3 Neutron Balance

7/27/2019 3 Neutron Balance

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2008 September 1

Neutron Balance

B. Rouben

McMaster University

EP 4D03/6D03Nuclear Reactor Analysis

2008 Sept-Dec


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2008 September 2

Contents

The Neutron-Transport Equation The Neutron-Diffusion Equation

Stages of practical neutronics calculations:

lattice calculations

full-core calculations


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2008 September 3

Reactor Statics: Neutron Balance

In reactor statics we study time-independent phenomena Independence of time means that there is (or is assumed to

be) neutron balance everywhere. Therefore, all phenomena which involve neutrons must

result altogether in equality between neutron productionand neutron loss (i.e., between neutron sources and sinks)at every position rin the reactor and for every neutronenergyE.

These phenomena are: Production of neutrons by induced fission Production of neutrons by sources independent of the neutron flux Loss of neutrons by absorption Scattering of neutrons to other energies or directions of motion Leakage of neutrons into or out of each location in the reactor


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2008 September 4

Neutron-Balance Equations

Neutron balance is expressed: essentially exactly, by the time-independent

neutron-transport (Boltzmann) equation, and

to some degree of approximation, by theneutron-diffusion equation


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2008 September 5

Co-Ordinate System

In reactor physics weusually use the co-ordinate systemshown in the figure.

The components ofare:

x = sincos

y = sinsin

z= cos

Instead of, we oftenuse cos.


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2008 September 6

Integrals Over Angle

Using the co-ordinate system above for angles,integrals over all angles can then be written

2

00

2

0

1

1

2

0

1

1cossin dddddd


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2008 September 8

Neutron-Transport Equation

From Eq. (4.43) in Duderstadt & Hamilton, I write

the time-independent balance equation (I use the

neutron energyErather than speed v as a variable):

wheres(r,E,) is the total source of new neutrons

appearing at rwith energyEand in direction .

What is the meaning of all the terms?

' '

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

,,,,,,,

E

s

t

ddEErEEr

ErErErsEr

.2

.inf

,,

slidesfollowingtheinshownisthisofderivationThe

ratvolumeinitesimalanofoutdirectioninmoving

EenergyofneutronsofleakagethegivesEr


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2008 September 9

Derivation of Expression for Leakage

The leakage of neutrons moving in direction out of a

differential area dS on a surface is given by:

contd

).(

,,,

angleanatearththehittingsunthefromfluxheatofefficacytosimilar

surfacethetonormaloutgoingtheisSdwhereSdErJ


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2008 September 10

Derivation of Expression for Leakage

The total leakage of neutrons moving in direction out

of the volume V bounded by the surface is then the

integral

,int""

,

,,,,

int

,',,,,,

rthenispoaofout

leakagethearbitraryisVvolumetheSince

dVErdVEr

egralvolumetheasthisrewritecanwe

theoremGaussybandSdErSdErJ

VV

S S


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2008 September 11


.)(4

1

.

.,.,

)2('',',',

:

)(

,4

1,,,,

,

'

isotropicassumedissourcef issionthebecauseappearsfactorThe

Eenergywithbornarewhich

neutronsoffractiontheeispectrumneutronfissiontheisE

dEErErEEErS

sourcefissiontheisSand

fluxneutrontheoftindependensourceneutronexternalanisSwhere

ErSErSErs

writecanwetermsourcetheFor

E

ff

f

f


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2008 September 12


I therefore rewrite the time-independent transportequation as

It is important to understand each term of the equation.

Explain the other 2 terms which I havent covered yet.

We can then see that the equation, at each position r

and for each energyE, equates the summed loss of

neutrons to the summed production of neutrons.

' '

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

,,,,4

1,,,,

E

s

tf

ddEErEEr

ErErErSErSEr


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2008 September 13

Neutron-Transport Equation (cont.)

Note how complicated the transport equation is: It involves both derivatives (first-order) and

integrals of the angular flux

It involves integrals over very large ranges in energy

(from several MeV to small fractions of 1 eV), withquantities (cross sections) which are very complex

functions, especially in the resonance range

It involves 6 independent variables: 3 for space (r),

2 for the neutrons direction of motion (), and 1for energy. Because the rates of absorption and

induced fission do not depend on , it would be

nice ifcould be removed as a variable.


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2008 September 14

Neutron-Diffusion Equation

The neutron-diffusion equation is an approximation tothe neutron-transport equation.

It is much simpler than the transport equation, because

it removes the neutron direction of motion from

consideration, i.e., the dependent variable is the totalflux at each energy rather than the angular flux, and

it is based on an approximate relationship between

the neutron current and the total (not the angular)

neutron flux for any given energyE; this relationshipis called Ficks Law:

)3(,,, ErErDErJ


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2008 September 15

Diffusion Coefficient

In Eq. (3), under the proportionality constant between the

current and the gradient of the flux is called the diffusion

constantD(r,E).

Under the approximations that the angular flux () is

only weakly dependent on angle, i.e., at most linear in ,and that the neutron sources are isotropic,D can be

shown to be (see derivation in Duderstadt & Hamilton in

the 1-speed approximation, pages 133-136):

.sec,

,cos

)4(,3

1

,,3

1,

0

0

tioncrosstransportthecalledisErand

anglescatteringtheofinetheofvalueaveragetheiswhere

ErErErErD

tr

trst


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2008 September 16

Neutron-Diffusion Equation (cont.)

Note: Ficks Law expresses the fact that in regions of totally free

neutron motion the net neutron current will be along thedirection of greatest decrease in the neutron density (or,equivalently, of flux), i.e., it will be proportional to the negativeof the gradient of the flux.

This is a consequence of the random nature of collisions in all

directions, and the greater number of collisions in regions ofgreater density. The approximation inherent in Ficks Law breaks down near

regions of strong sources or strong absorption, or nearboundaries between regions with large differences in properties,or external boundaries, because the motion and collisions of

neutrons are biased in or near such regions. This is why diffusion theory cannot be used in lattice physics, as

the fuel itself is a strong neutron absorber. Transport theorymust be used to homogenize properties (and therefore weakenabsorption, on the average) over (relatively large) lattice cells.


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2008 September 17

Neutron-Diffusion Equation (cont.)

The neutron-diffusion equation is derived in Duderstadt &

Hamilton on pages 124-140. Assignment: Read, or at least scan,this derivation, and ensure you understand the final result, Eq. (4-162).

The equation can also be derived simply by writing down theneutron balance at any given energyEwithin a differential volume

at r, and applying Ficks Law for the relationship between thecurrent and the flux. The neutron balance (neutron-diffusionequation) then has the following form, where I have left out thetime dependence:

Exercise: identify the meaning and structure of each term in thediffusion equation.

'

'

)5(,'',',

'',',,,,,

E

f

E

st

ErSdEErErE

dEErEErErErErErD


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2008 September 18

Application of Transport & Diffusion Equations

The transport equation is the most accurate (essentiallyexact) representation of neutronics in the reactor.

Therefore, ideally, it should be the equation to solve for

all problems in reactor physics.

However, because of its complexity, it is very difficult,or extremely time-consuming, to apply the transport

equation to full-core calculations.

So the neutronics problem is divided into stages, taking

advantage of the modular (or nearly modular)

geometry of the reactor lattice, as explained in the next

slides. contd


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2008 September 19

Basic CANDU Lattice Cell with 37-Element Fuel

D2OPrimaryCoolant

Gas Annulus

Fuel Elements

Pressure Tube

Calandria TubeModerator


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2008 September 20

Application of Transport & Diffusion Equations

For practicality, the transport equation is applied tosmall regions of the reactor (lattice basic cells): to find the detailed flux in space and energy in these

cells, and to derive homogenized properties (cross sections),

uniform over each lattice cell and which are collapsedonto a very small number of energy groups (as few as2 groups), for application over full-core models withdiffusion theory.

This is actually the strategy used most frequently, andsuccessfully, in the design and analysis of nuclearreactors.


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2008 September 21

Application of the Neutron-Diffusion Equation

As previously indicated, the neutron-diffusion equationis applied mostly in full-core calculations, because of

its much greater simplicity than the transport equation.

Full-core models (see example in next slide) consist of

homogeneous (uniform) properties over lattice cells, orlarge portions of cells, for a small number of neutron

energy groups.

The flux distribution (or flux shape) in the reactor (one

value per parallelepiped and energy group) is thenobtained by solving the diffusion equation - often in its

finite-difference form.


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2008 September 22

Full-Core Diffusion Model

The parallelepipeds

(cells) over whichthe flux iscalculated aredefined by theintersections of the

horizontal andvertical mesh lines,shown on the leftand top axes.


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2008 September 23

Interface & Boundary Conditions

To solve the transport or diffusion equation, wegenerally subdivide (as described earlier) theoverall domain into regions within which thecoefficients in the equations (i.e., the nuclear

properties) are constant (e.g., homogenized). The equation is then solved over each region,

and the solutions must be connected byinterface conditions at the interfaces (infinitely

thin virtual surfaces) between regions. We also generally needboundary conditions at

the external boundary of the domain.


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2008 September 24

Interface & Boundary Conditions for Transport

The Boltzmann transport equation has derivatives of first order we need one interface condition at each interface, and one

boundary condition

At interfaces the angular flux must be continuous (since there are

no sources or scatterers at an infinitely thin virtual interface):

where r+ and r- are the two sides of the interface

At rv, an outer boundary (assumed convex) with a vacuum, no

neutrons can enter, since the vacuum has no neutron sources orscatterers:

)6(,,,, allandEallforErEr

)7(intint0,, reactortheoingpoallforErv


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2008 September 25

Interface & Boundary Conditions for Diffusion

Interface conditions at each interface: The totalflux and the total current at each energy must be

continuous (since they are integrals of the

angular flux, which is continuous):

)8(,,,, EallforErJErJandErEr


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2008 September 27

Vacuum Boundary Condition for Diffusion

The boundary condition is written as a relation betweenthe flux and its gradient at the vacuum boundary - see

Eqs. (4.175)-(4.180) in Duderstadt & Hamilton.

If in our case the boundary is on the right-hand side and

atzs (see previous slide), the relationship is ultimatelywritten

)10(1

)'9(071.0

:,

)9(071.0

pathfreemeantransportwhere

dz

dz

rightonorleftonboundaryaforgenerallyand

dz

dz

tr

tr

z

trs

z

trs

z

s


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2008 September 28

Extrapolation Distance

The boundary condition Eq.(9) can be interpreted geometrically

as follows. If one extrapolates the diffusion flux linearly away from the

boundary, it would go to zero at an extrapolation point zex beyondthe boundary zs:

Note that the flux does not actually go to zero, but the boundarycondition is mathematically equivalent to flux = 0 atzex.

d0.71tris therefore called the extrapolation distance. The boundary condition can be applied as is in Eq. (9), i.e., as a

relationship between the flux and its derivative at the physicalboundaryzs,but it is also often applied by extending the reactorregion to a new boundary atzs +d, and forcing the flux to be zerothere. (This represents an approximation - usually small - since itmeans assuming the reactor is slightly larger than it really is.)

)11(71.0 trsex zz


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2008 September 29

1-Energy-Group Neutron-Diffusion Equation

Diffusion theory is applied mostly in 1 or 2 energygroups, or at most a few energy groups.

So lets start with the simplest case: 1 energy group.

In this case, the energy ranges in Eq. (5) are reduced to

a single distinct energy value, and therefore the energylabel can simply be removed.

If we assume that all neutrons have the same energy (or

speed), Eq. (5) reduces to the following [Eq. (4.149) in

Duderstadt & Hamilton, without the time dependence]:

)12(rSrrrrrrD fa


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2008 September 30

Interactive Discussion/Exercise

Derive Eq. (12) from Eq. (5); in particular,explain how the a arises, and why

disappears.


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2008 September 31

Derivation of Eq. (12) from Eq. (5)

.

,sinint

.1,1

.

,log1

)5(,'',',

'',',,,,,:)5.(

'

'

aast

ss

E

f

E

st

assimplywrittenbecanEEE

andEofvalueglethetoreducesoveregralThe

shownbenotneedandEenergyonlyisthereSince

droppedbethereforecanand

samethearelabelsenergyallymethodogrouptheIn

ErSdEErErE

dEErEErErErErErDfollowsasisEq


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2008 September 32

Operator Formulation

From Eq. (12) we can see that for the 1-group diffusion

equation, the flux vector and the operators take the form

and the diffusion equation in operator form is

)16(

)15(

)14(

)13(

rSr

rrDr

rr

rr

a

f

S

M

F

)17(rrrrr SFM


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2008 September 33

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