steady state diffusion equation

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).

1

Steady State Diffusion Equation

HW 20HW 20

Study example 5.3 and solve problem 5.8 in Lamarsh.


2

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

Steady State Diffusion Equation

A B

BA • At the interface

• What if A or B is a vacuum?• Linear extrapolation distance.• Bare slab with central infinite planar source (Lamarsh).• Same but with medium surrounding the slab. • Maybe we will be back to this after you try it!!

dx

dD

dx

dDJJ B

BA

ABA

x


3

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:(1)absorption

(1)leakage

)( rate absorptionneutron

rate productionneutron

A

(S)k

)( rate leakageneutron )( rate absorptionneutron

)( rate productionneutron

LEA

Skeff


4

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


5

More on One-Speed DiffusionHW 21HW 21

Show that for a critical homogeneous reactorcritical homogeneous reactor

DBDLBP

a

a

a

aleaknon 2222 1

1

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


6

HW 22HW 22

022

2

B

dx

dFor the infinite slab . Show that the general solution

With BC’sBxCBxAx sincos)(

0)(

0)2

(

0

0

xdx

xd

a

Flux is symmetric about

the origin.

0cos)( ABxAx

,...2

5,

2

3,

2)

2(0)

2(cos)

2( 000

aB

aBA

a

More on One-Speed Diffusion


7

,...2

5,

2

3,

2)

2( 0

a

BHW 22 HW 22 (continued)(continued)

,...5

,3

,0 BBBa

Fundamental mode, the only mode significant in critical reactors.

Buckling lGeometricacos)(00

0 a

Bxa

x

For a critical reactor, the geometrical buckling is equal to the material buckling.To achieve criticality

2

2

0

1

L

k

a



8

Spherical Bare Reactor (one-speed diffusion)Spherical Bare Reactor (one-speed diffusion)

334

2

3

2 46

a

a

a

a

Minimum leakage minimum fuel to achieve criticality.

xr

r0

0

2 22

2

B

dr

d

rdr

dHW 23HW 23

Brr

CBr

r

Asincos

Br

r

r

r

C 00

,sin

Continue!

Reactor



9

HW 24HW 24Infinite planer source in an infinite Infinite planer source in an infinite medium.medium.

LxeD

SLx /

2)(

D

xS

Ldx

xd )(1)(22

2

x

aa/2

a0/2

Source

)2/cosh(

2/2sinh

2 0

0

La

Lxa

D

SL

HW 25HW 25


Infinite planer source in a finite Infinite planer source in a finite medium.medium.


10


Infinite planer source in a multi-region medium.Infinite planer source in a multi-region medium.

FiniteInfinite Infinite

BCmore

dx

dD

dx

dD

aa

axax

2/

22

2/

11

21 )2/()2/(

Project 2Project 2


11

Back to Multiplication Factor

k = fp, leaknoneff P

k

k

leaknoneff Pfk

• Fast from thermal,• Fast from fast, .• Thermal from fast, p.• Thermal available for fission

Thinking QUIZThinking QUIZ• For each thermal neutron absorbed, how many fast neutrons are produced?

i

fa

ii )()(1

poisona

eratora

clada

fuela

fuelaf

mod


12

Two-Group Neutron Diffusion• Introductory to multi-group.• All neutrons are either in a fast or in a thermal energy group.• Boundary between two groups is set to 1 eV.• Thermal neutrons diffuse in a medium and cause fission (or are captured) or leak out from the system.• Source for thermal neutrons is provided by the slowing down of fast neutrons (born in fission).• Fast neutrons are lost by slowing down due to elastic scattering in the medium or leak out from the system (or fission or capture).• Source for fast neutrons is thermal neutron fission.


13

Two-Group Neutron Diffusion

ThermaldErEr

FastdErEr

eV

MeV

eV

1

0

2

10

1

1

),()(

),()(

221122

212

1

222111

aa

ffeff DD

k


14

)()()(0 12

1111 rDrrS a


Removal cross section = fission + capture + scattering to group 2

Depends on thermal flux.

Fast diffusion coefficient

)()()()(0 12

1112211 rDrrr aff

)()()(0 12

11122 rDrrk

aa

oror


15


)()()(0 22

2222 rDrrS a

Thermal diffusion coefficient

Thermal absorption cross section = fission

+ capture.

)()()(0 22

222121 rDrr as

Depends on fast flux.

)()()(0 22

22211 rDrr aa

oror


16


)()()(0 12

11122 rDrrk

aa

)()()(0 22

22211 rDrr aa

• A coupled system of equations; both depend on both fluxes.• For a critical, steady state system:

0)()(

0)()(

22

22

12

12

rBr

rBr

Geometrical

Review Cramer’s

rule!

steady state diffusion equation

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