Spatial neutronics modelling to evaluate the temperature reactivity feedbacks in a Lead-cooled Fast Reactor

Proceedings of ICAPP 2015 May 03-06, 2015 - Nice (France)

Paper 15288

SPATIAL NEUTRONICS MODELLING TO EVALUATE THE TEMPERATURE REACTIVITY FEEDBACKS IN A LEAD-COOLED FAST REACTOR

Stefano Lorenzi, Antonio Cammi, Lelio Luzzi Politecnico di Milano - Department of Energy CeSNEF (Enrico Fermi Center for Nuclear Studies)

via La Masa 34 - 20156 Milano (Italy) Tel:+39 022399 6333, Email: [email protected]

Abstract – The qualitative and quantitative assessment of the thermal reactivity feedbacks occurring in a nuclear reactor is a crucial issue for the time-dependent evolution of the system and, in turn, it has a great impact on the development and validation of advanced control techniques. In the present work, in order to overcome the limitations of the classic Point Kinetics adopted in the control simulators, a spatial neutronics model, representing the neutron flux as sum of a spatial basis weighted by time-dependent coefficients, has been considered. The reference reactor is ALFRED, the European demonstrator of the Lead-cooled Fast Reactor technology. Average cross-sections for each Fuel Assembly, calculated by means of a Monte Carlo code, have been used to solve the partial differential equations of the neutron diffusion, exploiting the capabilities of the COMSOL software. Once obtained the spatial functions, the set of equations for studying the reactivity effects has been implemented in the MATLAB environment. Among the several temperature reactivity feedbacks, specific attention has been paid to the Doppler effect in the fuel and to the lead density effect. Several spatial bases have been calculated and their capability of representing the reactivity variation have been assessed.

I. INTRODUCTION

The qualitative and quantitative assessment of the thermal reactivity feedbacks occurring in a nuclear reactor is a crucial issue for the time-dependent evolution of the system. Since these effects have a great impact on the dynamics of the core, they have to be fully characterized whether the development and validation of advanced control techniques are under investigation. In particular, in Generation IV Lead-cooled Fast Reactors (LFRs)1, the temperature field assumes a particular importance because of the corrosion phenomena due to the use of a chemically aggressive coolant as lead and the related technological constraints. Moreover, in this kind of nuclear system, the impact of the coolant density variation may act in different directions (i.e., with a positive or negative local coefficient) according both to the core zone involved and to the size of the reactor, leading to a different dynamics response and, in turn, to a different control strategy to consider2.

One of the main requirements of a control-oriented simulator is to accurately evaluate the reactivity variation following a temperature change. The modelling approach plays a relevant role because of its influence on the control techniques that can be employed. Indeed, a poor detailed modelling precludes the possibility to adopt an advanced control scheme, which usually requires a detailed knowledge of the system (i.e., not only integral quantities

of interest but also spatial information). At the same time, a simulation tool for control purposes has to fulfil some requirements, which are typical of this field. In particular, fast-running simulations, a comprehensive representation of the entire plant behaviour, the possibility to couple the plant dynamics simulator with the control system model are the main requests in this sense. These requirements have brought to the development of simulators that implement simple models (zero dimensional) for the description of the main physics of a nuclear reactor. As far as the reactor neutronics description and the thermal reactivity feedbacks are concerned, the zero dimensional Point Kinetics (PK) model3 is commonly employed in control-oriented tools. This lumped parameter approach describes the time dependence of the neutron population in the reactor and relates it to the flux by a constant of proportionality (single energy group approximation). The system reactivity feedback is usually expressed as a function of the mean values of characteristic temperatures and the relation usually adopted is linear with constant coefficients (both in time and temperature) calculated at nominal conditions. This model neglects the spatial dependence of the flux and, consequently, it does not allow considering the different contribution of each zone to the reactivity.

In order to overcome these limitations, a possible modelling improvement consists in separating the spatial and time dependence of the neutron flux, which can be

846

�� -��-��

��

�� -�� !� �� " �� -�� #�� $�� $�� #� �� !-%��&�� '�� (��)#*# �� !� ��&�� + �� +�� , �++� �� -��!� �� $�� . ��$��# �� "�� "��/�� (��#�� # � �� !� � �� #�� '��"�� !�� - �� # �� (��# �� /�� "��0�� #��!��"�� # ��0�� #��"��

(� �� # �� !��!��&��123�� # � �� ' � ��!�� 0�� 0�� (�4�� ((# �� 123 �� (�4�� (((# �� !�� ! �!�� 2��# �� 4�� (5�# �� 4�� 5�

((�464��+��473(��($8

�� -�!�� 1�� 2�� 3�� 9�-123��#��!�� 2�. 1�-��3 �1��-� �� -�!�� 3�� '��# �� -��0� �:;;+<�� -��123�(�� 2�� =� ��-123�� "� �� 2�� -�� 2-�� 2��=��=.=2-��!�� 0 �� !� � �� # �� !��

�� !� �� # ��#7 �� 3 �� 73�� 4�� 3 �� 43�� !�� #�� #�� >� (� �� (# �� ' � � �� 4�� (((��

�

��

2�� =�-123�� >��

�-?1�(

-123�� >�

�� :;; +<�� 2-��(��/$��0 �� =.=�*./==)� - ��2- =@. -7 ��/ �� );;/)9; A7��7�� =*-=*-�� - 2�� +$B - 7�� *�@*C=;-: �7�� )�,*C=;-: �� )�*;C=;-: �� =�;;C=;-: �-��!�� ;�, �

(((�+$��1��473(��($8

(��#�� 2- ��!�� -123�� !�� !��&� �� #�� $�� !�� # �� '�� -� �� 2�� @�# �� #� �� !�� "��!� � �� # �

847


Paper 15288

offline procedure has to be arranged in order to calculate the spatial basis, solving the neutron diffusion PDEs.

Fig. 2. Derivation of the spatial neutronics model: offline procedure and online calculation.

III.A. General Modelling Framework

In order to describe the neutron kinetics, the multi-group diffusion theory10, with seven energy groups and eight groups of precursors has been considered. In equations, it reads:

�� (1)

� �� ! � �� " #� (2)

where

��$% �� &��$% ��'�(�$% ��)�� *+,-. �/0 �$�1�� *+,-2�0�$�3�� *+,-24�0�$�3��- � � " 5�� 6 4��7�$� �4�87��$��4��78�$� 4�87�$�' ' ��9 �4�(7��$�9 �4�(78�$�: '�4��7(�$� �4�87(�$� 9 ��4�(7�$��;�� (3)

�� <��'��(=�� <��'��(=�� >?4@��$� 9 ?4@(�$�AThe neutron flux can be expressed as

��$% �� B �CD�$� EFDG� HD�� (4)

CD�$� � <CD��$� I II : II I CD(�$�= � *+,- JCD�$�K (5)

HD�� <HD��'HD(��= (6)

where i(r) is a spatial basis where the flux is projected and ni(t) are the time-dependent coefficients, which are the unknowns of the ODEs system.

In order to transform the multi-group diffusion PDEs into a set of ODEs involving only the time-dependent coefficient ni(t), the expression of Eq. (4) has to be substituted into Eq. (1) and Eq. (2), the latter have to be multiplied by test functions �i=diag(�g

i(r)) and integrated over the computational domain4. This procedure can be related to a Petrov-Galerkin projection. Finally, the ODE system for the time-dependent coefficients can be expressed, for each basis function, as

�L�DM HNMFMG� � � J�ODM � PODM � �� QDM � PQDM KF

MG� �HM��RD�S

�G� �� (7)

RND� � ��T &� �QDM � PQDM �HMFMG� ) � ��RD��! � �� " #� (8)

where

L�DM � UVD �� CM*W��ODM � UVD J� � � �� K�CM*W�PODM � UVD P J� � � �� KCM*W��Q+X � U V+ J�Y�ZKCX*W�PQDM � UVD P �� CM*W��RD� � UVD �� *W��T � <��[�� I II : II I ��([��(=�

(9)

Lim and Mim represent the contribution to the removal and production operator calculated in the unperturbed system. These quantities are calculated once in the “offline” process, and are kept constant during the transient simulation. On the other hand, �Lim and �Mim represent the variation of the removal and the production operators during the transients, for instance due to the temperature change of the cross-sections. Such effects constitute the reactivity feedbacks, which assume a particular relevance in the control-oriented perspective. According to this procedure, the variation is weighted on the spatial basis and test functions, allowing for the spatial characteristics of the perturbation and obtaining an accurate estimation of the reactivity evolution. This goal cannot be achieved with a PK approach since the reactivity variations are uniformly evaluated through the system.

III.B. Generation of Group Constants

In order to solve the neutron diffusion PDEs, the neutronic parameters (V-1, D, �a, �s, �p, �d, F

T) of the seven energy groups (the upper energy of the groups are 20 MeV, 2.23 MeV, 0.82 MeV, 0.3 MeV, 67.38 keV, 15.03 keV, 0.75 keV, respectively) have been assessed by means of the

Neutronics (Diffusion equation)+ Spatial basis calculationFinite Element software COMSOL

Generation of group constants(neutron cross-sections)

Monte Carlo simulation (MC) SERPENT

OBJECT-ORIENTED MODEL (OO)

Spatial neutronics model+ Reactivity evaluation

ODE solver MATLAB

��

��

CONTROL-ORIENTED MODEL

848

�� -��-��

��

� �� + �� 7�� 4�3��8�==# �� #��D�22:�==@�

-�� -123�� =.=2-�#==;��#=@73��)43�#��!�� 2��:��0 �� !�� !�0 �� 2�� :�� &� �� 73��43�� 0 �� !�� # �� 2 � �� -123�� # ��# � ��#� �� #�� >#=:#=)#=*�

� ��

2�� :� 3�� !�� !�� 4�3��8�� -123��

4�!�� !�� !� �� !�� &�� -123�� 0��((#�� 1�-��3�� '��>�� 3-8$4�

�-?1�((

7 �� !��

4�3��8� 3��>

� �� -*)>E=9 -*,,1��"�� /F� -;�:@.E;�;=>* -;�@,9†

-"��"�� /F� -;�=*@E;�;;, -;�=**3��"�� /F� -;�.9;E;�;;. -;�.9>

�� ,;; F# �� =*;;F� ��0 ��=@;;F� �� 0 �� 9;; F#�� ,.:F�� !� � �� # .=:F �� !� � �� .*: F �� !� �� !� � ��-�� !�� @G�� !��"��"�� 2��#

*7�� !��†7��

�� 2-�� ;�*G�� "��

��

�� #�� "�� "�� -� ��0�� -� �� #�� %��&�� '�� 8�!��# �� !-%��&�� '�� (��#� �� ' �� - �� -�� # �� !��&�� !��&�#��+ ��+�� $�� # �� # �� ' �� (� �� (((# �� # �� /�� # ��0��

�-?1�(((

4��

��

��

��

!! 7��- 7��?"#$ 7��7 7��

(((�7�=�+ ��+��

(� �� ++# �� !��

J� � � �� KCD � �D\�]��CD �=;�

�� %��!��"�� !�� !��# &'

%� -� �� # � �� 7��-� � ��' �� =;��7��?��!��!��

(((�7�@�� $��

-� �� $��

849

�� -��-��

��

�� =,� �� "�� "�� (��#�� #��#� �� 2��$��+ �� #��$�� 3��$��+ ��+ �� !��#�� !�� =.� (� �� !� �� # �� 3��$��+ �� $��

�� $�� =9#*H

=�7 ��8�� (%, ()

, …, (*s

@�?��" �� +,-(%, ()

, …, (*s.:� �� 4��5�� 45�� +�� H

^ � _�`�a� �==�

�� /,- %POD, )

POD, …, *POD. �� "

� �� $�� POD#�,��0�1��"

� �� !�� 1� �� !��-��!�� !�� (�� &#�� !�� !��&�# �� !� �� -123�� # ��!� � �� & ��# �� !�� -�� #��$�� * �7�� 7�� 8�!��# �� ' �� 7�� !�� #�� &�� -�' �� $�� -�$��H

=� 7 �� 8� �' �� (23%, (23)

, …, (23*srelated to the same configuration of the snapshot calculated for the POD modes @�?��" ��' �� ,-(23%

,

(23), …, (23*s.

:�7 ��" �� 4��

b � c�`�a� �=@�

�� I-�%APOD, �)

APOD, …, �*APOD. �� "

� �� APOD#4��%��

�� 45� � �� ==�� (� �� # �� #

��# �� $��

(� �� ++�� $� �� (5�# ��!�� =;� �� !�� 2�� 7$+4$1 +�� =>� �� -�� 4�3��8��!�� 0��0 ��2��)�# ��#* � �� #* � � �� #)� � ��43�#*� ��73��:� ��

�

��

��

2��)�J � ��0�� -�� -=� � � ��.�� !� �� -�� -=� � � �� =�� 7$+4$1� ��

�-?1�(5

2�� 0��

K �� 7��-#?�� 7��- �� 7��?� �� 7��7#�

�� 7��7� �� 7��

=

� � � �

@

� � � �

:

� � � ��

(� �� !��!��&�� -123�� # �� !�� !�� 7$+4$1 �� # �� -�� H

850


Paper 15288

TABLE V

Simulation cases carried out to reproduce the reactivity feedbacks.

Fuel temp. (K) (inner/outer)

Lead temp. (K) (below active zone & dummy/ active zone/above active zone)

Active length (cm) (inner/outer)

Fuel density (g/cm3) (inner/outer)

FA pitch (cm)

Unperturbed 1500 / 1200 673 / 713 / 753 60 / 60 10.443 / 10.47 17.1 Doppler (inner) 900 / 1200 673 / 713 / 753 60 / 60 10.443 / 10.47 17.1 Doppler (outer) 1500 / 600 673 / 713 / 753 60 / 60 10.443 / 10.47 17.1

Lead density 1500 / 1200 1473 / 1513 / 1573 60 / 60 10.443 / 10.47 17.1 Axial fuel expansion (inner) 1500 / 1200 673 / 713 / 753 61.2245 / 60 10.234 / 10.47 17.1 Axial fuel expansion (outer) 1500 / 1200 673 / 713 / 753 60 / 61.2245 10.443 / 10.261 17.1

Radial grid expansion 1500 / 1200 673 / 713 / 753 60 / 60 10.443 / 10.47 17.1855

4�Z� � d�e � b fg- JZZeKh (13)

As far as the boundary conditions are concerned, the albedo boundary conditions, previously calculated in the SERPENT model, are imposed at the axial and radial boundaries of the COMSOL model domain, namely:

i 2�0j�03 � �k��0��i 2�0j�03 � �kl�0 (14)

III.D. Spatial Neutronics Model and Reactivity Evaluation

The spatial neutronics model has been implemented in the MATLAB software20. As already mentioned in the previous sections, in this paper the attention is paid to the evaluation of the reactivity feedbacks. Therefore, the static formulation of the Eqs. (7) and (8) has been considered, neglecting the dynamic behaviour of the precursors as well. In a general and compact format, this (eigenvalue) problem can be expressed as:

�bm%n� � bm%� H� � �D\ �bo%n� � bo%� HH� � � p�H�q H8q r q HFs�bm%n� � � <O�� r O�F' : 'OF� r OFF=��bm%� � <PO�� r PO�F' : 'POF� r POF =bo%n� � � <Q�� r Q�F' : 'QF� r QFF=��bo%� � <PQ�� r PQ�F' : 'PQF� r PQF =

(15)

For the neutronics calculation, the geometry has been divided in each element assembly (FA, CR, SR, dummy element) and in 12 axial zones (10 evenly spaced ones for the active length, two zones of 30 cm above and below the active region).

For each region, the reactivity insertion is weighted on the spatial basis and test functions integrated over the zone, considering the temperature constant inside the zone. In this way, the calculation of the integral over the zone can be performed once during the offline process, and it is kept constant. Indeed, this quantity is multiplied by the removal (or production) variation, which is temperature dependent. According to this procedure, �Lim can be expressed as follows:

PODM�Z� ��UVD d� ��Zt� � ��Zt� � ��Zt�hCM*Wtt��d��Zt�UjVD jCM *Wt � ��Zt�UVD CM *Wtt ��Zt�UVD CM *Wth � kl U VD CM*`luvw

� k� U VD CM*`�uvx

(16)

where the Green’s first identity has been applied to the diffusion operator and, in the final form, the corresponding surface integrals can be computed only on the radial 56r

and axial boundary 56a of the domain. Moreover, the summation is carried out over the zones which the domain has been divided in.

IV. SIMULATION RESULTS

In order to assess the temperature reactivity feedbacks by means of the spatial neutronics model, seven simulation cases have been carried out at the conditions described in Section III.B and summarized in Table V. For each case, SERPENT and COMSOL simulations have been performed in order to obtain the cross-sections and to calculate the spatial basis and the test functions (see Fig. 2). The four pairs of spatial basis/test functions reported in Table III have been implemented and their capability to reproduce the reactivity feedbacks has been verified. The reactivity variation between the unperturbed case and the representative simulation of the temperature effect has been assessed in terms of global features and spatial distribution in the reactor. For the sake of brevity, only the results regarding the Doppler and the lead density effects are reported in the next subsections. Nevertheless, the other three simulations, regarding the axial and radial expansion, show similar results and lead to the same conclusions.

IV.A. Doppler Effect

Two separate simulations have been performed in order to take into account the different impact of a fuel temperature decrease in different zones of the core (and different enrichments). In Table VI, the results of the SERPENT and COMSOL simulations are reported. The discrepancies between the two simulations are acceptable considering the different neutronics approach (i.e.,

851

�� -��-��

��

�� 4�3��8� �� 7$+4$1��4�� 7$+4$1 �� # �� &��

�-?1�5(

� ��3��!��!�� #4�3��8�!��7$+4$1��

4�3��8��

7$+4$1��

�� G�

(��0 �� =@9�=E.�* =@=�= *�*$��0 �� @;,�.E.�* @@)�: 9�*

�� 5((�2�!�� "�"�� !�� 7��-#?#7#��(�� #�� !�� 7�� 7 �� !�� 7�� -�$�� 7��7��$�� !��"�� 8�!��# 7�� 7�� 7 ��# �"�� #�� #�� $�� -�$� �� $� �� -� �� + �� +�� # �� ' �� 7��?��

�-?1�5((

� ��3��!��!�� #��

7��-�8I*�

7��?�8I*�

7��7�8I*�

7��7�8I.�

7��8I*�

7$+4$1��

(��0 �� ),�. ==>�= >9�, =@=�= =@=�= ��$��0 �� .:�9 @@=�. @=@�; @@)�: @@)�: ��

�� ! �� !�� !�� ‡��2��*��*�� 0 ��#��!�� "�"�� #��!�� (��#� ��+ ��+�� 7��?�# �� # ��# �� -� � ��$��-�$�� # �� !�� "�� ) �� 7�� #� �!�� 7�� 7 �� !�� !� ��

$�� ' � �!�� &��

‡ ��7��-�� "�� !��

��!�� # ��# �� &� �� # �� !�� F��

��

��

2�� *�� 0 �� 0 ��! �� !��!�� !��

(�2��,�� !�0 �� !�� ,;;F��0 �� (�2��,�#� �� #� ��!�� !�� -123�� !��#��!��"��

�

��

��

2�� ,�� 4�� !�� !�� !�0 �� "�� -�$�� 7��

�7��8��$��94��

(� �� 5(((# �� 4�3��8� ��7$+4$1 �� -�� #��

1 2 3 4 5 6 790

100

110

120

130

140

# of functions

Rea

ctiv

ity (

pcm

)

Case BCase CCase DCOMSOL (reference)

1 2 3 4 5 6 7150

160

170

180

190

200

210

220

230

# of functions

Rea

ctiv

ity (

pcm

)


852

�� -��-��

��

�� #�� &��

�-?1�5(((

1��3��!��!�� #4�3��8�!��7$+4$1��

4�3��8��

7$+4$1��

�� G�

1�� -@,@E.�* -@.*�: *�@=

�� (B�2�!�� "�"�� !�� 7��-#?#7#��(�� #�� !�� 7�� 7 �� !�� $��#�� 7�� 7��7��!�� "�� # �!�� 7�� -�� + ��+�� #�� ' �� 7��?��

�-?1�(B

1��3��!��!�� #��

7��-�8I*�

7��?�8I*�

7��7�8I*�

7��7�8I.�

7��8I*�

7$+4$1��

-**>�= -@9,�@ -@..�) -@.*�: -@.*�: -��

�� ! �� !�� !�� 2��.��!��#�� !�� -�$�� 7�� $�� 7��7��(�� #�� 7��7 �� $�� # � �� !�� !��.�� 7 �!��#��-�$�� !�� !��

2�� .� 1�� ! �� !��!��

-� �� (�� # �� !� ��!�� 0 �� ! �!�� &�� 2��9��9�� (� ��# �� !� �� &��$�� #�� !�� !�0 ��#��#�� 2-� �� 0 ��#�� &� �� ' � ��

�

��

��

2��9�1��4��!��!�� !�0 �� "�� -�$�� 7��

�� !��:�: �� !� �� # �� !�� (� �� &# � �� # �� # �� 9;; F � �� 2 � �� # �� ! �!�� 9;; F �� !�� B�

�-?1�B

1�� 3��!��!�� #��

7��-�8I*�

7��?�8I*�

7��7�8I*�

7��7�8I.�

7��8I*�

7$+4$1��

-*,�@ :�> *)�> :=�9 .�9 ��

�� # � � �� # ��-�$�� !�� $� �� 7$+4$1�

1 2 3 4 5 6 7-500

-450

-400

-350

-300

-250

# of functions

Rea

ctiv

ity (

pcm

)


853


Paper 15288

IV.C. Discussion of Results

As far as the Modal Method is concerned, the results of the previous sections state that if the eigenvectors of the flux are employed as test functions (Case A), the spatial neutronics model is not able at all to reproduce the reactivity. On the other hand, if the adjoint eigenvectors are employed (Case B), the result is a little bit different from the reference but it can be considered acceptable. Even if the Modal Method is suitable for reproducing reactivity effects, it has been stated that this approach is not appropriate for considering strong perturbation of the flux as happens during the control rod movements5. Therefore, it is a good but not the ideal candidate to be employed in the spatial neutronics model.

The problem of strong perturbation can be overcome employing the POD method for the selection of the spatial basis. The results show very good results for Case C (the test functions are the same POD modes) and Case D (the test functions take into account the adjoint flux). In particular, the APOD method (Case D) shows better outcomes compared to the classic POD method for several reasons. Firstly, it reaches the best accuracy with less functions employed, meaning that the computational cost to run the model is reduced. Secondly, the APOD method is less affected by the dependence of the functions number on the convergence to the reference value, compared to the POD method. Thirdly, it obtains good results also in situations which are not included in the snapshots set.

Finally, the best performance of the APOD method can be explained considering the role that the adjoint flux takes on in the perturbation theory. In this context, it is used as weighting function for the evaluation of the reactivity variation21. Similarly, in our context, the test functions are used to “evaluate” the residual introduced with the approximation of Eq. (4) and constraining it to zero22.

V. CONCLUSIONS

In this paper, a spatial neutronics modelling to evaluate the temperature reactivity feedbacks in the ALFRED reactor has been studied. The main assumption of the model consists in separating the spatial and time dependence of the neutron flux, which can be represented as sum of a spatial basis calculated form the neutron diffusion PDEs weighted by time-dependent coefficients. The final set of equations is obtained multiplying the PDEs with suitable test functions. The selection of the pair spatial basis/test functions is therefore crucial. In this paper, two different methods for the selection of the spatial basis (the Modal Method and the Proper Orthogonal Decomposition) and two options for the test functions (the same functions that constitute the spatial basis or their adjoint) have been tested. As a major outcome of the simulation results, besides the capability of the model to take into account the spatial distribution of the reactivity, the Adjoint Proper

Orthogonal Decomposition, i.e., the POD method as spatial basis and the adjoint as test functions, has turn out to be the best choice in reproducing the reactivity effects in terms of both accuracy and number of functions employed. These results can be explained considering the role that the adjoint flux takes on in the perturbation theory as weighting function for the neutron importance.

As far as future work is concerned, further investigation about the APOD method is foreseen, i.e., the reproduction of a transient involving the control rod movement, in order to set up an accurate, though fast-running, spatial neutronics model aimed at being employed for control purposes.

NOMENCLATURE

b� coefficient used in Eq. (13), cm-1 � concentration of the jth precursor group, cm-3�0 neutron diffusion coefficient of the gth energy group, cm y number of employed functions in the spatial basis, (-) y� number of snapshots in the POD method, (-) y� number of employed POD for the spatial basis, (-) i surface normal unit vector, (-) HD0 time-dependent coefficient of the ith spatial function of the neutron flux of the gth energy group, (-) $ spatial coordinate, cm

S surface of the spatial domain, cm2Z temperature, K � time, s /0 neutron speed of the gth energy group, cm s-1

Greek Symbols

� total delayed neutron fraction, pcm �� delayed neutron fraction of the jth precursor group, pcm k albedo coefficient used in Eqs. (14), (-) �� decay constant of the jth precursor group, s-1�D\ ith eigenvalue, (-) ? average number of neutrons emitted per fission event, (-) VD0 ith test function of the gth energy group, cm-2 s-1z reactivity, pcm� generic macroscopic cross-section, cm-14�0 macroscopic absorption cross-section of the gth

energy group, cm-14@0 macroscopic fission cross-section of the gth energy group, cm-14�07 macroscopic cross-section including scattering out of the energy group g, cm-14�070{ macroscopic group transfer cross-section from energy group g to g´, cm-1�0 neutron flux of the gth energy group, cm-2 s-1

854


Paper 15288

��0 fraction of delayed neutrons generated in the gth

energy group, (-) ��0 fraction of prompt neutrons generated in the gth

energy group, (-) �]0 fraction of total neutrons generated in the gth energy group, (-) CD0 ith spatial function of the neutron flux of the gth energy group, cm-2 s-1| spatial domain, cm3

Subscript

0 reference value a axial r radial z zone

REFERENCES

1. GIF. Technology Roadmap Update for Generation IV. Nuclear Energy Systems. Available at https://www.gen-4.org/gif/upload/docs/application/pdf/2014-03/gif-tru2014.pdf (2014).

2. S. LORENZI et al. “Evaluation of the coolant reactivity coefficient influence on the dynamic response of a small LFR system”. International Topical Meeting on Advances in Reactor Physics (PHYSOR 2012), Knoxville, USA, April 15 - 20, 2012, American Nuclear Society (2012) (CD-ROM).

3. M.A. SCHULTZ. Control of Nuclear Reactors and Power Plants. McGraw-Hill, New York (1961).

4. S. LORENZI et al. “A control-oriented modeling approach to spatial neutronics simulation of a Lead-cooled Fast Reactor”. Accepted for publication in ASME J. of Nuclear Rad. Sci., Paper No: NERS-14-1050 (2015).

5. A. SARTORI et al. “Comparison of a Modal Method and a Proper Orthogonal Decomposition approach for multi-group time-dependent reactor spatial kinetics”. Ann Nuc. Energy, 71, 217–229 (2013).

6. W.M. STACEY. Space-Time Nuclear Reactor Kinetics. Academic Press (1969).

7. P. HOLMES et al. Turbolence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press (1996).

8. A. ALEMBERTI et al. “The Lead Fast Reactor Demonstrator (ALFRED) and ELFR Design”. Proc. of the IAEA International Conference on Fast Reactors and related Fuel Cycles: Safe Technologies and Sustainable Scenarios, Paris, France, March 4-7, 2013, IAEA (2013).

9. G. GRASSO et al. “The core design of ALFRED, a demonstrator for the European lead-cooled reactors”. Nucl. Eng. Des. 278, 287–301, (2014).

10. J.J. DUDERSTADT and L. J. HAMILTON. Nuclear Reactor Analysis. John Wiley and Sons, New York (1976).

11. SERPENT. PSG2/Serpent Monte Carlo Reactor Physics Burnup Calculation Code. URL http://montecarlo.vtt.fi (2011).

12. A. KONING et al. “The JEFF-3.1 Nuclear Data Library”. Technical Report NEA – OECD, JEFF Report 21 (2006).

13. V. SOBOLEV et al. “Design of a Fuel Element for a Lead-cooled Fast Reactor”. J. Nucl. Mater., 385, 392–399 (2009).

14. P. GAVOILLE et al. “Mechanical Properties of Cladding and Wrapper Materials for the ASTRID Fast-Reactor Project”. Proc. of the IAEA International Conference on Fast Reactors and related Fuel Cycles: Safe Technologies and Sustainable Scenarios, Paris, France, March 4-7, 2013, IAEA (2013).

15. NIST. Atomic Weights and Isotopic Compositions. http://physics.nist.gov/cgi-bin/Compositions/stand_alone.pl.

16. L. SIROVICH. “Turbulence and the dynamics of coherent structures”, Part I–III. Q. Appl. Math., 45, 561–590 (1987).

17. G. BERKOOZ et al. “The Proper Orthogonal Decomposition in the analysis of turbulent flows”. Annu. Rev. Fluid Mech, 25, 539-75 (1993).

18. S. VOLKWEIN. “Proper Orthogonal Decomposition and Singular Value Decomposition”. Karl-Franzens-Universität Graz & Technische Universität Graz.(1999).

19. COMSOL. COMSOL Multiphysics® 4.2a, User’s Guide. COMSOL Inc (2011).

20. MATLAB. MATLAB Software, The MathWorks, Inc. (2005).

21. A. F. HENRY. Nuclear reactor analysis. The MIT Press (1975).

22. D. AMSALLEM and C. FARHAT. “Stabilization of Projection-Based Reduced-Order Models”. Int. J. Numer. Meth. Engng, 91, 358–377 (2012).

855

Documents

Spatial neutronics modelling to evaluate the temperature reactivity feedbacks in a Lead-cooled Fast Reactor