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Nuclear Science and Engineering
ISSN: 0029-5639 (Print) 1943-748X (Online) Journal homepage: http://www.tandfonline.com/loi/unse20
A Data Identification Method for the Improvement
of a Cross-Section Model
Aldo Dall’Osso
To cite this article: Aldo Dall’Osso (2006) A Data Identification Method for the Improvement of a
Cross-Section Model, Nuclear Science and Engineering, 154:2, 241-246, DOI: 10.13182/NSE06A2630
To link to this article: http://dx.doi.org/10.13182/NSE06-A2630
Published online: 10 Apr 2017.
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Date: 25 October 2017, At: 11:44
NUCLEAR SCIENCE AND ENGINEERING: 154, 241–246 ~2006!
A Data Identification Method for the Improvement
of a Cross-Section Model
Aldo Dall’Osso*
Downloaded by [Australian Catholic University] at 11:44 25 October 2017
Framatome ANP
Tour Areva, 92084 Paris La Défense Cedex, France
Received April 21, 2005
Accepted November 3, 2005
Abstract – The accuracy of a neutronics model depends not only on the validity of the equations that are
solved but also on the quality of the cross-section model. This last is currently constituted by a set of
correlations, the parameterized tables, relating the data of the neutronics problem to the local conditions.
The more the correlations represent the local conditions, the more the results will be accurate. For a
simulation model, this means that the results will be closer to the measurements. The goal of the data
identification method presented is to solve a constrained inverse problem and to obtain the parameters of
some further correlations that will enhance the accuracy of the results. The constraint imposed minimizes
the error committed in solving the diffusion equation, using as reference the results of a more accurate
computer code or the measurements performed for in-core flux maps. Some purely numerical examples
and an application in conjunction with in-core measurements illustrate the method.
I. INTRODUCTION
2. moderator temperature, which modifies the Maxwellian distribution of the thermal neutrons
The accuracy of a neutronics model depends not only
on the validity of the equations that are solved but also
on the quality of the cross-section model. A crosssection model provides correlations relating the cross
sections to the local conditions of the reactor. As the
physical conditions vary inside the reactor, the physical
properties, the cross sections, are space dependent.
Cross sections depend on local conditions in two
ways:
3. moderator density, which acts directly as H 2O
and B ~soluble boron! number density change and
indirectly in influencing the neutron slowing down
4. fuel temperature, which acts mainly in the resonance capture of the heavy nuclides.
The influence of these parameters is instantaneous and
delayed via the nuclides buildup. Some historical parameter is used to represent this effect.
Multiparameterized tables technique enables relating the cross sections to the local physical parameters.
They are computed by detailed transport codes that model
all the aspects of neutron slowing down and spatial propagation. Nevertheless, despite the good accuracy of these
codes, cross-section tables contain some approximations. When used by a code representing the whole reactor, these approximations can cause a deviation of the
result with respect to a more accurate reference calculation or to the result of in-core measurements. The reference results can be taken as a target, and some functions
of core physical parameters can be taken as the means to
reach it. The goal of the method proposed in this paper is
1. directly, via the nuclides number density
2. indirectly, via the neutron flux shape versus space
and energy, on which is carried out the homogenization to obtain few-group constants.
The main local physical parameters representing the
local conditions are the following:
1. nuclide density, which acts directly via microscopic cross sections
*E-mail: aldo.dallosso@framatome-anp.com
241
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242
DALL’OSSO
to determine this kind of function. The principle at the
basis of the method is to determine correlation parameters that minimize the error on the neutron balance equations in the least-squares sense.
This method is an application of inverse problem
techniques of which the goal is to identify some input
parameters of the model. More precisely, since the operator ~or some of its coefficients! is not computed with
the objective of reproducing the measurements but is
only adjusted to minimize the error, we shall call it a
constrained inverse problem. This kind of problem was
already approached in neutronics in contexts rather close
to ours,1 to adjust the matrix of the system corresponding to a core calculation or to determine cross sections
using measurements by solving the diffusion equations,
where flux are the data and the cross sections are the
unknown parameters.2 Other works on parameter estimation approach the neutron kinetics problem in order to
tune a simplified model on a more detailed one.3 Adjustment techniques based on generalized perturbation theory
have also been proposed.4
II. METHOD
To describe the principle of our method, we write
the neutron balance equation in diffusion theory on its
discretized form, in which we express the cross sections,
such as they are produced by the cross-section model.
The equation in node n for the fast group is
(f ~an, n, g Fn,1 ⫺ af, n, g Ff,1 ! ⫹ ~S a, n,1 ⫹ S r, n !Fn,1
⫺
1
k eff
2
( nS f, n, g Fn, g ⫽ 0
,
~1!
g⫽1
and for the thermal,
⫺S r, n Fn,1 ⫹ ( ~an, n, g Fn,2 ⫺ af, n, g Ff,2 !
f
⫹ S a, n,2 Fn,2 ⫽ 0 ,
~2!
discretization used. In our case, they are calculated with
the nodal expansion method.
The solution of these equations provides the calculated flux distribution. It differs from the estimated flux
distribution coming from in-core measurements, which
we indicate by FM, n, g and will call pseudomeasured flux
in the remainder of the paper. If one introduces FM, n, g in
Eqs. ~1! and ~2!, these equations will not be satisfied
anymore. Even though it is not possible to satisfy the
equations in changing at the same time the cross-section
distribution, we can determine, by a least-squares approach, some corrective correlations to the cross sections to minimize the error.
II.A. The Least-Squares Formulation
In its basic formulation, the least-squares method
allows us to determine the function that best fits a set of
reference points, when the relationship between the dependent variables and the independent variables is known.
In our problem, this dependence is not known in a direct
way, and one cannot thus directly minimize the difference between calculated flux and pseudomeasured flux.
We regard it as an error to minimize the one that is made
on the application of the neutron balance operator. Indeed, one can consider the difference between the two
sides of Eqs. ~1! and ~2! like an estimator of the error
made by solving the neutron balance equation. This difference represents the excess or the defect of neutrons
produced in the unit of time and the unit of volume. This
deviation can be linked with inaccuracies in the calculation of the cross sections used in the calculation. These
inaccuracies could result in particular from the approximations in the relationships between the cross sections
and the core state parameters, relationships defined in
the multiparameterized tables. They can also be related
to the lack of modeling of some physical phenomena.
Let us indicate by dS the correction that one should
make to the cross sections to reduce these deviations
and introduce it, as well as the pseudomeasured flux,
in Eqs. ~1! and ~2! after having made the following
definition:
f
where
S a, n, g ⫽ absorption cross section ~g ⫽ 1,2!
S r, g ⫽ removal from fast to thermal cross section
nS f, n, g ⫽ fission neutron production cross section
af, n, g ⫽ coupling coefficients between the nodes
resulting from the discretization of the divergence of the neutron current.
Sums appearing in Eqs. ~1! and ~2! are applied to the
nodes f that have a common face with node n. In general,
the coupling coefficients af, n, g depend on the method of
冉
L n, g ⫽ ( an, n, g ⫺ af, n, g
Ff, g
Fn, g
冊
.
~3!
We thus obtain the expression of the error «n, g on the
application of the neutron balance operator:
«n,1 ⫽ L n,1 FM, n,1
⫹ ~S a, n,1 ⫹ dS a, n,1 ⫹ S r, n ⫹ dS r, n !FM, n,1
⫺
1
k eff
2
( ~nS f, n, g ⫹ dnS f, n, g !FM, n, g
,
~4!
g⫽1
«n,2 ⫽ ⫺~S r, n ⫹ dS r, n !FM, n,1 ⫹ L n,2 FM, n,2
⫹ ~S a, n,2 ⫹ dS a, n,2 !FM, n,2 .
NUCLEAR SCIENCE AND ENGINEERING
VOL. 154
~5!
OCT. 2006
IMPROVEMENT OF CROSS-SECTION MODEL
Although it is not possible to determine the dS’s for each
one of the nodes in the core with the information provided by a reference solution of the diffusion equation, it
is possible to determine correlations giving the corrections dS for the whole core. The relationships that express the dependence of the corrections to the cross
sections with respect to the physical parameters of the
core are expressed by the following order 2 polynomials
~but the choice could be extended to other kinds of
functions!:
3
2
j
a a, g, i, j X i, n
(
i⫽1 j⫽1
dS a, n, g ⫽ a a, g,0,0 ⫹ (
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3
dS r, n ⫽ a r,0,0,0 ⫹ (
,
and
3
2
j
a f,2, i, j X i, n
(
i⫽1 j⫽1
,
~6!
where X i, n indicates the values of the physical parameters ~burnup, water density, fuel temperature, etc.! in
node n and a k, n, g represents the required parameters of
the correlations.
We expressly avoided introducing a correction for
nS f,1 . Indeed, Eq. ~1! contains two terms multiplying
F1 , and one cannot find independent equations for the
parameters of the correlations relating to S a,1 and nS f,1 .
Although S r is also a term multiplying F1 , it appears at
the same time in the second equation and this removes
any ambiguity. In order to not leave nS f,1 uncorrected,
we introduce a relationship defining its correction. We
use the following relationship of proportionality between the cross sections and the corrections:
dnS f, n,1 ⫽ dnS f, n,2
nS f, n,1
nS f, n,2
.
~7!
To simplify the method, and in the current state of the
tests carried out, no correction is applied to the diffusion
coefficients. A correction of the diffusion coefficients
would affect the coupling coefficients appearing in Eq. ~3!.
If we substitute Eq. ~6! in Eqs. ~4! and ~5!, we raise
square and sum over all nodes n, we obtain an expression of the mean-square error as a function of the parameters a k, n, g appearing in Eq. ~6!:
N
«sq, g ⫽
«n,2 g
(
n⫽1
.
~8!
From Eq. ~8!, two systems of equations for the fast group
and for the thermal are obtained while imposing that the
derivative of esq, g with respect to each one of the paramNUCLEAR SCIENCE AND ENGINEERING
]«sq, g
]a k, g ', i, j
VOL. 154
⫽0 ,
~9!
where subscript k takes the values a ~absorption!, r ~removal from group 1 to 2!, and f ~fission!.
The solution of this system provides the parameters
of the correlations that will approach the model to the
measure.
II.B. Adjustment of the Cross-Section Model
by Residual Correlations
i⫽1 j⫽1
dnS f, n,2 ⫽ a f, g,0,0 ⫹ (
eters a k, n, g appearing in Eq. ~6! be null. This is expressed
by the relationship
,
2
( a r,0, i, j Xi,j n
243
OCT. 2006
In the choice of the correlations expressed by Eq. ~6!,
we can follow the same formulation established in the
multiparameterized tables ~order 2 polynomials!, a function of the physical parameters representing the core under the current operating conditions ~burnup, water
density, fuel temperature, etc.!, or add new relationships
relating the cross sections to new parameters. In the first
case, we have corrective correlations, while in the second case, we have improving model correlations.
When using our method to determine corrective functions, physical parameter X i, n indicates for i ⫽ 1 the
burnup, for i ⫽ 2 the water density, for i ⫽ 3 the square
root of the fuel temperature:
X 1, n ⫽ Bu n ,
X 2, n ⫽ rH2O, n ,
and
X 3, n ⫽ M Tfuel, n .
~10!
When using our method to determine model-improving
functions, X i, n indicates physical parameters not used in
the cross-section tables. In our analysis of the method,
we used the spectral index variation with respect to infinite medium conditions:
X 1, n ⫽
Fn,1
Fn,2
⫺
S a, n,2
S r, n
.
~11!
This physical parameter takes into account dependence
of the cross sections on the environment.
If we have three independent physical parameters,
the system relative to the fast group consists of 14 equations with 14 unknown parameters of the correlations.
The system relative to the thermal group also consists of
14 equations with 14 unknown parameters of the correlations. We initially solve the thermal system whose unknown parameters of the correlations are a r,0, i, j and
a a,2, i, j .The parameters a r,0, i, j thus calculated are then introduced into the other system, which is solved in its
turn to determine the parameters a a,1, i, j and a f,1, i, j .
244
DALL’OSSO
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III. METHODOLOGICAL ASPECTS
The method of adjustment presented in this paper is
based on the assumption of existence of a relationship
between the sought data and taken measurements. Good
results could be obtained only if the relationship indeed
exists. This method is to be used only when the deviations observed on the flux distribution are allotted to
approximations in the formalism of the multiparameterized tables. It asks many precautions in the implementation, because, like any best-fit method by least-squares,
it cannot be applied in a black box way. This remark is
even more important in our method because one does
not directly control the difference between the calculated flux and the reference. Indeed, control is carried
out indirectly while acting on the error made in the application of the neutron balance operator. We will therefore discuss some aspects of this method.
Concerning the robustness, as for any application of
the least-squares method, for the algorithm to work correctly, the set of points must be organized in a regular
way. It is necessary thus that the range of values of water
density and fuel temperature be rather broad and that
these variables not be correlated to each other. This requires that the calculation be made in a condition of full
power or that several sets of measurements be used at
the same time. Concerning the burnup, it is necessary
that the core be sufficiently depleted. In a contrary case,
the correlation on this physical parameter must be eliminated. Therefore, in this situation, the number of equations to be solved is reduced.
The correlations described in the previous sections
depend on the composition of the nodes. The sums on
the nodes have thus to be extended to the nodes of the
same composition. The parameters of the correlations
are, in general, a function of the composition.
In Sec. II.B, we showed the possibility of deducing
28 parameters of the correlations. If this is true from a
mathematical point of view, difficulties can arise from a
numerical point of view. Indeed, the greater the number
of parameters, the more the matrix of the systems approaches singularity.
This is a known behavior in least-squares problems.
The reason is that very often data to be fitted do not clearly
distinguish between two or more of the fit functions provided. The more functions that are provided, the greater
are the chances that two functions fit the data equally well
~see Ref. 5 for a discussion on this subject!. In our method,
this aspect is enhanced by the fact that the fit functions do
not enter directly in the least-squares process, but they are
filtered by the diffusion equations in which they are inserted, and regularization techniques such as singular value
decomposition do not significantly improve the method.
To prevent the matrix of the systems from being too close
to singularity, it is necessary to use the smallest number of
parameters in the correlations ~which reduces the number
of fit functions!. This recommendation is reinforced by
the fact that the number of points of measure in the core is
much lower than the number of calculation points. A big
part of the pseudomeasured flux distribution is an extension of measurement ~see Sec. IV! and thus does not bring
information in the best-fit process.
The calculated parameters satisfy a total minimization of the error in the core, and some local deviations
can continue to remain after the correction of the cross
sections.
It is obvious that reliable parameters of the correlations could be provided only if the error on the flux really
depends on approximations on the cross-section model.
This can be evaluated by analyzing the diagonal of the inverse of the matrix of the systems appearing in Eq. ~9!.
Indeed, according to the theory of the least-squares method,
each element of the diagonal represents the variance associated with the corresponding parameter. The error analysis by applying Eqs. ~4! and ~5! also allows evaluating the
relevance of the correlations. Other techniques can be used
to check the quality of the parameters of the correlations.
An immediate test is the reexecution of the flux calculation with application of the parameters of the correlations, which must approach the reference.
Note that even though the system composed by
Eqs. ~1! and ~2! is nonlinear, because the cross sections
depend on flux via the thermal feedback, the system composed by Eq. ~9! is in general linear. Therefore, the calculation of the parameters of the correlations is made in
a direct way. We could obtain a nonlinear system only if
the parameters of the correlations appear as an argument
of nonlinear functions.
A question about the use of the flux as a physical
parameter on which the identification process is based
could arise. Since the parameterized cross sections are
obtained from transport calculations by forcing the twogroup diffusion equation to preserve infinite-medium assembly reaction rates, the two-group diffusion fluxes are
not directly comparable with measured fluxes and it would
be more consistent to base the identification on reaction
rates. In our method, the consistence is granted by the
fact that pseudomeasured fluxes are inferred from measured reaction rates ~detector readings!, as described in
Sec. IV, using cross sections that have the property to
preserve infinite-medium detector reaction rates. Inserting the pseudomeasured fluxes into the node-by-node
diffusion equation enables preservation of the neutron
balance, a kind of global reaction rate, instead of preserving the detector reaction rate. Preserving the detector
reaction rate could not provide easy-to-use mathematical
conditions allowing correcting cross sections appearing
in the diffusion equation.
IV. ELEMENTS OF IMPLEMENTATION
The data input of the equations presented in Sec. II
is the pseudomeasured flux distribution deduced from
NUCLEAR SCIENCE AND ENGINEERING
VOL. 154
OCT. 2006
IMPROVEMENT OF CROSS-SECTION MODEL
theoretical flux by an interpretation of in-core measurements. Calculation of the parameters of the correlations
for the correction of the cross sections is as follows:
Downloaded by [Australian Catholic University] at 11:44 25 October 2017
1. calculation of the distribution of pseudomeasured power by extension of the deviations between calculation and measure
2. conversion of the distribution of pseudomeasured power into pseudomeasured flux distribution
3. solution of the equations described in Sec. II
4. calculation of flux with thermal feedback by taking into account the residual correlations.
Step 1 is carried out by computing the deviation between
predicted and measured reaction rates in the detector position, converting it into relative power deviation and
extending it to the whole core by polynomial interpolation. The extension is performed by determining the coefficients of an order 2 polynomial of the coordinates x
and y of each horizontal plane of nodes, by least-squares
fit of the deviations in the detector positions, and interpolating it in all the nodes of the core.
Step 2 is based on the assumption that the spectrum
of numerical neutron flux Fn, g is correct and is thus preserved in the distribution of pseudomeasured flux FM, n, g .
By indicating by Pn the pseudomeasured power in node
n, the pseudomeasured flux FM, n, g is thus given by solving the following system of equations:
and
FM, n,2
⫽
Fn,1
Fn,2
.
~12!
Step 4 is performed to verify the quality of the parameters of the correlations.
The procedure to obtain the pseudomeasured flux
~steps 1 and 2! is the same used by Framatome-ANP to
reset the three-dimensional on-line code for in-core
measurements.6
V. RESULTS
The method has been applied in two ways:
1. purely numerical experiments to verify the capability of the method to identify the data
2. application of the method to self-powered
neutron detector ~SPND! measurements from a
1300-MW reactor.
We have tested the capability of the method to identify
inaccuracies in the cross-section tables only by numerical experiments. Applications of the method using incore measurements did not identify any significant
NUCLEAR SCIENCE AND ENGINEERING
VOL. 154
discrepancy in cross-section tables. This shows that the
dependence of the cross sections on the classical physical parameters such as burnup, water density, fuel
temperature, etc., is correctly treated by the tables. Nevertheless, the application of the method to real measurements has revealed some physical effect that was not
taken into account by the cross-section tables ~see later
text!.
The method applied to the determination of corrective functions of the cross-section model was tested in
various situations, among them a very simple onedimensional subcritical core model and a threedimensional core model of the 1300-MW type. The
so-called pseudomeasured flux distribution is the result
of a reference calculation. The tests were carried out
according to the following procedure:
1. execution of a calculation with the initial distribution of cross sections simulating a distribution
affected by uncertainty
2. choice of a correlation and a set of values to be
applied to the parameters
3. calculation of the distribution of cross sections
with these correlations and determination of the
corresponding flux distribution ~reference distribution!, thus simulating a pseudomeasured flux
distribution
4. determination of the parameters of the correlations by solving Eq. ~9!; these parameters must
be very close to those imposed in step 2
kS f, n,1 FM, n,1 ⫹ kS f, n,2 FM, n,2 ⫽ Pn
FM, n,1
245
OCT. 2006
5. calculation ~for checking! of flux with the distribution of cross sections calculated with the parameters of the correlations determined in step 4;
this flux distribution must be very close to those
calculated with step 3.
These calculations ~numerical experiments! showed a
good behavior of the identification method. The parameters of the correlations obtained by solving system ~9!
were equal to the expected values, and the flux distribution was identical to the reference distribution.
The method applied to the determination of improving functions of the cross-section model was tested on a
three-dimensional core model of the 1300-MW type at
1260 MWd0tonne from the beginning of its 13th fuel
cycle. The pseudomeasured flux distribution comes from
in-core measurements. The procedure described in Sec. IV
was applied to obtain the pseudomeasured flux distribution from SPND. The core is at full power with regulation group ~group R! inserted at 9%.
A model improving correlation was sought on the
basis of the following eight parameters of the correlations:
a a,1,1,1 , a a,1,1,2 , a a,2,1,1 , a a,2,1,2 ,
a r,0,1,1 , a r,0,1,2 , a f,1,1,1 , a f,1,1,2 ,
246
DALL’OSSO
TABLE I
Core Parameters Relative to the Reference, Initial, and
Adjusted Power Distribution for a 1300-MW
Three-Dimensional Core Model
Core Parameter
Power axial offset ~%!
Power radial offset ~%!
Core power pic ~Fq !
Axial power pic ~Fz !
Enthalpy rise factor ~FDH !
Maximum radial power
pic ~Fxy, max !
Reference a Initial b Adjusted c
⫺1.77
12.6
1.56
1.15
1.33
⫺1.76
9.4
1.60
1.16
1.33
⫺1.75
13.2
1.55
1.15
1.32
1.54
1.49
1.50
a From
in-core measurements.
Predicted without application of residual correlations.
c Predicted with application of residual correlations.
Downloaded by [Australian Catholic University] at 11:44 25 October 2017
b
model. But the role of the method can be limited to determining which new correlation fits the observed behavior. The parameters of the new correlations can be
determined by adding new calculations in the building
of cross-section tables. Following this approach, Framatome ANP has recently improved the physical model in
its core simulator code: New correlations based on the
spectral index have been added in the multiparameterized tables.
If introduced in a three-dimensional on-line coremonitoring system, this method can be used as a tool to
gain experience from measurements and refine the model
on-line. Framatome ANP has successfully tested its own
core-monitoring system 7,8 and plans to use this method
to improve its cross-section model, namely, fission product cross sections as a function of burnup.
REFERENCES
corresponding to second-order correlations in the spectral index variation according to Eq. ~11! for all the cross
sections, with order 0 terms set to 0. The correlation on
dnS f,2 was dropped ~dnS f,2 was considered 0! and replaced by a correlation on dnS f,1 . A set of parameters for
each fuel type was calculated.
The results are presented in Table I, which shows
that with the application of the residual correlations on
the cross-section distribution, the power distribution
comes closer to the reference. The mean-square error
changed from 2.0 to 1.7%, and the maximum error passed
from 4.4 to 2.6%. The fact that the error is not reduced to
zero means that only a part of the error can be attributed
to the lack of modeling of the environmental spectrum
variation.
VI. CONCLUSION
We have presented a method of determining correlations relating the cross sections to some local core physical parameters ~burnup, water density, fuel temperature,
and spectral index! in order to approach a neutronics
model to a reference. This reference can result from measurements or calculations.
Numerical benchmarks have shown that the method
has good identification properties and can positively identify a correlation when this correlation exists.
When used to approach a core model to measurements, this method can be considered as a way to measure some physical effect that is not seen by our modeling.
In this context it can be used to explore new correlations
that could improve the cross-section model. The parameters of the correlations can be used as they are calculated, as an experimental adjustment of the cross-section
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