( ) 0 ( )

lim ( ) 0 0

lim ( )2 2

x L x L

x

x

dx x Ae Be

dx Lx B

Q Q LJ x A

D

Q0

0( )2

x LQ Lx e

D

Transport equation

3 s

Plane Infinite Source in Infinite Medium


Point Source in Infinite Medium

r

22

22 2

1( ) ( ) 0

1 1( ) 0 0

a

d dD r r Q xr dr drd dr r r

r dr dr L

2 00

0

( )

lim ( ) 0

lim 4 ( )4

r L r L

r

r

e er A B

r rr B

Qr J r Q A

D

0( )4

r LQ er

D r

2

20 0

( )4n abs. ( , )( ) r La r r drr r dr rp r dr e dr

Q Q L

2 2 2

0

( ) 6r r p r dr L


Plane Infinite Source in Slab Medium

( )x

Q0

0

2sinh

2( )2 cosh

2

a xQ L Lx

aDL

x = 0

2 22

1 1

0.71 a aa trx a

x = a/2x = -a/2

0( )2

x LQ Lx e

D

Slab:

Infinite:


Plane Infinite Source with Reflector

Q0

a

21

12 21

1( ) 0

dx

dx L

12 21

22

22 22

1( ) 0

dx

dx L

Reflector Reflector

Bare slab


• q(E) - number of neutrons, which per cubic-centimeter and second pass energy E.

• q(E) = [ncm-3 s-1]• X-sections depend on E: D(E),Σs(E),...

Energy

E q(E)

E0 Q0

2( )( )

( )

E

tE

D E dEE

Ecm

E

Slowing down medium: s a s t

log( )( )

( )

f

th

Ef th

th tht sE

E ED E dE DE

E E

1 ln1

21

6th s s mts sD n D L r

Mean Total Slowing down distance

Can be shown

Age of Neutrons


2( ) ( , ) ( , ) ( , ) 0aD E E dE E dE Q E dE r r r

( , ) is the number of neutrons at with energies in ( , )E dE E E dE r r

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

q EQ E dE q E dE q E dE

E

r

r r r

E+dE

E

q(E+dE)

q(E)

2 ( , )( ) ( , ) ( , ) 0a

q ED E E dE E dE dE

E

r

r r

( )Continuous slowing down: ( )

( )t

q E dEE dE

E E

2 ( , )( )( )( , ) ( , ) 0

( ) ( )a

t t

q EED Eq E q E

E E E E E

rr r

Fermi’s Age Equation

( ) ( ) ( )

( ) ( )

t

duu u du qu

u du E dE


0 ( )ˆ ˆ( , ) ( , ) exp ; ( , ) ( , ) ( ) 0

( )

Ea

atE

E dEq E q E q E q E E

E E

r r r r

2 ( , )( )( )( , ) ( , ) 0

( ) ( )a

t t

q EED Eq E q E

E E E E E

rr r

2ˆ( , ) ( )ˆ( , )

( )t

q E D Eq E

E E E

r

r

0 ( )new variable: ( )

( )

E

tE

D E dEE

E E

2ˆ( , )

ˆ( , )q

q

r

r

Fermi’s Age Equation II

τ ~ time


2

2

q qx

x = 0

No absorption

2

0 1 2

exp4

( , )4

pl

x

q x Q

r

22

1q qr

r r r

No absorption

2

0 3 2

exp4

( , )4

pt

r

q r Q

Solutions to the Age Equation


-6 -4 -2 0 2 4 60.00

0.02

0.04

0.06

0.08

=0.5 =1.0 =1.5

q(r,

)

r

2

0 3 2

exp4

( , )4

pt

r

q r Q

Slowing Down Density for Different Fermi’s Ages

0 ( ) ( )Q r


Migration Area (Length)Fast neutron borne

Thermal neutron absorbed

Fast neutron thermalized

r

rs rth

N N N

i s s i th th ii i i

r r r r r rN N N

2 2 2 2 2 2, ,

1 1 1

1 1 1; ;

2 21

6 thL r

M r2 216

s th

s th s s th th

s s th th s th

r r r

r r r r r

22 2 2

2 2 2 2 2

2

2

r r r

r r r r

r r

2 2

2 20

2

0

( , )4

6 6

( , )4

pt

s th

pt

r q r r dr

r r

q r r dr

th sM L L L 2 2 2 2


Diffusion and Slowing Down Parameters for Various Moderators

Moderator g/cm3 tr

cmL

cmtth

ms

tss

0

cm2

H2O 1.0 0.43 2.7 0.21 0.92 1 27

D2O(pure)

1.1 2.5 165 130 0.51 8 131

D2O(normal)

1.1 2.5 100 50 0.51 8 115

Be 1.8 1.5 22 3.8 0.21 10 102

BeO 2.96 1.4 31 8.1 0.17 12 100

C (puregraphite)

1.6 2.6 59 17 0.158 24 368

C (normal.graphite)

1.6 2.6 50 12 0.158 24 368


Neutrons in Multiplying Medium2

a

nD Q

t

2( , , )( ) ( , , ) ( ) ( , , ) ( , , )a

th th th th

n E tdE D E E t dE E E t dE Q E t dE

t

r

r r r

( , , ) ( ) ( ) ( )E t F G E T t r r

( , )( , , ) ( , ); ( , , ) ;

( ) ( , , ) ( , );

thth

avth th

a ac thth

tE t dE t n E t dE

v

E E t dE t

rr r r

r r

( ) ( , , )th

cth

D E E t dE

D

r

2( , )1( , ) ( , ) ( , )th

c th ac th thav

tD t t Q t

v t

rr r r

Assumption:


Principles of a Nuclear Reactor

1

2

N

Nk

n/

fissi

onN1

N2Leakage

Fast fission

Resonance abs.

Non-fuel abs.

Leakage

Non-fissile abs.

Fission

Slo

win

g d

ow

n

Ene

rgy

E

2 MeV

1 eV

200 MeV/fission

ν ≈ 2.5


Total number of fission neutronsFast fission factor 1.02

Number of fission neutrons from thermal neutrons

Ea

a sE

dEE

p E e

0

Resonance escape probability ( ) 0.87

F Fff

f F Fa a

P

Conditional probability

Ff

f Fa

P

Number of neutrons per absorption in fuel 1.65

Fa

a

f

Thermal utilization 0.71

FNL

TNL

NL FNL TNL

P

P

P P P

Fast non-leakage probability 0.97

Thermal non-leakage probability 0.99

Non-leakage probability

k fp

NL FNL TNLk k P fp P P


f thCore

a thCore

p dV

kdV

Rate of neutron production in coreRate of neutron absorption in core

a f th f th a thk p Q p k

2( , )1( , ) ( , ) ( , )th

c th ac th thav

tD t t Q t

v t

rr r r

22

( , ) 11( , ) ( , )th

th thcav c

t kt t

t Lv D

rr r 2 c

cac

DL

2

2

1m

c

kB

L

22 ( )

In the stationary case: ( )th

mth

B

r

r


Buckling as Curvature

B

2

aB 0

bBcB

a b cB B B

Large core

Small core

L SB B


2 2( , )1( , ) ( , )th

th m thav c

tt B t

tv D

r

r r

( , ) ( ) ( )th t F T t r r

221 ( ) ( )

( )( )m

av c

dT t FB

dt Fv DT t

r

r

22 2 2( )

( ) ( )( ) g g

FB F B F

F

r

r rr

2 22

1In a critical reactor: g m

c

kB B

L

2 ( ) ( )F F r r

Criticality Condition


2

2

1 2

1 2

2

2 2

12

matrix:

Differential operator: ( ) ( )

0 ( )

0 ( ) sin cos

BC1: (0) 0 0

BC2: ( ) 0 sin 0 ; 1,2,

; ( ) sin ; 1,2, ,

x x

n n

n n

dy x y x

dx

y x Ce C e

y x C x C x

y C

y a a n na

n ny x C x n

a a

Ax x

Eigenvalues

Transport operator

Differential operator

Matrix


Eigenfunctions

0 a

2

1 1 12; ( ) siny x C x

a a

Only one is physically meaningful


Solution of a Reactor Equation1

tr

tr

λ0.71RR

λ1.42LL

2 22

2 2

Φ 1 Φ ΦB Φ 0

r r r zΦ(r, z) F(r)G(z)

2( ) ( ) ( )2 2

2 2

2 22 2

2 2

1d F r 1 dF r 1 dG zB 0

F dr Fr dr G dz

1d F 1 dF 1 dGα β

F dr Fr dr G dz

2 2 2B α β

( ) sin cosG z A z C z

Symmetry: 0A


( ) n

L π πzG(z) Ccos z z β n G(z) Ccos

2 L L

22 2

2

d F dFx αr x x x F 0

dx dx

0 0

0 0

( ) ( ) ( )

( ) ( ) ( )

F x DJ x EY x

or

F r DJ r EY r

0

00( )

Rr

F r DJR

1 2 3 4 5 6 7 8

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 ( )J x

0 ( )Y x

0 2.405


00

max

0max 0

222 0

rπzΦ(r, z) A cos J

L R

A Φ

rπzΦ(r, z) Φ cos J

L R

πB

L R

B C

0max 0

rπzΦ(r, z) Φ cos J

L R

222 0π

BL R

max

ππ πΦ(r, z) Φ cos cos cos

yx za b c

2 2 22 π π π

Ba b c

max

πsin

Φ( ) Φ

rRrr

22 π

BR

Rectangular

Cylinder

Sphere


Critical Size of a ReactorWe assume bare homogenous reactorFor thermal neutrons we get:

2 ( ) ( ) ( , ) 0a thD r r qr

Slowing down neutrons:2 ( , )

( , )qr

qr

Assumption:Reactor is sufficiently big to treat neutron spectrum independently of space variables

2

2

2

22

0

2 2

0

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

( ) 1 ( )( )

( ) ( )

0

( , ) ( )

B

B

Tqr R r T T R r R r

R r dTB T T e

R r T d

R BR

qr R r T e

At the beginning slowing down density is =0 aR r T q r f p 0( ) ( ,0)

p1


For > 0 one has to take into account resonance capture through p – resonance passage factor.

2 2 2

2 2

B τ B τ B τ0 a a

R(r) Φ(r) Φ B Φ 0

q(r,τ) R(r)T pe Σ Φ(r)f ηpe Σ Φ(r)k e

2aD Σ Φ q 0

2

2

2

2 B τa a

B τ2

2 2

2 2 B τ

DB Σ Φ Σ Φk e 0

1 k eB 0

L L

(B L 1) k e 0

or

2

2 2 11

Bk eB L


Non-Leakage Probability

1NL FNL TNLk k P k P P 2

2 2 11

Bk eB L

2 2 22

1

1

a thV a

TNLaa th th

V V

dVA

PA L DB LBdV D dV

2

2 2

1

1

thBFNL

TNL

P e

PLB


Volume of an cylindrical reactor with buckling derived from a critical equation – the smallest critical size:

2 22

22 2

2

min 3

22

2.405We assume that L L and R R

(2.405)

3 2.405 3 1480 gives ; gives

2

1(s )

Generally:big reactor small B-value

BL R

LV R L V R L

BL

dVL R V

dL B B B

Bidesize


Minimum Volume

R

L

L = L(R)

V = V(R)

L

D = 1.08 L

BL R

RV

B R

222 0

2 2

220


Optimum Core Dimensions

Core shape

Optimum dimensions

Minimal volume

Cube

Cylinder

Sphere

3a b c

B

3

161V

B

03 3;

2L R

B B

3

148V

B

RB

3

130V

B


2

2 2

2 2 2 2 2 2 2 22

11

(1 )(1 ) 1 ( ) 11

6

Bk eB L

k k k kB L B B L BM r

B

Migration Area

11

1xe x

x


Improved Diffusion

(1) Isotropic Scattering: 1

(2) Boundary Condition: 0.66 0.71

(3) Migration Length:

s s

tr tr

L M


The END


CRITICALITY EQUATION - physical interpretation

reactorinfiniteinrateproduction ka

aBk e

2 production rate in the FINITE reactor

2

2 2

2 2 2 2 2 2 2 22

11

(1 )(1 ) 1 ( ) 11

6

Bk eB L

k k k kB L B B L BM r

B


e PBs

2 non leakage factor for all epithermal neutrons

Thermal leakage:

D

D a

Thermal non - leakage factor:

1

1

1

11

2

2 2

2 2

2

D

D DB

B LP

k e

B Lk P P

a

a

a

t

B

s t

for critical reactor


Z Z ZJ J J

;n vn J v

Derivation

Number of collisions in dV s dV

Neutrons scattered towards dA2

cos

4 4

Ω

s s

d dAdV dV

r

Neutrons through dA per 1 second 2

cos

4

srs

dAdV e

r2 2

0 0 0

( ) cos sin4

srs

r

J r e d drd


( ) n

L π πzG(z) Ccos z z β n G(z) Ccos

2 L L

22 2

2

d F dFx αr x x x F 0

dx dx

0 0

0 0

( ) ( ) ( )

( ) ( ) ( )

F x DJ x EY x

or

F r DJ r EY r

1

Rr

DJrF

R405.2

)(

405.2

0


Delayed Neutrons( , , , ) ( , , , ) ( , , , )f scE E E E E E r Ω Ω r Ω Ω r Ω Ω

6

10 4 0 4

1(1 )t sc fi i

i

d dE d dE C Qv t

Ω Ω Ω

0 4

6

1

0.0065

ii i i f

ii

CC d dE

t

Ω

( , , , ) ( , ) ( ; , , )

1( ; , , ) ( ; )

4( ; ) ( ; ) ( ; )

( ; ) 1; ( ; ) ( , )

( , )( , , , ) ( ; ) ( , )

4

r Ω Ω r r Ω Ω

r Ω Ω r

r r r

r r r

rr Ω Ω r r

f f f

f

f f

E E E f E E

f E E E E

E E E E E

E E dE E E E

EE E E E


1

1

Optimum dimensions and critical mass of acylindrical core

diffusion of neutrons overview basic physical assumptions generic transport equation diffusion...

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