4-1

Lesson 4 ObjectivesLesson 4 ObjectivesLesson 4 ObjectivesLesson 4 Objectives

• Development of source termsDevelopment of source terms• Review of Legendre expansionsReview of Legendre expansions

• Resulting full Boltzmann EquationResulting full Boltzmann Equation• Source vs. Eigenvalue calculationsSource vs. Eigenvalue calculations

• Four eigenvalue formulationsFour eigenvalue formulations

4-2

BE so farBE so farBE so farBE so far

• The time-independent Boltzmann Equation we have derived to this point The time-independent Boltzmann Equation we have derived to this point is:is:

• But you will remember that we swept a bunch of terms under the rug by But you will remember that we swept a bunch of terms under the rug by wrapping them up into the q termwrapping them up into the q term• (Which ones?)(Which ones?)

ˆ ˆ ˆ ( , , , ) , ( , , , )

ˆ ( , , , )

tr E t r E r E t

q r E t

4-3

Transport with Secondary ParticlesTransport with Secondary ParticlesTransport with Secondary ParticlesTransport with Secondary Particles

• We will now “unwrap” the source terms:We will now “unwrap” the source terms:• External fixed sourcesExternal fixed sources• Scattering sourcesScattering sources• Fission sourcesFission sources

sources particle Fission

sources Scattering

sources (fixed) External

),ˆ,(

),ˆ,(

),ˆ,(

),ˆ,(),ˆ,(),ˆ,(),,ˆ,(

Erq

Erq

Erq

ErqErqErqtErq

f

s

ex

fsex

4-4

External Fixed SourceExternal Fixed SourceExternal Fixed SourceExternal Fixed Source

• This source term comprises particle sources that This source term comprises particle sources that do notdo not depend on flux (e.g., depend on flux (e.g., radioactive isotopes, cosmic rays)radioactive isotopes, cosmic rays)

• These are simply These are simply specifiedspecified for the calculation as: for the calculation as:

• In many cases of interest there is no or dependence.In many cases of interest there is no or dependence.

E about dE and , about d ,r aboutdV in emitted

particles source of Number

ˆ

),ˆ,(

dEdVdErqex

r

4-5

Use of Legendre expansionsUse of Legendre expansionsUse of Legendre expansionsUse of Legendre expansions

• Using the cosine of the deflection angle, we Using the cosine of the deflection angle, we can represent the angular dependence of the can represent the angular dependence of the distribution in a Legendre expansion:distribution in a Legendre expansion:

00

0 )(12),(

PEEEE ss

• This allows us to represent the scattering This allows us to represent the scattering distribution by determining the Legendre distribution by determining the Legendre coefficients:coefficients:

..., 1, 0, ),( EEs

4-6

Use of Legendre expansions (2)Use of Legendre expansions (2)Use of Legendre expansions (2)Use of Legendre expansions (2)

• Using the orthogonality of the Legendre Using the orthogonality of the Legendre polynomials:polynomials:

1

1

2

2 1m md P P

• We can operate on both sides of the We can operate on both sides of the expansion (1expansion (1stst eqn. previous slide) with: eqn. previous slide) with:

1

0 0

1

md P

4-7

Use of Legendre expansions (3)Use of Legendre expansions (3)Use of Legendre expansions (3)Use of Legendre expansions (3)

• And remembering that a Kronecker delta works pulls out a single element like this:And remembering that a Kronecker delta works pulls out a single element like this:

• To get:To get:

),(2

1)( 00

1

1

0

EEPdEE smsm

• Work this out for yourself (Prob. 4-1)Work this out for yourself (Prob. 4-1)

0 1 1 20

0 0 ... 1 0 0 ...i ik k k k ki

f f f f f f f

4-8

Scattering SourceScattering SourceScattering SourceScattering Source

• This source term comprises all particle reactions (other than fission) from which particles that we are interested in are This source term comprises all particle reactions (other than fission) from which particles that we are interested in are emittedemitted

• The basic cross section is:The basic cross section is:

• Note that it is a distribution in Note that it is a distribution in destinationdestination energy and direction energy and direction

. about E, about dE into scatter

will direction ,energy , position at particles that

path unit pery Probabilit

ˆ

ˆ

),ˆˆ,(

d

Er

dEdEErs

4-9

Scattering Source (2)Scattering Source (2)Scattering Source (2)Scattering Source (2)

• The scattering source is:The scattering source is:

• Again, Legendre expansions are normally Again, Legendre expansions are normally used for the scattering cross section:used for the scattering cross section:

),ˆ,(),ˆˆ,(),ˆ,(

),ˆ,(

40

ErEErdEdErq

dVdEdErq

ss

s

: where,

ˆˆ),(12),ˆˆ,(0

PEErEEr ss

4-10

Scattering Source (3)Scattering Source (3)Scattering Source (3)Scattering Source (3)

• The coefficients are given by:The coefficients are given by:

• Substituting this expression gives:Substituting this expression gives:

0

00

1

1

0 ),,(2

1),(

where

EErPdEEr ss

),ˆ,(ˆˆ

),(12),ˆ,(

4

00

ErPd

EErEdErq ss

4-11

Scattering Source (4)Scattering Source (4)Scattering Source (4)Scattering Source (4)

• However, we can use the Legendre However, we can use the Legendre addition theorem, which says:addition theorem, which says:

wherewhere

ˆˆ12

1ˆˆ *m

mm YYP

imPml

mY mm exp

!

!12ˆ2/1

/221 1 ( 0)m

mmmm

dP P m

d

!

1!

mm mmP P

m

4-12

Scattering Source (5)Scattering Source (5)Scattering Source (5)Scattering Source (5)

• Substituting this expression gives:Substituting this expression gives:

wherewhere

0 0

* ),(),(ˆ),ˆ,(

ErEErEdYErq m

sm

ms

4

ˆ ˆ( , ) ( , , )mmr E d Y r E

4-13

Scattering Source (5)Scattering Source (5)Scattering Source (5)Scattering Source (5)• Conveniently, this term is also the coefficient of the angular flux expansion in spherical coordinates:Conveniently, this term is also the coefficient of the angular flux expansion in spherical coordinates:

• The scattering terms are therefore implemented through these flux momentsThe scattering terms are therefore implemented through these flux moments

0

* ˆ),(),ˆ,(

m

mm YErEr

4-14

Fission SourceFission SourceFission SourceFission Source• The text develops the time-dependent fission source term, including prompt and delayed fission neutron terms.The text develops the time-dependent fission source term, including prompt and delayed fission neutron terms.• Our primary concern is not time-dependent, so we will use:Our primary concern is not time-dependent, so we will use:

• Note that there is no angular dependence and that it is “per unit solid angle”Note that there is no angular dependence and that it is “per unit solid angle”

),(),(),ˆ,(

),ˆ,(

40

ErErdEdEErq

dVdEdErq

ff

f

: where,

4-15

Complete SourceComplete SourceComplete SourceComplete Source• Combining the three parts of the source:Combining the three parts of the source:

where the “double zero” moment is equivalent to the normal scalar flux.where the “double zero” moment is equivalent to the normal scalar flux.

),(),(

),(),(ˆ

),ˆ,(),ˆ,(

00

40

0 0

*

ErErdEdE

ErEErEdY

ErqErq

f

ms

mm

ex

4-16

Full EquationFull EquationFull EquationFull Equation• The full time-independent Boltzmann Equation is:The full time-independent Boltzmann Equation is:

),ˆ,(

),ˆ,(),(

),ˆ,(),ˆˆ,(

),ˆ,(,),ˆ,(ˆ

40

40

Erq

ErErdEdE

ErEErdEd

ErErEr

ex

f

s

t

4-17

Source vs. Eigenvalue CalculationsSource vs. Eigenvalue CalculationsSource vs. Eigenvalue CalculationsSource vs. Eigenvalue Calculations• The nature of the source terms divides the solution into two categories:The nature of the source terms divides the solution into two categories:

• Source problems: subcritical with:Source problems: subcritical with:

• Eigenvalue problems with:Eigenvalue problems with:

• The subcriticality requirement is because there is no time-independent The subcriticality requirement is because there is no time-independent physicalphysical solution for critical or super-critical systems with sources solution for critical or super-critical systems with sources• Mathematical solutions would have negative fluxesMathematical solutions would have negative fluxes

0),ˆ,( Erqex

0),ˆ,( Erqex

4-18

Eigenvalue CalculationsEigenvalue CalculationsEigenvalue CalculationsEigenvalue Calculations• Without external souces, we get the homogeneous equation:Without external souces, we get the homogeneous equation:

• Two characteristics of the solution:Two characteristics of the solution:1.1. Any constant times a solution is a solution.Any constant times a solution is a solution.2.2. There probably isn’t a meaningful solutionThere probably isn’t a meaningful solution

0),ˆ,(),(

),ˆ,(),ˆˆ,(

),ˆ,(,),ˆ,(ˆ

40

40

ErErdEdE

ErEErdEd

ErErEr

f

s

t

4-19

Eigenvalue solution normalizationEigenvalue solution normalizationEigenvalue solution normalizationEigenvalue solution normalization• For the first point, we generally either normalize to 1 fission neutron:For the first point, we generally either normalize to 1 fission neutron:

or to a desired power level:or to a desired power level:

where k is a conversion constant (e.g., 200 MeV/fission)where k is a conversion constant (e.g., 200 MeV/fission)

1),ˆ,(),(40

ErErdEddV f

V

PErErdEddVk f

V

),ˆ,(),(40

4-20

Eigenvalue approachEigenvalue approachEigenvalue approachEigenvalue approach

• For the second problem (i.e., no meaningful solution), we deal with it by adding a term with a constant that we can adjust to achieve balance in the equation.For the second problem (i.e., no meaningful solution), we deal with it by adding a term with a constant that we can adjust to achieve balance in the equation.• We will discuss four different eigenvalue formulations:We will discuss four different eigenvalue formulations:

1.1. Lambda (k-effective) eigenvalueLambda (k-effective) eigenvalue2.2. Alpha (time-absorption) eigenvalueAlpha (time-absorption) eigenvalue3.3. BB22 (buckling) eigenvalue (buckling) eigenvalue4.4. Material search “eigenvalue”Material search “eigenvalue”

4-21

Lambda (k-effective) eigenvalueLambda (k-effective) eigenvalueLambda (k-effective) eigenvalueLambda (k-effective) eigenvalue• The first (and most common) eigenvalue form involves dividing The first (and most common) eigenvalue form involves dividing , the number of neutrons emitted per fission:, the number of neutrons emitted per fission:

• Keep largest of multiple eigenvalues Keep largest of multiple eigenvalues

),ˆ,(),(1

),ˆ,(),ˆˆ,(

),ˆ,(,),ˆ,(ˆ

40

40

ErErdEdE

ErEErdEd

ErErEr

f

s

t

...210

4-22

Lambda eigenvalue (2)Lambda eigenvalue (2)Lambda eigenvalue (2)Lambda eigenvalue (2)

• The criticality state is given by:The criticality state is given by:

• Advantages:Advantages:1.1. Everybody uses itEverybody uses it2.2. Guaranteed real solutionGuaranteed real solution3.3. Fairly intuitive (if you don’t take it too seriously)Fairly intuitive (if you don’t take it too seriously)4.4. Good measure of distance from criticality for reactorsGood measure of distance from criticality for reactors5.5. Very straightforward calculation (no search required)Very straightforward calculation (no search required)

• Disadvantages:Disadvantages:1.1. No physical basisNo physical basis2.2. Not a good measure of distance from criticality for CSNot a good measure of distance from criticality for CS

calitysupercriti denotes

ycriticalit denotes

litysubcritica denotes

1

1

1

4-23

Alpha (time-absorption) eigenvalueAlpha (time-absorption) eigenvalueAlpha (time-absorption) eigenvalueAlpha (time-absorption) eigenvalue• The second eigenvalue form involves adding a term to the removal term:The second eigenvalue form involves adding a term to the removal term:

• Keep largest of multiple eigenvalues Keep largest of multiple eigenvalues

),ˆ,(),(

),ˆ,(),ˆˆ,(

),ˆ,(,),ˆ,(ˆ

40

40

ErErdEdE

ErEErdEd

ErErEv

Er

f

s

t

...210

4-24

Alpha eigenvalue (2)Alpha eigenvalue (2)Alpha eigenvalue (2)Alpha eigenvalue (2)

• Physical basis is the representation of the time dependence as exponential:Physical basis is the representation of the time dependence as exponential:

tf

ts

tt

tt

edEdedEdeet

e

v

4040

ˆ1

tf

ts

tt

tt edEdedEdeeev

4040

ˆ

fst dEddEdv 4040

ˆ

4-25

Alpha eigenvalue (3)Alpha eigenvalue (3)Alpha eigenvalue (3)Alpha eigenvalue (3)

• The criticality state is given by:The criticality state is given by:

• Advantages:Advantages:1.1. Physical basisPhysical basis

2.2. Intuitive for kinetics workIntuitive for kinetics work

• Disadvantages:Disadvantages:1.1. No guaranteed real solutionNo guaranteed real solution

2.2. Not intuitive for reactor design or CS workNot intuitive for reactor design or CS work

3.3. Search required (to make k-effective go to 1)Search required (to make k-effective go to 1)

calitysupercriti denotes

ycriticalit denotes

litysubcritica denotes

0

0

0

4-26

BB22 (buckling) eigenvalue (buckling) eigenvalueBB22 (buckling) eigenvalue (buckling) eigenvalue• The third eigenvalue form also involves adding a term to the removal term:The third eigenvalue form also involves adding a term to the removal term:

• Physical basis is the diffusion theory approximation of leakage byPhysical basis is the diffusion theory approximation of leakage by

),ˆ,(),(

),ˆ,(),ˆˆ,(

),ˆ,(,,),ˆ,(ˆ

40

40

2

ErErdEdE

ErEErdEd

ErErBErDEr

f

s

t

ErBED ,)( 2

4-27

BB22 eigenvalue (2) eigenvalue (2)BB22 eigenvalue (2) eigenvalue (2)

• Mathematical basis is the representation of spatial dependence of flux as Fourier transform:Mathematical basis is the representation of spatial dependence of flux as Fourier transform:

• This substitutes to give us:This substitutes to give us:

rBiEFEr exp,ˆ,ˆ,

EFdEdE

EFdEdEFEFBi

f

st

,ˆ

,ˆ,ˆ,ˆˆ

40

40

4-28

BB22 eigenvalue (3) eigenvalue (3)BB22 eigenvalue (3) eigenvalue (3)

• The criticality state is given by:The criticality state is given by:

• Advantages:Advantages:1.1. Physical basisPhysical basis

2.2. Good measure of distance from criticalityGood measure of distance from criticality

• Disadvantages:Disadvantages:1.1. No guaranteed real solutionNo guaranteed real solution

2.2. Not intuitive for kinetics or CS workNot intuitive for kinetics or CS work

3.3. Search required (to make k-effective go to 1)Search required (to make k-effective go to 1)

2

2

2

0 denotes subcriticality ( is imaginary)

0 denotes criticality

0 denotes supercriticality ( is real)

B B

B

B B

4-29

Homework 4-1Homework 4-1Homework 4-1Homework 4-1

Group constantsGroup constants Group 1Group 1 Group 2Group 2 Group 3Group 3 Group4Group4

11 00 00 00

ff0.00550.0055 0.0680.068 2.482.48 2.002.00

aa.072.072 .20.20 2.022.02 1.01.0

D (cm)D (cm)22 .7.7 0.20.2 0.30.3

V (cm/sec)V (cm/sec)2.2e92.2e9 3.2e83.2e8 2.7E72.7E7 2.5e52.5e5

•Find the lambda, BFind the lambda, B22, and alpha eigenvalues for infinite , and alpha eigenvalues for infinite medium cross sections (Use EXCEL, MatLab, whatever)medium cross sections (Use EXCEL, MatLab, whatever)

•Find the resulting group fluxes.Find the resulting group fluxes.•Use Use Scattering 1->2=.06 2->3=.09 3->4=.6 (others 0)

•Base eqn: Base eqn:

•Find the lambda, BFind the lambda, B22, and alpha eigenvalues for infinite , and alpha eigenvalues for infinite medium cross sections (Use EXCEL, MatLab, whatever)medium cross sections (Use EXCEL, MatLab, whatever)

•Find the resulting group fluxes.Find the resulting group fluxes.•Use Use Scattering 1->2=.06 2->3=.09 3->4=.6 (others 0)

•Base eqn: Base eqn: 4 4

1 1'

g gag g g s g eff g g

g gg g

k