The collision probability method in 1D –part 2
Alain Hebert
Institut de genie nucleaire
Ecole Polytechnique de Montreal
ENE6101: Week 9 The collision probability method in 1D – part 2 – 1/25
Content (week 9) 1
Calculation of 1D collision probabilities
plane 2D geometry
cylindrical 1D geometry
spherical 1D geometry
ENE6101: Week 9 The collision probability method in 1D – part 2 – 2/25
The collision probability method 1
Consider an infinite lattice of unit cells, each of them represented as⋃
i
Vi
V ∞i represents the infinite set of regions Vi belonging to all the cells in the lattice
The sources of secondary neutrons are uniform and equal to Qi on each region Vi.
The collision probability Pij,g is the probability for a neutron born uniformally and
isotropically in any of the regions Vi of the lattice to undergo its first collision in
region Vj of a unit cell or assembly.
If the total cross section Σ(r) is constant and equal to Σj in region Vj , reduced CPs are
pij =Pij
Σj=
1
4π Vi
∫
V ∞
i
d3r′∫
Vj
d3re−τ(s)
s2(1)
where the optical path τ(s) is given by
τ(s) =
∫ s
0ds′ Σ(r − s′ Ω) .(2)
ENE6101: Week 9 The collision probability method in 1D – part 2 – 3/25
Plane 2D geometry 1
We will now study the case of a 2D geometry defined in the x–y plane and homogeneous
along the z–axis. A cylinder or tube of infinite length is an example of such a geometry. In
this case, it is possible to integrate analytically the integral transport equation along the axial
angle θ. This angle is use to define the direction cosine of the particle relative to the z–axis.
(x,y)
θ
ε
x
y
z
Ω
r = (x,y,z)
The direction of the particle is
Ω = sin θ cos ǫ i+ sin θ sin ǫ j + cos θ k.
Integration over θ is possible by taking the
projection of each particle free path on the
x–y plane. We write
d2Ω = dθ dǫ sin θ
τ(ρ) = τ(s) sin θ
dρ = ds sin θ(3)
where ρ is the projection of s on the x–y
plane.
ENE6101: Week 9 The collision probability method in 1D – part 2 – 4/25
Plane 2D geometry 2
We will limit ourself on the integration of the integral transport equation, defined on the
domain of a unique unit cell. We use the relations r = r′ + sΩ and d3r = s2 d2Ω ds and
obtain
pij =1
4π Vi
∫
Vi
d3r′∫
4πd2Ω
∫
Ij
ds e−τ(s)(4)
where the quantity Ij is the set of points belonging simultaneously
to the half straight line of origin r and direction −Ω;
to volume Vj of the unit cell.
With the help of figure, Eq. (4) can be rewritten as
pij =1
4π Vi
∫ 2π
0dǫ
∫
Vi
d2r′∫ π
0dθ
∫
Ij
dρ e−
τ(ρ)sin θ(5)
where Vi now represents a surface in the x–y plane and Ij is the set of points belonging
simultaneously
to the half straight line of origin r′ and direction ǫ;
to surface Vj of the unit cell.
ENE6101: Week 9 The collision probability method in 1D – part 2 – 5/25
Bickley functions 1
Bickley functions are defined as
Kin(x) =
∫ π/2
0dθ sinn−1 θ e−
xsin θ =
∫ π/2
0dθ cosn−1 θ e−
xcos θ(6)
with n ≥ 1.
0 0.5 1 1.5 2 2.50
0.5
1
1.5
Ki1(x)
Ki2(x)
Ki3(x)
Ki4(x)
Ki5(x)
x
The following equations are relating Bickley
functions of different orders:
∫ x′
xdu Kin(u) = Kin+1(x)−Kin+1(x
′) ,
andd
dxKin(x) = −Kin−1(x)
They can be evaluated efficiently using the matlab script akin.
ENE6101: Week 9 The collision probability method in 1D – part 2 – 6/25
Plane 2D geometry 1
Equation (5) can be simplified by introducing the Bickley functions. We obtain
pij =1
2π Vi
∫ 2π
0dǫ
∫
Vi
d2r′∫
Ij
dρKi1[τ(ρ)] .(7)
Sets of integration lines, referred as tracks, are draw over the complete domain. Each set is
characterized by a given angle ǫ and contains parallel tracks covering the domain. Tracks in
a set can be separated by a constant distance ∆h or can be placed at optimal locations, as
depicted in sub-figures (a) and (b), respectively. Each track is used forward and backward,
corresponding to angles ǫ and −ǫ.
∆h
(a) Rectangular quadrature (b) Gaussian quadrature
Three-point formula
Three-point formula
Three-point formula
ENE6101: Week 9 The collision probability method in 1D – part 2 – 7/25
Plane 2D geometry 2
In the case of convex volumes i and j, Eq. (7) can be rewritten as
pij =1
2π Vi
∫ 2π
0dǫ
∫ hmax
hmin
dh
∫ ℓi
0dℓ′
∫ ℓj
0dℓ Ki1(τij + Σi ℓ
′ + Σj ℓ) if i 6= j(8)
where τij is the optical path of the materials between regions i and j and
pii =1
2π Vi
∫ 2π
0dǫ
∫ hmax
hmin
dh
∫ ℓi
0dℓ′
∫ ℓi
ℓ′dℓ Ki1[Σi (ℓ− ℓ′)] .(9)
x
y
ε
ViVj
h
r
r'
ll ij
l i
l j
l'
ENE6101: Week 9 The collision probability method in 1D – part 2 – 8/25
Cylindrical 1D geometry 1
In the case of 1D cylindrical geometry, the
tracks are identical for any value of the
angle ǫ.
The tubular volumes are concave, making
extra terms to appear. The corresponding
geometry is depicted in figure. Two tracks
are represented, both contributing to
collision probability components pij with
i < j and pii. Track 1© corresponds to the
integration domain where region i is concave
and track 2© corresponds to the integration
domain where it is convex. In this figure, τij
and τii are the optical paths of the materials
located between regions i and j or between
two instances of region i.
h
Vi
Vj
ri-1/2 ri+1/2
rj-1/2 rj+1/2
τij
τii
li
lj
li
li
lj
lj
lj
track 2
1track
ENE6101: Week 9 The collision probability method in 1D – part 2 – 9/25
Cylindrical 1D geometry 2
In this case, Eq. (8) can be rewritten as
pij =2
Vi
∫ ri−1/2
0dh
∫ ℓi
0dℓ′
∫ ℓj
0dℓ
[
Ki1[(τij + τii +Σi ℓi) + Σi ℓ′ + Σj ℓ]
+ Ki1(τij +Σi ℓ′ +Σj ℓ)
]
+
∫ ri+1/2
ri−1/2
dh
∫ ℓi
0dℓ′
∫ ℓj
0dℓ Ki1(τij +Σi ℓ
′ +Σj ℓ)
if i < j(10)
where ri±1/2 are the radii bounding region i and Eq. (9) can be rewritten as
pii =2
Vi
∫ ri−1/2
0dh
∫ ℓi
0dℓ′
[
2
∫ ℓi
ℓ′dℓ Ki1[Σi (ℓ− ℓ′)] +
∫ ℓi
0dℓ Ki1[τii + Σi (ℓ
′ + ℓ)]
]
+
∫ ri+1/2
ri−1/2
dh
∫ ℓi
0dℓ′
∫ ℓi
ℓ′dℓ Ki1[Σi (ℓ− ℓ′)]
(11)
where the first and second terms are the contributions from tracks 1© and 2©, respectively.
ENE6101: Week 9 The collision probability method in 1D – part 2 – 10/25
Cylindrical 1D geometry 3
These equations can be simplified by introducing the following definitions:
Cij(τ0) =
∫ ℓi
0dℓ′
∫ ℓj
0dℓ Ki1(τ0 + Σi ℓ
′ + Σj ℓ)(12)
and
Di =
∫ ℓi
0dℓ′
∫ ℓi
ℓ′dℓ Ki1[Σi (ℓ− ℓ′)](13)
so that
pij =2
Vi
∫ ri−1/2
0dh [Cij(τij + τii +Σi ℓi) + Cij(τij)] +
∫ ri+1/2
ri−1/2
dh Cij(τij)
(14)
if i < j, and
pii =2
Vi
∫ ri−1/2
0dh [2Di + Cii(τii)] +
∫ ri+1/2
ri−1/2
dhDi
.(15)
ENE6101: Week 9 The collision probability method in 1D – part 2 – 11/25
Cylindrical 1D geometry 4
Integration in ℓ and ℓ′ is next performed, leading to the following relations:
a) Σi 6= 0 and Σj 6= 0 :
Cij(τ0) =1
Σi Σj
[
Ki3(τ0)−Ki3(τ0 +Σi ℓi)−Ki3(τ0 + Σj ℓj)
+ Ki3(τ0 +Σi ℓi +Σj ℓj)]
b) Σi = 0 and Σj 6= 0 :
Cij(τ0) =ℓi
Σj[Ki2(τ0)−Ki2(τ0 +Σj ℓj)]
c) Σi 6= 0 and Σj = 0 :
Cij(τ0) =ℓj
Σi[Ki2(τ0)−Ki2(τ0 + Σi ℓi)]
d) Σi = Σj = 0 :
Cij(τ0) = ℓi ℓj Ki1(τ0)
ENE6101: Week 9 The collision probability method in 1D – part 2 – 12/25
Cylindrical 1D geometry 5
e) Σi 6= 0 :
Di =ℓi
Σi−
1
Σ2i
[Ki3(0)−Ki3(Σi ℓi)]
f) Σi = 0 :
Di =πℓ2i
4.
ENE6101: Week 9 The collision probability method in 1D – part 2 – 13/25
Cylindrical 1D geometry 6
The above relations are implemented in Matlab using the following two scripts:
function f=cij_f(tau0,sigi,sigj,segmenti,segmentj)
if sigi ˜= 0 && sigj ˜= 0
f=(akin(3,tau0)-akin(3,tau0+sigi*segmenti)- ...
akin(3,tau0+sigj*segmentj)+ ...
akin(3,tau0+sigi*segmenti+sigj*segmentj))/(sigi*sigj) ;
elseif sigi == 0 && sigj ˜= 0
f=(akin(2,tau0)-akin(2,tau0+sigj*segmentj))*segmenti/sigj ;
elseif sigi ˜= 0 && sigj == 0
f=(akin(2,tau0)-akin(2,tau0+sigi*segmenti))*segmentj/sigi ;
else
f=akin(1,tau0)*segmenti*segmentj ;
end
function f=di_f(sig,segment)
if sig ˜= 0
f=segment/sig-(akin(3,0)-akin(3,sig*segment))/sigˆ2 ;
else
f=pi*segmentˆ2/4 ;
end
ENE6101: Week 9 The collision probability method in 1D – part 2 – 14/25
Tracking 1
The tracking of 1D cylindrical and spherical geometries is similar and can be
generated with the same algorithm.
A matlab script f=sybt1d(rad,lgsph,ngauss) is presented to produce a tracking
object in these cases.
The figure presents an example with I = 2 regions and K = 2 tracks per regions.
Tracks 1© and 2© have two segments, identified as ℓ1 and ℓ2. Tracks 3© and 4© have
only one segment, identified as ℓ2.
h
1
2
3
4
φ
l1
l2
l1
l2
l2
l2
h1 h3h2 h4
ENE6101: Week 9 The collision probability method in 1D – part 2 – 15/25
Tracking 2
A K point Gauss-Jacobi quadrature is
used
The four tracks are obtained using
rad=[0.075 0.15] ;
track=sybt1d(rad,false,2) ;
track(i,j) is a 2D cell array
containing I ×K elements, one per
track. The i index refers to the tracks
located in ri−1/2 < h < ri+1/2. The
(i, j)–th cell is also a cell array
containing three elements:
the weight wk of the track,
the product wk cosφ,
a 1D array of dimension I − i+ 1
containing the track segments.
The ℓ1 segment length of track 2© is
l2=track1,23(1)
leading to l2 = 0.07186585.
Any length trackik,il3(1)
corresponding to a segment that cross
the h axis must be multiplied by 2.
h
1
2
3
4
φ
l1
l2
l1
l2
l2
l2
h1 h3h2 h4
ENE6101: Week 9 The collision probability method in 1D – part 2 – 16/25
Tracking 3
The following script will perform a numerical integration of p11 in 1D cylindrical tube of radius
rad and macroscopic total cross section sigt:
function f=p11_cyl(rad,sigt)
% compute the p11 component in 1D cylindrical geometry
track=sybt1d(rad,false,5) ;
val=0 ;
for i=1:5
segment=2*track1,i3(1) ;
if sigt ˜= 0
d=segment/sigt-(akin(3,0)-akin(3,sigt*segment))/sigtˆ2 ;
else
d=pi*segmentˆ2/4 ;
end
val=val+track1,i1*d ;
end
f=val/(pi*radˆ2) ;
ENE6101: Week 9 The collision probability method in 1D – part 2 – 17/25
Boundary conditions 1
The external boundary condition of a 1D cylindrical geometry is introduced by
choosing an albedo β+ set to zero for representing a voided boundary, or set to one
for representing reflection of particles.
The voided boundary condition is similar to the vacuum condition used for 1D slab
geometry.
The reflecting boundary condition is implemented in the context of the Wigner-Seitz
approximation for lattice calculations in reactor physics. The Wigner-Seitz
approximation consists of replacing the exact boundary with an equivalent cylindrical
boundary, taking care to conserve the amount of moderator present in the cell. The
correct condition in this case is the white boundary condition in which the particles are
reflected with an isotropic angular distribution.
Fuel Fuel
ENE6101: Week 9 The collision probability method in 1D – part 2 – 18/25
Boundary conditions 2
The application of a white boundary condition is based on the transformation of the reduced
collision probability matrix P = pij , i = 1, I and j = 1, I corresponding to a vacuum
boundary condition. We first compute the escape probability vector P iS = PiS , i = 1, I,
a column vector whose components are obtained from
PiS = 1−I
∑
j=1
pij Σj(16)
representing the probability for a neutron born in region i to escape from the unit cell without
collision.
The probability for a neutron entering in the unit cell by its boundary, with an isotropic
angular distribution, to first collide in region i is represented by the component PSi. It is
possible to show that
pSi =PSi
Σi=
4Vi
SPiS(17)
where S is the surface of the boundary.
ENE6101: Week 9 The collision probability method in 1D – part 2 – 19/25
Boundary conditions 3
The transmission probability PSS is therefore given as
PSS = 1−I
∑
i=1
pSi Σi .(18)
The reduced collision probability matrix P = pij , i = 1, I and j = 1, I corresponding to a
white boundary condition can now be obtained using
P = P+ P iS
[
β+ + (β+)2 PSS + (β+)3 P 2SS + (β+)4 P 3
SS + . . .]
p⊤Sj
or, using a geometric progression, as
P = P+β+
1− β+ PSSP iS p⊤
Sj .(19)
The collision probability matrix P can be scattering-reduced and used in the matrix flux
equation of the CP method.
ENE6101: Week 9 The collision probability method in 1D – part 2 – 20/25
Spherical 1D geometry 1
Computing the tracking and collision probabilities in 1D spherical geometry is similar to
the cylindrical geometry case.
The tracking is obtained by the sybt1d Matlab script
The expression of reduced collision probabilities is written
pij =1
Vi
∫
Vi
d3r′∫
Ij
ds e−τ(s)(20)
ENE6101: Week 9 The collision probability method in 1D – part 2 – 21/25
Spherical 1D geometry 2
which can be rewritten as
pij =2π
Vi
∫ ri−1/2
0dh h
∫ ℓi
0dℓ′
∫ ℓj
0dℓ
[
e−[(τij+τii+Σi ℓi)+Σi ℓ′+Σj ℓ]
+ e−(τij+Σi ℓ′+Σj ℓ)
]
+
∫ ri+1/2
ri−1/2
dh h
∫ ℓi
0dℓ′
∫ ℓj
0dℓ e−(τij+Σi ℓ′+Σj ℓ)
if i < j(21)
where ri±1/2 are the radii bounding region i and
pii =2π
Vi
∫ ri−1/2
0dh h
∫ ℓi
0dℓ′
[
2
∫ ℓi
ℓ′dℓ e−[Σi (ℓ−ℓ′)] +
∫ ℓi
0dℓ e−[τii+Σi (ℓ′+ℓ)]
]
+
∫ ri+1/2
ri−1/2
dh h
∫ ℓi
0dℓ′
∫ ℓi
ℓ′dℓ e−[Σi (ℓ−ℓ′)]
(22)
where the first and second terms are the contributions from tracks 1© and 2©, respectively.
ENE6101: Week 9 The collision probability method in 1D – part 2 – 22/25
Spherical 1D geometry 3
These equations can be simplified by introducing the following definitions:
Cij(τ0) =
∫ ℓi
0dℓ′
∫ ℓj
0dℓ e−(τ0+Σi ℓ′+Σj ℓ)
(23)
and
Di =
∫ ℓi
0dℓ′
∫ ℓi
ℓ′dℓ e−Σi (ℓ−ℓ′)
(24)
so that
pij =2π
Vi
∫ ri−1/2
0dh h [Cij(τij + τii +Σi ℓi) + Cij(τij)] +
∫ ri+1/2
ri−1/2
dh h Cij(τij)
(25)
if i < j and
pii =2π
Vi
∫ ri−1/2
0dh h [2Di + Cii(τii)] +
∫ ri+1/2
ri−1/2
dh hDi
.(26)
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Spherical 1D geometry 4
Integration in ℓ and ℓ′ is next performed, leading to the following relations:
a) Σi 6= 0 and Σj 6= 0 :
Cij(τ0) =1
Σi Σj
[
e−τ0 − e−(τ0+Σi ℓi) − e−(τ0+Σj ℓj) + e−(τ0+Σi ℓi+Σj ℓj)]
b) Σi = 0 and Σj 6= 0 :
Cij(τ0) =ℓi
Σj
[
e−τ0 − e−(τ0+Σj ℓj)]
c) Σi 6= 0 and Σj = 0 :
Cij(τ0) =ℓj
Σi
[
e−τ0 − e−(τ0+Σi ℓi)]
d) Σi = Σj = 0 :
Cij(τ0) = ℓi ℓj e−τ0
ENE6101: Week 9 The collision probability method in 1D – part 2 – 24/25
Spherical 1D geometry 5
e) Σi 6= 0 :
Di =ℓi
Σi−
1
Σ2i
[
1− e−Σi ℓi]
f) Σi = 0 :
Di =ℓ2i
2.
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