The exact interpretation of neutron transport in heterogeneous domains is so complex

Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).

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• The exact interpretation of neutron transport in heterogeneous domains is so complex.• Assumptions and approximations.• Simplified approaches.• Simplified but accurate enough to give an estimateestimate of the average characteristics average characteristics of neutron populationneutron population.• Numerical solutions.• Monte Carlo techniques.

Fick’s Law

MCNPMCNP


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Fick’s LawAssumptions:1.The medium is infinite.2.The medium is uniform 3.There are no neutron sources in the medium.4.Scattering is isotropic in the lab coordinate system.5.The neutron flux is a slowly varying function of position.6.The neutron flux is not a function of time.

)(rnot

Restrictive! Applicability??

Restrictive! Applicability??

http://www.iop.org/EJ/article/0143-0807/26/5/023/ejp5_5_023.pdf

http://www.iop.org/EJ/article/0143-0807/26/5/023/ejp5_5_023.pdf


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Lamarsh puts it more bluntly:“Fick’s Law is invalid: a) in a medium that strongly absorbs neutrons; b) within three mean free paths of either a neutron source or the surface of a material; and c) when neutron scattering is strongly anisotropic.”

Fick’s Law


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Fick’s Law


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Fick’s LawCurrent Jx

x

Con

cent

ratio

n C

dC/dx

x

(x)

High flux

More collisions

Low flux

Less collisions

Negative Flux GradientCurrent Jx

• Diffusion: random walk of an ensemble of particles from region of high “concentration” to region of small “concentration”.• Flow is proportional to the negative gradient of the “concentration”.

xDJ x

• From larger flux to smaller flux!• Neutrons are not pushed!• More scattering in one direction than in the other.

Recall:


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x

y

z

rdAz

Fick’s Law

der

dAr rz

st

24

cos)(

Number of neutrons scatteredscattered per second from d at rr and going through dAz

Slowly varying)(rnot ss

Isotropic

Removed en route

(assuming no buildup)

d


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Fick’s Law


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Fick’s Law

2

0

2/

0 0

sincos)(4 r

rzszz ddrder

dAdAJ t

HW 14HW 14

023

zJJJ

t

szzz

?

zz dAJ

and show that

and generalize23 t

sDDJ

Diffusion Diffusion coefficientcoefficient

Fick’s lawFick’s law

The current density is proportional to the negative of the gradient of the neutron flux.

s

D

3

1

Total removal


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Fick’s LawValidity:1. The medium is infinite. Integration over all space. after few mean free paths 0 corrections at the surface are still required. 2. The medium is uniform.

and are functions of space re-derivation of Fick’s law? locally larger s extra JJ cancelled by iff ???

Note: assumption 5 is also violated!3. There are no neutron sources in the medium.

Again, sources are few mean free paths away and corrections otherwise.

rte

)(rnot ss

)(rs

rr sat ee )( HW 15HW 15


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Fick’s Law4. Scattering is isotropic in the lab. coordinate system.

If reevaluate D.

For “practical” moderators:

5. The flux is a slowly varying function of position.a variation in .

03

2)(cos

A

33

1

)(3

1 tr

trst

D

1

str

)(2

2

rr

HW 16HW 16

Weekly absorbing t = s.


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Estimate the diffusion coefficient of graphite at 1 eV.

The scattering cross section of carbon at 1 eV is 4.8 b.

Fick’s LawHW 17HW 17

Scattering

Absorption

Other materials?


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Fick’s Law6. The neutron flux is not a function of time.

Time needed for a thermal neutron to traverse 3 mean free paths 1 x 10-3 s (How?).If flux changes by 10% per second!!!!!!

Very small fractional change during the time needed for the neutron to travel this “significant” distance.

43

1

101101.011

/

xxmss

ms

DJ


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Back to the Continuity Equation

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

trJtrrtrStrtv a

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

trDtrrtrStrtv a


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The Diffusion Equation

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

trDtrrtrStrtv a

If D is independent of r (uniform medium)

),(),()(),(),(1 2 trDtrrtrStr

tv a

)()()()(0 2 rDrrrS a

or scalar Helmholtz equation.

)()()(0 2 rDrra

Buckling equation.

Laplacian


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)()()()(0 2 rDrrrS a

a

DL

2Define L Diffusion Length

L2 Diffusion Area Moderation Length

D

S

L

22 1

Steady State Diffusion Equation

01

22

L

Boundary ConditionsBoundary Conditions• Solve DE get .• Solution must satisfy BC’s.• Solution should be real and non-negative.


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Steady State Diffusion EquationOne-speed neutron diffusion in infinite mediumOne-speed neutron diffusion in infinite medium

Point source

0)(1

)(2

2 rL

r

0)(1

)(2

)(22

2

rL

rdr

d

rr

dr

d HW 18HW 18

r

eC

r

eA

LrLr //

General solution

A, C determined from BC’s.

BC BC r 0 C = 0.

Show that

neutrons per second absorbed in the ring.

Show that


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r

eA

Lr /

D

SA

4

r

e

D

S Lr /

4

HW 18 HW 18 (continued)(continued)

a

DL

2

rdr

22 6Lr r

adrr 24


18


HW 19HW 19

Study example 5.3 and solve problem 5.8 in Lamarsh.

Multiple Point Sources?


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One-speed neutron diffusion in a finite mediumOne-speed neutron diffusion in a finite medium


A B

BA • At the interface

• What if A or B is a vacuum?• Linear extrapolation distance.

dx

dD

dx

dDJJ B

BA

ABA

x


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One-speed neutron diffusion in a multiplying mediumOne-speed neutron diffusion in a multiplying medium

More realistic multiplying medium

The reactor core is a finite multiplying medium.• Neutron flux?• Reaction rates?• Power distribution in the reactor core?Recall:• Critical (or steady-state):Number of neutrons produced by fission = number of neutrons lost by:absorptionandleakage

)( rate absorptionneutron

rate productionneutron

A

(S)k

)( rate leakageneutron )( rate absorptionneutron

)( rate productionneutron

LEA

Skeff


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More realistic multiplying medium

yprobabilit leakage-nonleaknoneff P

LEA

A

k

k

aa

a

V

SA

S

LE

VolumeVS

SALE

1

area surface

3

2

)()()(0 2 rDrrk aa

Steady state homogeneous reactorSteady state homogeneous reactor

2222 1

0)()(L

kBrBr

Material buckling

For a critical reactor:Keff = 1K > 1

Recall:

multiplying m

edium

multiplying m

edium

Things to be used later…!


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More realistic multiplying medium0)()( 22 rBr

)(

)(22

r

rB

• The buckling is a measure of extent to which the flux curves or “buckles.”• For a slab reactor, the buckling goes to zero as “a” goes to infinity. There would be no buckling or curvature in a reactor of infinite width. • Buckling can be used to infer leakage. The greater the curvature, the more leakage would be expected.


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More on One-Speed DiffusionHW 20HW 20

Show that for a critical homogeneous reactorcritical homogeneous reactor

DBDLBP

a

a

a

aleaknon 2222 1

1

Infinite Bare Slab Reactor Infinite Bare Slab Reactor (one-speed diffusion)(one-speed diffusion)

x

aa/2

d da0/2

• Vacuum beyond.• Return current = 0.d = linear extrapolation distance = 0.71 tr (for plane surfaces) = 2.13 D.

z

Reactor

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The exact interpretation of neutron transport in heterogeneous domains is so complex