The aim of this work is to test and compare the performance of three APOLLO3 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.

The APOLLO3 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. APOLLO3 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 that is to calculate the flux by solving the neutron transport equation, each of these solver uses different numerical methods for an angular variable and for a spatial variable

Extracto

Contents

Introduction

Neutron Transport Equation

ZPR 1D Benchmark

3.1

Description of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Stepanek Benchmark

4.1

Description of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2

Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Takeda Benchmark

5.1

Takeda model 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1.1

Description of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . .

16subsection.5.1.2

5.1.3

Results for case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2

Takeda model 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2.1

Description of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2.2

Results for case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2.3

Results for case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3

Takeda model 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3.1

Description of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3.2

Results for case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3.3

Results for case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3.4

Results for case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Conclusion and Recommendations

A Example of Apollo3 input data file

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

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.

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

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.

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

(discrete ordinates method) which calculate the angular flux

(r, ) for a

set of angular directions

= 1, · · · , N

· The second method is called

(spherical harmonic method) which expand the angular flux in

spherical harmonics according to:

(r, ) =

n=0

m=n

m=-n

(r)y

(),

where the functions y

() are the real spherical harmonics and

(r) are the flux moments.

A variant of the

method is the

method (Simplified

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

POLLO

3 code:

· The

MINOS

solver ([1], [11]) uses the

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

conditions: zero flux and specular reflection. Currently, the vacuum boundary condition is not

available in the solver.

MINOS

solves the

equations for cartesian and hexagonal geometries.

For hexagonal geometry each hexagon is refined using four trapezoids.

· The

IDT

solver [14] is based on the

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

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,

flux distribution and error.

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.

TEPANEK

This is a mono-kinetic test case with cartesian geometry (2D).

AKEDA

These are 3D problems descriped in two energy groups for model 1 and in four energy

groups for model 2 and 3.

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.

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

is referred to as the inner iteration.[9]

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

(r) for each benchmark problem has been normalized as

following:

(r) =

(r),

(1.1)

where

is the maximum scalar flux and ¯

(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

ef f

) =

ref

ef f

- k

ef f

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

= |

(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.

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

the unit sphere the set of all angular directions. We use the variables

D and S

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

= {(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

(r, )

g=1,··· ,G

solutions of problem (2.1):

( · +

)

= H

in D

× S

(2.1a)

(2.1b)

where

(

)(r, ) =

(r)

(r, ),

(2.2)

)(r, ) =

g,g

(r, ·

)

(r,

) d

(2.3)

)(r) =

(r)

(r,

) d

(2.4)

In the above equations

(r) is the total macroscopic cross-section,

g,g

(r, ·

) is the macroscopic

differential scattering cross-section,

(r) is the number of emitted neutrons per fission multiplied

by macroscopic cross section (production macroscopic cross-section),

(r) is the fission spectrum

and

(r, ) is the incoming flux.

The vacuum boundary condition is represented as

= 0 in equation (2.1b). The specular reflection

boundary condition consisits in replacing the equation (2.1b) by:

(r, ) = (r,

)

(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 + .

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.

MPTY

ILLER

TEEL

ODIUM

U3O8

ODIUM

U3O8

B4C

ODIUM

U3O8

ODIUM

x (cm)

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

take

IDT

(

) results as a reference. Table 3.2 gives the absolute error of the multigroup flux between

the three solvers. The angular approximations used in each solver are:

NEMO

(

IDT

(

) and

MINOS

(

Table 3.1: keff results

solver

order

keff

err. (pcm)

outer iterations

CPU (s)

1.02081

0.05

IDT

1.02164

81.3

0.06

1.02214

130

0.06

MINOS

1.01777

297.8

0.01

1.02091

9.8

0.06

1.00302

1742.7

NEMO

1.01796

279.2

1.02084

2.9

Table 3.2: flux error per group

Groups

NEMO

MINOS

(%)

IDT

NEMO

(%)

IDT

MINOS

(%)

0.04

0.10

0.15

0.09

0.15

0.22

0.07

0.09

0.15

0.29

0.25

0.34

0.97

0.29

0.93

3.10

0.30

3.12

100

150

200

250

300

x (cm)

0,05

0,1

0,15

0,2

scalar flux

NEMO

IDT

MINOS

Figure 3.2: flux distribution in group 1

100

150

200

250

300

x (cm)

0,2

0,4

0,6

0,8

scalar flux

NEMO

IDT

MINOS

Figure 3.3: flux distribution in group 6

Fig. 3.2

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.

Fig. 3.2 to 3.4 are plotted for solvers orders:

NEMO

(

IDT

(

MINOS

(

100

150

200

250

300

x (cm)

0,01

0,02

0,03

0,04

flux error

IDT-NEMO

IDT-MINOS

NEMO-MINOS

100

150

200

250

300

x (cm)

0,001

0,002

0,003

0,004

flux error

IDT-NEMO

100

150

200

250

300

x (cm)

0,01

0,02

0,03

0,04

flux error

IDT-MINOS

100

150

200

250

300

x (cm)

0,01

0,02

0,03

0,04

flux error

NEMO-MINOS

Figure 3.4: flux error in group 6

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

x (cm)

y (cm)

Figure 4.1: geometry for Stepanek benchmark

Table 4.1: cross section for Stepanek benchmark

medium

MED1

0.6

0.53

0.079

MED2

0.48

0.2

MED3

0.7

0.66

0.043

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

(

) 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

(

IDT

(

)

and

MINOS

(

Table 4.2: keff results

solver

order

keff

err. (pcm)

outer iterations

CPU (s)

1.00884

8.18

IDT

1.00886

8.57

1.00886

11.34

1.00663

223

0.04

MINOS

1.00872

0.1

1.00879

0.19

1.00653

233

133

NEMO

1.00653

233

133

1.00851

33.9

131

1.00857

131

Table 4.3: flux error per group

Groups

NEMO

MINOS

(%)

IDT

NEMO

(%)

IDT

MINOS

(%)

0.93

0.45

0.47

Fig. 4.2

gives the scalar flux resulting from

NEMO

solver calculation with

approximation. Fig.

4.3, 4.4 and 4.5 illustrate the absolute error in scalar flux between the three solvers.

Fig. 4.2 to 4.9 are plotted for solvers orders:

NEMO

(

IDT

(

MINOS

(

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

Figure 4.2: scalar flux (

NEMO P

)

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

0.292345E-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

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

0.227872E-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

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

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.

y (cm)

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

100

x (cm)

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

y (cm)

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

100

x (cm)

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

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)

CORE

REFLECTOR

CRP

VOID

x (cm)

z (cm)

Figure 5.1: geometry for Takeda model 1

5.1.2

Results

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

ef f

error (pcm)

outer iterations

CPU (s)

0.9774

61.3

29.89

IDT

0.9775

51.5

35.02

0.9775

51.5

43.38

0.9163

6308.8

500

3.01

MINOS

0.9425

3629.9

500

8.51

0.9431

3568.5

500

16.15

0.9399

3895.7

0.9300

4908

NEMO

0.9740

409

186

0.9764

163.6

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

(

IDT

(

) and

MINOS

(

Table 5.2: flux error per group (Takeda 1 case 1)

Groups

NEMO

MINOS

(%)

IDT

NEMO

(%)

IDT

MINOS

(%)

23.5

4.58

26.3

8.71

2.81

10.9

Table 5.3 shows the error between solvers for

and

orders.

Table 5.3: flux error per group (Takeda 1 case 1) with

Groups

NEMO

MINOS

(%)

IDT

NEMO

(%)

IDT

MINOS

(%)

24.8

3.16

26.3

9.39

1.93

10.9

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.

In Fig. 5.2

to 5.7 we plot the scalar flux evaluated by the three solvers:

NEMO

IDT

and

MINOS

. for

case 1.

z (cm)

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

z (cm)

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

Fig. 5.2 to 5.11 are plotted for the solvers orders:

NEMO

(

IDT

(

MINOS

(

y (cm)

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

y (cm)

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

z (cm)

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

z (cm)

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

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

ef f

error (pcm)

outer iterations

CPU (s)

0.962663

27.3

27.23

IDT

0.962644

25.4

34.02

0.962625

23.4

42.54

0.921611

4231

406

2.14

MINOS

0.948322

1455

388

5.95

0.949128

1371.9

378

10.99

0.938457

2487.8

0.931811

3178.4

NEMO

0.962117

29.4

141

0.962800

41.6

826

Table. 5.5 shows the error between solvers for

and

orders.

Table 5.5: flux error per group (Takeda 1 case 2)

Groups

NEMO

MINOS

(%)

IDT

NEMO

(%)

IDT

MINOS

(%)

3.33

1.35

3.62

3.46

0.72

3.28

y (cm)

0,2

0,4

0,6

0,8

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

y (cm)

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

y (cm)

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

y (cm)

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

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)

CORE

RADIAL BLANK ET

AXIAL BLANK ET

RADIAL REFLECTOR

CRP

x (cm)

z (cm)

35 45 55

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,

ref

ef f

=0.9736(

±0.0002).

Table 5.6: k

ef f

for Takeda 2 case 1

solver

order

ef f

error (pcm)

outer iterations

CPU (s)

0.973522

33.08

14.19

IDT

0.97354

34.6

16.78

0.973548

35.7

21.35

0.963551

991.4

139

0.31

MINOS

0.967678

567.4

137

0.87

0.967747

560.3

138

1.67

0.971882

135.4

0.968019

532.3

NEMO

0.972783

42.8

0.972808

40.2

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

(%)

0.34

0.37

2.66

0.28

2.62

2.40

0.18

2.37

0.20

0.07

0.20

Figs. 5.13 to 5.16

shows the flux distribution along the z-axis for the case when the control rods

are withdrawn.

Figures 5.13 to 5.19 have been plotted for the solvers orders:

NEMO

(

IDT

(

MINOS

(

z (cm)

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

z (cm)

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

z (cm)

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

z (cm)

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

5.2.3

Results for case 2

The reference Monte Carlo calculation for this case when the control rods are half inserted gives,

ref

ef f

=0.9599(

±0.0002).

Table 5.8: k

ef f

for Takeda 2 case 2

solver

order

ef f

error (pcm)

outer iterations

CPU (s)

0.964116

491.5

38.17

IDT

0.964139

494

47.81

0.964148

495

60.74

0.954672

493

238

1.69

MINOS

0.959522

12.71

234

4.76

0.959629

23.8

234

9.11

0.960176

80.88

0.957268

222.2

NEMO

0.962544

327.7

0.962612

334.8

350

Table 5.9: flux error per group (Takeda 2 case 2)

Groups

NEMO

MINOS

(%)

IDT

NEMO

(%)

IDT

MINOS

(%)

0.35

0.52

0.38

2.70

2.13

2.68

2.42

1.43

2.42

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

y (cm)

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

y (cm)

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

y (cm)

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

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,

calculation is not carried out due to limitaion on the computer memory.

Morever the

NEMO

calculations are not converged toward the reference eigenvalue

CRP

INTERNAL BLANKET

CORE

RADIAL BLANKET

RADIAL REFLECTOR

EMPTY MATRIX

x (cm)

y (cm)

0 5

35 45

120

145 160

120

145

160

CRP

INTERNAL BLANKET

CORE

RADIAL BLANKET

AXIAL BLANKET

RADIAL REFLECTOR

AXIAL REFLECTOR

EMPTY MATRIX

x (cm)

z (cm)

0 5

35 45

120

145 160

INTERNAL BLANKET

CORE

RADIAL BLANKET

RADIAL REFLECTOR

EMPTY MATRIX

x (cm)

y (cm)

120

145 160

120

145

160

INTERNAL BLANKET

CORE

RADIAL BLANKET

AXIAL BLANKET

RADIAL REFLECTOR

AXIAL REFLECTOR

EMPTY MATRIX

x (cm)

z (cm)

120

145 160

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:

ref

ef f

=0.9709(

±0.0002).

Table 5.10: k

ef f

for Takeda 3 case 1

solver

order

ef f

error (pcm)

outer iterations

CPU (s)

0.97040

177.57

IDT

0.970441

217.18

0.970458

272.31

0.960639

1057

256

3.76

MINOS

0.966736

429

258

10.58

0.966989

403

254

19.83

0.978948

829

500

362

0.983032

1250

500

1005

NEMO

0.990128

1980

500

1434

The maximum error between each two solvers is shown in 5.11 for each energy group:

Table 5.11: flux error per group (Takeda 3 case 1)

Groups

NEMO

MINOS

(%)

IDT

NEMO

(%)

IDT

MINOS

(%)

0.62

0.69

0.57

3.37

2.73

2.95

2.65

1.90

2.34

0.50

0.21

0.56

In Fig. 5.21

to 5.26 we plot the scalar flux evaluated by the three solvers:

NEMO

IDT

and

MINOS

for cases 1, 2 and 3.

z (cm)

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

z (cm)

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

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).

Figs. 5.21 to 5.26 are plotted for the solvers orders:

NEMO

(

IDT

(

MINOS

(

Table 5.12: k

ef f

for Takeda 3 case 2

solver

order

ef f

error (pcm)

outer iterations

CPU (s)

0.998635

186

144.61

IDT

0.998667

183

186.68

0.998681

182

220.40

0.990142

1035

211

3.08

MINOS

0.995251

525

210

8.58

0.995466

503

210

16.23

0.995499

500

346

0.995945

455

500

533

NEMO

0.998802

170

500

1429

Table 5.13: flux error per group (Takeda 3 case 2)

Groups

NEMO

MINOS

(%)

IDT

NEMO

(%)

IDT

MINOS

(%)

0.45

0.60

0.52

3.12

2.09

3.00

2.28

1.76

2.20

0.50

0.21

0.52

z (cm)

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

z (cm)

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

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

ef f

error (pcm)

outer iterations

CPU (s)

1.02295

152

178.58

IDT

1.02296

153

220.27

1.02297

154

273.30

1.01716

415

185

2.74

MINOS

1.02079

187

7.66

1.02095

187

14.54

1.00138

1960

500

348

1.00050

2046

500

983

NEMO

1.00066

2030

500

1264

Table 5.15: flux error per group (Takeda 3 case 3)

Groups

NEMO

MINOS

(%)

IDT

NEMO

(%)

IDT

MINOS

(%)

10.6

10.5

0.64

17.9

16.9

2.88

12.2

11.1

2.22

3.81

3.48

0.53

z (cm)

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

z (cm)

0,2

0,4

0,6

0,8

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

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

, this is because the

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.

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

and

so generally using these

orders of angular approximation is sufficient. However it is sufficient to use

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

for Takeda 1). It is worth to note that

calculations are sufficient enough

to get a satisfactory results.

· In general the simplified spherical harmonics is better than the diffusion calculations (

) but

this cannot be guaranteed in all problems.

· For a heterogeneous geometry (even in 1D) we need a high orders

and

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

method is equivelant to

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

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.

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

)

#-----------------------------------------------------------------------

# Definition of a 2D Cartesian geometry

|---------|---------------|---------------|---------|

18.

|---------|---------------|---------------|---------|

25.

|---------|---------------|---------------|---------|

25.

|---------|---------------|---------------|---------|

18.

|---------|---------------|---------------|---------|

18.

30.

18.

#-----------------------------------------------------------------------

GeometryGrid(

Name: idGG

Type: XY

DeltaX:

18.0

30.0

18.0

DeltaY:

18.0

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

GridRefinY: 18

)

#-----------------------------------------------------------------------

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:

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

)

#-----------------------------------------------------------------------

# 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

)

#-----------------------------------------------------------------------

# 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:

200

1.E-5

Inners:

1000

1.E-4

)

#-----------------------------------------------------------------------

# Define Minos options for calculation

#-----------------------------------------------------------------------

SolverOptionMINOS(

Name:

idSO_MINOS

OptionGeneral:

idSO_Gen

FemType:

FemxyOrder:

FemzOrder:

SPN:

FemIntegration:

gauss

)

#-----------------------------------------------------------------------

# Define NEMO options for calculation

#-----------------------------------------------------------------------

SolverOptionNEMO(

Name:

idSO_NEMO

OptionGeneral:

idSO_Gen

Approximation:

PnOrder:

FemType: continous

)

#-----------------------------------------------------------------------

# Define IDT options for calculation

#-----------------------------------------------------------------------

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

#-----------------------------------------------------------------------

Bibliography

[1] Baudron A. M., Lautard J.J. (2007). MINOS: A Simplified P

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

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.

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Título: Validation of a new numerical neutron flux solver in APOLLO3 code
Autor: Hamza Ayyash (Autor)
Año de publicación: 2011
Páginas: 45
No. de catálogo: V370187
ISBN (Ebook): 9783668495692
ISBN (Libro): 9783668495708
Idioma: Inglés
Etiqueta: Neutron Transport Equotation Phsysics Mathematics Minos Apollo3 IDT NEMO

Citar trabajo: Hamza Ayyash (Autor), 2011, Validation of a new numerical neutron flux solver in APOLLO3 code, Múnich, GRIN Verlag, https://www.grin.com/document/370187

Validation of a new numerical neutron flux solver in APOLLO3 code

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