2015 September 1

7: Neutron Balance

B. Rouben

McMaster University

Course EP 4D03/6D03

Nuclear Reactor Analysis

(Reactor Physics)

2015 Sept.-Dec.

2015 September 2

Contents

The Neutron-Transport Equation

The Neutron-Diffusion Equation

Stages of practical neutronics calculations:

lattice calculations

full-core calculations

2015 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, in reactor statics, all phenomena which involve

neutrons must result altogether in equality between neutron production and neutron loss (i.e., between neutron sources and sinks) at every position r in the reactor and for every neutron energy E.

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

2015 September 4

Neutron-Transport (Boltzmann) Equation

Neutron balance is expressed:

essentially exactly, by the neutron-transport (Boltzmann)

equation – see Section 4.II in Duderstadt & Hamilton, ending

with Eq. (4-43)

to some degree of approximation, by the neutron-diffusion

equation - see Section 4.IV.D in Duderstadt & Hamilton,

ending with Eq. (4-162)

2015 September 5

Neutron Balance

Both the Transport and the Diffusion time-independent

equations express the neutron balance at a point

(actually, in a differential volume, but since this can be

assumed as small as desired, it’s really at a point)

The Transport equation expresses the balance in the

angular flux, whereas

The Diffusion equation expresses the balance in the total

flux

To write down the balance, terms for all the events that

can take place are included (these were listed 2 slides

back)

Neutron Production Rate By Source

Production of neutrons at in a given direction of

motion and at a given energy E (per differential

volume, solid angle and energy interval):

From an external (independent) source (assumed

isotropic) =

From fission

where (E) = the fission neutron spectrum (fraction

of fission neutrons born with energy E)2015 September 6

ErS ,4

1

'

' 'ˆ

'',',4

''ˆ'ˆ,',',4

E

f

E

f

dEErErE

dEdErErE

r

Production Rate of Neutrons By Scattering In

The rate of neutrons entering the differential

volume (of space, energy, and direction of

motion) by scattering from other neutron

directions of motion or other neutron energies =

2015 September 7

''ˆ'ˆ,',ˆ'ˆ,',' 'ˆ

dEdErEErE

s

Loss Rate of Neutrons

Loss rate of neutrons:

by absorption and scattering:

where = Total cross section

by physical leakage out:

2015 September 8

ˆ,,, ErErt

ErErEr sat ,,,

ˆ,, ErJ

2015 September 9

Neutron-Transport Equation

We can now write the time-independent neutron-

transport equation, which expresses the balance

between the production and loss rates of neutrons:

The left-hand-side of Eq. (1) gives, per differential

volume at r, direction of motion and energy E, the

total production of neutrons minus the total loss of

neutrons.

This is the integro-differential form of the equation.

' '

'

)1(0ˆ,,ˆ,,,''',',',',

'',',4

,4

1

E

ts

E

f

ErJErErddEErEEr

dEErErE

ErS

2015 September 10

Neutron-Transport Equation (cont.)

Note how complicated the transport equation is: It involves both derivatives and integrals of the flux It involves integrals in energy, over very large

ranges in energy (from several MeV to small fractions of 1 eV), with quantities (cross sections) which are very complex functions of energy, especially in the resonance range

It involves 6 independent variables: 3 for space (r), 2 for the neutron’s direction of motion (), and 1 for energy E.

Note: it is first-order in terms of derivatives.

2015 September 11

Neutron-Transport Equation (cont.)

The transport equation is the most accurate (essentially

exact) 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.

Because cross sections do not depend on the initial angle

of motion , it would be “nice” if could be removed

as a variable.

Integrating over Angle

We can try to remove the angle by

integrating the equation [Eq. (1)] over it, to

see if we can obtain an equation in the

angle-integrated flux only.

However, integration of over presents a challenge to our plan, since the

angle-integrated current bears in general no

algebraic relationship to the angle-

integrated.

2015 September 12

Er ,

,, ErJ

2015 September 13

Fick’s Law

But we can use an approximation often used in

diffusion problems, Fick’s Law.

This is an approximate relationship between the

neutron flux and the neutron current:

where is called a “diffusion coefficient”.

Physically, Fick’s law says that the overall

neutron current (at a given neutron energy) is in

the direction of maximum decrease of the total

flux of neutrons of that energy.

)3(,,, ErErDErJ

ErD ,

2015 September 14

Significance of Fick’s Law

Fick’s Law expresses the expectation/fact that in

regions of totally free neutron motion the overall net

neutron current will tend to be from regions of high

density to regions of low density.

Mathematically speaking, the net overall current should

flow along the direction of greatest decrease in the

neutron density (or, equivalently, of flux), i.e., it will be

proportional to the negative of the gradient of the flux.

This is a consequence of the greater number of

collisions in regions of greater density, with collisions

allowing neutrons to go off freely in all directions.

2015 September 15

Breakdown of Fick’s Law

The approximation inherent in Fick’s Law

breaks down near regions of strong sources or

strong absorption, or near boundaries between

regions with large differences in properties, or

near external boundaries, because the motion of

neutrons is biased in or near such regions.

Here “near” a region or boundary means within,

say, 2 or 3 neutron mean free paths of the region

or boundary.

2015 September 16

Neutron-Diffusion Equation

By integrating the transport equation over angle, and

making use of Fick’s Law, we get the (here, time-

independent) diffusion equation [I will leave you to

study the full derivation in Duderstadt & Hamilton]:

Identify and make sure you understand each term in the

neutron-diffusion equation.

Why is there a + sign in front of the ?

)4(0,,,,'',',

'',',,

'

ErErDErErdEErEEr

dEErErEErS

E

ts

E

f

2015 September 17

Neutron-Diffusion Equation

The neutron-diffusion equation is much simpler

than the transport equation, because it removes the

neutron direction of motion from consideration,

i.e., the dependent variable is the total flux at each

energy rather than the angular flux.

However, it is based on an approximate

relationship between the angle-integrated neutron

current and flux.

2015 September 18

Discretizing the Energy

The equation 2 slides back is over continuous energy E.

To simplify the equation further, we discretize the energy

variable (i.e., subdivide the range [0, ) into a number of G of

subintervals.

All the neutrons of any energy in subinterval g (g = 1,…,G) are

considered to be in the same “energy group” g and the nuclear

properties are uniform over energy in any single energy group.

By convention, group 1 is the group with highest energy, and

group G is the one with lowest energy (the thermal group)

Group: G G-1 G-2 … 1

0 EG EG-1 E1

2015 September 19

Multigroup Neutron-Diffusion Equation

The diffusion equation in the discretized energy is called the

multigroup diffusion equation. It is actually a set of equations,

one for each energy group g. Time-independent equation:

Gg

rrDrrrrrrrS ggggt

G

g

gggsg

G

g

gfgg

,...,1

0,

1'

'','

1'

',

Fission from all groups;g = fraction of fission neutrons appearing in group g.

Scattering into group g

Total cross section for group g, including scattering out of g

Leakage from group g

External source in group g

2015 September 20

Solution of Neutronics Problem

The neutron-diffusion equation cannot be

used to calculate the flux in the basic

lattice cell (see figure in next slide),

because the fuel itself is a strong neutron

absorber and the cell is very

heterogeneous.

Therefore, the overall neutronics problem

is solved in 2 stages, as explained further

below.

2015 September 21

D2O

Primary

Coolant

Gas Annulus

Fuel Elements

Pressure Tube

Calandria Tube Moderator

CANDU BASIC-LATTICE CELL WITH 37-ELEMENT FUEL

Face View

of a Bundle

in a Fuel

Channel

2015 September 22

2-Stage Solution of Neutronics Problem

Stage 1: The transport equation is applied to the basic lattice cells: to find the detailed flux in space and energy (a large

number of energy groups) in a basic cell, and to derive “homogenized” (average) properties over each

cell (therefore weakening absorption, on the average) and “collapse” onto a very small number of energy

groups (often 2 groups). Stage 2: These homogenized lattice-cell properties are then

applied in full-core reactor models using diffusion theory. See a simplified diffusion model in the next slide.

This is the strategy used most frequently (and successfully) in the design and analysis of nuclear reactors.

2015 September 23

Face View of Diffusion Reactor Model

Legend

Each square is a

homogenized lattice cell.

Different-colour cells

have different properties,

mostly on account of

different fuel ages

(burnups).

2015 September 24

Interface & Boundary Conditions

To solve the transport or diffusion equation, we generally subdivide (as described earlier) the overall domain into regions within which the coefficients in the equations (i.e., the nuclear properties) are constant (homogenized).

The equation is then solved over each region, and the solutions must be connected by interface conditions at the interfaces (infinitely thin virtual surfaces) between regions.

We also generally need boundary conditions at the external boundary of the domain.

2015 September 25

Interface & Boundary Conditions for Transport

The neutron-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 or

scatterers:

)5(,,,, allandEallforErEr

)6(intint0,, reactortheoingpoallforErv

2015 September 26

Interface & Boundary Conditions for Diffusion

Interface conditions at each interface: The total

flux and the total current must be continuous

(since they are integrals of the angular flux,

which is continuous):

)7(,,,, EallforErJErJandErEr

2015 September 27

Boundary Condition for Diffusion

The boundary condition with a vacuum, in plane

geometry and in 1 energy group, is written as a relation

between the flux and its gradient at the boundary xv:

tr is called the “transport cross section”.

)10(cos

)9(11

)8(071.0

anglescatteringelasticofineaverageand

where

dx

dx

s

ssttr

tr

x

trv

v

2015 September 28

Extrapolation Distance

The boundary condition Eq.(8) can be interpreted geometrically as follows.

If one were to extrapolate the diffusion flux linearly away from the boundary, it would go to zero at an extrapolation point xex

beyond the boundary:

Note that the flux does not actually go to zero, but the boundary condition is mathematically equivalent to flux = 0 at xex.

0.71*tr is therefore called the “extrapolation distance”. The boundary condition can be applied as is in Eq. (8), i.e., as a

relationship between the flux and its derivative at the physical boundary xv, but it is also often applied by “extending” the reactor region to a new boundary at xex+tr, and forcing the flux to be zero there. (This represents an approximation - usually small - since it means assuming the reactor is slightly larger than it really is.)

)11(71.0 trex xx

2015 September 29

1-Energy-Group Neutron-Diffusion Equation

Diffusion theory is applied mostly in 1 or 2 energy

groups, or at most a few energy groups.

So let’s start with the simplest case – 1 energy group.

In this case, the energy ranges in Eq. (4) are reduced to

a single distinct energy value, and therefore the energy

label can simply be removed.

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

speed), Eq. (4) reduces to the following :

[Exercise: Where do the and the scattering terms go?]

)12(rSrrrrrrD fa

2015 September 30

Derivation of Eq. (12) from Eq. (4)

Solution to Exercise:

.

,)!sin(""""

.1,1

.

,log1

at withleftarewefrom

andcancelenergygleaofoutandinscatteringThe

shownbenotneedandEenergyonlyisthereSince

droppedbethereforecanand

samethearelabelsenergyallymethodogrouptheIn

2015 September 31

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

Φ

)17(rrrrr SΦFΦM

2015 September 32

END