Validation of a new numerical neutron flux solver in APOLLO3 code


Research Paper (postgraduate), 2011

45 Pages


Excerpt


Contents
1
Introduction
2
2
Neutron Transport Equation
5
3
ZPR 1D Benchmark
7
3.1
Description of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
3.2
Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
4
Stepanek Benchmark
11
4.1
Description of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
4.2
Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
5
Takeda Benchmark
15
5.1
Takeda model 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
5.1.1
Description of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
16subsection.5.1.2
5.1.3
Results for case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
5.2
Takeda model 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
5.2.1
Description of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
5.2.2
Results for case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
5.2.3
Results for case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
5.3
Takeda model 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
5.3.1
Description of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
5.3.2
Results for case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
5.3.3
Results for case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
5.3.4
Results for case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
6
Conclusion and Recommendations
32
A Example of Apollo3 input data file
34
1

Résumé
L'objectif de ce travail est de tester par comparaison avec des valeurs de référence les coefficients
de multiplication (keff) obtenues par la méthode Monte Carlo, les performances des trois solveurs
de flux (
MINOS
,
IDT
et
NEMO
) du code de calcul déterministe A
POLLO
3. Il permet en particulier de
démontrer les potentialités du nouveau solveur
NEMO
. Cette étude numérique permet aussi une com-
paraison entre les trois solveurs sur d'autres paramètres comme la distribution du flux neutronique et
le temps de calcul CPUs. A cet effet, les trois solveurs de flux sont confrontés à trois benchmarks :
ZPR 1D, Stepanek et Takeda. De façon globale, les solveurs démontrent un très bon accord avec la
référence Monte Carlo du keff, avec parfois des temps CPU très raisonnables. La comparaison des
solveurs entre eux montrent que le solveur
NEMO
donne des résultats corrects mais les performances
de ce solveur restent à améliorer. Ce travail a été réalisé au CEA Saclay dans le Service d'Etudes
des Réacteurs et de Mathématiques Appliquées (SERMA) du Département de Modélisation des Sys-
tèmes et des Structures (DM2S). Ce rapport est requis pour l'obtention du diplôme de Master dans la
spécialité Physique et ingénierie des réacteurs nucléaires.
2

Abstract
The aim of this work is to test and compare the performance of three A
POLLO
3 flux solvers which are:
MINOS IDT
and
NEMO
against reference Monte Carlo and to test the new neutron transport equation
solver
NEMO
. This testing is limited to numerical aspects by comparing the differences between:
keff, flux and CPU time resulting of the three solvers. We will compare the value of keff against
reference Monte Carlo. To do this we will expose these solvers to three benchmark problems: ZPR 1D,
Stepanek, Takeda. In general the three solvers shows a very good agreements with Monte Carlo and
sometimes with much less CPU time. Good results were founded by
NEMO
This work was performed
at CEA, Saclay, DM2S/SERMA. This report is required to obtain the Master degree in the specality
of Nuclear Reactor Physics and Engineering.
Key words: Neutron transport equation; Benchmarck; A
POLLO
3,
MINOS
,
IDT
,
NEMO
.

Acknowledgments
I want to thanks professor C. Diop for giving me this opportunity to have my internship at SERMA
and thanks to C. Calvin and R. Lenain for introducing me to SERMA. Also I express my pleasures
and thanks to: L. Bourhrara, F. Damian, F. Gabriel and I. Zmijarevic who have help me to do this
work. Without their patient and kind support it was difficult to complete it.
1

Chapter 1
Introduction
The A
POLLO
3 [8] code is being developed by SERMA (Service d'Etudes de Réacteurs et de
Mathématiques Appliquées du CEA). Its purpose is to provide a neutronic codes and studies for both
thermal spectrum reactors and fast spectrum reactors. A
POLLO
3 code has four deterministic solvers
MINOS
,
IDT
,
MINARET
,
TDT
and recently a new solver called
NEMO
is introduced on an experimental
basis. These different solvers have a common object thats it to calculate the flux by solving the neutron
transport equation, each of these solver uses different numerical methods for angular variable
and
for spatial variable r.
For the angular variable, there are mainly two methods:
· The first is called
S
n
(discrete ordinates method) which calculate the angular flux
(r, ) for a
set of angular directions
d
(d
= 1, · · · , N
).
· The second method is called
P
n
(spherical harmonic method) which expand the angular flux in
spherical harmonics according to:
(r, ) =
N
n=0
m=n
m=-n
m
n
(r)y
m
n
(),
where the functions y
m
n
() are the real spherical harmonics and
m
n
(r) are the flux moments.
A variant of the
P
n
method is the
SP
n
method (Simplified
P
n
method), see [11].
For the spatial variable, several methods are used: finite element method, finite difference method,
nodal method.
The purpose of this document is to provide a comparison of three solvers
MINOS
,
IDT
and
NEMO
of
A
POLLO
3 code:
· The
MINOS
solver ([1], [11]) uses the
SP
n
method for the angular variable and the finite element
method RT (Raviart-Thomas) for space variable. It can take into account two types of boundary
2

conditions: zero flux and specular reflection. Currently, the vacuum boundary condition is not
available in the solver.
MINOS
solves the
SP
n
equations for cartesian and hexagonal geometries.
For hexagonal geometry each hexagon is refined using four trapezoids.
· The
IDT
solver [14] is based on the
S
n
method for angular variable and has three different
methods to treat the space variable: the nodal method, the characteristic method and the dia-
mond differences method.
IDT
solver can take into account two types of boundary conditions:
specular reflection and vacuum. Currently
IDT
solves the transport equation only for cartesian
geometries.
· The
NEMO
solver ([3], [4]) is based on
P
n
method for angular variable and Lagrange finite
element method in space variable. It considers two types of boundary conditions: reflection
and vacuum.
NEMO
solves the transport equation for cartesian and hexagonal geometries. For
hexagonal geometry each hexagon is refined as three lozenges or six triangles.
We will perform a comparison study between these three solvers by using different benchmark prob-
lems which cover fast and thermal reactor. In this study We will present k
ef f
, CPU time, number of
outer iterations,
1
flux distribution and error.
Z
PR
1
D
This is a test case in plane geometry (1D) with six energy groups and with heteroge-
neous material properties involving cross sections for fast-spectrum reactor.
S
TEPANEK
This is a mono-kinetic test case with cartesian geometry (2D).
T
AKEDA
These are 3D problems descriped in two energy groups for model 1 and in four energy
groups for model 2 and 3.
A
POLLO
3 has a user text interface based on the technique of keywords and associated values [5]. This
interface allows the user to describe the data of his problem, such as geometry, boundary condition,
etc. by using a number of commands and keywords. Test cases studied in this work have been made
using this interface. As an example in Appendix (A) we illustrate A
POLLO
3 test corresponding to
Stepanek benchmark.
1
In the multiplication eigenvalue problems the power iteration solution to obtain k
ef f
is referred to as the outer iteration.
While the evaluation of the inverse scattering operator H
-1
gg
is referred to as the inner iteration.[9]
3

The following equations and conditions were used to calculate error between
NEMO
,
IDT
and
MINOS
solvers in order to produce graphs and tables of comparison. Morever we illustrate specific conditions
for each solver.
Flux Normalization
The scalar flux
g
(r) for each benchmark problem has been normalized as
following:
¯
g
(r) =
1
M
g
(r),
(1.1)
where
M
is the maximum scalar flux and ¯
g
(r) is the normalized flux in group g.
Error Calculations
To calculate the relative error between a reference value and a value obtained
by a solver we use:
Error
(k
ef f
) =
|k
ref
ef f
- k
ef f
|
k
ref
ef f
(1.2)
To calculate the absolute error in flux between any two solvers like solver i and j we used:
i-j
= |
i
-
j
|
(1.3)
Outer iteration convergence criteria
In the three solvers we use a maximum of 500 outer iterations
as a maximum number with the following criteria to achieve the convergence:
· 10
-5
for k
ef f
.
· 10
-4
for point wise fission source.
Calculations conditions
We used a linux based system to perform the calculations with the follow-
ing hardware configurations:
- 1 CPU Intel Xenon 2.66GHz.
- 4Gb memory.
4

Chapter 2
Neutron Transport Equation
To recall the neutron transport problem ([6], [9]), let us set a number of notations. We denote by
D the space domain and S
2
the unit sphere the set of all angular directions. We use the variables
r
D and S
2
to denote the spatial and angular variables. The boundary of D will be denoted by
D. We define the incoming boundary
-
and the outgoing boundary
+
as:
= D × S
2
,
-
= {(r, ) , · n(r) < 0},
+
= {(r, ) , · n(r) > 0},
where n
= n(r) is the outward unit normal vector at boundary D.
The resolution of the neutron transport equation consists in determining the largest eigenvalue and
the associated multigroup angular flux
=
g
(r, )
g=1,··· ,G
solutions of problem (2.1):
( · +
g
)
g
= H
g
+
1
F
g
in D
× S
2
,
(2.1a)
g
=
g
-
on
-
,
(2.1b)
where
(
g
g
)(r, ) =
g
(r)
g
(r, ),
(2.2)
(H
g
)(r, ) =
G
g
=1
S
2
g,g
s
(r, ·
)
g
(r,
) d
,
(2.3)
(F
g
)(r) =
g
G
g
=1
g
f
(r)
S
2
g
(r,
) d
.
(2.4)
5

In the above equations
g
(r) is the total macroscopic cross-section,
g,g
s
(r, ·
) is the macroscopic
differential scattering cross-section,
g
f
(r) is the number of emitted neutrons per fission multiplied
by macroscopic cross section (production macroscopic cross-section),
g
(r) is the fission spectrum
and
g
-
(r, ) is the incoming flux.
The vacuum boundary condition is represented as
g
-
= 0 in equation (2.1b). The specular reflection
boundary condition consisits in replacing the equation (2.1b) by:
(r, ) = (r,
)
on
(r, )
-
,
(2.5)
where
is the direction resulting by reflection of direction
with respect to the normal plane to the
unit vector n
(r):
= -2( · n)n + .
6

Chapter 3
ZPR 1D Benchmark
3.1
Description of the problem
This problem represents a simplified zero power fast reactor. We describe this problem by using 6-
groups energy. There are 13 media along x-axis in this problem. We have vacuum boundary condition
on both sides. Fig. 3.1 illustrates the geometry for this benchmark.
E
MPTY
F
ILLER
R
R
S
TEEL
R
R
S
ODIUM
R
B
U3O8
R
B
S
ODIUM
O
C
UP
U
M
O
O
C
S
ODIUM
O
C
U3O8
C
I
B4C
C
I
S
ODIUM
I
C
U3O8
I
C
S
ODIUM
I
C
UP
U
M
O
x (cm)
0
255
Figure 3.1: geometry for ZPR 1D benchmark
3.2
Result
Table 3.1 shows the calculated k
ef f
for this problem. To calculate the relative error in k
ef f
we
take
IDT
(
S
8
) results as a reference. Table 3.2 gives the absolute error of the multigroup flux between
7

the three solvers. The angular approximations used in each solver are:
NEMO
(
P
13
),
IDT
(
S
8
) and
MINOS
(
SP
13
).
Table 3.1: keff results
solver
order
keff
err. (pcm)
outer iterations
CPU (s)
S
8
1.02081
-
23
0.05
IDT
S
12
1.02164
81.3
22
0.06
S
16
1.02214
130
22
0.06
MINOS
SP
3
1.01777
297.8
18
0.01
SP
13
1.02091
9.8
18
0.06
P
0
1.00302
1742.7
29
1
NEMO
P
2
1.01796
279.2
29
1
P
13
1.02084
2.9
30
1
Table 3.2: flux error per group
Groups
NEMO
-
MINOS
(%)
IDT
-
NEMO
(%)
IDT
-
MINOS
(%)
1
0.04
0.10
0.15
2
0.09
0.15
0.22
3
0.07
0.09
0.15
4
0.29
0.25
0.34
5
0.97
0.29
0.93
6
3.10
0.30
3.12
8

0
50
100
150
200
250
300
x (cm)
0
0,05
0,1
0,15
0,2
scalar flux
NEMO
IDT
MINOS
Figure 3.2: flux distribution in group 1
0
50
100
150
200
250
300
x (cm)
0
0,2
0,4
0,6
0,8
1
scalar flux
NEMO
IDT
MINOS
Figure 3.3: flux distribution in group 6
Fig. 3.2
1
and 3.3 show the scalar fluxes versus the space variable x for thermal and fast energy groups.
From Fig. 3.2 and 3.3 the scalar flux resulting from the three solvers are almost the same for thermal
group while for fast group we notice some differences specially at the boundaries.
Fig. 3.4 gives the absolute error for 6-energy groups for the three solvers and also between each two
solvers separately.
1
Fig. 3.2 to 3.4 are plotted for solvers orders:
NEMO
(
P
13
),
IDT
(
S
8
),
MINOS
(
SP
13
).
9

0
50
100
150
200
250
300
x (cm)
0
0,01
0,02
0,03
0,04
flux error
IDT-NEMO
IDT-MINOS
NEMO-MINOS
0
50
100
150
200
250
300
x (cm)
0
0,001
0,002
0,003
0,004
flux error
IDT-NEMO
0
50
100
150
200
250
300
x (cm)
0
0,01
0,02
0,03
0,04
flux error
IDT-MINOS
0
50
100
150
200
250
300
x (cm)
0
0,01
0,02
0,03
0,04
flux error
NEMO-MINOS
Figure 3.4: flux error in group 6
10

Chapter 4
Stepanek Benchmark
4.1
Description of the problem
The geometry of the Stepanek benchmark [12] is that of a small reactor core comprising four
central regions and an external moderator with vacuum boundary conditions. The geometry of the
problem is shown in Fig. 4.1. The mesh size used is 1cm
× 1cm. The cross sections for five media
are given in table 4.1.
MED
1
MED
2
MED
3
MED
4
MED
5
x (cm)
y (cm)
0
18
48
78
96
0
18
43
68
86
Figure 4.1: geometry for Stepanek benchmark
11

Table 4.1: cross section for Stepanek benchmark
medium
t
s
f
MED1
0.6
0.53
0.079
1.
MED2
0.48
0.2
MED3
0.7
0.66
0.043
1.
MED4
0.65
0.5
MED5
0.9
0.89
4.2
Result
Table 4.2 lists different results regarding k
ef f
for the three solvers and for different angular ap-
proximation. In this table the results of
IDT
(
S
16
) is taken as reference. The absolute error in flux
between the solvers are shown in Table 4.3. The used orders for each solver are:
NEMO
(
P
2
),
IDT
(
S
8
)
and
MINOS
(
SP
3
).
Table 4.2: keff results
solver
order
keff
err. (pcm)
outer iterations
CPU (s)
S
8
1.00884
2
83
8.18
IDT
S
12
1.00886
-
83
8.57
S
16
1.00886
-
84
11.34
SP
1
1.00663
223
68
0.04
MINOS
SP
3
1.00872
14
66
0.1
SP
5
1.00879
7
68
0.19
P
0
1.00653
233
133
4
NEMO
P
1
1.00653
233
133
8
P
2
1.00851
33.9
131
15
P
4
1.00857
29
131
47
Table 4.3: flux error per group
Groups
NEMO
-
MINOS
(%)
IDT
-
NEMO
(%)
IDT
-
MINOS
(%)
1
0.93
0.45
0.47
Fig. 4.2
1
gives the scalar flux resulting from
NEMO
solver calculation with
P
2
approximation. Fig.
4.3, 4.4 and 4.5 illustrate the absolute error in scalar flux between the three solvers.
1
Fig. 4.2 to 4.9 are plotted for solvers orders:
NEMO
(
P
2
),
IDT
(
S
8
),
MINOS
(
SP
3
).
12

Xd3d 8.3.1 (6 Oct 2007)
16/05/11 lb224222
idFS_NEMO__WN_2_g1_moment_0_0.xyz
idFS_NEMO__WN_2_g1_moment_0_0.z
Triangles 3D P1
nodes : 8256
faces : 16150
20 colors
-0.1311302E-05
0.1019918
0.2039848
0.3059779
0.407971
0.509964
0.6119571
0.7139502
0.8159432
0.9179363
1.019929
1.121922
1.223915
1.325909
1.427902
1.529895
1.631888
1.733881
1.835874
1.937867
2.03986
x
y
Figure 4.2: scalar flux (
NEMO P
2
)
Xd3d 8.3.1 (6 Oct 2007)
16/05/11 lb224222
idt_minos_g1.xyz
idt_minos_g1.z
Triangles 3D P1
nodes : 8256
faces : 16150
20 colors
0
0.292345E-01
0.58469E-01
0.877035E-01
0.116938
0.1461725
0.175407
0.2046415
0.233876
0.2631105
0.292345
0.3215795
0.350814
0.3800485
0.409283
0.4385175
0.467752
0.4969865
0.526221
0.5554554
0.58469
x
y
Figure 4.3: flux error
IDT
-
MINOS
Xd3d 8.3.1 (6 Oct 2007)
16/05/11 lb224222
nemo_idt_g1.xyz
nemo_idt_g1.z
Triangles 3D P1
nodes : 8256
faces : 16150
20 colors
0
0.227872E-01
0.455744E-01
0.683616E-01
0.911488E-01
0.113936
0.1367232
0.1595104
0.1822976
0.2050848
0.227872
0.2506592
0.2734464
0.2962336
0.3190208
0.341808
0.3645952
0.3873824
0.4101696
0.4329568
0.455744
x
y
Figure 4.4: flux error
NEMO
-
IDT
Xd3d 8.3.1 (6 Oct 2007)
16/05/11 lb224222
nemo_minos_g1.xyz
nemo_minos_g1.z
Triangles 3D P1
nodes : 8256
faces : 16150
20 colors
-0.205636E-05
0.398455E-01
0.7969305E-01
0.1195406
0.1593882
0.1992357
0.2390832
0.2789308
0.3187784
0.3586259
0.3984734
0.438321
0.4781685
0.5180161
0.5578637
0.5977112
0.6375588
0.6774063
0.7172539
0.7571014
0.796949
x
y
Figure 4.5: flux error
NEMO
-
MINOS
Fig. 4.6, 4.7, 4.8 and 4.9 show the scalar flux (for one-energy group) for a given positions.
13

0
20
40
60
80
y (cm)
0
0,02
0,04
0,06
0,08
scalar flux
NEMO
IDT
MINOS
Figure 4.6: flux distribution along the line parallel to
y-axis at x=48.5 cm
0
20
40
60
80
100
x (cm)
0
0,05
0,1
0,15
0,2
scalar flux
NEMO
IDT
MINOS
Figure 4.7: flux distribution along the line parallel to
x-axis at y=43.5 cm
0
20
40
60
80
y (cm)
0
1e-06
2e-06
3e-06
4e-06
5e-06
6e-06
7e-06
scalar flux
NEMO
IDT
MINOS
Figure 4.8: flux distribution along the line parallel to
y-axis at x=95.5 cm
0
20
40
60
80
100
x (cm)
0
5e-06
1e-05
1,5e-05
2e-05
scalar flux
NEMO
IDT
MINOS
Figure 4.9: flux distribution along the line parallel to
x-axis at y=85.5 cm
14

Chapter 5
Takeda Benchmark
5.1
Takeda model 1
5.1.1
Description of the problem
This model represents a small Light Water Reactor (Small LWR). The core configuration is shown
in Fig. 5.1. The reference mesh size is 1cm
× 1cm × 1cm. The core calculation is performed in 2
groups. The cross section set and energy group structure are given in [13]. For each axis the boundary
condition is reflective at the minimum and vacuum at the maximum. Two cases are considered:
· case 1: the control rod position is empty (void).
· case 2: the control rod is inserted.
x (cm)
y (cm)
0
15
20
25
0
5
15
25
CORE
REFLECTOR
CRP
/
VOID
x (cm)
z (cm)
0
15
20
25
0
15
25
Figure 5.1: geometry for Takeda model 1
15

5.1.2
Results
1
for case 1
The reference calculations by Monte Carlo for this case when there is only a void gives k
ref
ef f
=0.9780(
±0.0006)
[13].
Table 5.1: k
ef f
for Takeda 1 case 1
solver
order
k
ef f
error (pcm)
outer iterations
CPU (s)
S
8
0.9774
61.3
11
29.89
IDT
S
12
0.9775
51.5
11
35.02
S
16
0.9775
51.5
11
43.38
SP
1
0.9163
6308.8
500
3.01
MINOS
SP
3
0.9425
3629.9
500
8.51
SP
5
0.9431
3568.5
500
16.15
P
0
0.9399
3895.7
10
6
P
1
0.9300
4908
18
45
NEMO
P
2
0.9740
409
18
186
P
4
0.9764
163.6
20
3318
Tables 5.2 and 5.5 shows the maximum absolute error between each two solvers for a corresponding
energy group. The used orders for each solver are:
NEMO
(
P
2
),
IDT
(
S
8
) and
MINOS
(
SP
3
).
Table 5.2: flux error per group (Takeda 1 case 1)
Groups
NEMO
-
MINOS
(%)
IDT
-
NEMO
(%)
IDT
-
MINOS
(%)
1
23.5
4.58
26.3
2
8.71
2.81
10.9
Table 5.3 shows the error between solvers for
S
8
,
P
4
and
SP
5
orders.
Table 5.3: flux error per group (Takeda 1 case 1) with
P
4
Groups
NEMO
-
MINOS
(%)
IDT
-
NEMO
(%)
IDT
-
MINOS
(%)
1
24.8
3.16
26.3
2
9.39
1.93
10.9
1
In this benchmark model
MINOS
was not optomized specially for the void case as it give a big error with respect to the
reference Monte Carlo.
16

In Fig. 5.2
2
to 5.7 we plot the scalar flux evaluated by the three solvers:
NEMO
,
IDT
and
MINOS
. for
case 1.
0
5
10
15
20
25
30
z (cm)
0
0,1
0,2
0,3
0,4
scalar flux
NEMO
IDT
MINOS
Figure 5.2: flux distribution (group-1) along the line
parallel to z-axis at x=15.5 cm and y=4.5
cm
0
5
10
15
20
25
30
z (cm)
0
0,05
0,1
0,15
0,2
scalar flux
NEMO
IDT
MINOS
Figure 5.3: flux distribution (group-2) along the line
parallel to z-axis at x=15.5 cm and y=4.5
cm
2
Fig. 5.2 to 5.11 are plotted for the solvers orders:
NEMO
(
P
2
),
IDT
(
S
8
),
MINOS
(
SP
3
).
17

0
5
10
15
20
25
30
y (cm)
0
0,1
0,2
0,3
0,4
scalar flux
NEMO
IDT
MINOS
Figure 5.4: flux distribution (group-1) along the line
parallel to z-axis at x=15.5 cm and z=4.5
cm
0
5
10
15
20
25
30
y (cm)
0
0,05
0,1
0,15
0,2
0,25
scalar flux
NEMO
IDT
MINOS
Figure 5.5: flux distribution (group-2) along the line
parallel to z-axis at x=15.5 cm and z=4.5
cm
0
5
10
15
20
25
30
z (cm)
0
0,05
0,1
0,15
0,2
scalar flux
NEMO
IDT
MINOS
Figure 5.6: flux distribution (group-1) along the line
parallel to z-axis at x=15.5 cm and y=15.5
cm
0
5
10
15
20
25
30
z (cm)
0
0,1
0,2
0,3
0,4
scalar flux
NEMO
IDT
MINOS
Figure 5.7: flux distribution (group-2) along the line
parallel to z-axis at x=4.5 cm and y=4.5 cm
18

5.1.3
Results for case 2
In this case the control rod was inserted and the Monte Carlo calculation gives k
ref
ef f
=0.9624(
±0.0006).
Table 5.4: k
ef f
for Takeda 1 case 2
solver
order
k
ef f
error (pcm)
outer iterations
CPU (s)
S
8
0.962663
27.3
14
27.23
IDT
S
12
0.962644
25.4
14
34.02
S
16
0.962625
23.4
14
42.54
SP
1
0.921611
4231
406
2.14
MINOS
SP
3
0.948322
1455
388
5.95
SP
5
0.949128
1371.9
378
10.99
P
0
0.938457
2487.8
12
6
P
1
0.931811
3178.4
26
46
NEMO
P
2
0.962117
29.4
24
141
P
4
0.962800
41.6
28
826
Table. 5.5 shows the error between solvers for
S
8
,
P
4
and
SP
5
orders.
Table 5.5: flux error per group (Takeda 1 case 2)
Groups
NEMO
-
MINOS
(%)
IDT
-
NEMO
(%)
IDT
-
MINOS
(%)
1
3.33
1.35
3.62
2
3.46
0.72
3.28
19

0
5
10
15
20
25
30
y (cm)
0
0,2
0,4
0,6
0,8
1
scalar flux
NEMO
IDT
MINOS
Figure 5.8: flux distribution (group-1) along the line
parallel to z-axis at x=3.75 cm and z=3.75
cm
0
5
10
15
20
25
30
y (cm)
0
0,1
0,2
0,3
0,4
scalar flux
NEMO
IDT
MINOS
Figure 5.9: flux distribution (group-2) along the line
parallel to z-axis at x=3.75 cm and z=3.75
cm
0
5
10
15
20
25
30
y (cm)
0
0,05
0,1
0,15
0,2
0,25
scalar flux
NEMO
IDT
MINOS
Figure 5.10: flux distribution (group-1) along the line
parallel to z-axis at x=17.25 cm and
z=3.75 cm
0
5
10
15
20
25
30
y (cm)
0
0,05
0,1
0,15
0,2
0,25
scalar flux
NEMO
IDT
MINOS
Figure 5.11: flux distribution (group-2) along the line
parallel to z-axis at x=17.25 cm and
z=3.75 cm
20

5.2
Takeda model 2
5.2.1
Description of the problem
This benchmark problem represents a small Fast Breeder Reactor (FBR). The reactor configura-
tion is shown in Fig. 5.12. The mesh size used in the calculations is 5cm
× 5cm × 5cm. In this model
4-energy groups was used to perform core calculations. For x and y axes the boundary condition is re-
flective at the minimum and vacuum at the maximum while for z axis we have only vacuum boundary
condition at the minimum and at the maximum. Two cases are considered:
· case 1: Control rods are withdrawn.
· case 2: Control rods are half inserted.
x (cm)
y (cm)
0
35
45
55
70
0
55
70
CORE
RADIAL BLANK ET
AXIAL BLANK ET
RADIAL REFLECTOR
CR
CRP
CR
/
CRP
x (cm)
z (cm)
0
35 45 55
70
0
20
75
130
150
Figure 5.12: Takeda 2
5.2.2
Results for case 1
The reference Monte Carlo calculation for this case when the control rods are withdrawn gives,
k
ref
ef f
=0.9736(
±0.0002).
21

Table 5.6: k
ef f
for Takeda 2 case 1
solver
order
k
ef f
error (pcm)
outer iterations
CPU (s)
S
8
0.973522
33.08
19
14.19
IDT
S
12
0.97354
34.6
17
16.78
S
16
0.973548
35.7
17
21.35
SP
1
0.963551
991.4
139
0.31
MINOS
SP
3
0.967678
567.4
137
0.87
SP
5
0.967747
560.3
138
1.67
P
0
0.971882
135.4
21
4
P
1
0.968019
532.3
22
43
NEMO
P
2
0.972783
42.8
43
41
P
4
0.972808
40.2
47
167
In Table. 5.7 we evaluate the absolute error between solvers for a corresponding energy group:
Table 5.7: flux error per group (Takeda 2 case 1)
Groups
NEMO
-
MINOS
(%)
IDT
-
NEMO
(%)
IDT
-
MINOS
(%)
1
0.34
0.37
0.37
2
2.66
0.28
2.62
3
2.40
0.18
2.37
4
0.20
0.07
0.20
Figs. 5.13 to 5.16
3
shows the flux distribution along the z-axis for the case when the control rods
are withdrawn.
3
Figures 5.13 to 5.19 have been plotted for the solvers orders:
NEMO
(
P
2
),
IDT
(
S
8
),
MINOS
(
SP
3
).
22

0
20
40
60
80
z (cm)
0
0,02
0,04
0,06
0,08
0,1
scalar flux
NEMO
IDT
MINOS
Figure 5.13: flux distribution (group-1) along the line
parallel to z-axis at x=37.5 cm and y=2.5
cm
0
20
40
60
80
z (cm)
0
0,1
0,2
0,3
0,4
0,5
scalar flux
NEMO
IDT
MINOS
Figure 5.14: flux distribution (group-2) along the line
parallel to z-axis at x=37.5 cm and y=2.5
cm
0
20
40
60
80
z (cm)
0
0,1
0,2
0,3
0,4
scalar flux
NEMO
IDT
MINOS
Figure 5.15: flux distribution (group-3) along the line
parallel to z-axis at x=42.5 cm and y=2.5
cm
0
20
40
60
80
z (cm)
0
0,005
0,01
0,015
0,02
scalar flux
NEMO
IDT
MINOS
Figure 5.16: flux distribution (group-4) along the line
parallel to z-axis at x=42.5 cm and y=2.5
cm
23

5.2.3
Results for case 2
The reference Monte Carlo calculation for this case when the control rods are half inserted gives,
k
ref
ef f
=0.9599(
±0.0002).
Table 5.8: k
ef f
for Takeda 2 case 2
solver
order
k
ef f
error (pcm)
outer iterations
CPU (s)
S
8
0.964116
491.5
24
38.17
IDT
S
12
0.964139
494
24
47.81
S
16
0.964148
495
24
60.74
SP
1
0.954672
493
238
1.69
MINOS
SP
3
0.959522
12.71
234
4.76
SP
5
0.959629
23.8
234
9.11
P
0
0.960176
80.88
35
11
P
1
0.957268
222.2
47
47
NEMO
P
2
0.962544
327.7
45
88
P
4
0.962612
334.8
49
350
Table 5.9: flux error per group (Takeda 2 case 2)
Groups
NEMO
-
MINOS
(%)
IDT
-
NEMO
(%)
IDT
-
MINOS
(%)
1
0.35
0.52
0.38
2
2.70
2.13
2.68
3
2.42
1.43
2.42
4
0.17
0.15
0.17
In the following three figures the scalar flux has been evaluated at the different positions of control
rod. However the three solvers shows a very good agreements
24

0
10
20
30
40
50
60
70
y (cm)
0
0,02
0,04
0,06
0,08
0,1
scalar flux
NEMO
IDT
MINOS
Figure 5.17: flux distribution (group-1) along the line
parallel to y-axis at x=37.5 cm and z=72.5
cm
0
10
20
30
40
50
60
70
y (cm)
0
0,1
0,2
0,3
0,4
0,5
scalar flux
NEMO
IDT
MINOS
Figure 5.18: flux distribution (group-2) along the line
parallel to y-axis at x=37.5 cm and z=52.5
cm
0
10
20
30
40
50
60
70
y (cm)
0
0,1
0,2
0,3
0,4
scalar flux
NEMO
IDT
MINOS
Figure 5.19: flux distribution (group-3) along the line parallel to y-axis at x=37.5 cm and z=52.5 cm
25

5.3
Takeda model 3
5.3.1
Description of the problem
This benchmark problem represents an axially heterogeneous Fast Breeder Reactor (FBR). The
reactor configuration is shown in Fig. 5.20. In this model 4-energy groups was used to perform core
calculations and the mesh size is 5cm
× 5cm × 5cm. For each axis the boundary condition is reflective
at the minimum and vacuum at the maximum. Three cases are considered:
· case 1: Control rods are inserted.
· case 2: Control rods are withdrawn.
· case 3: Control rods are withdrawn and the space is replaced with core and/or blanket.
Note: In Takeda model 3,
P
4
calculation is not carried out due to limitaion on the computer memory.
Morever the
NEMO
calculations are not converged toward the reference eigenvalue
CR
CRP
INTERNAL BLANKET
CORE
RADIAL BLANKET
RADIAL REFLECTOR
EMPTY MATRIX
x (cm)
y (cm)
0 5
35 45
80
120
145 160
0
5
35
45
80
120
145
160
CRP
/
CR
CRP
INTERNAL BLANKET
CORE
RADIAL BLANKET
AXIAL BLANKET
RADIAL REFLECTOR
AXIAL REFLECTOR
EMPTY MATRIX
x (cm)
z (cm)
0 5
35 45
80
120
145 160
0
10
55
80
90
26

INTERNAL BLANKET
CORE
RADIAL BLANKET
RADIAL REFLECTOR
EMPTY MATRIX
x (cm)
y (cm)
0
80
120
145 160
0
80
120
145
160
INTERNAL BLANKET
CORE
RADIAL BLANKET
AXIAL BLANKET
RADIAL REFLECTOR
AXIAL REFLECTOR
EMPTY MATRIX
x (cm)
z (cm)
0
80
120
145 160
0
10
55
80
90
Figure 5.20: Takeda 3
5.3.2
Results for case 1
The reference Monte-Carlo calculations for this case when the control rods are inserted gives:
k
ref
ef f
=0.9709(
±0.0002).
Table 5.10: k
ef f
for Takeda 3 case 1
solver
order
k
ef f
error (pcm)
outer iterations
CPU (s)
S
8
0.97040
51
32
177.57
IDT
S
12
0.970441
47
32
217.18
S
16
0.970458
45
32
272.31
SP
1
0.960639
1057
256
3.76
MINOS
SP
3
0.966736
429
258
10.58
SP
5
0.966989
403
254
19.83
P
0
0.978948
829
500
362
P
1
0.983032
1250
500
1005
NEMO
P
2
0.990128
1980
500
1434
P
4
-
-
-
-
The maximum error between each two solvers is shown in 5.11 for each energy group:
27

Table 5.11: flux error per group (Takeda 3 case 1)
Groups
NEMO
-
MINOS
(%)
IDT
-
NEMO
(%)
IDT
-
MINOS
(%)
1
0.62
0.69
0.57
2
3.37
2.73
2.95
3
2.65
1.90
2.34
4
0.50
0.21
0.56
In Fig. 5.21
4
to 5.26 we plot the scalar flux evaluated by the three solvers:
NEMO
,
IDT
and
MINOS
.
for cases 1, 2 and 3.
0
20
40
60
80
z (cm)
0
0,05
0,1
0,15
0,2
scalar flux
NEMO
IDT
MINOS
Figure 5.21: flux distribution (group-1) along the line
parallel to z-axis at x=67.5 cm and y=52.5
cm
0
20
40
60
80
z (cm)
0
0,2
0,4
0,6
0,8
scalar flux
NEMO
IDT
MINOS
Figure 5.22: flux distribution (group-2) along the line
parallel to z-axis at x=67.5 cm and y=52.5
cm
5.3.3
Results for case 2
The reference Monte-Carlo calculations for this case when Control rods are withdrawn and the
space is filled by coolant gives: k
ref
ef f
=1.0005(
±0.0002).
4
Figs. 5.21 to 5.26 are plotted for the solvers orders:
NEMO
(
P
2
),
IDT
(
S
8
),
MINOS
(
SP
3
).
28

Table 5.12: k
ef f
for Takeda 3 case 2
solver
order
k
ef f
error (pcm)
outer iterations
CPU (s)
S
8
0.998635
186
26
144.61
IDT
S
12
0.998667
183
28
186.68
S
16
0.998681
182
26
220.40
SP
1
0.990142
1035
211
3.08
MINOS
SP
3
0.995251
525
210
8.58
SP
5
0.995466
503
210
16.23
P
0
0.995499
500
500
346
P
1
0.995945
455
500
533
NEMO
P
2
0.998802
170
500
1429
P
4
-
-
-
-
Table 5.13: flux error per group (Takeda 3 case 2)
Groups
NEMO
-
MINOS
(%)
IDT
-
NEMO
(%)
IDT
-
MINOS
(%)
1
0.45
0.60
0.52
2
3.12
2.09
3.00
3
2.28
1.76
2.20
4
0.50
0.21
0.52
0
20
40
60
80
z (cm)
0
0,2
0,4
0,6
0,8
scalar flux
NEMO
IDT
MINOS
Figure 5.23: flux distribution (group-3) along the line
parallel to z-axis at x=82.5 cm and y=22.5
cm
0
20
40
60
80
z (cm)
0
0,01
0,02
0,03
0,04
scalar flux
NEMO
IDT
NEMO
Figure 5.24: flux distribution (group-4) along the line
parallel to z-axis at x=82.5 cm and y=22.5
cm
29

5.3.4
Results for case 3
The reference Monte-Carlo calculations for this case (when the control rods are withdrawn and
the space is replaced by blanket regions), gives: k
ref
ef f
=1.0214(
±0.0002). Errors between solvers for
each energy groups are tabulated in Table. 5.15.
Table 5.14: k
ef f
for Takeda 3 case 3
solver
order
k
ef f
error (pcm)
outer iterations
CPU (s)
S
8
1.02295
152
33
178.58
IDT
S
12
1.02296
153
33
220.27
S
16
1.02297
154
32
273.30
SP
1
1.01716
415
185
2.74
MINOS
SP
3
1.02079
60
187
7.66
SP
5
1.02095
44
187
14.54
P
0
1.00138
1960
500
348
P
1
1.00050
2046
500
983
NEMO
P
2
1.00066
2030
500
1264
P
4
-
-
-
-
Table 5.15: flux error per group (Takeda 3 case 3)
Groups
NEMO
-
MINOS
(%)
IDT
-
NEMO
(%)
IDT
-
MINOS
(%)
1
10.6
10.5
0.64
2
17.9
16.9
2.88
3
12.2
11.1
2.22
4
3.81
3.48
0.53
30

0
20
40
60
80
z (cm)
0
0,05
0,1
0,15
0,2
scalar flux
NEMO
IDT
MINOS
Figure 5.25: flux distribution (group-1) along the line
parallel to z-axis at x=82.5 cm and y=22.5
cm
0
20
40
60
80
z (cm)
0
0,2
0,4
0,6
0,8
1
scalar flux
NEMO
IDT
MINOS
Figure 5.26: flux distribution (group-2) along the line
parallel to z-axis at x=82.5 cm and y=22.5
cm
31

Chapter 6
Conclusion and Recommendations
The main objective of this report is to validate the following three flux solvers of A
POLLO
3 code:
IDT
,
MINOS
and a new solver called
NEMO
by comparing their results against reference Monte Carlo
in several international benchmark problems. In general each solver showed a good performance
and agreement with respect to reference Monte Carlo calculation but this is strongly depends on the
physics of the problem. For example
MINOS
shows a good results for fast reactor benchmark(Takeda
models 2 and 3) otherwise it was not optomized for small LWR (Takeda model 1).
In the voided regions of Takeda benchmark models we see that the solvers results are not as good as
when the control rods were presented. This is because the streaming effect in the voided region is
not very well taken into account into second order deterministic codes as some neutrons are able to
escape and then not taken into account.
A new A
POLLO
3 neutron flux solver
NEMO
is a new A
POLLO
3 flux solver. It based on a finite element method in space and on the spherical
harmonics in angle. In general as we have seen in the different benchmark models results, increasing
the order of angular approximation give a more satisfactory results but on account of the CPU time.
However when we use a high order angular approximation the calculations then required a big memory
and consumes a lot of time and this is one of the obstacle that
NEMO
can some times overcome it by
using a function called preconditioner to manipulate a given matrix in a way that reduce memory and
the CPU time. Two of the preconditioners called dsa(does not needs big memory or CPU time and the
other is called ilu0(needs a big memory and CPU time). Despite this limitation,
NEMO
results shows
a good agreement against the reference Monte Carlo.
One of the advantage of shperical harmonics method is the mitigation of ray-effect which present
usually in the discrete ordinates method
S
n
, this is because the
P
n
equations are invariant under rotation
of the coordinates and do not depend on the direction of the coordinates and this should give no ray
effect.
On the other hand the used
MINOS
solver in this report does not use vacuum boundary condition yet
and we recommend to use it in the future to take into account a nonzero flux near a boundaries.
32

In the following I summarize the results and the recommendations from this report.
· Systems containing void regions present a considerable challanges to deterministic transport
codes. This is because of the streaming in the voided regions (due to the skewed angular distri-
bution) which leads to highly anisotropic flux and then we need high-order transport approxi-
mation.
·
IDT
and
NEMO
are in good agreement specially between
S
8
and
P
2
so generally using these
orders of angular approximation is sufficient. However it is sufficient to use
S
8
for
IDT
solver
in the previous benchmarks problems (this depends strongly on the physics and the geometry
of the problem).
· For
NEMO
incresing the order of angular approximation makes the calculations much longer
(For example see
P
4
for Takeda 1). It is worth to note that
P
2
calculations are sufficient enough
to get a satisfactory results.
· In general the simplified spherical harmonics is better than the diffusion calculations (
P
0
) but
this cannot be guaranteed in all problems.
· For a heterogeneous geometry (even in 1D) we need a high orders
P
n
and
SP
n
to be used in
order to achieve convergence (ZPR problem).
· In Takeda 1 problem
NEMO
and
IDT
solvers are in good agreements almost at all positions but
near control rods we see differences in the flux distribution between these solvers.
· We see that in 1D problem such as ZPR-1D that
SP
n
method is equivelant to
P
n
method for all
values of n.
· We do not need fine mesh geometry refinment for fast reactors because the mean free path is
large compared to thermal reactors.
· the
SP
n
approximation do not converge to reference eigenvalue sometimes even with increased
order of angular approximation. This results from the incomplete set of angular trial functions,
so we note that
MINOS
is not adequate for a small LWR of Takeda 1.
· In Takeda 2 case 2 and Takeda 3 case 3,
MINOS
solver shows an excellent performance both for
the convergence of the problem (k
ef f
) and for CPU time consuming.
· To get a very quick result on a certain problem it seems that
MINOS
is the best choich regarding
the CPU time although the result may be a rough estimate sometimes.
· The deterministic codes are very important and powerful tools in neutronics and to illustrate this
if we take the Takeda 1 case 1, the calculation by
GMVB
code (with 10,000
×20,000 histories)
needed 4 hours in 3GHz CPU and the result was, k
ef f
=0.96238
±0.00006.
33

Appendix A
Example of Apollo3 input data file
########################################################################
#
IAEA BENCHMARK (Stepanek )
2D a 1 GROUPE
########################################################################
#-----------------------------------------------------------------------
Title(
Name: Stepanek
Title:
Stepanek
Category: 1
Summary: Stepanek
)
#-----------------------------------------------------------------------
CrossSection(
Name: MED1
NGroup: 1
Anisotropy: 0
Total: 0.6
Transfer: 0.53
NuFission: 0.079
Spectrum: 1.
)
CrossSection(
Name: MED2
NGroup: 1
Anisotropy: 0
Total: 0.48
Transfer: 0.2
)
CrossSection(
Name: MED3
NGroup: 1
Anisotropy: 0
Total: 0.7
Transfer: 0.66
NuFission: 0.043
Spectrum: 1.
)
CrossSection(
Name: MED4
NGroup: 1
Anisotropy: 0
Total: 0.65
Transfer: 0.5
)
CrossSection(
Name: MED5
NGroup: 1
Anisotropy: 0
Total: 0.9
Transfer: 0.89
)
34

#-----------------------------------------------------------------------
# Definition of a 2D Cartesian geometry
#
#
|---------|---------------|---------------|---------|
#
|
m5
|
m5
|
m5
|
m5
|
18.
#
|---------|---------------|---------------|---------|
#
|
m5
|
m4
|
m3
|
m5
|
25.
#
|---------|---------------|---------------|---------|
#
|
m5
|
m1
|
m2
|
m5
|
25.
#
|---------|---------------|---------------|---------|
#
|
m5
|
m5
|
m5
|
m5
|
18.
#
|---------|---------------|---------------|---------|
#
18.
30.
30.
18.
#
#-----------------------------------------------------------------------
GeometryGrid(
Name: idGG
Type: XY
DeltaX:
18.0
30.0
30.0
18.0
DeltaY:
18.0
25.0
25.0
18.0
Grid:
m5 m5 m5 m5
m5 m4 m3 m5
m5 m1 m2 m5
m5 m5 m5 m5
Legend: m1 MED1
Legend: m2 MED2
Legend: m3 MED3
Legend: m4 MED4
Legend: m5 MED5
)
#-----------------------------------------------------------------------
# Define the macro library
#-----------------------------------------------------------------------
MacroLibrary(
Name: idML
MacroSection: MED5 MED1 MED2 MED4 MED3
)
#-----------------------------------------------------------------------
GeometryGridRefined(
Name: idGGR
Geometry: idGG
GridRefinX: 18
30
30
18
GridRefinY: 18
25
25
18
)
#-----------------------------------------------------------------------
35

ZoneAssignment(
Name:
idZA
Geometry:
idGGR
)
#-----------------------------------------------------------------------
# Setting boundary condition
#-----------------------------------------------------------------------
BoundaryCondition(
Name: idBC
Vacuum: xmin
Vacuum: xmax
Vacuum: ymin
Vacuum: ymax
)
#-----------------------------------------------------------------------
# Define SN quadrature
#-----------------------------------------------------------------------
SnQuadrature(
Name:
idSQ
Order:
2
2
Type:
gaussLegendre
)
#-----------------------------------------------------------------------
# define the problem
#-----------------------------------------------------------------------
ProblemStationary(
Name:
idPS
ZoneAssign:
idZA
MacroLib:
idML
BoundaryCondition:
idBC
)
#-----------------------------------------------------------------------
# run IDT solver
#-----------------------------------------------------------------------
FluxSolver(
Name:
idFS_IDT
Problem:
idPS
SolverName:
IDT
OptionSolver:
idSO_IDT
Instrumenter:
idInst_IDT
)
36

#-----------------------------------------------------------------------
# run MINOS solver
#-----------------------------------------------------------------------
FluxSolver(
Name:
idFS_MINOS
Problem:
idPS
SolverName:
MINOS
OptionSolver:
idSO_MINOS
Instrumenter:
idInst_MINOS
)
#-----------------------------------------------------------------------
# run NEMO solver
#-----------------------------------------------------------------------
FluxSolver(
Name:
idFS_NEMO
Problem:
idPS
SolverName:
NEMO
OptionSolver:
idSO_NEMO
Instrumenter:
idInst_NEMO
)
#-----------------------------------------------------------------------
# IDT tester
#-----------------------------------------------------------------------
Instrumenter(
Name:
idInst_IDT
Description:
stepanek IDT
Reference:
1.00884
ResultType:
keff
Tolerance:
1.E-4
Comment:
linux-x86-32-centos-ifc
)
#-----------------------------------------------------------------------
# MINOS tester
#-----------------------------------------------------------------------
Instrumenter(
Name:
idInst_MINOS
Description:
stepanek MINOS
Reference:
1.00872
ResultType:
keff
Tolerance:
1.E-4
Comment:
linux-x86-32-centos-ifc
)
37

#-----------------------------------------------------------------------
# NEMO tester
#-----------------------------------------------------------------------
Instrumenter(
Name:
idInst_NEMO
Description:
stepanek NEMO
Reference:
1.00854
ResultType:
keff
Tolerance:
1.E-4
Comment:
linux-x86-32-centos-ifc
)
#-----------------------------------------------------------------------
Sequence(
Name: main
FluxSolver: idFS_IDT
FluxSolver: idFS_MINOS
FluxSolver: idFS_NEMO
)
#-----------------------------------------------------------------------
# Setting general options for calculation
#-----------------------------------------------------------------------
SolverOptionGeneral(
Name:
idSO_Gen
Outers:
1
200
1.E-5
1.E-5
Inners:
1
1000
1.E-4
)
#-----------------------------------------------------------------------
# Define Minos options for calculation
#-----------------------------------------------------------------------
SolverOptionMINOS(
Name:
idSO_MINOS
OptionGeneral:
idSO_Gen
FemType:
RT
FemxyOrder:
0
FemzOrder:
1
SPN:
3
FemIntegration:
gauss
)
#-----------------------------------------------------------------------
# Define NEMO options for calculation
#-----------------------------------------------------------------------
SolverOptionNEMO(
Name:
idSO_NEMO
OptionGeneral:
idSO_Gen
Approximation:
WN
PnOrder:
4
FemType: continous
)
#-----------------------------------------------------------------------
# Define IDT options for calculation
#-----------------------------------------------------------------------
38

SolverOptionIDT(
Name:
idSO_IDT
OptionGeneral:
idSO_Gen
SnQuadrature:
idSQ
SpatialApproximation:
characteristics linear
KeepAngularFlux:
false
)
#-----------------------------------------------------------------------
END END END END END END END END END END END END END END END END END END
#-----------------------------------------------------------------------
39

Bibliography
[1] Baudron A. M., Lautard J.J. (2007). MINOS: A Simplified P
n
Solver for Core Calculation. Nu-
clear Science and Engineering, 155:250-263.
[2] Bell, G.I. and Glasstone, S. (1970). Nuclear Reactor Theory, Van Nostrand.
[3] Bourhrara L. (2004). New Variational Formulations for the Neutron Transport Equation, Trans.
Theory Statist. Phys., 33:93-124.
[4] Bourhrara L. (2009). W
N
Approximations of Neutron Transport Equation, Trans. Theory Statist.
Phys., 38:195-227.
[5] Bourhrara L., Zmijarevic I. (2010). Guide utilisateur des commandes de données du code
Apollo3, DRAFT
[6] Duderstadt, J. J., Martin, W. R. (1979). Transport theory, John Wiley & Sons.
[7] Go Chiba, Kazuyuki (2007). Neutron transport benchmark problem proposal for fast critical
assembly without homogenizations, Annals of Nuclear Energy 34:443-448.
[8] H.Golfier et al. (2009). APOLLO3: a common project of CEA, AREVA and EDF for the de-
velopment of a new deterministic multi-purpose code for core physics analysis, International
conference on Mathematics, Computational methods & Reactor Physics (M&C 2009), Saratoga
Springs, NY, 2009.
[9] Lewis, E. E., Miller, W.F. Jr. (1993). Computational Methods of Neutron Transport, American
Nuclear Society.
[10] Paul Ruess (2008). Nuetron Physics, EDP SCIENCES.
[11] Rayan G. McClarren (2011). Theoretical aspects of the simplified P
n
Equations, Trans. Theory
Statist. Phys., 39:73-109.
[12] Stepanek T., Auerbach T., Halg W. (1982) Calculation of four thermal benchmarks in xy geom-
etry, EIR-Bericht Nr 464, Federal Institue of Technology (ETH).
40

[13] TAKEDA T., IKEDA H. (1991). 3-D Neutron Transport Benchmarks, OECD/NEA Committe
on Reactor Physics, NEACRP-L-330.
[14] Zmijarevic I. (1999). Multidimensional Discrete Ordinates Nodal and Characterstics Methods
for the Apollo2 Code, M&C'99 Madrid, Spain.
41
Excerpt out of 45 pages

Details

Title
Validation of a new numerical neutron flux solver in APOLLO3 code
Author
Year
2011
Pages
45
Catalog Number
V370187
ISBN (eBook)
9783668495692
ISBN (Book)
9783668495708
File size
1529 KB
Language
English
Keywords
Neutron, Transport, Equotation, Phsysics, Mathematics, Minos, Apollo3, IDT, NEMO
Quote paper
Hamza Ayyash (Author), 2011, Validation of a new numerical neutron flux solver in APOLLO3 code, Munich, GRIN Verlag, https://www.grin.com/document/370187

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