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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