analytical versus zed solutions of four-group

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2009 International Nuclear Atlantic Conference - INAC 2009 Rio de Janeiro,RJ, Brazil, September27 to October 2, 2009 ASSOCIAÇÃO B RASILEIRA DE E NERGIA N UCLEAR - ABEN ISBN: 978-85-99141-03-8

ANALYTICAL VERSUS DISCRETIZED SOLUTIONS OF FOUR-GROUPDIFFUSION EQUATIONS TO THERMAL REACTORS

Fernando da Silva Melo1, 2

, Ronaldo Glicério Cabral1, Paulo Conti Filho

3

1Seção de Engenharia Nuclear (SE/7)

Instituto Militar de Engenharia (IME)

Praça General Tibúrcio 80 – Praia Vermelha22290-270 Rio de Janeiro

[email protected]

2Programa de Engenharia Nuclear (PEN)

Instituto Alberto Luiz Coimbra de Pos-Graduação e Pesquisa de Engenharia (COPPE)Universidade Federal do Rio de Janeiro (UFRJ)

Avenida Horácio Macedo, 2030, Centro de Tecnologia, Bloco G sala 206 Ilha do Fundão Rio de [email protected]

3Comissão Nacional de Energia Nuclear (CNEN)Rua General Severiano, 90 – Botafogo

22294-900 Rio de Janeiro

[email protected]

ABSTRACT

This paper presents the application of four-group Diffusion theory to thermal reactor criticality calculation. Thefour-group diffusion equations are applied to the spherical nucleus and reflector of an example reactor. Theneutrons fluxes depend upon the radial coordinate. The simultaneous linear ordinary differential equations are

solved given the solutions for the fluxes. The neutron fluxes for the nucleus are functions of the eight functions

linearly independent consisting of sin, cos, sinh, cosh, sin sinh, sin cosh, cos sinh , and cos cosh .

The analytical and discretized calculations of eff k value give excellent agreement, an error around 0,03%.

1. INTRODUCTION

A large number of neutronic studies have been performed on thermal reactors. Some of these

studies have concentrated on neutron multiplication factor, eff k , and flux distribution

calculations. The standard diffusion theory has been applied to neutronic calculations for the

thermal reactors with an excellent performance.

2. ANALYTICAL SOLUTION OF DIFFUSION THEORY

In the present work, it was considered a spherical reactor of radius R, involved by a reflector

of thickness T, surrounded by vacuum. Four-groups of energy were used, being three fast andone thermal. The energy groups of fast neutrons were represented by subscripts 1, 2, 3 and

the one of thermal neutrons by the subscript 4 [1, 2, 5].The four-group diffusion equations for the core fluxes can be seen as follows:

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INAC 2009, Rio de Janeiro, RJ, Brazil.

1

2 1

1 1 1 R

eff

D Sk

(1)

2 12

2 2

2 2 2 1 R s

eff

D Sk

(2)

3 13 23

2 3

3 3 3 1 2 R s s

eff D Sk

(3)

4 14 24 34

2

4 4 4 1 2 3 R s s s D (4)

The four-group diffusion equations for the reflector fluxes can be written as follow:

1

2

1 1 10

r r r R r D (5)

2 12

2

2 2 2 1r r

r r R r s D (6)

3 13 23

2

3 3 3 1 2r r r r r R r s r s r D (7)

4 14 24 34

24 4 4 1 2 3r r r r r R r s r s r r s r D (8)

Solving the simultaneous linear differential equations with constants for the core fluxes, on

one can write the eight routs of the characteristic equation given by: i , , a ib [2,

3, 4]. Thus, one can write the core flux solutions as,

1 2 3 4 5

6 7 8

cos cos cos

cos cos cos , (9), (10), (11), (12)

g g g g g g

g g g

sen r r senh r h r senh ar br r c c c c c

r r r r r

senh ar sen br h ar sen br h ar br c c cr r r

One can write the expressions forr g

as follows,

1 1

1

1 1 2 2

2

3 31 1 2 2

3

1 1

1 1 2 1

1

2 1 1 1 2 3 4 2

2

3 2 1 2 2 3 3 3 4 5 6 3

3

4 4 1 4 2

,

,

,

r

r

r

k r k r R

r

r

k r k r k r k r R

r

r

k r k r k r k r k r k r R

r

r

k r k r

r

e er d d k

r r D

e e e er Y d Y d d d k

r r r r D

e e e e e er Y d Y d Y d Y d d d k

r r r r r r D

e er Y d Y d

r r

3 32 2

4 4

4

5 3 5 4 6 5 6 6

7 8 4

4

, r

k r k r k r k r

k r k r R

r

e e e eY d Y d Y d Y d

r r r r

e ed d k

r r D

(13), (14), (15), (16)

Seventeen boundary conditions for the determination of the core and the reflector fluxes and

eff k were considered:




(i) At r = 0, g are finite;

(ii) At r = R, g = gr , and g gr

g g

d d D D

dr dr

;

(iii) At r = R+T=H, ;

(iv) ;

The expressions of the probabilities of neutron absorption in the core,gc

A , in the reflector,

gr A , and the probabilities of neutron leakage to the vacuum, vg A , can be given as follows [5].

2

04

g g

R

c a gr

A r dr

(17)

24g gr

H

r a gr r R

A r dr (18)

24

4 2

r r r

g

g g g

v

r H

D d A H

dr

(19)

Where4

11

g g gc r vg

A A A

3. RESULTS

3.1. Introduction

As a numeral application it was considered a thermal reactor of homogenized spherical

nucleus of radius R equal to 60 cm and involved by a reflector of thickness T equal to 120cm.For this example, the code XSDRNPM [6] was used in the generation of the four energy

group constants.

In the table 1 is presented the composition of thermal reactor.

Table 1. Composition of the thermal reactor

REGION MATERIAL

ATOMIC

DENSITIES

(Atoms / barn . cm)

CORE

Uranium – 235 0.12200 E-03

Uranium – 238 0.59700 E-02

Oxygen – 16 0.34420 E-01

Natural chrome 0.93460 E-03

Manganese – 55 0.94200 E-04

04 2

gr gr gr D d

dr

4

2 2

01

4 1 neutron/sg

R

a g g gr

g

D r dr




Natural irom 0.33470 E-02

Natual nickel 0.47110 E-03

Hydrogen – 1 0.44470 E-01

REFLECTOR Water 0.33430 E-01

In the table 2 are presented the group constants for the core.

Table 2. Four group constants of the thermal reactor from the XSDRNPM code.

In the table 3 are presented the group constants for the reflector.

Table 3. Four group constants of the thermal reactor from the XSDRNPM code.

3.2. Analytical versus discretized results.

The analytical results were obtained using a diffusion code, developed in this study,

ALBD4G and compared with the discretized solution [3] of the code CITATION.

Table 4 shows the results of eff k .

CORE

g = 1

g` = 2

g` = 3

g = 4

D 0.17607E+01 0.80339E+00 0.47001E+00 0.19923E+00

Σa 0.33928E-02 0.18935E-02 0.17635E-01 0.57172E-01

νΣf 0.72250E-02 0.51635E-03 0.59613E-02 0.66730E-01χ 0.74415E+00 0.25565E+00 0.20189E-02 0.12480E-08

Σs g g`

g → g` g` = 2

g` = 3

g` = 4

1 0.89651E-01 0.46418E-03 0.15529E-06

2 - 0.95330E-01 0.31330E-04

3 - - 0.98090E-01

REFLECTOR

g = 1 g` = 2 g` = 3 g = 4

D 0.18109E+01 0.78453E+00 0.50770E+00 0.14915E+00

Σa 0.31290E-03 0.95302E-05 0.57242E-03 0.15539E-01

Σs g g`

g → g` g` = 2

g` = 3

g` = 4

1 0.11270E+00 0.69381E-03 0.23278E-06

2 - 0.14163E+00 0.46992E-04

3 - - 0.14601E+00




Table 4. eff k results.

Table 5 shows the absorption and leakage probabilities for the thermal reactor.

Table 5. Absorption and Leakage Probabilities.

The analytical and discretized solutions presented a very good agreement.

4. CONCLUSIONS

The results of eff k and absorptions in the core and the reflector with the analytical method

presented very good agreement with the discretized results of the code citation, used as

comparative bases. It was obtained relative error of eff k smaller than 0.03% between the

analytical and discretized solutions of the four-group diffusion equations applied to a thermal

reactor. The analytical solution shows the core fluxes consisting of sin, cos, sinh, cosh, sin sinh,

sin cosh, cos sinh , and cos cosh what it is not possible to see using the discretized solution.

REFERENCES

1. BARBOSA, T. N. Cálculo neutrônico de reatores térmicos a quatro grupos de energiaaplicando o Método do Albedo e da Difusão („CITATION‟). Dissertação (Mestrado emEngenharia Nuclear) - Instituto Militar de Engenharia - IME, Brasil, 2008.

2. CABRAL, R. G. Multigroup albedo theory with application to neutronic calculation for agas core reactor . Dissertation (Doctor of Philosophy) - The University of Florida, USA,

1991.3. CONTI, F. P. Avaliação e Aprimoramento de Metodologia de Cálculo Neutrônico.

Dissertação (Mestrado em Engenharia Nuclear) - Instituto Militar de Engenharia - IME,

Brasil, 1984.

4. DUDERSTADT, J. J., HAMILTON, L. J. Nuclear reactor analysis. New York: John

CODE keff ERROR

CITATION 0,99088

0,03%ALBD4G 0,99055

CODE 1

c A

2c A 3

c A 4c A

ALBD4G 0,24872E-01 0,17959E-01 0,13699E+00 0,75942E+00

CITATION 0,25476E-01 0,17648E-01 0,13472E+00 0,75933E+00

1r

A 2r

A 3r

A 4r

A

ALBD4G 0,13873E-03 0,49002E-05 0,31542E-03 0,60302E-01

CITATION 0,11557E-03 0,43019E-05 0,28251E-03 0,62414E-01

1v A

2v A 3

v A 4v

A

ALBD4G 0,29755E-07 0,23847E-07 0,21540E-07 0,15984E-06

CITATION 0,26095E-07 0,21155E-07 0,19236E-07 0,14413E-06




Wiley & Sons Inc., 1976. 650p.

5. MELO, F. S. Análise de Criticalidade de reatores térmicos a quatro grupos de energiacom coeficientes variáveis de núcleo usando o Método do Albedo. Dissertação (Mestrado em

Engenharia Nuclear) - Instituto Militar de Engenharia - IME, Brasil, 2009.

6. PETRIE, L. M., GREENE, N. M., “XSDRNPM”: AMPX Module with One-Dimensional

Sn Capability for Spatial Weighting, “AMPX: A Modular Code System for Generating Coupled Multigroup Neutron-Gamma Libraries from ENDF/B, ORNL-TM-3706,Oak RidgeNational Laboratory, Oak Ridge, TN (March 1976).

analytical versus zed solutions of four-group

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