ch 5 neutron diffusion(2)

8/6/2019 Ch 5 Neutron Diffusion(2)

1/22

Neutron Diffusion andNeutron Diffusion and

ModerationModeration


2/22

Energy dependent fluxEnergy dependent flux

F

! 7Jq

Fa ! 7a E 0g

J E dE


3/22

Ficks LawFicks Law

Luckily for you you still dont have to solveLuckily for you you still dont have to solve

the previous equationthe previous equation

FIRST ASSUMPTION THE NEUTRONSFIRST ASSUMPTION THE NEUTRONSARE MONOENERGETICARE MONOENERGETIC

x

J(x)Jx

Jx

! DdJ

dx


4/22

Current is a VectorCurrent is a Vector

y

x

z

r

dr

d]

dU

Jz ! Jz Jz

dJz y dS! 7sJdV cos] dS4Tr2

e 7s r

dS

Jz-

Rearrange and substitute in spherical coordinates


5/22

Magic HappensMagic Happens

Jz

!7s4T

Je7s r cos]sin]dUd]dr? A0

g

0

T

2

02T

Use the Maclaurin series then integrate .

Jz ! 1

37s

xJxz


6/22

Use the GradientUse the Gradient

Solve for the neutron current in the x and ySolve for the neutron current in the x and y

direction.direction.

Jz ! 137s

xJxz

Jx ! 1

37s

xJxx

Jy ! 1

37s

xJxy

J! 1

37s

xJxx

xJxy

xJxz

J! DJ


7/22

Whats ImportantWhats Important

D !

1

37tr !

1

37s 1 cosH !Ptr

3

Use the average angle of scatter to correct for the anisotropic scattering

cosH!2

3A


8/22

ContinuityContinuity

Accumulation = SourceAccumulation = Source --SinksSinks

As neutrons diffuse through a reactor core theyAs neutrons diffuse through a reactor core they

maymay Be absorbed by fuel, moderator, coolant,structureBe absorbed by fuel, moderator, coolant,structure

Leak out of the core boundariesLeak out of the core boundaries

Act as a source for more new fission neutronsAct as a source for more new fission neutrons


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IndividuallyIndividually

SourceSource

AbsorbedAbsorbed

S! kgJ7a

# abs ! J7a


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What about leakage?What about leakage?

Jy

Jy+dy

Ly net ! Lydy Ly ! D xJxy x

2

Jxy 2 dy

dxdz DxJxy dxdz

Ly net ! Dx2J

xy2dxdydz


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LeakageLeakage

Same derivation for the x and y directionsSame derivation for the x and y directions

Add the leakage terms togetherAdd the leakage terms together

L ! Dx2Jxx 2

x2Jxy 2

x2Jxz2

! D2J


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Back to ContinuityBack to Continuity

Accumulation=SourceAccumulation=Source -- leakleak -- lostlost

At Steady StateAt Steady State

dndt

! g7aJ D2J 7aJ

0 ! kg7aJ D2J 7aJ


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ManipulationManipulation

Divide by macroscopic absorption crossDivide by macroscopic absorption cross--

sectionsection

RearrangeRearrange

0 ! kgJ D2J

7a J

0 ! kg 1 J D2J

7a


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Thermal Diffusion LengthThermal Diffusion Length

Define a term LDefine a term L -- thermal diffusion lengththermal diffusion length

Substitute into Continuity equation andSubstitute into Continuity equation and

rearrangerearrange

L !

D

7a

0 !kg 1 J

L2

2J


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Material BucklingMaterial Buckling

0 !kg 1 JL

2 2J

0 ! B2J 2J

B2 ! kg 1L2


16/22

Steady State coreSteady State core

The material buckling has an inverse relationThe material buckling has an inverse relation

to the size of a critical core.to the size of a critical core.

The overall neutron production must justThe overall neutron production must justbalance the lost and the leakingbalance the lost and the leaking

B2

!k

g

1L2 1! kg

1

B2L2 1


17/22

Infinite and EffectiveInfinite and Effective

1! kg1

B

2

L

2

1keffPnl

Pnl !1

B2L

2 1

Remember our original assumption? Mono energetic neutrons

So this is for thermal neutrons only.


18/22

Solving the Diffusion EquationSolving the Diffusion Equation

Infinite slabInfinite slab

Point sourcePoint source

Great Math in the text and I am not going to reproduceGreat Math in the text and I am not going to reproduce

it here.it here.


19/22

Crows FlightCrows Flight

We now have enoughWe now have enoughinformation to get ainformation to get abetter estimatebetter estimate

Instead of using theInstead of using themean free path tomean free path toestimate distanceestimate distancetraveled for neutronstraveled for neutronsuse diffusion lengthuse diffusion length

r

L2 !

1

6r2


20/22

Correcting the mono energeticCorrecting the mono energetic

To correct the flux for thermal energies use eqTo correct the flux for thermal energies use eq

5.585.58

To correct the absorption cross sections useTo correct the absorption cross sections useeq 5.59eq 5.59

To correct D and L for Temperature use 5.63To correct D and L for Temperature use 5.63

and 5.64and 5.64


21/22

Two GroupsTwo Groups

Ok we can solve for two energy groupsOk we can solve for two energy groups

Fast and SlowFast and Slow

M2 ! L2 X


22/22

Two groupsTwo groups

M is migration length andM is migration length and XX is neutron ageis neutron age

Substitute M for L and you have the two groupSubstitute M for L and you have the two groupsolution.solution.

ch 5 neutron diffusion(2)

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