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INSTITUTO DE ENGENHARIA NUCLEAR

RT-IEN- 02/2011

NOTA ESTE RELATÓRIO É PARA USO EXCLUSIVO DO INSTITUTO DE

ENGENHARIA NUCLEAR

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Adaptive remeshing in 2D Neutron Diffusion using external

Programs: GenMesh and Triangle.

by

Debora Rufino Senra

Reinaldo Jacques Jospin

Luis Osório de Brito Aghina

Maio/2011

INSTITUTO DE ENGENHARIA NUCLEAR

Título: Remalhagem Adaptativa em Difusão Neutrônica utilizando Programas Externos: GenMesh e Triangle Autor(es): Debora Rufino Senra Reinaldo Jacques Jospin Luis Osório de Brito Aghina

e-mail: [email protected] [email protected] [email protected]

Identificação:

No de páginas: 92

Tipo de Divulgação: Irrestrita (x) Restrita ( )

Divulgar para: Localização:

Publicação externa associada (congresso/periódico): Palavras chave: Cinética Espacial, Difusão Neutrônica, Remalhagem Adaptativa, Elementos Finitos Resumo: O objetivo deste trabalho é apresentar não só uma ferramenta que propicie uma melhor precisão na solução no caso da análise the difusão neutronica mas também apresentar alguns resultados numa bem simplificada análise de cinética espacial. Para este primeiro objetivo, um indicador de erro baseado na continuidade da corrente de neutrons nas interfaces inter-elçementos é apresentada e alguns resultados baseados na remalhagem adaptativa são apresentados no caso 2D utilizando programas externos tais como GenMesh (com limitações no numero de materiais) e Triangle. Para alcançar o segundo objetivo foi necessário inserir a cinética pontual no pograma MEF. Alguns exemplos de cinética pontual são apresentados no intuit de certificar que os resultados apresentados pelo programa MEF estão em acordo com aqueles apresentados por outros autores. Finalmente, uma análise bem simplificada de cinética espacial é realizada em um reator cujas seções de choque, correspondentes a duas regiões diferentes do núcleo, variam de um forma senoidal com fases diferentes.

Abstract: The objective of this work is not only to present a tool to give more precision in the case of the neutron diffusion analysis but also to give some results in a very simplified spatial kinetic analysis. For the first objective, an error indicator based in the neutron current is presented and some results of adaptive remeshing are presented in the 2D case using external programs like GenMesh (with some limitations in the number of materials) and Triangle. To reach the second objective it was necessary to add the point kinetic analysis in the MEF program. Some examples in point kinetics are presented to certify that the results presented by MEF are in accordance with those presented by other authors. Finally a spatial kinetic analysis was done in a rector where the cross section varies in a non phase sinusoidal form applied in two different core regions.

Emissão Data:

Elaboração:

Nome Rubrica Data Débora Rufino Senra Reinaldo Jacques Jospin Luis Osório de Brito Aghina

Divisão:DIRE

Revisão:

Paulo Augusto Berquó de Sampaio

Serviço:SETER

Aprovação :

Maria de Lourdes Moreira

Instituto de Engenharia Nuclear: Via 5 s/n, Cidade Universitária, Ilha do Fundão, CEP 21945-970, CP 68.550, Rio de Janeiro – RJ - Brasil . Tel.: 00 55 21 2209-8080 Internet: www.ien.gov.br

RT-IEN-02/2011

Relatório de Atividades ITI-1A

Remalhagem Adaptativa em Difusão Neutrônica utilizando

Programas Externos: GenMesh and Triangle.

Débora Rufino Senra

Reinaldo Jacques Jospin

Luis Osório de Brito Aghina

2011

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Index

Introdução 6 Spatial Kinetic Equation 6 Integral Formulation 8 Approximate Solution 10 Static Neutron Diffusion Equation 14

Point Kinetic Equations 15 Reactor Power Calculation 18 Error indicator 19 Continuity on the neutron current: 19 Numerical Results 22 Spatial Neutron Diffusion Examples (1D and 2D) 22

AIEA Numerical Benchmark using GenMesh 23 AIEA Numerical Benchmark using Triangle 24 Argonauta Reactor (2D) using original plate fuel. 26 New Argonauta Reactor (2D) using Angra I fuel rods 27 Analytical results: 29

Point Kinetic Numerical Examples: Step Reactivity in Argonauta Reactor (Adiabatic Reactor) Subcritical reactor with a step Reactivity of � = �. �� % ⁄ � Critical reactor with a step reactivity of � = �. �� % ⁄ � Supercritical reactor with a step reactivity of � = �. �� % ⁄ � Ramp Reactivity of 0.1 [$/s]: Ramp reactivity insertion in Argonauta reactor using a non null heat transfer

coefficient (subcritical reactor). 39 A non null neutron source without reactivity insertion in Argonauta reactor. A hypothetic accident of coolant lost in Argonauta reactor. Point kinetic with a constants neutron source and reactivity Accelerated Driven System reactors (ADS) using pulsing neutron source.

Quasi-Static Spatial Kinetic Conclusions: Acknowledgments: References: Annex A: PSLG data file to calculate, in Mef program, the neutron diffusion (criticality factor and neutron flux distribution) for the IAEA numerical benchmark reactor example. Annex B: PSLG data file to calculate, in Mef program, the neutron diffusion (criticality factor and neutron flux distribution) for the Argonauta reactor. Annex C: Cross sections obtained from Hammer program to the new Argonauta core using Angra I rod fuel. Annex D: Cross sections obtained from Hammer program to the new Argonauta core using Angra I rod fuel. Annex E: The GID graphical interface to calculate the point kinetic in ramp reactivity insertion in Argonauta Reactor using MEF program (subcritical reactor?). Annex F: Criticality equation solution using 2 neutron groups energy for a finite cylindrical reactor without neutron reflectors Annex G: Extrapolated distance calculation in Argonauta core diffusion theory

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Introdução

The objective of this work is to present a numerical solution of the spatial kinetic equation using the finite element method. It is intended to use in this objective the quasi static approach (QS) or the improved quasi static approach (IQS) largely employed to solve this equation. First of all, the initial step is toward a steady neutron diffusion solution. For this, an exercise of calculating, in analytical way, the criticality factor of a slab reactor was realized. Using a finite element formulation, this factor was also calculated using the numerical method and verified using the BSS-6-A numerical benchmark analysis. In the sequel, the reactor criticality coefficient of the Argonauta reactor is obtained in a simplified analytical way. The same reactor is then calculated using the MEF-DIFU program developed at IEN. Two configuration of the reactor are used: one using the original plate fuel element and the other one using the fuel tube element. To improve the numerical solution quality, the adaptive remeshing is used to calculated the criticality factor and the neutron flux distribution of 2D reactor in two case: the first one is the Argonauta reactor and the second one the AIEA Numerical Benchmark. Finally, a first step to calculate the reactor kinetic is given using the point kinetic analysis in the benchmark numerical cases presented by Kinard, and all. Some results in a spatial reactor kinetic are given using a very simple trick: a time sequence of static neutron diffusion calculation followed by a point kinetic solution. The first one gives us the possibility to calculate the reactor reactivity and the second one gives us a temporal evaluation of the neutron population of the reactor.

Spatial Kinetic Equation

Here the time and spatial evolution of the neutron flux of the reactor are analysed.

Differential Equation (Strong Formulation)

The basic space-time kinetics equation of a nuclear reactor are:

1��

�� , �� = ∇ ∙ ��∇ϕ� − Σ�ϕ� + Σ!,�"→�ϕ�"$

�%&'�"(�+ �1 − β*++�χ-� νΣ+�"ϕ�"

$

�%&'

+ λ0χ0�C02

0&'+ Q� for g = 1, … , G

and

�:;��, �� = <; =Σ+�

>

�&',…,>�� − ?;:; @A� B = 1, … , C

Where:

��, ��: neutron flux for neutron energy group D, position � and time �.

:;��, ��: concentration of delayed neutron precursors in group B, position � and time �.

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=�: mean velocity of neutrons in group D

��: diffusion coefficient in group D and at position � and time �.

Σ��r, t�: removal cross section [1/cm] in group D at position � and time �.

Σ!,�"→��r�: scattering cross section [1/cm] at position �, from group D′ to group D.

β*++: total effective fraction of delayed neutrons. χ-�: spectrum of prompt neutrons in group D.

H;�: spectrum of delayed neutrons in group D.

ν: mean number of fission neutrons. Σ+�": fission cross section [1/cm] in group D′ at position r.

λ0: decay constant [1/s] of group B precursors. <;: fraction of delayed neutrons in group B. Q�: external source of neutrons in group D.

This multi-group spatial kinetic equation system can be written in reaction-diffusion form, as

the parabolic system [12]:

�I�� − ∇ ∙ J∇I + KI = @

where I�L, �� is the column vector of neutron flux and precursors density,

I�L, �� = ��', … �> , :', … , :M��>NM�×'

J�L, �� is a diagonal matrix of diffusion cross sections,

J�L, �� = PBQD�='�', … =>�> , 0, … ,0��>NM�×�>NM�

K�L, �� = ST ΓV ΛX�>NM�×�>NM�

T is the matrix of absorption, scattering and fission cross sections,

T = Y−=�" Σ�" Z��" + [1 − Z��"\=� Σ!,�"→� + �1 − <�=�χ-� νΣ+�"]�>×>�

Γ and V are the matrices of neutron precursors,

Γ = Y=�H;�?;]�>×M�

V = Y<;�=Σ+��]�M×>�

Λ is the diagonal matrix of decay constant,

Λ = −PBQD�?' ?M��M×M�

and @is the column vector of group sources:

@ = �='^', … , =>^> , 0, … ,0��>NM�×'

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Integral Formulation (Weak Formulation)

Multiplying the differential equation by a weight function _, the integral formulation can be

written in the following form:

` �I�� _PΩ

Ω− ` �∇ ∙ J∇I + KI�_

ΩPΩ = ` @_

ΩPΩ

Using the derivative by parts:

∇ ∙ �J�∇I�_� = �∇ ∙ J∇I�_ + J∇I ∙ ∇_

and the Green theorem result:

` �I�� _PΩ

Ω− ` �p∇u ∙ ∇_ + KI_�

ΩPΩ = ` J�∇I�_. dee

PΓ + ` @_Ω

PΩ

Factorization Method:

Supposing that the neutron flux shape function varies slowly than the amplitude function, the

neutron flux variable can be separated into a product of a shape function ��L, �� updated at

each macro time step ∆� and an amplitude function d�� uptaded at each micro time step Z�

with ∆� > Z�:

u�L, �� = d��L, ��

Results:

` h�d�� L, �� + d�� L, ��

�� i _�L, ��PΩΩ

− ` �pn�t�∇��L, �� ∙ ∇_�L, �� + Kd��L, ��_�L, ��Ω

PΩ= ` J�∇��_�L, ��. dee

PΓ + ` @_�L, ��Ω

PΩ

Improved Quasi-Static Method:

Using an implicit method for the time integration:

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�� = ��L, �� − ��L, � − ∆��

∆�

Results:

` h�d�� L, �� + d�� L, �� − ��L, � − ∆��

∆� i _�L, ��PΩΩ

− ` �pn�t�∇��L, �� ∙ ∇_�L, �� + Kd��L, ��_�L, ��Ω

PΩ= ` J[∇��L, ��\_�L, ��. dee

PΓ + ` @_�L, ��Ω

PΩ

Dividing the equation by the medium neutron population d��:

` h 1d��

�d�� + 1

∆� + Ki ��L, ��_�L, ��PΩΩ

− ` �p∇��L, �� ∙ ∇_�L, ��Ω

PΩ= ` 1

∆� ��L, � − ∆��_�L, ��PΩΩ

+ ` J�∇��L, ��_�L, ��. deePΓ

+ ` @_�L, ��Ω

PΩ

Classical Quasi Static Method:

In the classical quasi-static approach, the term kl�m,n�

kn is neglected justified by the argument

that it varies slowly.

` h 1d��

�d�� + Ki ��L, ��_�L, ��PΩ

Ω− ` �p∇v�L, �� ∙ ∇_�L, ��

ΩPΩ

= ` J�∇��L, ��_�L, ��. deePΓ + ` @_�L, ��

ΩPΩ

Adiabatic Method

In the adiabatic method, the quantity '

p�n�kp�n�

kn = kqrp�p�n��skn is also neglected producing the

following shape equation:

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` K��L, ��_�L, ��PΩΩ

− ` �p∇v�L, �� ∙ ∇_�L, ��Ω

PΩ= ` J�∇��L, ��_�L, ��. dee

PΓ + ` @_�L, ��Ω

PΩ

where: ��L, �� = ��', … �> , :', … , :M��>NM�×'

J�L, �� = �='�', … =>�> , 0, … ,0��>NM�×�>NM�

K�L, �� = ST ΓV ΛX�>NM�×�>NM�

��L, �� = ��', … �>��>×'� J�L, �� = �='�', … =>�>��>×>�

K�L, �� = �T Γ��>×>�

It can be stated, in this method, that the delayed neutron source is in equilibrium with the

shape and the new problem is once again an eigenvalue problem.

Approximate Solution

Here we are interested in developing a numerical procedure to solve the Neutron Diffusion

Equation. From the integral formulation of the differential equation it is possible to use for

example, a Ritz, Galerkin, Collocation methods to put this formulation in a form that can be

easily discretized by the Finite Element Method. Then, the Gauss numerical integration can be

used to evaluate all the integral value required in the problem.

Neutron Energy Discretization

Before to proceed to the geometric and spatial field discretization, urge to think about neutron

energy groups. Since the first objective in this work is to analyze PWR thermal reactors, the

neutron flux is discretized only in few groups: one fast, one thermal and two epithermals. For

reactors like the Fast Breeder Reactor (FBR) or Advanced Driven System (ADS) is necessary to

use much more energy groups but they can be condensated in few groups.

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Geometric and Field Discretization

For the geometric and spatial discretization, the use of the Finite Element Method is a suitable tool to be used in the integral formulation. Then, the geometry and the neutron flux field can be discretize as the follow:

Ω = Ω*pt

t&'

Lt�L� = u;�v�L;pp

;&'

_t�L, �� = u;�v�_;��pp

;&'

where the sub-domains Ω* are the new support for the variables _t�L, �� and are the non-overlapping domains that compose the full domain Ω. The functions u;�v� and _;�� are the finite element interpolation function defined in the sub-domains and the neutron flux nodal value respectively for the node B. The number of nodes in each element is dd. Using a linear triangular finite element (nn=3), the following matrices can be stated: �p = q�'' �w' �x' �'w �ww �xw … �x> :'' :w' :x' … �Mxsx×�>NM�

( )GG

GGG

G

NNN

NNN

NNN

N

××

=

3321

23

22

21

13

12

11

..............................

...

...

( )II

III

I

NNN

NNN

NNN

N

××

=

3321

23

22

21

13

12

11

..............................

...

...

( ) ( )IGIG

I

G

N

NN

+×+

=

30

0

� = u�p

Substituting this approximation in the integral formulation results:

y` �uzKu�PΩ*Ω{_p − ` [B}pB\

Ω{PΩ*_p

pt

t&'= ` J�∇��L, ��_�L, ��. dee

PΓ + ` uz@uΩ{

PΩ*~

If the rector does not have neutron sources, the stationary condition is given by:

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�� 00 0� ��:� + ST� Γ�V� Λ�X ��:� = �00�

Where:

�� = ` u> ∙ Ju>Ω{

PΩ*

T� = ` u> ∙ Hu>Ω{

PΩ*

Γ� = ` u> ∙ ΓuMΩ{

PΩ*

V� = ` uM ∙ Bu>Ω{

PΩ*

Λ� = ` uM ∙ ΛuMΩ{

PΩ*

Using the Gauss numerical integration, the above integrals can be written in the following

form:

T� = ` u> ∙ Hu>Ω{

P��tPvP�P� = u> ∙ Hu>P��;t_;tp;p�;&'

Γ� = ` u> ∙ ΓuMΩ{

P��tPvP�P� = u> ∙ ΓuMP��;t_;tp;p�;&'

V� = ` uM ∙ Bu>Ω{

PvP�P� = uM ∙ Bu>P��;t_;tp;p�;&'

Λ� = ` uM ∙ ΛuMΩ{

PvP�P� = uM ∙ ΛuMP��;t_;tp;p�;&'

where v, � QdP � are the reference coordinates defined in the elementary finite element,

P��;t is the Jacobian transformation from the reference coordinates to the global coordinates:

L = qL, �, �s and _;t is the weight of the numerical integration point B and element �.

Then, the above equation system can be written as the following:

V�� = Λ�C ↦ : = �Λ� ��'V�� + T�� = �� = Γ�:

Here the concentration of delayed neutrons is given by the first equation and the flux associate

is given by the second equation. These equations did not provide any information about the

evolution of the criticality factor and consequently about the reactor reactivity. In this case, it

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is necessary to calculate the adjoint flux to obtain the reactor reactivity. Although the use of

the method given by Scal & others[13] provide this factor, it did not correct the spatial flux

with the delayed neutrons.

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Static Neutron Diffusion Equation

In this topic, the differential equation that governs the neutron balance in a nuclear reactor is

presented. Basically this equation represents the balance among the absorption, fission and

scattering neutrons. If the flux neutron is slow down that means the absorption is bigger than

the production and the reactor have a criticality coefficient less than one. If the production is

bigger than the absorption the criticality factor of the reactor will be greater than one and the

rector will have an incursion of power.

∇ ∙ J∇I + KI = @

Using the derivative by parts defined before and multiplying the differential equation by a test

function _ and using the Green theorem result:

− ` �p∇u ∙ ∇_ + KI_�Ω

PΩ = ` J�∇I�_. deePΓ + ` @_

ΩPΩ

where I�L, �� is the column vector of neutron flux:

I�L, �� = ��', … �>��>�×'

J�L, �� is a diagonal matrix of diffusion cross sections:

J�L, �� = PBQD�='�', … =>�>��>�×�>�

K�L, �� = �T��>�×�>�

T is the matrix of absorption, scattering and fission cross sections:

T = Y−=�" Σ�" Z��" + [1 − Z��"\=� Σ!,�"→� + �1 − <�=�χ-� νΣ+�"]�>×>�

and @is the column vector of group sources:

@ = �='^', … , =>^>��>�×'

The weight function _ could be chosen as the same function used to the neutron flux I

(Galerkin method) producing:

− ` �p∇u ∙ ∇u + KII�Ω

PΩ = ` J�∇I�I. deePΓ + ` @I

ΩPΩ

Using a two neutron energy groups, the equation can be written in a generalized eigenvalue

problem defined by the following equation:

` Y∇�'�'∇�' + �'[Σ�' + Σ!w→'\�']dΩ = λΩ

` Y�'H'[νΣ+'�' + νΣ+w�w\ + �'��']Ω

PΩ

` Y∇�w�w∇�w + �wΣ�w�w]dΩ =Ω

` ��wHwYνΣ+'�' + [νΣ+w + Σ!w→'\�w] + �w��w�Ω

PΩ

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Using the finite element approximations for the I field results:

�T' 00 Tw� ��'�w� = S? 00 C X � �' �w�w→' 0 � ��'�w� + ��'��w�

Supposing now that the neutron sources are null (f=0), a generalized eigenvalue problem can

be defined by the following equation:

` Y∇�'�'∇�' + �'[Σ�' + Σ!w→'\�']dΩ = λΩ

` �'H'[νΣ+'�' + νΣ+w�w\Ω

PΩ

` Y∇�w�w∇�w + �wΣ�w�w]dΩ =Ω

` �wHwYνΣ+'�' + [νΣ+w + Σ!w→'\�w]Ω

PΩ

that can be put in a matrix form:

�T' 00 Tw� ��'�w� = S? 00 CX � �' �w�w→' 0 � ��'�w�

where the matrices T', Tw, �', �w and �w→' are defined in the work of Scal et all.[13].

Point Kinetic Equations

The point kinetic equation supposes that the reactor can be treated geometrically as a point.

This means that the flux geometric distribution is not take into account (eigenvector). The

interest is only the evolution of the neutron population in this reactor taking into account the

prompt and the delayed neutrons (eigenvalue). This evolution can be stated as []:

( ) ∑ ++−= Scn

dt

dniiλβρ

l

with ii

ii cndt

dcλ

β−=

l

and where:

• n is the number of neutrons;

• ρ is the reactivity;

• β is the proportion of delayed neutrons;

• l is the neutron lifetime in a critical reactor;

• λ is the radioactive decay constant of the precursor;

• c is the precursors concentration of delayed neutrons;

• the index i is the number of the neutron group.

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Spatial Kinetic Equations

The spatial kinetic equations can be used to analyse the reactor behavior to a suddenly non

expected modification of the geometric or material reactor configuration. For example, these

new configuration can be stated by the quickly lift of the control bar or by the cross section

modification due to the temperature variations and so on. As the interest here is not to follow

in details the temporal spatial neutron distribution, a very simplified kinetic analysis can be

done [13]. The spatial distribution flux in each step, when a perturbation is imposed, suffer a

very quickly modification in its form and stabilizes in a new spatial distribution. As explained

before, the main interest is to obtain only the new steady state spatial neutron distribution

(step stabilization). This can be done by a very simple static neutron diffusion analysis in each

step that gives the new spatial neutron distribution and the criticality factor. Then, the reactor

reactivity can be calculated, in each step, from the criticality factor variation. The new neutron

population in the reactor can finally be obtained from the point kinetics. The scheme of this

calculation is presented in the scheme 1 below.

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Simplified quasi-static spatial Kinetics method

Scheme 1: Simplified Kinetic Analysis in Nuclear Reactors

Time: t=0 Step: n=0

Φ$�N' = Φ$�N'�Σ��N'� �t��pN' = �t��pN'�Σ��N'�

neutron flux calculation and criticality factor:

�pN' = �p + Δ� Time:

Λ�N' = 1�>�Σ�

Prompt neutron lifetime:

η�N' = η�N'[Λ�N', ρ�N'\

Global neutron population:

ρ�N' = �t��pN' − �t��pN'�t��pN'

Reactivity calculation:

ΣpN' = Σp + ΔΣ Cross section evolution:

Static neutron diffusion

equation:

Point kinetic equation:

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Improved Quasi-static spatial Kinetics method (IQS)

Reactor Power Calculation

(Como calcular a potência numa célula do reator à partir do fluxo de neutrons)

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Error indicator

Continuity on the neutron current:

First of all, we will use the neutron current definition given by Reuss[6]:

��, �� = −��∇Φ��, ��

where � is the diffusion coefficient defined in the neutron diffusion equation and Φ is the

neutron flux inside the nuclear reactor. The real neutron current � inside a nuclear reactor is

known to be continuous in all over the reactor domain including the interface frontiers on the

sub-domains Ω*(finite element sides). To ensure this continuity in the approximated solution,

we try to vanish, on the interface at the sub-domains, the discontinuities that appears when

using for example, an :¡ approximation of the neutron flux Φ based in the linear triangular

finite element. In this kind of approximation, the neutron current � is constant by sub-domains

with discontinuities on the interfaces of these sub-domains.

Defining the real and approximated scalar flux distribution by Φ and Φ¢ respectively and

remembering that the last one can be obtained from a solution of the integral neutron

diffusion equation using, for example, a linear finite element approximation:

�1� Φ* = ∑ u;Φ;tpp;&' then, the neutron current can be calculated from this solution by:

�1� �¤ = −� ∑ ∇u;Φ;tpp;&' where dd is the number of nodes of the finite element, u; is the interpolation functions

defined on the node B of the sub-domain Ω* and Φ;t is flux defined at node B on the element �.

It is clear that the current �¤ is constant by sub-domain Ω* since the interpolation functions are

linear. Defining another neutron current distribution J¦, better approximation to the real flux J than the approximated flux J¤, by:

�2� J¦ = ∑ u;pp;&' J¦0 it is possible to establish a tensor error measure defined from the distance between the two

pre defined neutron currents:

�3� ℇ = J¦ − �¤

Φ Φ �' = −�'∇Φ �w = −�w∇Φ

Material 1 Material 2

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It is important to note that the neutron current �¦ is unknown since the nodal values are also

unknowns. A scalar measure of the error, over all the domain Ω, can be calculated from the

expression:

�4� «¬« = ℰwPΩ Ω where ℰw = ℰ;¯ℰ;¯ .

Using the definition of the domain discretization by finite element:

�5� Ω = ∑ Ωtptt&' the error can now be obtained from the equation:

�6� «¬« = ∑ ℰwPΩ{ Ω*ptt&' where d� is the number of finite elements used to discretize all the domain.

Now, the unknowns values �¦; could be obtained from the error minimization:

�7� k«³«

´µ¦¶ = ∑ 2[J¦ − �¤\ · ´µ¦´µ¦¶ − k ¤

´µ¦¶¹Ω{ PΩ*ptt&' =0 since:

�8� ´µ¦´µ¦¶ = u;

�9� k ¤´µ¦¶ = 0

resultando:

�10� k«³«

´µ¦¶ = ∑ [J¦ − �¤\u PΩtΩ¼ptt&' = 0

Recaindo num sistema de equações em J¦i:

�11� ∑ u;J¦0u PΩtΩ¼ptt&' − ∑ u J¦PΩtΩ¼

ptt&' =0 Or in a condensed form:

½¾¦¿ = À

where the matrix ½ and À are defined in Jospin[] for the linear triangular and tetrahedron

finite elements. We have to have in mind that in the diffusion equation the neutron flux is

discretize in groups of energy (2 or 4 groups) before to discretize the reactor geometry and the

neutron flux itself. That means that I have to choice to minimize the error in function of the

neutron current derived from one of the neutron flux groups associate to the groups of

neutron energy:

��, �� = −��∇Φ��, ��

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Numerical Results

The objective of this chapter is to compare the results of a stationary neutron diffusion problem obtained using the Mef program with the ones presented in the bibliography.

Spatial Neutron Diffusion Examples (1D and 2D)

BSS-6-A1 Numerical Benchmark (1D)

In this example, we are only interested to compare the precision of MEF program solutions

with the solutions of the BSS-6-A1 numerical benchmark calculated by various authors and

using different methods (nodal methods, finite differences, etc…). The example is a very simple

slab reactor. The adaptive remeshing is not used here and the discretization is performed using

a one-dimensional quadratic finite element with the same length for all the elements. The

thermal and fast flux variations, calculated by the MEF-DIFU program using 24 one-

dimensional quadratic F.E., are presented in Figure 1. and in Figure 2 respectively. The Table 1

presents a comparison among the results of the reactor criticality factor obtained by various

authors and by MEF-DIFU program.

Figure 1: Thermal Flux of BSS -6-A1 using 24 one-dimensional quadratic F.E.

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23

Figure 2: Fast Flux of BSS -6-A1 using 24 one-dimensional quadratic F.E.

It can be observed that the results of MEF-DIFU program present a good precision even with a

low number of quadratic finite element.

Table 1: Multiplication factor value for the BSS-6-A1 Numerical Benchmark.

Multiplication Factor Value for the BSS-6-A1

Feyzy Inanc (nodal method)[1]

Y. Nagaya & K.Kobayashi [2]

Mef-Difu (1D Quadratic F.E)

Zelmo (Finite Diferences)[9]

Benchmark Multiplication Factor

ANL-7416(1977)

0.9105223 0.9000870 (6 F.E.)

0.9361254 (6 Pontos)

0.9015507

0.9015320 (24 F.E.)

0.9039435 (24 Pontos)

0.9015870 (48 F.E.)

0.902285 (48 Pontos)

0.9015960 (96 F.E.)

0.901772 (96 Pontos)

0.9015960 (192 F.E.)

0.901540 (192 Pontos)

AIEA Numerical Benchmark using GenMesh (2D)

The following example is considered an important numerical benchmark to check numerical programs that solve the neutron diffusion equation. A lot of commercial programs like Bosor made use of this numerical example to check their precision in obtaining the flux and criticality factor. This example is used, in this work, to check the solution convergence when an adaptive remeshing procedure is used. If the neutron flux is discretized in two energy groups: the fast and thermal groups, the predefined error also can be calculated on the fast or thermal flux. The mesh refinement is produced by the program named GenMesh, developed by Sampaio [4]. In the figures 1 and 2, the fast flux is presented first for a basic discretization of 504 linear triangular elements and then, for a refined discretization of 1090 elements. It’s clear that the

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24

refinement is based on the error of the thermal neutron flux since the remeshing is more accentuated where the thermal neutron flux gradient is bigger. The neutron thermal fluxes, for the basic and refined mesh, are presented in the figures 3 and 4.

If the refinement is used in the .....

AIEA Numerical Benchmark using Triangle (2D).

The mesh refinement is now produced by the program named Triangle, developed by

Shewchuc [2]. The geometric model can be obtained from the data file *.poly presented in the

Annex A and visualized by the program ShowMe[3] as presented inFigure 3. In the Figure 5, the

fast flux is presented first, for a basic discretization of 504 linear triangular elements and then,

Figure 2: Fast Flux obtained from a 1000 linear triangular elements discretization

Figure 1: Fast Flux for 506 linear triangular elements

Figure 3: Thermal Flux for 506 linear triangular elements

Figure 4: Thermal flux obtained for 1000 linear triangular elements

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for a refined discretization of 1090 elements. It is clear that the refinement is based on the

error of the thermal flux since the remeshing is more accentuated where the neutron current

(flux gradient is bigger). The thermal flux, for the basic and refined mesh, is presented in the

Figure 5 and in the Figure 4.

Figure 3: Generated Mesh with Triangle

Figure 5: Fast neutron Flux Figure 4: Thermal neutron flux

RT-IEN-02/2011 Argonauta Reactor (2D) using original plate fuel.

The next example is based on the model proposed by Aghina[1] for the Argonauta reactor. This

geometric model can be obtained from the data file *.poly presented in the Annex B and

visualized by the program ShowMe[3] as presented in Figure 6. The advantage to use a

piecewise linear data file (PSLG) to represent the model is the possibility to use the open

program named Triangle to execute the basic and refined mesh based in linear triangle finite

element. The first discretization of the model (basic mesh) is presented in Figure 7 and is

obtained using the Triangle program with the following option: -pcBev[].

Using the adaptive remeshing based on the error in the neutron current continuity at the element interfaces we obtain the mesh presented at the Figure 9. and the corresponding fast and thermal neutron flux at the Figure 10 and Figure 8 respectively.

The criticality coefficient �t�� is given

Figure 6: Model generated from the *.poly file

Figure 7: Basic linear triangular finite element mesh discretization

Figure 9: Generated Mesh by the Program Triangle Figure 8: Thermal neutron flux

Figure 10: Fast neutron flux

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New Argonauta Reactor (2D) using Angra I fuel rods

Below a new geometry for the Argonauta core is presented using the fuel rods very similar to

the one used in Angra I. First of all, it is supposed that the homogenization of the core can be

done with the definition of an elementary cell that can represent all core domain by a simple

cell repetition. An error can be expected near the core domain boundary. This fuel rod is

composed of two materials: ÁÂw and Ã� surrounded by the water.

Since the diffusion theory is used, it is necessary to define two types of distance values for the

reactor core: the physical and the geometric distances. The difference between these two

distances is exactly the named extrapolated distance, necessary to define correctly or more

approximated boundary conditions at the reactor boundary. This is due to the fact that in the

diffusion theory, derived from the transport theory, the flux is approximated null only at the

extrapolated distance. Although in the diffusion theory, to impose a correct boundary

condition (neutron flux null), an extrapolated distance has to be used for the core dimensions.

The reactor physical height is given by the geometric height plus the extrapolated distance in

his two extremities:

( )extzG dZZ ×+= 2

Pitch=1.5 [cm]

TwÂ

ÁÂw Ã�

gap

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Supposing that the Argonauta reactor model is a cylindrical one with the geometric height

][60 cmZG = and the extrapolated distance ][5.7 cmdextz = , then the geometric height is

given by:

][75 cmZ = .

The reactor physical radius is given by:

( )2

2

405,2

−

=

ZBg

R

π

Since the relation between the physical and geometric reactor radius is defined by:

extrG dRR +=

and the reactor physical radius is given by:

cmR 814,23=

then, the reactor geometric radius is given by:

cmRG 314,16=

where the extrapolated radius was supposed to be ][5,7 cmdextr = .

A resume of the material and geometric data that define the fuel rod is presented in Table 2.

Table 2: Material and geometric properties of the Argonauta fuel rod

External diameter [cm]

Atomic concentration

( ) barncmatoms 3

Region 1: Pellets

( )2UO

235N

238N ON

805.0 4107688.7 −⋅ 2102073.2 −⋅ 2105699.4 −⋅ Region 2: Cladding

( )Zr

ZrN

950.0 2102914.4 −⋅

Region 3: Moderator

( )OH 2

HN ON

693.1 2106740.6 −⋅ 2103370.3 −⋅

Using the above data as the input data to Hammer program (Annex C.1), the cross sections can

be obtained from the Hammer output data (Annex C.2) where the enhanced terms correspond

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to the fast and thermal neutron groups respectively. These terms are presented in the Table 3

and are used in the calculation of the reactor criticality factor effk .

Table 3: Cross Section for Argonauta core using rod fuel elements

Cross Sections Neutron energy group g=1 (fast neutron group)

Neutron energy group g=2 (Thermal neutron group)

D 1.16082800 225495.0

fΣ 00213392.0 5893610.0

fΣν 00542189.0 1432150.0

aΣ 00744684.0 8945540.0

rΣ 02869630.0 0

Analytical results:

Using the formulation presented in equation 1 from Annex E, a solution of the criticality

equation of the Argonauta reactor could be given by:

�12� ( ) ( ) 0222 =++ βα BgBg if a cylindrical geometry core model without blanket is supposed to be used. The Ä and < coefficients are function of the cross sections and expressed by:

( )

21

2111112

DD

aDfraD

⋅

Σ⋅+Σ−Σ+Σ⋅=

να

( )

21

12211112

DD

rffraa

⋅

Σ⋅Σ−Σ−Σ+Σ⋅Σ=

ννβ

Using the given values presented in Table 3 where the Argonauta core geometry using Angra I

fuel rods was considered, the values of Ä and < for the Argonauta core reactor are:

42317175,0=α and 00520154−=β

Substituting these values in the equation 29 results:

01195410,02 =Bg

Remembering from Annex E that the neutron equilibrium equation could be obtained from a ?

multiplicator value defined by:

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Aff

a

⋅Σ+Σ

Σ=

2211

11

ννλ

where 22

1

a

rA

Σ

Σ= , 2

11111 BgDraa ⋅+Σ+Σ=Σ and 22222 BgDaa ⋅+Σ=Σ .

Substituting the values given in Table 3 in the above equation, results:

05001970,011 =Σa ;

09215099,022 =Σa ;

31140523,0=A .

Sowe obtain:

99999820,0=λ

Finally, the criticality factor of this Argonauta reactor model is:

00000180,11

==λ

efK

that confirms the Argonauta reactor as a critical reactor.

Point Kinetic Numerical Examples:

Step Reactivity in Argonauta Reactor (Adiabatic Reactor)

In this example, the Argonauta reactor is supposed to be critical with a power of KW1,0 . The

reactor is supposed to be adiabatic (boundary condition), that means a heat transfer

coefficient CKWk °= 0 , and a negative temperature variation coefficient

C°−= %017,0α . The reactivity slope is supposed to be positive sec%002,0=ρ with

sec60 of an actuation time. The data, for this example, is presented in table 1.

Table 4: Data for the Argonauta reactor point kinetic analysis:

Analysis data: unity value

number of neutron precursors 6 neutron source [n/s.cm²] 0.0 reactor reactivity slope [%/s] 0.002 total time of analysis [s] 1500.0 actuation time of reactor reactivity slope [s] 60.0 time step [s] 1.0 initial reactor power [kW] 1.0

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equivalent reactor thermal capacity [kW s/°C] 600.0 linear temperature variation coefficient [%/ºC] -0.017 heat transfer coefficient (capacity) [kW/°C] 0.0 Neutron lifetime [s] 2. 5 × 10�Å. In figure 9 we can see that, initially, the neutrons population and the temperature grow. But at

high temperatures, the neutron population decreases and the temperature remains constant.

We can also notice, respectively, in Figures 10 and 11 the reactivity of Argonauta reactor when

the heat transfer coefficient is null and the variation of fast and slow neutron groups.

Figure 9: Variance of neutrons population (red) and the reactor temperature (green)

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Figure 10: Insert function of the reactor reactivity.

Figure 11: Variation of slow and fast groups.

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Subcritical reactor with a step Reactivity of � = �. �� % ⁄ �

In this example, a prompt subcritical with the time-dependent reactivity function ρ = 0.003[%/seg]. The reactor is supposed to be critical with a power of 1.0[kW]. For the boundary condition, the reactor is supposed to be adiabatic, that means a heat transfer coefficient is null k = 0 [�Æ/°:]. The linear temperature variation coefficient is also null α = 0.0 [%/°:]. The insertion has an actuation time of 2 [s]. A comparison among the results obtained by Kinard and those calculated by this numerical analysis is presented in the table 1.

Time Analytical solution (Kinard)

Numerical Solution MEF


1 2.2098

10 8.0192

20 2.8297x101

Figure 11: Inserted reactivity variation

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Critical reactor with a step reactivity of � = �. �� % ⁄ �

In this example, a prompt critical with the time-dependent reactivity function ρ = 0.007[%/sec]. The reactor is supposed to be critical with a power of 1.0[kW]. For the boundary condition, the reactor is supposed to be adiabatic, that means a null heat transfer coefficient (k

Figure 12:Neutron population and temperature variations

Figure 13: 1st and 2nd Precursors

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= 0 [�Æ/°:]). The linear temperature variation coefficient is also null (α = 0.0 [%/°:]). The insertion has an actuation time of 2 [sec]. A comparison of the results is presented in

Figure 14: Inserted reactivity variation

Figure 15: Neutron population variation

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Supercritical reactor with a step reactivity of � = �. �� % ⁄ �

In this example, a prompt supercritical with the time-dependent reactivity function ρ = 0.008[%/seg]. The reactor is supposed to be critical with a power of 1.0[kW]. For the boundary condition, the reactor is supposed to be adiabatic, that means a null heat transfer coefficient (k = 0 [�Æ/°:]). The linear temperature variation coefficient is also null (α = 0.0 [%/°:]). The insertion has an actuation time of 2 [seg]. A comparison of the results is presented in

Figure 16: Temperature variation

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Ramp Reactivity of 0.1 [$/s]:

The objective of this section is to compare the numerical results obtained in the case of a remp reactivity with those obtained by Kinard using an analytical solution.

Analysis data:

total time of analysis [s] 9 time step [s] 0.01, 0.001, 0.0001 Initial power in the critical state [kW] 1.0 equivalent reactor thermal capacity [kW s/°C]: 100. linear temperature variation coefficient [%/ºC]: 0.0 heat transfer capacity [kW/°C] 0.0 Neutron lifetime [s] 2.× 10�É.

Some care has to be taken when the heat transfer capacity is null. To avoid numerical

problems of division by zero, the MEF program user have to change it to a very low value.

Time Analytical solution (Kinard)


(∆� = 0.01 �Ê��


(∆� = 0.001 �Ê��


(∆� = 0.0001 �Ê��

2 1.3379 1.3383 4 2.2283 2.2285 6 5.5815 5.5817 8 4.2781x101 4.2769x101 9 4.8745x102 4.8699x102

Figure 17: Reactivity insertion

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Figure 18: Neutron population

Figure 19: Water reactor temperature variation

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Ramp reactivity insertion in Argonauta reactor using a non null heat transfer

coefficient (subcritical reactor).

Para uma capacidade de transferência de calor não nula e uma mesma variação temporal da

reatividade apresentada anteriormente (reatividade em rampa) resulta:

Tabela 1: dados gerais do problema:

tempo total de análise: Ë= 3000.0[s]

passo de tempo: Δ�= 1.0[s]

capacidade térmica do reator: : = 600.0[kW.s/°C]

coeficiente de retroação da temperatura: -0.017[kW/°C]

coeficiente da variação linear de temperatura 0.0 [°C]

coeficiente de transferência de calor do reator P= 0.1[kW/°C]

potencia inicial do reator P= 0.1[kW]

tempo de vida dos neutrons: Λ= 250.0 × 10�Ì[s]

fonte de neutrons: Ê��= 0 [n/cm²]

número de neutrons precursores: n= 6

fator de criticalidade do reator: �t��= 0.984

A non null neutron source without reactivity insertion in Argonauta

reactor.

Figure 20: 1st and 2nd neutron precursors

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fonte de neutrons: Ê��= 200 [n/(s.cm²)]



A hypothetic accident of coolant lost in Argonauta reactor.








Inserção de reatividade Í= 0.0024[%/s]



fonte de neutrons: Ê��= 200 [n/(s.cm²)]



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Point kinetic with a constants neutron source and reactivity

Lambda=1.d-5 (reator rápido) Reactivity=-0.01626 (constante)

Figure 21

Source=1000.(constante)

Figure 22

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Total time=1.0 [s] Step=1.d-4 [s]

Figure 23

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Time=1.d+1 Step=1.d-3 Source=1000.

Figure 24

Source=500.

Figure 25

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Sem neutrons retardados: Time=1.d+1 Step=1.d-2 Flux=1000.

Figure 26

Flux=500.

Figure 27

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Time=1.d+2 Step=1.d-2

Figure 28:

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Accelerated Driven System reactors (ADS) using pulsing neutron source.

The objective of this example is to simulate a reactor named Accelerated Driven System (ADS) using the Point Kinetic analysis. It is know that this kind of reactor is a subcritical one and uses a neutron source directed to a plumb target localized in the center of the reactor core to reach the criticality. In this kind of problem with a criticality factor less than one (subcritical) from t=0, it can be considered that for t<0 the reactor is also subcritical (stable) with the same negative reactivity that will have for t>0 and with a neutron density normalized to 1 at t=0. In this case, for t<0 the source is determined to specific conditions that not necessarily, has to be calculated since for t≥0 a new source will be introduced continuing with the same negative reactivity. The delayed neutron precursors density for t≤0 is calculated supposing that the neutron density is 1 for t≤0. As a matter of fact, for t≤0, Í < 0, d¡ = 1 and the source is defined by:

Ê�� = ÍÑ

where: Ñ: is the neutron generation time in the reactor

é o tempo de geração dos nêutrons no reator (no Argonauta G=250 microseg),

:; = ÒÓ>ÔÓ é a concentração do precursor de ordem i (i de 1 a 6) sendo Õ; a fração do de

ordem i com V = ∑ Õ;Ì;&' e Ö; a constante de decaimento do de ordem i. Deve-se então desprezar o que acontece para t<0 e deve-se tomar para as condições iniciais

(t≥0), d¡ = 1 e Í como Ê�� = ×> impostos (Í é considerado o mesmo para t≤0), o programa

da cinética pontual faz o resto (calcula a concentração dos precursores em t=0). Tabela 4: dados gerais do problema:


passo de tempo: Δ�= 1.0 × 10�É[s]

capacidade témica do reator: : = 100.0[kW.s/°C]

coeficiente de retroação da temperatura: 0.0[kW/°C]






Tabela 5: dados dos neutrons precursores

i ?;�Ê�'� :;�× 10�x� 1 0.01246 0.1824

2 0.03040 1.2450

3 0.11390 1.0970

4 0.30610 2.1530

5 1.13300 0.6280

6 2.95000 0.2346

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total 5.5400

Supondo que o reator é sub-crítico com um fator de criticalidade �t�� = 0.984, a reatividade

do reator e dada pela expressão:

Í = �t�� − 1�t�� = 0.984 − 1

0.984 = −0.01626

Inserindo portanto no reator uma reatividade negativa de −1.626 × 10�w% constante no

tempo (Figura 1), o reator manterá o fator de criticalidade �t�� = 0.984. Para levá-lo a ser

crítico adiciona-se uma fonte de neutrons pulsante com onda quadrada apresentada na Figura

2.

5

t[ØÊ�

Ë = 25 �ØÊ�

50 37.5 25 12.5

Fonte de neutrons

1000[n/cm²]

5

t[ØÊ�

Ë = 25 �ØÊ�

50 37.5 25 12.5

Reatividade Í =

-0.01626

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Pd��P� = Í¡Λ d�� − <

Λ d�� + ?;:;Ì

;&'+ Ê��

P:;��P� = <

Λ d�� + ?;:; ; B = 1, … ,6

onde:

d�� [n/area]: população de neutrons

×ÚÛ : parcela relativa a variação dos neutrons prompts na população de neutrons

onde Λ [s] representa o tempo de vida dos neutrons prompts. Para Í < 0 há

uma perda de neutrons, para Í = 0 a população de neutrons se mantém

estável, para Í > 0 há um aumento na população de neutrons.

ÜÛ: parcela relativa a contribuição dos neutrons retardados na população de

neutrons

?;:; [n/vol.s] parcela relativa a contribuição dos neutrons retardados para o grupo i onde

?;[1/s]representa o decaímento e :; [n/vol] os precursores – fonte de

neutrons relativa ao decaímento dos produtos de fissão.

Ê��[n/area]: fonte de neutrons externa

Fazendo com que a variaçãop dos neutrons precursores e da população de neutrons sejam

nulos, resulta :

P:��P� = 0

Pd��P� = 0

e portanto teremos para o reator uma população de neutrons constante. Da primeira equação

resulta para a fonte externa de neutrons:

Ê�� = − Í¡Λ d��

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Figura 1: Reatividade constante � = −�. �ÝÞßÞ inserida no reator.

Figura 2: Fonte de neutrons pulsante com 20 ciclos de onda quadrada

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Figura 3: População total de neutrons no reator [n/cm²] para uma fonte de neutrons pulsante s=1000[n/cm².s] e 20 ciclos.


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Figura 5: População total de neutrons no reator [n/cm²] para uma fonte de neutrons pulsante s=500[n/cm².s].


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Figura 8: População total de neutrons no reator [n/cm²] para uma fonte de neutrons pulsante s=1.5 [n/cm².s] e 60 ciclos.

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Figura 10: População total de neutrons no reator [n/cm²] para uma fonte de neutrons pulsante s=0.5[n/cm².s] e 60 ciclos.

Verifica-se que a população de neutrons não chega a se estabilizar com fontes 1.0 > Ê >1.5[n/cm²s]. Portanto a fonte ótima para a estabilização de um reator encontra-se nesta faixa.

Pode-se até fazer uma programação de otimização para se achar o ponto ótimo.

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Quasi-Static Spatial Kinetic Numerical Examples

The objective of this example is to simulate the spatial kinetic example presented by Hansen

Kang where the variable absorption cross section, represented by a temporal sinusoidal form,

is defined in two different cells with a à fase angle.

Figure 29: Reactor core geometry

Figure 30: Reactor core boundary conditions (á = �)

The variable cross section perturbation is introduced in the cells âw and âx and are defined by

the following functions:

�Σã�w��, �� = �Σã�w��, 0� �1 − 0.1 sin �æ2àË ç �� ¬âw

�Σã�w��, �� = �Σã�w��, 0� �1 + 0.1 sin �æ2àË ç �� ¬âx

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where the sinusoidal period is Ë = 0.001 �Ê�è�.

Figure 31: Variable absorption cross section (éê) in reactor core regions ëßand ë�.

Figure 32: Material distribution in a 2 regions reactor core

âw

âx

â'

A

B

C

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Figure 33: Material distribution in a 2 regions reactor core

The reactor core constant cross sections are presented in Table 5. . The fission cross section

[νΣ�\w = 0.2376606, instead of 0.218, is used to render the reactor initially critical.

Table 5: Cross Section Constants for one and two regions

Neutron Group -> Fast Group (g=1)

�' = 1.0 × 10ì�èØÊ �

Thermal Group (g=2)

�w = 2.2 × 10Ì�èØÊ �

Cross Section Constants

Fuel Reflector Fuel Reflector

Diffusion �� 1.5 1.2 0.4 0.15

Total Fission �Σn�� 0.0623 0.101 0.2 0.02

Fission [νΣ�\� 0.0 0.0 0.218 0.0

Absorption �Σã�� 0.0023 0.001 0.2 0.02

Scattering �Σ'→w�� 0.06 0.1 - -

The absorption cross section is obtained from the relation:

�Σ�� = �Σ��D − �Σ1→2�D

From the neutron speed, the following approximated neutron lifetime can be obtained:

For fast group:

�' ≅ 1�'νΣ�'

= 11.0 × 10ì�èØÊ � × 2.3 × 0.0� 1èØ� = ∞�Ê�

For thermal group:

�w ≅ 1�wνΣ�w

= 12.2 × 10É�èØÊ � × 2.3 × 0.23766006� 1èØ� = 8.3156 × 10�Ì�Ê�

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The neutron population lifetime can vary between 10�x�Ê� for thermal reactor to 10�ï�Ê� for

the fast reactors.

The quasi-static spatial kinetic analysis is based in the supposition that the flux variable can be

separated in spatial and temporal variables. The spatial variable, defined by the criticality

coefficient and the flux distribution, is obtained using the diffusion analysis and the temporal

one, defined by the medium neutron population, reactor reactivity and water temperature, is

obtained by using the point kinetic analysis. The delayed neutron constant, for the point

kinetics, are presented in Table 6.

Table 6: Delayed Neutron Constants

Group i <; ?; 1 0.0002850 0.0127 2 0.0015975 0.0317 3 0.0014100 0.1150 4 0.0030525 0.3110 5 0.0009600 0.0140 6 0.0001950 0.0387

< = <;Ì;&'

0.0075000

For the point kinetics analysis, the following reactor characteristic properties are used: Table 7: Reactor characteristics

total time of analysis: Ë= 1.0[s] time step: Δ�= 1.25 × 10�É[s] reactor thermal capacity: : = 1.00 × 10�Ì[kW.s/°C] coeficiente de retroação da temperatura: 0.0[kW/°C] linear coefficient of temperature variation 0.0 [°C] initial reactor power P= 1.0[kW] neutron lifetime: Λ= 250.0 × 10�Ì[s] Neutron sources: Ê��= 0.0 [n/cm²] fator de criticalidade do reator: �t��= 1.0

and the reactivity variation Í is calculated by the following relation:

Í = �t��nN∆n − �t��n�t��nN∆n

where the criticality coefficient �t�� is obtained from the spatial diffusion analysis calculated

at time � and � + ∆�. The neutron flux distribution obtained from MEF program is presented in

RT-IEN-02/2011

58

Figure 34: Fast neutron flux in a 2 regions reactor core

Figure 35: Thermal neutron flux in a 2 regions reactor core

Figure 36: Thermal neutron flux in a one region reactor core

RT-IEN-02/2011

59

and compared with those obtained by Hansen-Kang[] and Zelmo[] presented in Table 8 at the points ð, V and : defined in Figure 21. Table 8: Flux comparison among the numerical results obtained by Hansen, Zelmo and Mef.

Hansen-Kang¹ Zelmo² Mef³

Time t �ñ �ò �ó �ñ �ò �ó �ñ �ò �ó

0.0 1.0000 0.43294 0.43294 1.0000 1,00000 0,43178 0,43466

T/8 1.0000 0.42492 0.44152 1.0001 0,99939 0,42075 0,44680

T/4 1.0016 0.40329 0.47167 1.0005 0,99873 0,41589 0,45281

3T/8 1.0069 0.40691 0.47747 1.0022 0,99939 0,42075 0,44680

T/2 1.0111 0.42380 0.45635 1.0032 1,00000 0,43321 0,43322

5T/8 1.0133 0.45048 0.42714 1.0034 0,99939 0,44685 0,42080

3T/4 1.0150 0.47743 0.40981 1.0040 0,99878 0,45289 0,41596

7T/8 1.0195 0.48352 0.41223 1.0053 0,99939 0,44685 0,42080

T 1.0245 0.46295 0.37714 1.0062 1,00000 0,43321 0,43322

¹ - Finite Element Method using a Hermite time discretization. ² - Finite difference using an implicit time integration. ³ - Finite Element method using a quasi-static solution.

From the reactor criticality level, the absorption cross section of the reactor suffers a variation in regions âw and âx in a sinusoidal form with the period Ë = 0.003 �Ê�. This absorption cross section evolution and the evolution of the thermal neutron flux at the line represented by the equation � = −L + 30 are presented for time t=0.(lime green), t=T/4 (red) and t=3T/4 (moss green ) in the Figure 37(a) and Figure 37(b) respectively. In the Figure 38 (a) and (b) are presented the evolution of the reactor criticality factor and the reactivity evolution for this cross section variation. The reactor neutron population and reactor temperature is presented in Figure 39(a) and (b) respectively. These figures show an increase in the population and temperature even though the medium variation of the cross section is null.

Figure 37: (a)Cross section evolution and (b)thermal neutron flux in the path y=-x+30 for time t=0.(lime green),

t=T/4 (red) and t=3T/4 (moss green )

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Figure 38: (a)Criticality factor and (b)reactivity evolution.

Figure 39: Neutron population and temperature evolution

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QStep-by-Step spatial kinetic

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Conclusions:

This work presents an overview of analysis capacity that MEF can realize passing through a

neutron diffusion solution for 1D and 2D reactor core, a point kinetic analysis of the Argonauta

research reactor and an Accelerated Driven System (ADS) and a very simplified spatial kinetic

analysis based in the point kinetic and neutron diffusion analysis and so on. This last analysis

had been presented before [13], but some mistakes are here corrected. The use of the

adaptive remeshing shows us the performance of this method to improve the spatial neutron

distribution without increasing too much the cpu cost of the solution. A formulation based on

[10] of the quasi-static method (adiabatic method) to calculate the spatial reactor kinetics

using the finite element method is under development.

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63

Acknowledgments:

We are grateful to CNEN/CNPq for the financial support given via a PCI scholarship that makes

possible to develop this work, to Shewchuk[64] for making available the Triangle and ShowMe

program, to Frey [64] for making available the Medit visualizer program, to Sampaio[4] for

making available the GenMesh program.

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64

References:

1. Aghina”

2. Shewchuc, J. R.; “Triangle - A Two Dimensional Quality Mesh Generator and Delaunay

Triangulator”, http://www.cs.berkeley.edu/~jrs,

http://www.cs.cmu.edu/~quake/triangle.html

3. Shewchuc, J.R.; “ShowMe – A display program for Meshes and More”,

http://www.cs.berkeley.edu/~jrs, http://www.cs.cmu.edu/~quake/showme.html .

4. Sampaio, P.A.B; “GenMesh – A Two Dimensional Mesh Generator and Delaunay

Triangulator”, IEN – CNEN,

5. Frey, P.J.; ”Medit - An interactive mesh visualization software”, INRIA-National Institut

in Automation and Informatic Research, Technical Report n. 0253, France, December,

2001.

6. Reuss, P;”Neutron Physics”

7. Jospin,R.J.;””; Relatório técnico IEN, nº 12, 2006.

8. Hansen,K.F.; Kang,C.M.; “Finite Elements Methods in Reactor Analysis”, MIT.

9. Zelmo;””,

10. Roson, D.; “Introduction to Nuclear Reactor Kinetics”, Polytechnic International Press,

ISBN 2-553-00700-0, 1998

11. Yasinsky, J.B.; Henry, A.F.; “Some Numerical Experiments Concerning Space-Time

Reactor kinetics Behavior”, Nuclear Science and Engineering, 22. pp. 171-181 (1965)

12. Grossman, L.M.; Hennart, J.P.; “Nodal Diffusion Methods for Space-Time Neutron

Kinetics”, Progress in Nuclear Energy, 49(2007), pp. 181-216

13. Scal, D.; Jospin, R.J.; Aghina, L.O.B; dos Santos, R.S.; “Finite Element Formulation on

Reactor Kinetics Problem”; Relatório técnico IEN, nº 12, 2006.

14.

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65

Annex A: PSLG data file to calculate, in Mef program, the neutron

diffusion (criticality factor and neutron flux distribution) for the IAEA

numerical benchmark reactor example.

# 2DIAEAReactor - 2 energy groups

48 2 1 0

# First row in the x direction

1 0. 0.

2 10. 0.

3 30 0.

4 50 0.

5 70 0.

6 90. 0.

7 110. 0.

8 130. 0.

9 150. 0.

10 170. 0.

# Second row in the x direction

11 10. 10.

12 30 10.

13 50 10.

14 70 10.

15 90. 10.

16 110. 10.

17 130. 10.

18 150. 10.

19 170. 10.

# Third row in the x direction

20 30 30.

21 50 30.

22 70 30.

23 90. 30.

24 110. 30.

25 130. 30.

26 150. 30.

27 170. 30.

# Fourth row in the x direction

28 50 50.

29 70 50.

30 90. 50.

31 110. 50.

32 130. 50.

33 150. 50.

34 170. 50.

# Fifth row in the x direction

35 70 70.

36 90. 70.

37 110. 70.

38 130. 70.

39 150. 70.

40 170. 70.

# Sixth row in the x direction

41 90. 90.

42 110. 90.

43 130. 90.

44 150. 90.

# Seventh row in the x direction

45 110. 110.

46 130. 110.

47 150. 110.

#

48 130. 130.

# Nine segments with boundary markers.

46 1

1 10 19 2

2 19 27 2

3 27 34 2

4 34 40 2

5 40 39 2

6 39 44 2

7 44 47 2

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8 47 46 2

9 46 48 2

# Thirty seven segments without boundary markers.

10 1 2

11 2 3

12 3 4

13 4 5

14 5 6

15 6 7

16 7 8

17 8 9

18 9 10

#

19 1 11

20 11 20

21 20 28

22 28 35

23 35 41

24 41 45

25 45 48

#

26 2 11

27 8 17

28 17 25

29 25 24

30 24 31

31 31 37

32 37 36

33 36 41

34 45 42

35 42 43

36 43 38

37 38 32

38 32 33

39 33 26

40 26 18

41 18 9

#

42 5 14

43 14 15

44 15 6

#

45 35 36

46 36 41

# number of holes

0

# attribute list

6

1 8.0 5.0 10

2 80. 5.0 10

3 88. 80. 10

4 80. 50.0 9

5 140. 15. 8

6 160. 15. 15

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Annex B: PSLG data file to calculate, in Mef program, the neutron

diffusion (criticality factor and neutron flux distribution) for the

Argonauta reactor.

# 2DArgonautaReactor-2 energy groups

7 2 1 0

# Outer box has these vertices;

1 0 0.

2 0 80

3 58 0.

4 58 80

# Inner square has these vertices;

5 8 0

6 0 30

7 8 30

# Four segments with boundary markers.

8 1

1 1 6

2 6 2

3 2 4 2

4 4 3 2

# Four segments without boundary markers.

5 3 5

6 5 1

7 5 7

8 7 6

# number of holes

0

# attribute list

2

1 4.0 15.0 11

2 29. 40.0 13

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68

Annex C: Cross sections obtained from Hammer program to the new

Argonauta core using Angra I rod fuel.

A resume of data to be used in the calculation of the cross section of the Argonauta reactor

using Angra I fuel rods is presented below:

REGION 1 ( 2UO ):

Match the region of fuel in the tablets form.

a) Geometry:

A circle with diameter cmd 805,01 = .

b) Atomic concentration:

The fuel is uranium oxide in a tablets form, with 3,4% enriched uranium into 235U . Each tablet

has a diameter of 0,805cm and 1 cm of height. The volume of the tablet is

32

96,508104

05,8mm=×

×π. The volume of each ‘dish’ is 3945,3 mm .

Thus, the volume of a tablet becomes 307,501945,3296,508 mmVt =×−= . The specific

mass of 2UO is 34,10 cmg , but for equivalent tablets without blanks, apparent specific

mass is 324,104,10

96,508

07,501cmga =×=ρ .

The molecular mass of 2UO is given by

( ) gAUO 898,269162238966,0235034,02 =×+×+×= . Since =0N Avogadro's number

barn60225,01060225,0 24 =×= , the molecular concentration of 2UO will be

898,269

60225,024,102

×=UOM or ( ) barncmmolMUO

342 104952,228 −×= , where

24101 −=barn .

So for each isotope atomic concentrations are

- ( ) barncmatomsN 34

235 107688,7 −⋅=

- ( ) barncmatomsN 32

238 102073,2 −⋅=

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69

- ( ) barncmatomsNO

32105699,4 −⋅=

REGION 2 ( )Zr :

Corresponds to the region of zircaloy tube as encasement of the 2UO tablets.

Geometry:

A zircaloy tube with the following characteristics

- outward diameter: cm950,0

- inward diameter: cm818,0

The diameter used in the computation is cmd 950,02 = .

Atomic concentration:

Specific mass of Zr: 35,6 cmg=ρ

Atomic mass: 022,91 NgAZr =

Atomic contration:

- ( ) barncmatomsN Zr

32102914,4 −⋅=

REGIÃO 3 ( )OH 2 :

Corresponds to the region's water as a moderator/refresher.

a) Geometry:

Tubes step: cms 5,1=

The outward diameter of this region is given by 4

32 ds

×=

π.

Where we find cmd 693,13 =.

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70

b) Atomic concentration:

Specific mass of OH 2 (at 20°C): 39982,0 cmg=ρ

Molecular mass: 02 0152,18 NgA OH =

Molecular concentration: ( ) barncmgN

N OH

302 03337,0

0152,18

9982,0=

×=

Thus, for each isotope the atomic concentration is

- ( ) barncmatomsNN OHH

322 106740,62 −×=×=

- ( ) barncmatomsNN OHO

322 10337,3 −×==

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71

Annex D: Cross sections obtained from Hammer program to the new

Argonauta core using Angra I rod fuel.

D.1. Hammer Data:

2 11

HAMMER.OUT

LITHELIB.BIN

HELPLIB.BIN

1 1 1 11 26

1 1 1 REATOR ARGONAUTA

1 1 Combustivel Angra I

2 3 3 3 1 100.

3 1 1 3 30 0.316929 20.

4 92235. 0. 7.7688e-4

5 92238. 0. 2.2073e-2

6 8000. 29. 4.5699e-2

7 2 2 1 10 0.374016 20.

8 40000. 0. 4.2914e-2

9 3 3 2 51 0.666535 20.

10 1001. 1029. 6.6740e-2

1 11 8000. 29. 3.3370e-2

D.2. Hammer Output:

***** HAMMER VERSAO SVS-FORTRAN77 *****

0*HAMMER* BNL-SRL LATTICE ANALYSIS CODE

0OPTIONS

CAPN 1 0 0 0 0 0 0 0 0 0 0 0

THERMOS 1 0 0 0 0 0 0 0 0 0 0 0

HAMLET 1 0 0 1 1 0 0 0 0 0 0 0

FLOG 0 0 0 0 0 0 0 0 0 0 0 0

DIED 2 6 0 0 0 0 0 0 0 0 0 0

0BATCH 1 1 LATTICES 0 CORES REATOR ARGONAUTA

0CASE 1 Combustivel Angra I

0LIBRARY TAPE LABELED LITHE 11-66

0EPITHERMAL LIBRARY TAPE LABELED HELP 11-66 +

1Combustivel Angra I CASE 1 THERMOS

0LIBRARY TAPE LABELED LITHE 11-66

0CYLINDRICAL GEOMETRY - NO LEAKAGE

0IT= 20 NORM= 1.0000 CRIT= 2.5321E-04 RES= 2.3448E-04

0ETA= 1.85829 F= .86156 ETA*F= 1.60102 XA= .896289E-01

NU*XF= .143498E+00 MOD.ABS.FRACT= .13315

0ABSORPTION AND PRODUCTION BY ISOTOPE AND REGION

ISOTOPE T/M TYPE REG 1 REG 2 REG 3 REG

92235. 0. ABS 7.700E-01 0.000E+00 0.000E+00

PROD 1.601E+00 0.000E+00 0.000E+00

92238. 0. ABS 9.155E-02 0.000E+00 0.000E+00

PROD 0.000E+00 0.000E+00 0.000E+00

8000. 29. ABS 0.000E+00 0.000E+00 0.000E+00

PROD 0.000E+00 0.000E+00 0.000E+00

40000. 0. ABS 0.000E+00 5.293E-03 0.000E+00

PROD 0.000E+00 0.000E+00 0.000E+00

1001. 1029. ABS 0.000E+00 0.000E+00 1.332E-01

PROD 0.000E+00 0.000E+00 0.000E+00

0NEUTRON DENSITY, VBAR, AND FOIL ACTIVATION

0 N R DENSITY VBAR FLUX FLUXVOL ACT-

1 .000 2.8047E+00 1.4318 4.0157E+00 8.1752E-02

2 .161 2.8734E+00 1.4236 4.0905E+00 6.6620E-01

3 .322 3.0867E+00 1.4003 4.3223E+00 1.4079E+00

0 AVERAGE 3.0072E+00 1.4086 4.236E+00 2.156E+00

4 .439 3.4284E+00 1.3673 4.6877E+00 9.3690E-01

0 AVERAGE 3.4284E+00 1.3673 4.688E+00 9.369E-01

5 .512 3.6334E+00 1.3499 4.9046E+00 1.1726E+00

temperature

Atomic concentration

ÁÂw

TwÂ

Ã�

isotopes

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72

6 .586 3.8249E+00 1.3329 5.0981E+00 1.3957E+00

7 .661 3.9704E+00 1.3234 5.2545E+00 1.6208E+00

8 .735 4.0745E+00 1.3185 5.3724E+00 1.8436E+00

9 .809 4.0857E+00 1.3160 5.3769E+00 2.0316E+00

0 AVERAGE 3.9437E+00 1.3258 5.229E+00 8.064E+00

0AVERAGE CROSS SECTIONS

MACROSCOPIC (1/CM) MICROSCOPIC (BARNS)

ISOTOPE ABSORPTION FISSION NU*FISSION ABSORPTION FISSION NU*FISSION

92235. 6.90146E-02 5.90525E-02 1.43498E-01 4.59746E+02 3.93382E+02 9.55919E+02

92238. 8.20579E-03 0.00000E+00 0.00000E+00 1.92393E+00 0.00000E+00 0.00000E+00

8000. 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

40000. 4.74399E-04 0.00000E+00 0.00000E+00 1.31644E-01 0.00000E+00 0.00000E+00

1001. 1.19341E-02 0.00000E+00 0.00000E+00 2.47392E-01 0.00000E+00 0.00000E+00

0B1 APPROX. L2= 2.52075E+00 XA= 8.94554E-02 D= 2.25495E-01 BSQD= 1.00000E-02 IT=

20


0EPITHERMAL LIBRARY TAPE LABELED HELP 11-66 +MO

0MAXIMUM ITERATIONS ON THE FLUX SHAPE IN ANY MACRO GROUP = 2 GROUP 1

MAXIMUM RENORMALIZATION IN ANY GROUP = .99987 GROUP16


OVERALL FEW GROUP DATA

0OUTPUT GROUP MUFT GROUPS SIGMA-A SIGMA-R SIGMA-F NU SIGMA-F P

AGE DIFF.COEF.

1 OF 1 1 TO 54 .00744684 .02869630 .00213392 .00542189 .79396250

32.1175 1.1608280

1 OF 2 1 TO 25 .00245449 .05263450 .00123582 .00329238 .95544490

24.9262 1.3734210

2 OF 2 26 TO 54 .01875315 .09368557 .00416788 .01024465 .83321450

30.9683 .6793633

1 OF 3 1 TO 10 .00332499 .08979148 .00231648 .00623336 .96429210

14.7474 1.8231940

2 OF 3 11 TO 25 .00169597 .09849875 .00029416 .00072970 .98307330

24.5415 .9815005

3 OF 3 26 TO 54 .01875315 .09368556 .00416788 .01024465 .83321450

30.5836 .6793633


0 P = 7.939625E-01 NU(1-PF) = 1.500116E-01 TAU = 3.211752E+01

MATERIAL BUCKLING = 1.182989E-02 K-INFINITY = 1.420769E+00 K-EFFECTIVE AT INPUT

BUCKLING = 1.051990E+00

ASYMPTOTIC FOUR GROUP FLUXES, 6.567574E+00, 7.480135E+00, 6.117210E+00, 6.220970E+00

1CELL-HOMOGENIZED ATOM NUMBER DENSITIES AND FOUR-GROUP MICROSCOPIC CROSS SECTIONS

(BARNS) = ARRAYS SCHAND AND SBALTC. CASE 0

ISOTOPE N (CELL-HOMOG) GRP XA XF NU*XF XTR

XREM XCAPT

------- -------------- --- ----------- ----------- ----------- -----------

----------- -----------

92235 1.75643E-04 1 1.3517E+00 1.2247E+00 3.3933E+00 6.7882E+00

7.6051E-01 1.2700E-01

2 2.1155E+00 1.6618E+00 4.1234E+00 1.0210E+01

1.2050E-02 4.5370E-01

3 3.5934E+01 2.3729E+01 5.8326E+01 4.1752E+01

3.7150E-03 1.2205E+01

4 3.9216E+02 3.3554E+02 8.1537E+02 3.3886E+02

0.0000E+00 5.6616E+01

92238 4.99045E-03 1 4.8663E-01 4.2108E-01 1.1296E+00 5.7258E+00

7.9958E-01 6.5551E-02

2 2.4738E-01 4.5508E-04 1.0922E-03 8.6488E+00

1.4618E-02 2.4692E-01

3 2.3124E+00 0.0000E+00 0.0000E+00 1.0492E+01

4.0986E-03 2.3124E+00

4 1.6414E+00 0.0000E+00 0.0000E+00 7.3884E+00

0.0000E+00 1.6414E+00

8000 3.31947E-02 1 1.2456E-02 0.0000E+00 0.0000E+00 2.5523E+00

1.8628E-01 1.2456E-02

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73

2 0.0000E+00 0.0000E+00 0.0000E+00 3.4648E+00

1.0005E-01 0.0000E+00

3 0.0000E+00 0.0000E+00 0.0000E+00 3.3692E+00

2.0390E-02 0.0000E+00

4 0.0000E+00 0.0000E+00 0.0000E+00 3.2282E+00

0.0000E+00 0.0000E+00

40000 3.81007E-03 1 6.4015E-02 0.0000E+00 0.0000E+00 4.3235E+00

2.1536E-01 6.4015E-02

2 2.1391E-02 0.0000E+00 0.0000E+00 6.9669E+00

2.6727E-02 2.1391E-02

3 7.9365E-02 0.0000E+00 0.0000E+00 6.0657E+00

6.1480E-03 7.9365E-02

4 1.2428E-01 0.0000E+00 0.0000E+00 4.9232E+00

0.0000E+00 1.2428E-01

1001 4.57254E-02 1 3.6788E-05 0.0000E+00 0.0000E+00 1.1343E+00

1.7203E+00 3.6788E-05

2 1.8332E-04 0.0000E+00 0.0000E+00 3.3483E+00

2.0776E+00 1.8332E-04

3 1.3111E-02 0.0000E+00 0.0000E+00 6.4736E+00

2.0331E+00 1.3111E-02

4 2.6047E-01 0.0000E+00 0.0000E+00 2.7467E+01

0.0000E+00 2.6047E-01

5 ISOTOPES, RECIPROCAL VELOCITIES (G=1,4): 5.2370E-10 2.5551E-09

1.8089E-07 3.3742E-06 (S/CM)

RECIPROCAL VELOCITY (1 OF 2): 5.6523E-08 (S/CM)

1*HAMMER* EDIT2

0LISTING OF LATTICE LIBRARY

0EDIT OPTION 6

1LATTICE NUMBER 1, RUN AT , BY *HAMMER*

*********************

TITLE...Combustivel Angra I ...

0THERMOS LIBRARY LITHE 11, HAMLET LIBRARY -66 HELP 11-

0MISCELLANEOUS PARAMETERS

INPUT BUCKLING= 100.00000 (M-2) ETA= 1.85829

THERMAL UTIL.= .86156 FRACT. THERMAL ABS. IN MOD.= .13315

P= .79396 NU*(1-PF)= .15001

AGE= 32.11752 MATERIAL BUCKLING(M-2)= 118.29890

K INFINITY= 1.42077 EIGENVALUE AT INPUT BUCKLING= 1.05199

FEW GROUP ASYMPTOTIC FLUXES

PHI1= 6.5676E+00 PHI2= 7.4801E+00 PHI3= 6.1172E+00 PHI4= 6.2210E+00

0FEW GROUP CROSS SECTIONS BY ISOTOPE, ARRAY BALT CASE 1

ISOTOPE GRP XA(TOT) XF(TOT) NU*XF(TOT) XTR(REN) XREM(REN)

------- --- ---------- ---------- ---------- ---------- ----------

92235. 1 2.3742E-04 2.1511E-04 5.9602E-04 1.1923E-03 1.3358E-04

2 3.7157E-04 2.9188E-04 7.2424E-04 1.7933E-03 2.1165E-06

3 6.3116E-03 4.1679E-03 1.0245E-02 7.3334E-03 6.5251E-07

4 6.8880E-02 5.8936E-02 1.4321E-01 5.9518E-02 0.0000E+00

92238. 1 2.4285E-03 2.1014E-03 5.6373E-03 2.8574E-02 3.9903E-03

2 1.2345E-03 2.2710E-06 5.4505E-06 4.3161E-02 7.2949E-05

3 1.1540E-02 0.0000E+00 0.0000E+00 5.2361E-02 2.0454E-05

4 8.1912E-03 0.0000E+00 0.0000E+00 3.6872E-02 0.0000E+00

8000. 1 4.1348E-04 0.0000E+00 0.0000E+00 8.4723E-02 6.1834E-03

2 0.0000E+00 0.0000E+00 0.0000E+00 1.1501E-01 3.3212E-03

3 0.0000E+00 0.0000E+00 0.0000E+00 1.1184E-01 6.7684E-04

4 0.0000E+00 0.0000E+00 0.0000E+00 1.0716E-01 0.0000E+00

40000. 1 2.4390E-04 0.0000E+00 0.0000E+00 1.6473E-02 8.2052E-04

2 8.1502E-05 0.0000E+00 0.0000E+00 2.6544E-02 1.0183E-04

3 3.0239E-04 0.0000E+00 0.0000E+00 2.3111E-02 2.3424E-05

4 4.7350E-04 0.0000E+00 0.0000E+00 1.8758E-02 0.0000E+00

1001. 1 1.6822E-06 0.0000E+00 0.0000E+00 5.1866E-02 7.8664E-02

2 8.3822E-06 0.0000E+00 0.0000E+00 1.5310E-01 9.5001E-02

3 5.9949E-04 0.0000E+00 0.0000E+00 2.9601E-01 9.2964E-02

4 1.1910E-02 0.0000E+00 0.0000E+00 1.2559E+00 0.0000E+00

0AVE FOIL CROSS SECTION BY GROUP AND MESH POINT CASE 1

FOIL REGION GROUP 1 GROUP 2 GROUP 3 GROUP 4

-------- ------ ---------- ---------- ---------- ----------

0FEW GROUP DATA FOR FLOG CALCULATIONS, ARRAY FOGX CASE 1

GROUP D XA NU*XF XR XF CHI

----- ----------- ----------- ----------- ----------- ----------- -----------

1 1.82319E+00 3.32499E-03 6.23336E-03 8.97915E-02 2.31648E-03 7.53200E-01

2 9.81501E-01 1.69597E-03 7.29696E-04 9.84987E-02 2.94155E-04 2.46610E-01

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3 6.79363E-01 1.87532E-02 1.02447E-02 9.36856E-02 4.16788E-03 1.90020E-04

4 2.25495E-01 8.94554E-02 1.43215E-01 0.00000E+00 5.89361E-02 0.00000E+00

0AVERAGE FLUX BY GROUP AND SPACE MESH POINT, ARRAY FLX CASE 1

MESH POINT GROUP 1 GROUP 2 GROUP 3 GROUP 4

---------- ---------- ---------- ---------- ----------

1 3.2280E+00 3.4910E+00 2.7045E+00 2.2390E+00

2 3.2105E+00 3.4772E+00 2.7057E+00 2.2808E+00

3 3.1374E+00 3.4327E+00 2.7093E+00 2.4100E+00

4 2.9489E+00 3.3478E+00 2.7156E+00 2.6137E+00

5 2.8767E+00 3.3052E+00 2.7185E+00 2.7347E+00

6 2.8466E+00 3.2870E+00 2.7200E+00 2.8426E+00

7 2.8274E+00 3.2752E+00 2.7209E+00 2.9298E+00

8 2.8161E+00 3.2681E+00 2.7215E+00 2.9956E+00

9 2.8108E+00 3.2649E+00 2.7218E+00 2.9980E+00

0OVERLAPPING THERMAL GROUP DATA, ARRAYS OTGC AND OTGA CASE 1

AVE S(V)= 7.3052E-02 AVE V*S(V)= 2.4333E-01

GROUP V V*N(V)*DV XA(V) D(V) NU*XF(V) V*X-(V*P)

----- ---------- ---------- ---------- ---------- ---------- ----------

1 1.0000E-01 1.1805E-04 4.6264E-01 2.1172E-02 4.9158E-01-6.2209E+00

2 2.0000E-01 9.2925E-04 3.5955E-01 4.0276E-02 4.9544E-01-2.6072E+00

3 3.0000E-01 3.0477E-03 2.9685E-01 5.5912E-02 4.4153E-01-1.4879E+00

4 4.0000E-01 6.8791E-03 2.5075E-01 6.8676E-02 3.8584E-01-9.5214E-01

5 5.0000E-01 1.2532E-02 2.1585E-01 7.9792E-02 3.3828E-01-6.3603E-01

6 6.0000E-01 1.9815E-02 1.8954E-01 9.0414E-02 3.0058E-01-4.2554E-01

7 7.0000E-01 2.8247E-02 1.6860E-01 1.0131E-01 2.6947E-01-2.7654E-01

8 8.0000E-01 3.7142E-02 1.5131E-01 1.1302E-01 2.4303E-01-1.6745E-01

9 9.0000E-01 4.5750E-02 1.3688E-01 1.2601E-01 2.2054E-01-8.5781E-02

10 1.0000E+00 5.3341E-02 1.2461E-01 1.4060E-01 2.0111E-01-2.4088E-02

11 1.1000E+00 5.9324E-02 1.1356E-01 1.5699E-01 1.8409E-01 2.2347E-02

12 1.2000E+00 6.3224E-02 1.0402E-01 1.7487E-01 1.6858E-01 5.9162E-02

13 1.3000E+00 6.4922E-02 9.5931E-02 1.9348E-01 1.5556E-01 9.4686E-02

14 1.4000E+00 6.4279E-02 8.8773E-02 2.1050E-01 1.4392E-01 1.4153E-01

15 1.5000E+00 6.1407E-02 8.2390E-02 2.2333E-01 1.3346E-01 2.1203E-01

16 1.6050E+00 6.2063E-02 7.6497E-02 2.3154E-01 1.2357E-01 3.1074E-01

17 1.7200E+00 5.9547E-02 7.0867E-02 2.3771E-01 1.1413E-01 4.1981E-01

18 1.8450E+00 5.4508E-02 6.5612E-02 2.4684E-01 1.0508E-01 5.0246E-01

19 1.9800E+00 4.7826E-02 6.0660E-02 2.6166E-01 9.6378E-02 5.4630E-01

20 2.1225E+00 3.9307E-02 5.5932E-02 2.8061E-01 8.8150E-02 5.8004E-01

21 2.2775E+00 3.3926E-02 5.1914E-02 2.9672E-01 8.0983E-02 6.4614E-01

22 2.4550E+00 2.8796E-02 4.8286E-02 3.1565E-01 7.4700E-02 7.2022E-01

23 2.6600E+00 2.4931E-02 4.5276E-02 3.4105E-01 7.0088E-02 7.7742E-01

24 2.8975E+00 2.1124E-02 4.3859E-02 3.5362E-01 6.8699E-02 8.6633E-01

25 3.1725E+00 2.1539E-02 4.5961E-02 3.8468E-01 7.2316E-02 9.5668E-01

26 3.4900E+00 2.0526E-02 4.4319E-02 3.8628E-01 7.1853E-02 1.0712E+00

27 3.8550E+00 2.0979E-02 3.2120E-02 3.8961E-01 5.1914E-02 1.1380E+00

28 4.2725E+00 2.1771E-02 2.4201E-02 3.8420E-01 3.7147E-02 1.2073E+00

29 4.7475E+00 2.2202E-02 1.9228E-02 3.6566E-01 2.8213E-02 1.3123E+00

0LATTICE INPUT DATA BLOCK LIMP CASE 1

MESH POINTS= 9 MIXTURES= 3 REGIONS= 3 ISOTOPES= 5 FOILS= 0

GEOMETRY= 3 P1(0) OR B1(1)= 1 BOUND.COND.= 0 MAT. BUCKLING= 1.0000E-02

0REGION DATA CASE 1

REGION THICKNESS PTS MIX H-L BREAK TEMP VOLUME FISS/CC

------- ---------- ----- --- ----- ------- ---------- ----------

1 .40250 3 1 0 3 20.00 5.0896E-01 1.8280E+00

2 .07250 1 2 0 4 20.00 1.9987E-01 1.8744E+00

3 .37150 5 3 1 9 20.00 1.5423E+00 2.0186E+00

0ISOTOPE CONCENTRATION DATA CASE 1

ISOTOPE T/M MIXTURE 1 MIXTURE 2 MIXTURE 3 MIXTURE 4 MIXTURE 5

------- ----- ---------- ---------- ---------- ---------- ----------

92235. 0. 7.7688E-04 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

92238. 0. 2.2073E-02 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

8000. 29. 4.5699E-02 0.0000E+00 3.3370E-02 0.0000E+00 0.0000E+00

40000. 0. 0.0000E+00 4.2914E-02 0.0000E+00 0.0000E+00 0.0000E+00

1001. 1029. 0.0000E+00 0.0000E+00 6.6740E-02 0.0000E+00 0.0000E+00

0MESH POINT DATA CASE 1

POINT REGION MIXTURE R(INNER) R(AVE) R(OUTER) VOLUME

----- ------ ------- ---------- ---------- ---------- ----------

1 1 1 0.0000E+00 0.0000E+00 8.0500E-02 2.0358E-02

2 1 1 8.0500E-02 1.6100E-01 2.4150E-01 1.6287E-01

3 1 1 2.4150E-01 3.2200E-01 4.0250E-01 3.2573E-01

4 2 2 4.0250E-01 4.3875E-01 4.7500E-01 1.9987E-01

5 3 3 4.7500E-01 5.1215E-01 5.4930E-01 2.3909E-01

6 3 3 5.4930E-01 5.8645E-01 6.2360E-01 2.7378E-01

7 3 3 6.2360E-01 6.6075E-01 6.9790E-01 3.0846E-01

8 3 3 6.9790E-01 7.3505E-01 7.7220E-01 3.4315E-01

9 3 3 7.7220E-01 8.0935E-01 8.4650E-01 3.7784E-01

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1LIBRARY CONTAINS 1 LATTICES

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Annex E: The GID graphical interface to calculate the point kinetic in

ramp reactivity insertion in Argonauta Reactor using MEF program

(subcritical reactor?).

Analisa-se aqui alguns problemas básicos de cinética Pontual no reator Argonauta:


1. tempo total de análise: Ë= 1500.0[s]









Abaixo apresenta-se a variação temporal da reatividade:

5

t[Ê�

1500 60.0

Reatividade Í�%ô �

0.12

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Figura 11: Função de inserção de reatividade no reator.

Figura 12: Fonte the neutrons externa nula

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Figura 13: Variação da população de neutrons (vermelho) e temperatura (verde) do reator

Figura 14: Variação da população de neutrons retardados ( 1 e 2) no reator

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Annex F: Criticality equation solution using 2 neutron groups energy

for a finite cylindrical reactor without neutron reflectors

SOLUÇÃO DA EQUAÇÃO DE CRITICALIDADE, PELA TEORIA DE 2 GRUPOS DE ENERGIA, PARA

UM REATOR CILINDRICO FINITO E SEM REFLETORES DE NÊUTRONS (“NU”).

Autor: Luis Osório de Brito Aghina

I) Equação da Criticalidade - Teoria de 2 Grupos de Energia :

Seja um reator cilíndrico nu (sem refletor) e finito, caracterizado por um raio RG (geométrico)

e uma altura ZG (geométrica).

Nota: Denomina-se limites físicos (R e Z) os limite geométricos acrescidos das “distâncias

extrapoladas” (dextr e dextz) , distâncias essas obtidas da teoria do transporte e igual a: 0.71 x

“comprimento médio de transporte” . Assim R = RG+ dextr e Z = ZG +2 x dextz. Nos limites

físicos, é imposto que os fluxos de nêutrons φ (rápido e térmico) se anulam. É uma

aproximação que melhora os resultados da distribuição dos fluxos de nêutrons, caso se anular

os fluxos nos limites geométricos.

Pela Física de Reatores e pela teoria da Difusão de Nêutrons é visto que a forma dos fluxos de

nêutrons φ (rápido e térmico ) em um reator “nu” tem o mesmo aspecto (são proporcionais

ponto a ponto). Esta forma dos fluxos, é a da solução homogênea da equação diferencial do

seu estado de equilíbrio neutrônico), e é dado por :

∆φ∆φ∆φ∆φ +Bm2φφφφ = 0

onde ∆ é o operador Laplaciano e Bm2

é uma constante que depende das propriedades

neutrônicas dos materiais do reator. Esta equação é a mesma que a “Equação de Onda”

estudada em vários campos da Física:

∆φ∆φ∆φ∆φ +Bg2φφφφ = 0

onde a função φ, depende das coordenadas espaciais, e no caso da geometria cilíndrica, de r

(distância radial) e de z (direção de seu eixo vertical). O Bg2

desta “Equação de Onda” para um

corpo cilíndrico é um auto-valor e dado por:

Bg2=.

2 22.405

RG ZG

π +

onde o valor 2.405 é a raiz dada pela função de Bessel Jo.

Nota: É fácil demonstrar que o Bm2

é o mesmo para as equações de equilíbrio neutrônico

tanto para os nêutrons rápidos como térmicos. Razão de ser a mesma forma dos fluxos, para

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ambos os grupos de energia, rápido e térmico. Se a Equação de Onda genérica tem solução

com o: Bg2 fornecido pelas propriedades geométricas, então a Equação do Equilíbrio

neutrônico, tem solução se Bm2 = Bg

2. (neste caso Bg

2 com R e Z)

Nota: A função φ, como solução da Equação de Onda para geometria cilíndrica, ou seja φ(r,z)

varia na direção radial na forma da função de Bessel (Jo (r)) e na direção do eixo do cilindro na

forma trigonométrica do Cos(z) A função φ(r,z) é expressa como ( método da separação de

variáveis) : em φ(r,z) = φ(r) x φ(z).

É adotada a seguinte convenção:

i : grupo (G) de energia. Rápido i=1; térmico i=2.

Σai : seção de choque macroscópica (só) de absorção do grupo i (cm-1)

Σr1 : seção de choque macroscópica de remoção (por espalhamento) do grupo rápido (cm-

1), responsável pelo ingresso dos nêutrons rápidos para o grupo térmico, pelo processo da

moderação.

νιΣfi : ν vezes a seção macroscópica de fissão do grupo i (cm-1) (Produção de nêutrons)

Di : coeficiente de difusão do grupo i (cm)

φi : fluxo de nêutrons do grupo i (nêutrons/cm2.seg)

λ : autovalor. É igual a 1/Kef Kef = Coeficiente de multiplicação efetiva do reator.

Nota: Reações nucleares por nêutrons, por unidade de volume e de tempo, é dado por:

j,i i.Σ φΣ φΣ φΣ φ

sendo j,i

ΣΣΣΣ a seção de choque macroscópica para a reação j e para nêutrons do grupo i de

energia

Seja o sistema de equações a 2 grupos de energia de nêutrons, representativo do estado de

equilíbrio neutrônico de um reator (equação da criticalidade), cuja formulação é baseada na

condição de balanço de nêutrons. Para cada grupo de energia dos nêutrons, em um local

(qualquer)do reator :

{FUGAS} +{ABSORÇÕES} de nêutrons = {PRODUÇÃO} de nêutrons

sempre por unidade de volume e tempo. Assim:

- Fonte de nêutrons para o grupo (G = 1) (rápidos): são os nêutrons provenientes das fissões

no U235 por nêutrons térmicos (G=2) e por nêutrons rápidos (G=1) no (U235 e U238):

(νννν1111ΣΣΣΣf1.φφφφ1 + νννν2222ΣΣΣΣf2.φ.φ.φ.φ2)

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- Fonte de nêutrons térmicos: corresponde aos nêutrons do G=1 (rápidos) que por moderação

ingressam no espectro dos nêutrons térmicos (G=2) (limiar superior de 0,625 ev):

ΣΣΣΣr1.φφφφ1.

- Absorção dos neutrons rápidos (G=1): corresponde aos nêutrons que por moderação

desaparecem neste grupo G=1 e ingressam no grupo térmico (G=2), acrescida dos nêutrons

absorvidos durante a moderação (absorção normal e em ressonâncias):

(ΣΣΣΣa1 + ΣΣΣΣr1).φφφφ1

- Absorção dos neutrons térmicos (G=2): é caracterizada pela desaparecimento dos nêutrons

térmicos pela reação nuclear de absorção.:

ΣΣΣΣa2.φ.φ.φ.φ2

- Fugai dos neutrons (G=1 e 2): É o escape de nêutrons por difusão, na unidade de tempo, da

superficie de um volume unitário em um local genérico do reator.. Pelo teorema da

Divergência aplicado ao vetor da corrente líquida de nêutrons (Lei de Fick), tens-se que a fuga

de nêutrons como especificada, corresponde a:

-Di∆φ∆φ∆φ∆φι

onde ∆∆∆∆ = operador Laplaciano= Div(Grad).

Reunindo os termos definidos acima, formam-se o sistema de equações (1) composta pelas

equações de balanço neutrônico para cada grupo (G) de energia de nêutrons.

-D1∆φ∆φ∆φ∆φ1 + (ΣΣΣΣa1 + ΣΣΣΣr1).φφφφ1 = λλλλ.(νννν1111ΣΣΣΣf1.φφφφ1 + νννν2222ΣΣΣΣf2.φ.φ.φ.φ2) (r) para o grupo rápido ( i=1)

-D2∆φ∆φ∆φ∆φ2 + ΣΣΣΣa2.φ.φ.φ.φ2= ΣΣΣΣr1.φφφφ1 (t) para o grupo térmico ( i=2) (1)

Onde λλλλ é um auto-valor para dar a solução estável ao sistema de equações.

Deste sistema obtém-se a expressão relativa a solução homogêneas da equações do grupo

rápido (G=1) que é:

-D1∆φ∆φ∆φ∆φ1 + (ΣΣΣΣa1 + ΣΣΣΣr1).φφφφ1 =0

Comparando com a expressão da Equação de Onda multiplicada por –D1, pode-se escrever:

-D1∆φ∆φ∆φ∆φ1 –D1Bg2.φφφφ1 =0 ou -D1∆φ∆φ∆φ∆φ1 = D1Bg2.φφφφ1

Nota: Esta equação representa as fugas dos nêutrons rápidos, pela superfície duma unidade de

volume no reator e na unidade de tempo. Para um reator de grandes dimensões Bg2 é muito

pequeno é a fuga do volume unitário representativo do reator seria praticamente nula, ou seja

o fluxo de nêutrons varia pouco no corpo do reator.

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Nota: . D1Bg2 é uma constante, mas a fuga vai depender do valor e da forma de φφφφ1 para o

volume unitário considerado.

Substituindo esta útima expressão na equação (1-r) tem-se

(ΣΣΣΣa1 + ΣΣΣΣr1 + D1Bg2 ).φφφφ1 = λλλλ.(νννν1111ΣΣΣΣf1.φφφφ1 + νννν2222ΣΣΣΣf2.φ.φ.φ.φ2

O que se fez foi transformar a fuga de nêutrons rápidos de um volume unitário qualquer em

uma absorção “fictícia” neste volume unitário.

Usando o mesmo raciocínio para o grupo de nêutrons térmicos, no mesmo volume unitário, e

tendo em vista (já mencionados anteriormente) que o Bg2 é o mesmo tanto para o grupo de

nêutrons rápidos como térmicos ( ambos os fluxos variam da mesma forma e são

proporcionais ponto a ponto), obtem-se:

(Σ(Σ(Σ(Σa2+ D2Bg2) . φφφφ2= ΣΣΣΣr1.φφφφ1

forma-se então o sistema (2) representativo da criticalidade do reator:

(ΣΣΣΣa1 + ΣΣΣΣr1 + D1Bg2 ).φφφφ1 = λλλλ.(νννν1111ΣΣΣΣf1.φφφφ1 + νννν2222ΣΣΣΣf2.φ.φ.φ.φ2 (r) (2)

(Σ(Σ(Σ(Σa2+ D2Bg2) . φφφφ2= ΣΣΣΣr1.φφφφ1 (f)

Fazendo:

ΣΣΣΣa11 = (ΣΣΣΣa1 + ΣΣΣΣr1 + D1Bg2 )

ΣΣΣΣa22 = (Σ(Σ(Σ(Σa2+ D2Bg2)

onde ΣΣΣΣa11 é a seção de choque macroscópica TOTAL para o grupo rápido e ΣΣΣΣa22 é a seção de

choque macroscópica TOTAL para o grupo térmico.

À partir do sistema de equações (2f,2r) pode-se montar a equação matricial (3):

2

1 11 2

2

a11 0 f f. . .

r1 a22 0 0

φφφφ φφφφΣ νΣ νΣΣ νΣ νΣΣ νΣ νΣΣ νΣ νΣ = λ= λ= λ= λ φφφφ φφφφ−Σ Σ−Σ Σ−Σ Σ−Σ Σ

(3)

Dado os valores do raio e altura de um reator nu e obtido os valores das Seções de choque

macroscópica dos materiais (homogêneo) constituinte do reator, calcula-se qual o valor de λλλλ

que permite a solução do sistema (2) ou da equação matricial (3). O inverso de λλλλ é o Kef do

reator é a sua “Constante de Multiplicação Efetiva”. Quando Kef =1, o reator é dito Critico ou

seja a população de nêutrons se mantém em equilíbrio temporal sem o auxilio de fontes

externas de nêutrons, quando Kef < 1 o reator é Sub- Critico, precisando de fontes externas de

nêutrons para manter sua população de nêutrons estável. Quando Kef >1, o reator é Super-

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Critico, ou seja, sua população cresce indefinidamente independente de fontes externas de

nêutrons. Encontrado λλλλ diferente de 1, indica que se deve modificar a geometria do reator ou

sua composição material para torná-lo Crítico.

2) Solução Analítica para o Cálculo de λλλλ ou Kef:

A equação 2-f, pode ser escrita:

2 1A.φ = φφ = φφ = φφ = φ

onde 1

22

rA

a

ΣΣΣΣ====

ΣΣΣΣ é a constante de proporcionalidade entre os fluxos rápidos e térmicos),

donde;

Substituindo esta expressão de 2

φφφφ na equação (2-r), obtêm-se para o valor de λλλλ :

11

1 2

a

f f .A

ΣΣΣΣλ =λ =λ =λ =

νΣ + νΣνΣ + νΣνΣ + νΣνΣ + νΣ

e. 1

Kef ====λλλλ

3) Determinação da variação espacial do fluxos Rápidos e Térmicos:

O fluxo de nêutrons (rápido ou térmico) i

φφφφ é expresso como o produto dos fluxos para as

direções radial e paralela ao eixo do cilindro, ou seja:

i i i(r, z) (r). (z)φ = φ φφ = φ φφ = φ φφ = φ φ

A forma da variação dos fluxos nas coordenadas r e z é determinada pela solução da Equação

de Onda para a geometria cilíndrica quando se separam as variáveis em r e z. Para a direção

radial os fluxos rápidos e térmicos (são proporcionais) variam com a coordenada r, de zero a R,

segundo a função Bessel de primeira espécie e de ordem zero (Jo). O fluxo térmico é

proporcional ao fluxo rápido pela constante A ( ou seja , mantém a forma). Esta função Bessel

é a que satisfaz a equação de Onda:

∆φi + Bg2.φ1=0

aplicada a seção reta do cilindro do reator, com seu valor nulo em r = R e unitário

(normalizado) em r=0. Assim:

r varia de 0 até R

Para o fluxo rápido:

1

2,405.r(r) Jo

R

φ =φ =φ =φ =

2,405 = raiz de Jo

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1 1(0) 1 e (R) 0φ = φ =φ = φ =φ = φ =φ = φ =

Para o fluxo térmico:

2

2,405.r(r) A.Jo

R

φ =φ =φ =φ =

2 2(0) A e (R) 0φ = φ =φ = φ =φ = φ =φ = φ =

Para a direção no eixo do cilindro, a função da forma dos fluxos, é obtida da Equação de

Onda:

∆φi + Bg2.φi =0

na coordenada z, e neste caso, φφφφi varia com a função trigonométrica cos(z). Normaliza-se o

fluxo rápido em z=0 e anula-se o fluxo em +H/2 e –H/2, Assim:

z varia de 0 a Z/2 ou –Z/2

Para o fluxo rápido:

1

z(z) cos .

Z

φ = πφ = πφ = πφ = π

1 1(0) 1 e (Z / 2 ou Z / 2) 0φ = φ − =φ = φ − =φ = φ − =φ = φ − =

Para o fluxo térmico:

2

z(z) A.cos .

Z

φ = πφ = πφ = πφ = π

2 2(0) 1 e (Z / 2 ou Z / 2) 0φ = φ − =φ = φ − =φ = φ − =φ = φ − =

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Annex G: Extrapolated distance calculation in Argonauta core

diffusion theory


When a cylindrical reactor is defined without blanket, commonly the pure water or graphite, it

is supposed that adjacent to the cylindrical external surface you have the vacuum. From the

Transport theory the neutron flux (fast and thermal one) reach zero at an approximated

extrapolated distance:

Ptmn ≅ 0.71?nõãpô�öõn

where ?nõãpô�öõn = 3� and � is the neutron diffusion coefficient.

Supposing that the neutron flux is discretize in 2 energy groups: the fast and the

thermal ones, the extrapolated distance is given by:

Ptmn' ≅ 0.71 × 3�' = 0.71 × 3 × 1.1608 = 2.47 �èØ� Ptmnw ≅ 0.71 × 3�w = 0.71 × 3 × 0.2255 = 0.48 �èØ�

As can be seen the extrapolated distance are different for each neutron group. Based in the

transport theory the flux in both groups has to be null at the same point. In reality, in a precise

theory a different boundary condition is used for each neutron energy group. In an

approximate solution for thermal reactors, the extrapolated distance of the thermal energy

group is used in both. The thermal neutron flux are more important in the neutron reactions

than the fast neutron in this kind of reactor (. e também se garante a proporcionalidade entre

os fluxos rápidos e térmicos exigido na teoria??).

Then, for a cylindrical reactor without blanket the extrapolated distance for both energy

groups can be assumed to be:

Ptmn ≅ 0.5 �èØ�

Para o núcleo do reator que você calculou, sem refletores ( lateral, inferior e superior), suas

dimensões geométricas “criticas” seriam: HG = 60 cm (dado) e RG = 23,81(raio físico calculado)

– 0,50 (dist. extr [vacuo] = 23,31 [cm].

In the case of a reactor surrounded by a reflector composed of a pure water, the extrapolated

distance is calculated in a different process or it is obtained in a laboratory. Using this last

method in the Argonauta reactor, Aghina arrives, for the extrapolated distance, to an

approximated value of 7.5 [cm] where is supposed that the reactor is entirely reflected by the

water, that means, in the horizontal surfaces (superior and inferior) and in the cylindrical

surface of the reactor.

Supposing that the geometric dimensions of the reactor, surrounded by water, has the height

of 60 [cm], the geometric radius is equal the physical radius minus the extrapolated distance:

23,81 – 7,5 = 16,31 [cm].

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In the Argonauta reactor case, a new core composed by rod fuel can be used. Extracting the

aluminum central tank that contains the graphite column, extracting also the graphite wedges

and prism that are situated between the internal and external tanks and finally, extracting the

plate fuel elements composed by aluminum plates filled with with ÁxÂì, where the uranium is

enriched in 20% in ÁwxÉ atoms, it is possible to create a new core using the rod fuel elements

juxtaposed in a way that the mean radius is approximately of 17 [cm]. Similar to Angra I these

rod fuel elements with a pitch of ---- are placed in the center of the external tank with a radius

of 46 [cm]. Filling this tank with water is possible to obtain a critical core reactor taking into

consideration that the uranium is enriched in 3.4% in ÁwxÉ. NOTA: Calculei corretamente (processo analítico) o reator Argonauta :

AGHINA/DEBORA (núcleo envolto por água) e obtive para o núcleo, o raio critico (no caso o

geométrico) de 16,5 cm.

Assim, neste caso a Dist.Extrapolada deveria ser : 23,31 – 16,5 = 6,81 cm, valor próximo aos

7,5 cm usado para um cálculo aproximado como o executado, e que deu um BOM resultado.

Observe, o reator sem refletor teria que ter um raio de 23,31 cm e com o refletor de nêutrons

(água) ficou com 16,5 cm . A diferença (no cálculo aproximado chamada de dist extrapolada),

quando se faz o calculo correto ela é chamada de Economia de Refletor, pois é quanto o

refletor faz o reator economizar em combustível

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Annex H: Sinusoidal reactivity variation device to be used in the

Argonauta reactor.


A device to simulate a sinusoidal reactivity variation is presented in the figure 1. This device

is composed of a fixed part A where is inserted the B part that can turn around his axis. This

device is then placed inside the Argonauta core reactor. The rotation of the B part, around

his axis with a w rotation, of the complete device (A+B) permits to simulate a reactivity

variation since the cadmium is an absorber of neutrons. The total absorption is independent

of the thickness of these sheets but dependent of the sheet areas: one in a rectangular form

placed inside a hole cylinder(part A) and the other with a sinusoidal form placed outside the

graphite cylinder(Part B).

Figure 1: Device A and B respectively to simulate a sinusoidal source

Internal sheet of

Cadmium

External sheet of

Cadmium in a

sinus form

Al Gr

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Frequencies [c/s] Amplitude Experimental Analysis

Amplitude Numerical Analysis

.05 20.0

.15 16.0

.216 15.0

.366 14.0

.586 13.0

.834 12.0

1.22 10.0

1.87 7.5

2.7 5.0

3.95 2.5

6.45 1.5

10. 1

Table 9: Variation of the neutron population with the frequency

It can be noticed in table 4 that more the frequency is high more the amplitude is low (neutron

population).

0

5

10

15

20

25

0 2 4 6 8 10 12

A

A

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0

0,2

0,4

0,6

0,8

1

1,2

1,4

-1,5 -1 -0,5 0 0,5 1 1,5

logA

logA

0

0,5

1

1,5

2

2,5

3

3,5

-4 -3 -2 -1 0 1 2 3

lnA

lnA

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